A REGULARIZATION APPROACH FOR RECONSTRUCTION AND VISUALIZATION OF 3-D DATA. Hebert Montegranario Riascos, M.Sc.

Transcription

1 A REGULARIZATION APPROACH FOR RECONSTRUCTION AND VISUALIZATION OF 3-D DATA Hebert otegraaro Rascos,.Sc. Thess submtted partal fulfllmet of the requremets for the Degree of Doctor of Phlosophy Advsor Prof. Jaro Esposa Ph. D. UNIVERSIDAD NACIONAL DE COLOBIA Facultad de as Doctorado e Igeería - Sstemas edellí, Colomba 010

2 Foreword Esta vestgacó se llevó a cabo gracas a ua comsó de estudos que me cocedó la Uversdad de Atoqua como profesor de su Departameto de atemátcas. De otra maera, habría sdo mucho más dfícl realzarla. Debo agradecer a alguas persoas que de ua u otra forma ha fludo e esta avetura. A m drector Jaro Esposa por sus oportuos cometaros y su apoyo codcoal, a Jame Valeca y Carlos aro Jaramllo, colegas de la Uversdad de Atoqua, a los profesores Herá Darío Álvarez, Joh W. Brach y su grupo de vestgacó por amarme a ser parte de la Facultad de as de la Uversdad Nacoal, como també al profesor Carlos Cadavd y a Clara y La e la Uversdad EAFIT. Este trabajo está dedcado a m esposa Zorada y a uestros ños Haa y Satago Hebert otegraaro Rascos, Juo 010, edellí.

3 Summary Computer vso s a ter-dscplary research feld that has as prmary objectve to address vsual percepto through mathematcal modelg of vsual uderstadg tasks. The frst part of vso from mages to surfaces- has bee termed early vso ad cossts of a set of processes that recover physcal propertes of vsble three-dmesoal surfaces from the two dmesoal mages. Dfferet argumets suggest that early vso processes correspod to coceptually depedet modules that a frst approxmato ca be studed solato. Surface recostructo s oe of these modules. Ths thess s about surface recostructo from rage mages usg some extesos of Tkhoov regularzato that produces sples applcable o -dmesoal data. The cetral dea s that these sples ca be obtaed by regularzato theory, usg a trade-off betwee fdelty to data ad smoothess propertes; as a cosequece, they are applcable both terpolato ad approxmato of exact or osy data. We propose a varatoal framework that cludes data ad a pror formato about the soluto, gve the form of fuctoals. We solve optmzato problems whch are extesos of Tkhoov theory, order to clude fuctoals wth local ad global features that ca be tued by regularzato parameters. The a pror s thought terms of geometrc ad physcal propertes of fuctoals ad the added to the varatoal formulato. The results obtaed are tested o data for surface recostructo, showg remarkable reproducg ad approxmatg propertes. I ths case we use surface recostructo to llustrate practcal applcatos; evertheless, our approach has may other applcatos. I the core of our approach s the geeral theory of verse problems ad the applcato of some abstract deas from fuctoal aalyss. The sples obtaed are lear combatos of certa fudametal solutos of partal dfferetal operators from elastcty theory ad o pror assumpto s made o a statstcal model for the put data, so t ca be thought terms of oparametrc statstcal ferece. They are mplemetable a very stable form ad ca be appled for both terpolato ad smoothg problems. Well-posedess of a verse problem depeds o the topologcal propertes, the s very mportat the fucto spaces whch our optmzato problems are gog to be formulated ad solved. Here we show how Schwartz dstrbuto theory s a fudametal tool for modelg ad uderstadg of recostructo problems. Dstrbutoal spaces, such as Sobolev ad Beppo-Lev spaces provdes a abstract settg for cludg dscrete ad cotuous fucto the same framework. I ths way we obta explct expressos for a famly of terpolatg ad smoothg sples. Ths famly cludes the well-kow Th Plate Sple (TPS), whch s used as performace crtero for testg the other members of the class. Numercal tests appled wth these sples yeld very promsg ad successful results, showg that t s possble to desg ew sples able to mprove the propertes of TPS by a approprate selecto of parameters ad regularzato fuctoals. These results also have cosequeces other approaches that ca be appled for solvg recostructo problems as Learg Theory, Noparametrc Statstcal Iferece ad Neural Networks. We thk that ths way we are makg cotrbutos to fll the gap betwee some abstract mathematcal deas ad ther correspodg egeerg applcatos

4 Lst of symbols Natural umbers Iteger umbers The Euclda dmesoal space k C ( ) The space of cotuously dfferetable fuctos of order k ( ) Hlbert space of square tegrable fuctos 0 ( C ) Iftely dfferetable fuctos wth compact support Space of tempered dstrbutos ' Space of dstrbutos BL ( ) Beppo-Lev spaces A t 1 A A Traspose of a matrx Iverse of a matrx Geeralzed verse u x x u x Partal dervatves Euclda orm v

5 Acroyms TPS GCV CAGD PDE CV RS AX AD AE ATLAB SVD RKHS PD CPD wp1 Well-posed codto 1 wp Well-posed codto Th Plate Sple Geeralzed Cross-Valdato Computer Aded Geometrc desg Partal Dfferetal Equato Cross-Valdato Root ea Square Error axmum Error mum absolute devatos mum absolute errors atrx Laboratory Sgular Value Decomposto Reproducg Kerel Hlbert Space Postve Defte Codtoally Postve Defte wp3 Well-posed codto 3 RBF Radal bass fucto v

6 Cotets page 1 Itroducto Surface Recostructo from rage mage data 1. Applcatos of 3D recostructo Requremets of surface recostructo The thess objectves Geeral Specfc Proposed model Cotets ad cotrbutos Bref cotets by chapter 11 Revew ad classfcato of methods for surface recostructo Itroducto 13. Geometrc methods 15.3 Statstcal methods 16.4 Regularzato methods 18 3 The seve trsc problems of surface recostructo Itroducto 3 3. The problems of surface recostructo Reproducto-geeralzato dlemma Restrctos o the choce of spaces The Curse of dmesoalty Nose Error bouds The Computatoal complexty Local ad global propertes 30 4 Regularzato ad verse theory surface recostructo Itroducto 3 4. Regularzato ad verse problems Ill-posed problems ad regularzato Surface recostructo as a verse problem Regularzato of surface recostructo Iterpolato ad approxmato by regularzato 36 v

7 4.5. Proposed model 36 5 Regularzato fuctoals ad ther physcal terpretatos Itroducto Plate ad membrae models Weak formulato Fudametal solutos Crtera for selecto of regularzato fuctoals The Resoluto of the fuctoal Addg more fuctoals 45 6 Dstrbuto spaces ad regularzato for surface recostructo 6.1 Dstrbutos ad verse problems The dervatve problem Dstrbutos Covoluto The Schwartz Space Fudametal solutos Sobolev ad Beppo Lev spaces Dstrbutos ad ercer codto o kerels 54 7 Iterpolato of surfaces from pot clouds Itroducto m D Sples or Th Plate Sple m Geeralzatos of D sples ad semorms Sples teso ( -sples) Coclusos 66 8 Surface approxmato from osy data Itroducto Soluto of smoothg 3D Extedg Tkhoov regularzato The Relato wth Radal bass fuctos 7 9 Applcatos, mplemetatos ad umercal expermets Itroducto Represetato of surfaces odels ad objects aalyzed 74 v

8 9.4 Iterpolato ad Smoothg wth surface sples How to choose a terpolat Recostructo of data wthout ose Codto umbers ad error bouds Evaluato Crtera for the approxmato methods Accuracy Vsual Aspects Sestvty to parameters Tmg Ease of mplemetato Recostructo from osy data. Cross valdato Comparsos wth other methods Coclusos Coclusos ad ope problems 10 Appedx 105 Refereces 11 v

9 Chapter 1 Itroducto Through vso, we obta a uderstadg of what s the world, where objects are located, ad how they are chagg wth tme. Because we obta ths uderstadg mmedately, effortlessly, ad wthout coscous trospecto, we may beleve that vso should therefore be a very smple task to perform. A mportat cotrbuto that computatoal studes have made s to show how dffcult vso s to perform, ad how complex are the processes eeded to perform vsual tasks successfully. Computer vso s a ter-dscplary research feld that has as prmary objectve to address vsual percepto through mathematcal modelg of vsual uderstadg tasks. It has evolved durg the last four decades wth two ma goals: to develop mage uderstadg systems ad to uderstad bologcal vso. Its ma focus s o theoretcal studes of vso, cosdered as a formato processg task. I the secod half of the tweteth cetury, there were two mportat attempts to provde a theoretcal framework for uderstadg vso, by Davd arr [113] ad James Gbso [68]. We shall focus at the level arr called computatoal theory of vso. arr emphaszed that vso was othg more tha a formato-processg task. Ay such task, he argued, could be descrbed o three levels: () computatoal theory; () specfc algorthms; ad () physcal mplemetato. The mportat pot s that the levels ca be cosdered depedetly. Ths cocept of depedet levels of explaato remas a paradgm of vso research. arr attempted to set out a computatoal theory for vso as a whole. He suggested that vsual processg passes through a seres of stages, each correspodg to a dfferet represetato, from retal mage to 3D model represetato of objects. Today, forty years o, most would agree that arr s framework for vestgatg huma vso ad [ ], partcular, hs strategy of dvdg the problem to dfferet levels of aalyss, has become uquestoed [74,75]. The frst part of vso from mages to surfaces- has bee termed early vso ad cossts of a set of processes that recover physcal propertes of vsble three-dmesoal surfaces from the two dmesoal mages. Dfferet argumets [114,116] suggest that early vso processes correspod to coceptually depedet modules that as frst approxmato ca be studed solato. Surface recostructo s oe of these modules. Other early vso modules are edge detecto, spato-temporal terpolato ad approxmato, 1

10 computatoal optcal flow, shape from cotours, shape from texture, shape from shadg ad bocular stereo. A very remarkable fact s that most early vso problems are ll-posed the sese defed by Hadamard [81]. A problem s well posed whe ts soluto (a) exsts, (b) s uque, ad (c) depeds cotuously o the tal data. Ill-posed problems fal to satsfy oe or more of these crtera. Bertero, Poggo ad Torre [18] show precsely the mathematcally llposed structure of these problems. Ths fact suggests the use of regularzato methods developed mathematcal physcs for solvg the ll-posed problems of early vso. The ma dea supportg Tkhoov regularzato theory [179,13,8,83] s that the soluto of a ll-posed problem ca be obtaed usg a varatoal prcple, whch cotas both the data ad pror smoothess formato. If we cosder, for stace, 3D data, these two features are take to accout, assumg z f ( a) ad mmzg the fuctoal 1 (1.1) m [ f ] : ( f ( a) z ) R[ f ]. f 1 Wth ths approach, we are lookg for a approxmato that s smultaeously close to the data ad smooth. Smoothess s cluded wth the smoothess fuctoal or regularzer R[ f ] such a way that lower values of the fuctoal correspods to smoother fuctos, ad s a postve umber called regularzato parameter. 1.1 Surface Recostructo from rage mage data I the area of surface recostructo the goal s to obta a dgtal represetato of a real, physcal object or pheomeo descrbed by a cloud of pots, whch are sampled o or ear the object s surface. Recetly there has bee a growg terest ths feld motvated by the creased avalablty of pot-cloud data obtaed from medcal scaers, laser scaers, vso techques (e.g. rage mages), ad other modaltes. Rage mages [36,61,0] are a specal class of dgtal mages. Each pxel of a rage mage expresses the dstace betwee a kow referece frame ad a vsble pot the scee. Therefore, a rage mage reproduces the 3D structure of a scee. Rage mages are also referred to as depth mages, depth maps, xyz maps, surface profles ad.5d mages. 3 Rage mages ca be represeted two basc forms. Oe s a lst A { a1, a,, a } of scattered pots 3D coordates a ( x, y, z ), 1,, a gve referece frame (pot clouds), for whch o specfc order s requred. The other s a matrx of depth values of pots alog the drectos of the xy, mage axes, whch makes spatal orgazato explct. Itesty mages are of lmted use terms of estmato of surfaces. Pxel values are related to surface geometry oly drectly. Rage mages ecode the posto of surface drectly; therefore, the shape ca be computed reasoably easy.

11 Apart from beg ll-posed, the problem of surface recostructo from uorgazed pot clouds s challegg because the topology of the real surface ca be very complex, ad the acqured data may be o-uformly sampled ad cotamated by ose. oreover, the qualty ad accuracy of the data sets deped upo the methodologes whch have bee employed for acqusto (.e. laser scaers versus stereo usg ucalbrated cameras). Furthermore, recostructg surfaces from large datasets ca be prohbtvely expesve terms of computatos. Our approach to surface recostructo wll be based the followg multvarate terpolato problem: Gve a dscrete set of scattered pots A { a1, a,, a } ad a set of possble osy measuremets{ z }, fd a cotuous or suffcetly dfferetable fucto 1 f :, such that f terpolates ( f ( a ) z ) or approxmates ( f ( a ) z ) the data D {( a, z ) } Applcatos of 3D recostructo Surface recostructo s a mportat problem computatoal geometry, computer aded desg (CAD), computer vso, graphcs, ad egeerg. The problem of buldg surfaces from uorgazed sets of 3D pots has recetly gaed a great deal of atteto. I fact, addto to beg a terestg problem of topology extracto from geometrc formato, ts applcatos are becomg more ad more umerous. For example, the acqusto of large umbers of 3D pots s becomg easer ad more affordable usg, for example, 3D-scaers [0]. There are a umber of other applcatos where objects are better descrbed by ther exteral surface rather tha by uorgazed data (clouds of pots, data slces, etc.). For example, medcal applcatos based o CAT scas or NRs t s ofte ecessary to vsualze some specfc tssues such as the exteral surface of a orga startg from the acqured 3D pots. Ths ca be acheved by selectg the pots that belog to a specfc class (orga boudary, tssue, etc.) ad the geeratg the surface from ther terpolato. I most cases the defto of ths surface s a ll-posed problem as there s o uque way to coect pots of a dataset to a surface; therefore t s ofte ecessary to troduce costrats for globally or locally cotrollg the surface behavor. As a matter of fact, the resultg surface ofte turs out to exhbt a complex topology due to ose the acqured data or ambgutes the case of o-covex objects [00]. I order to overcome such problems, surface recostructo algorthms eed to corporate specfc costrats o the qualty of the data fttg (surface closeess to the acqured pots), o the maxmum surface curvature ad roughess, o the umber of resultg tragles, etc. Three dmesoal object surface recostructo ad modelg plays a very mportat role reverse egeerg [146]. For example, f we wat to buld a geometrcal model to a 3

12 3D object for reproducto, the oly way s to sca the object wth a dgtal-data acqusto devce ad perform surface recostructo to get ts geometrcal model. Whle covetoal egeerg trasforms egeerg cocepts ad models to real parts, reverse egeerg real parts are trasformed to egeerg models ad cocepts. The exstece of a computer model provdes eormous gas mprovg the qualty ad effcecy of desg, maufacture ad aalyss. Reverse egeerg typcally starts wth measurg a exstg object so that a surface or sold model ca be deduced order to explot the advatages of CGD/CA techologes. There are several applcato areas of reverse egeerg. It s ofte ecessary to produce a copy of a part, whe o orgal drawgs or documetato are avalable. I other cases we may wat to re-egeer a exstg part, whe aalyss ad modfcatos are requred to costruct a ew mproved product. I areas where aesthetc desg s partcularly mportat such as the automoble dustry, real-scale wood or clay models are eeded because stylsts ofte rely more o evaluatg real 3D objects tha o vewg projectos of objects o hgh resoluto D screes at reduced scale. Aother mportat area of applcato s to geerate custom fts to huma surfaces, for matg parts such as helmets, space suts or prostheses. It s mportat to say that there exst other meags for reverse egeerg but we refer here to surface recostructo. Sce surface-based represetato of a 3D object s crucal ot oly data rederg, but also 3D object aalyss, modelg ad reproducto, may surface recostructo algorthms have bee proposed recet years. I 3D object recostructo ad dsplay, 3D surface pots ca be collected ether by tactle methods such as Coordate easurg aches (C), or va o-cotact methods, such as magetc feld measuremet maches ad optcal rage scaers. After surface pot acqusto, the ext step of 3D object recostructo wll be data fuso (patch regstrato ad tegrato) to traslate the surface pot sets captured at dfferet vew-agles to a commo coordate system, kow as the World Coordate System (WCS), ad merge the overlappg pots of ay two eghborg data sets. The last step of the process s surface recostructo (surface meshg/tragulato) ad rederg. From the pot of vew of techology, 3D recostructo methods ca be collected uder two groups: actve ad passve. Actve methods make use of calbrated lght sources such as lasers or coded lght most typcal example of whch s the shape from optcal tragulato method. Passve methods o the other had, extract surface formato by the use of D mages of the scee. Amog the most commo that fall to ths category are the techques kow as shape from slhouette, shape from stereo, ad shape from shadg. ay results are avalable cocerg relable recostructo of objects usg these methods. However, there s stll a eed for mproved recostructos sce each specfc method, actve or passve, has ts ow drawbacks ad defceces. All techques that recover shape are commoly called shape-from-x, where X ca be shadg, stereo, texture, or slhouettes, etc. (see [9,58,93,98,148,166] ad the refereces there). For example, the stereo problem, oe frst extracts features (e.g., corers, les, etc.) from a collecto of put mages, ad the solves the so-called correspodece problem,.e., matchg features across mages. After obtag depth formato at the 4

13 locatos of the extracted features, oe eeds to recostruct the surfaces of the objects preset the scee. Oe way of achevg ths s by usg techques that recostruct surfaces from pot clouds. Recostructg a surface from uorgazed pot clouds s a challegg problem because the topology of the real surface ca be very complex, the acqured data may be o-uformly sampled ad the data may be cotamated by ose. I addto, the qualty ad accuracy of the data sets strogly deped upo the acqusto methodology. Furthermore, the computatoal cost of recostructg surfaces from large datasets ca be prohbtve. ost of the exstg recostructo methods were developed postulatg that precse ad ose-free data s avalable. Therefore, they caot meet the demads posed by osy ad/or sparse data. 1.3 Requremets of surface recostructo A prmary goal of early vso s to recover the shapes ad motos of 3D objects from ther mages. To acheve ths goal, we must sythesze vsual models that satsfy a bewlderg varety of costrats. Some costrats derve from the sesory formato cotet of mages. Others reflect backgroud kowledge about mage formato ad about the shapes ad behavors of real-world objects. Explotg dverse costrats combato has prove to be a challege. We eed models whch ot oly tegrate costrats, but whch escape the cofes of covetoal represetatos that mpose smplfyg assumptos about shape ad moto. Computatoal vso calls for geeral-purpose models havg the capablty to accurately represet the free-form shapes ad orgd motos of atural objects---objects wth whch the huma vsual system copes routely. Clearly, we are lookg for models that ca accommodate deformato, ocovexty, oplaarty, exact symmetry, ad a gamut of localzed rregulartes. ay results are avalable cocerg relable recostructo of objects usg dfferet methods. However, there s stll a eed for mproved recostructos sce each specfc method has ts ow drawbacks ad defceces. A recostructed surface s a termedate represetato to brdge the gap betwee sesor data ad a symbolc descrpto of a surface. A deal framework or algorthm for recostructo should have several propertes [97] that arr eloquetly expresses [113] the followg codtos: R1. Recostructo must be varat wth respect to vewpot, that s, to rotatos ad traslatos of the surfaces beg recostructed. Ths s especally mportat whe recostructo s part of a object recogto system. I ths case a chage ths termedate represetato may cause a chage ay symbolc descrpto that s derved, resultg falure to detfy objects a scee. R. It s desrable to fd dscotutes both depth ad oretato. A recostructo algorthm, f detecto of dscotutes s ot smultaeously carred out the recostructo process, should at least sharply preserve regos ear dscotutes for a later stage of dscotuty detecto. R3. A recostructo framework should coduct to computatoally effcet algorthms 5

14 We use these requremets as a gude for our proposal of surface recostructo, although t s extremely dffcult, f ot mpossble, to fully satsfy them. Nevertheless, from the early work of arr [113] t has bee show that problems of computer vso are a set of llposed problems, so we have cosdered regularzato theory as the most atural tool for dealg wth these problems, partcular surface recostructo. The purpose of ths thess s ot oly to derve a method for surface recostructo but to provde a framework ad mathematcal formulato for thkg ad corporatg kowledge about the problem. 1.4 The thess objectves Geeral Our goal s to gve a geeral approach ad a theoretcal framework for solvg the problem of surface recostructo from rage mage data, corporatg both local ad global formato about the surface Specfc 1. Formulato of surface recostructo as a varatoal problem terms of verse theory.. To establsh a regularzato approach ad a mathematcal framework for surface recostructo able to satsfy data ad smoothess costrats o the surface. 3. To obta crtera for the selecto of smoothg fuctoals ad regularzato parameters that satsfes local or global features. 4. Valdato of the proposed framework for surface recostructo wth pot clouds from dfferet objects, usg crtera as accuracy ad computatoal complexty. Surface recostructo s oe of the fudametal problems of computer vso, a class of extremely ll-posed verse problems that has orgated two braches of fuctoal aalyss: (a) The theory of geeralzed verses, whch s a exteso of the oore-perose verse of a matrx ad (b) The regularzato theory of verse problems. Gve the dffculty of the problem we are dealg wth, t s ot eough to use the theory of geeralzed verses. As a cosequece we apply regularzato theory as a varatoal problem o fte-dmesoal fucto spaces. 6

15 Naturally, tryg to represet a completely arbtrary surface wth a fucto f s ext to mpossble. Therefore oe should make some assumptos about ths fucto. The most commo assumptos are explct assumptos about the shape, or the form of the fucto. We fulfll these objectves cosderg our fuctos ad our data as lear fuctoals o Schwartz s dstrbutoal spaces (geeralzed fuctos) [40,65,66,15,163,164,0] ad aalyzg the problem uder the pot of vew of some geeralzatos of Tkhoov regularzato. From the practcal pot of vew, we to obta explct represetatos for a surface the form f :, makg terpolato or approxmato of rage data D {( a, ) } z. The fucto 1 f may the be used as a model or represetato of the real object or pheomeo from whch data are take. We approach the problem of surface recostructo wth a geeralzato of the stadard regularzato (1.1) that satsfy the above metoed requremets (R1, R, R3). The regularzato fuctoals come from a semorm R[ f ] Pf, where P s a dfferetal operator ad s a approprate fucto space, so we ca make dfferet choces for R[ f ]. 1.5 Proposed model Our framework uses two regularzato fuctoals R 1, R, oe fuctoal for global smoothess ad other for localty; wth ther correspodg regularzato parameters 1,, cluded the varatoal problem 1 (1.) m [ f ] ( f ( a) z ) 1 R1 [ f ] R [ f ], f 1 where R 1, R are fuctoals take from the famly of semorms m! R[ f] Jm[ f] f ( x ) dx. (1.3)! m These semorms clude ad geeralze to dmesos the Tkhoov orgal proposal p m d f [170] gve by the orm R[ f ] wm ( ) dx. m dx m0 A semorm has the same propertes of a orm wth the dfferece that ts ull space ca be dfferet from zero. Ths property makes semorms less restrctve tha orms. Furthermore, t ca be proved that a semorm s a eough codto for solvg the optmzato problem [78,79,48,1]. 7

16 We provde a abstract framework for the soluto S( x) of our proposal (1.) ad obta these solutos as traslates of a bass fucto :[0, [, the form S( x) j ( x a j ) p( x). (1.4) j1 These sples are lear combatos terms of the fudametal solutos (Gree s m fuctos) of operators formed by dfferet combatos of, where s the Laplaca m m1 operator ad f ( f). They clude Th Plate Sple (TPS) ad sples teso [63, 178]. For example, whe m, Jm[ f ] represets the potetal eergy of deformato of a th plate, so f a elastc materal s deformed to satsfy the costrats f ( a ) z, the plate fts these pots, mmzg ts potetal eergy f f f J[ f ] ( ) ( ) ( ) dx dy. (1.5) x xy y I ths approach, we satsfy requremets (R1, R, R3) of surface recostructo the followg way: R1. Our solutos to regularzato framework have radal form gve by (1.4). As a cosequece they are varat to rotatos ad traslatos. R. We use local ad global fuctoal wth two regularzato parameters oe for localty ad other for global smoothess. R3. Ad they have robust mplemetato because they have explct mathematcal expressos as lear combatos of traslates of a bass fucto whose weghts are foud solvg a system of lear equatos. Ths work s maly spred arr ad Poggo [ ] wth ther formulato of computatoal vso as a set of verse problems; D. Terzopoulos [ ], the addto of proper fuctoal for mprovg stadard regularzato, ad J. Ducho [47-50] hs dstrbutoal approach for the soluto of varatoal problems. We preserve the ma vrtues of these three vewpots ad we also provde some cotrbutos that make a dfferece wth them. For example, Terzopoulos [167] do ot gve demostratos of our explct expressos S( x) for the soluto of the varatoal problem, stead, he used fte elemet methods for maagg dscotutes; we do t by tug the regularzato parameters. O the other sde, Ducho [48] hadle varatoal problems a dstrbutoal approach but he does ot cosder osy data. 1.6 Cotets ad cotrbutos Ths work s based ad at the same tme s a geeralzato of these prevous publcatos of the author [19-131] 8

17 otegraaro, H., Esposa, J. (007). Regularzato approach for surface recostructo from pot Clouds. Appled athematcs ad Computato.188, otegraaro, H. (008). A ufed framework for 3D recostructo. PA - Proc. Appl. ath. ech. 7, otegraaro, H., Esposa, J. (009). A geeral framework for regularzato of - dmesoal verse problems. VII Cogreso Colombao de odelameto Numérco. Bogota.009 I the frst paper [19] we made a proposal of regularzato for surface recostructo from pot clouds the framework of radal bass fuctos. I the secod paper [130] we make a geeralzato to several kds of data ad the thrd oe we cotrbute some practcal applcatos. Now, ths thess our work goes further order to formulate the geeral problem of surface recostructo terms of verse problems theory. We are showg sples ad radal bass fuctos as a partcular case of our framework. The key dea s the exteso of Tkhoov regularzato to clude global ad local characterstcs of data to be recostructed. We propose a regularzato approach uder a ufed varatoal framework that cludes fdelty to data ad geeralzato of the surface ad makes a proper choce of the a pror kowledge the form of regularzato fuctoals. Tradtoally, surface recostructo problems ad ther solutos are cosdered for partcular problems, ad dfferet methods have bee developed for each case. Ulke ths, we face the problem wth a tegrated framework. The advatage of ths approach s that provde the egeerg commuty wth a tool that helps to brdge the dstace betwee theory ad applcatos. Our approach les heavly o verse problems theory ad approxmato theory, so we ca obta robust methods ad algorthms. Up to ow, may researchers do ot have pad eough atteto to crtera for choce of regularzato fuctoals. We tackle ths problem selectg these fuctoals by physcal ad geometrc crtera, provdg a heurstc reasog that erches the mathematcal formulato. I chapter 1, we state the problem of surface recostructo from rage data ad ts mportace for egeerg ad techology. We the detfy the questo as a verse problem wth may applcatos the so called verse egeerg ppele, makg emphass recostructo from pot clouds. We the expla the purpose ad cotrbutos of the thess, maly spred : arr ad Poggo wth ther formulato of computatoal vso as a set of verse problems; D. Terzopoulos, the addto of proper fuctoal for mprovg stadard regularzato, ad J. Ducho hs dstrbutoal approach for the soluto of regularzato. Next, chapter we place our work to the past ad preset of surface recostructo. We revew the state of the art ad make a classfcato of methods for surface recostructo. Curretly there s a wde varety of methods; a stuato whch may produce cofuso amog developers of applcatos; so ths problem deserves a coceptually based 9

18 taxoomy. We propose to classfy the dfferet procedures as: geometrc, statstcal ad regularzato methods. I chapter 3 we state oe of our cotrbutos to the phlosophcal bass of verse problems, showg a set of problems as ose, complexty ad others, whch should be detfed ad tackled surface recostructo problems. We have called them the seve trsc problems of surface recostructo, ad ts statemet s doe terms of how regularzato theory cotrbutes to ther soluto. I chapter 4 we do the mathematcal statemet of ths thess problem: The problem of surface recostructo from pot clouds. Usg deas from fuctoal aalyss, we make a tegrated treatmet of the problem, showg how er product spaces ad semorms provde a abstract settg for surface recostructo. Our framework uses two regularzato fuctoals, oe for global smoothess ad other for localty; wth ther correspodg regularzato parameters, we dstgush two basc problems: terpolato ad smoothg. The frst alteratve ca be appled o exact data, the secod oe, for osy data. Oe mportat fact s that our proposal of regularzato cludes ad solves both problems a ufed varatoal framework. I chapter 5, we show how the study of semorms, provde us a set of crtera for choosg the fuctoals whch later wll be used as regularzers. Although theoretcally may fuctoals could be used for ths task, some extra crtera or costrat should be mposed to ts choce. We show that physcal ad geometrcal crtera provde a frm gude ths decso. Semorms assocated wth classcal problems of cotuum mechacs, as for stace the cocept of deformato potetal eergy, are show to be very useful ad erch the solutos of varatoal problems wth very suggestve heurstc terpretatos. Ths terpretato eables us for hadlg global ad local features terms of regularzato fuctoals. Chapters 6, t s the most theoretcal oe ad provdes strog ad very refed mathematcal tools. Here we use Schwartz dstrbuto theory (geeralzed fuctos) for -dmesoal surface recostructo. Dstrbutoal fucto spaces combe cotuous ad dscrete fuctos a ufed framework, makg possble the recostructo of scattered data whch are the bass of moder meshless methods. Wth dstrbuto theory, we ca develop a successful geeralzato of mmum orm problems from oe to dmesos. Although ths treatmet of verse theory s more abstract tha usual ad requres a bg amout of fuctoal aalyss, our reward s a theory that provdes costructve proofs ad mplemetable algorthms. Followg ths framework, we fd that some deas of classcal aalyss, as reproducg kerel Hlbert spaces ad Gree s fuctos yeld very practcal results for costructg terpolatg ad smoothg sples. Chapter 7 cotas applcatos ad algorthmc mplemetatos; here we apply the deas developed the former chapters to obta cocrete ad explct solutos to surface recostructo. Here s show the way whch varatoal methods are appled to terpolato of pot clouds data ad ts solutos are obtaed as lear combato of traslates of radal bass fuctos, mplemetable as the soluto of a lear system. The 10

19 deas from dstrbuto theory are combed wth Hlbert space propertes a geeral abstract framework that cludes ( ms, ) -sples ad -sples. I chapter 8 we preset geeral results for our proposed model, cosderg data wth or wthout ose. The most terestg aspect s that these results provde geeral solutos the form of radal fuctos that clude two regularzato parameters. I the same way, we obta practcal algorthms for terpolato ad approxmato of rage data. Fally chapter 9, the propertes ad applcablty of our models are aalyzed usg exact ad osy data. We use dfferet crtera order to fd the best performer amog a famly of sples that resulted as a cosequece our proposed model of regularzato. We fd that these sples ft the propertes we are lookg for, about the detecto of local ad global features the data. From these sples oly th plate sple s well kow. The aalyss shows that our results are a good alteratve for problems of recostructo, performg equally well to TPS ad better tha t complex stuatos (for example, osy data), where models better tha TPS are ecessary. I chapter 10 we make coclusos ad future research. 1.7 Bref cotets by chapter Chap 1. Chap. Chap 3. Chap 4. Chap 5. Chap 6. Chap 7. Chap 8. Chap 9. Chap 10. Statemet of thess problem ad ts relevace egeerg ad scece The state of the art ad methods of surface recostructo The key problem terms of regularzato theory Regularzato theory surface recostructo The physcal meag The math tools Iterpolato Smoothg Applcatos Coclusos 11

20 Fg. 1.1 Structure of the chapters The logcal depedece of chapters (Fg. 1.1) ca be used by dfferet readers, the followg maer: Iverse problems applcatos: Chapters 1,,3,5,8 Dstrbuto theory ad ts applcatos: Chapters 6, 7, 8 Applcatos of sples: Chapters 4,6,7,9 1

21 Chapter Revew ad classfcato of methods for surface recostructo. 1 Itroducto The computer graphcs, computer-aded desg ad computer vso lteratures are flled wth a amazgly dverse array of approaches to surface descrpto. The reaso for ths varety s that there s o sgle represetato of surfaces that satsfes the eeds of every problem every applcato area. Apart from beg ll-posed, the problem of surface recostructo from uorgazed pot clouds s challegg because the topology of the real surface ca be very complex, ad the acqured data may be o-uformly sampled ad cotamated by ose. oreover, the qualty ad accuracy of the data sets deped upo the methodologes whch have bee employed for acqusto. Furthermore, recostructg surfaces from large datasets ca be prohbtvely expesve terms of computatos. ethods to dgtze ad recostruct the shapes of complex three dmesoal objects have evolved rapdly recet years. The speed ad accuracy of dgtzg techologes owe much to advaces the areas of physcs ad electrcal egeerg, cludg the developmet of lasers, CCD s, ad hgh speed samplg ad tmg crcutry. Such techologes allow us to take detaled shape measuremets wth precso better tha 1 part per 1000 at rates exceedg 10,000 samples per secod. To capture the complete shape of a object, may thousads, sometmes mllos of samples must be acqured. The resultg mass of data requres algorthms that ca effcetly ad relably geerate computer models from these samples. Surface recostructo from uorgazed data set s very challegg three ad hgher dmesos. Furthermore the orderg or coectvty of data set ad the topology of the real surface ca be very complcated three ad hgher dmesos. A desrable recostructo procedure should be able to deal wth complcated topology ad geometry as well as ose ad o-uformty of the data to costruct a surface that s a good approxmato of the data set ad has some smoothess (regularty). oreover, the recostructed surface should have a represetato ad data structure that s ot oly good for statc rederg but also good for deformato, amato ad other dyamc operato o surfaces. Noe of the preset approaches possess all of these propertes. 13

22 I geeral, surfaces ca be expressed a mplct or explct form. A explct form of a surface s the graph of a fucto of two varables. 1 Defto.1. Let f :. The graph of f t s the subset of cosstg of the pots ( x1,, x, f ( x1,, x)), for ( x1,, x ) a ope subset of. I symbols graph f {( x,, x, f ( x,, x )) : ( x,, x ) }, for the case 1 the graph s, tutvely speakg, a curve, whle for, t s a surface. I the cotext of vso, the depth maps produced from stereo algorthm or a rage fder ca be terpreted as the sample data set from a surface the explct form z f ( x, y), where z s the dstace from the vewer to the object pots a scee ad ( xy, ) are the mage plae coordates. The mplct form of a surface 3 f : ; f ( x, y, z) costat, 3 s expressed as the fucto where ( x, y, z) are the Cartesa coordates of the surface pots. Of course, the explct form s a specal case of the mplct equato. I fact, all surfaces the explct form ca be trasformed to a mplct form but ot vce versa. Aother useful represetato of a 3 surface s the parameterzed form. A parametrc surface s a smooth map 3 x :, where s a coected ope set such that d x p has rak for each p. I ths way the surface wll be a set of pots x of the form x( u1, u ) ( x1 ( u1, u ), x ( u1, u ), x3 ( u1, u )), ( u1, u). Explct (boudary) represetatos descrbe the surface terms of pot coectos, ad tradtoal approaches are based o Delauay tragulato ad Voroo dagrams [6,5]. Aother well kow explct approach s a parametrc surface descrpto based o NURBS [140,149]. Oe example of surface-oreted soluto, proposed [140,90] s based o the computato of the sged Eucldea dstace betwee each sample pot ad a learly regressed plae that approxmates the local taget plae. Curless ad Levoy [38] developed a explct algorthm tued for laser rage data, whch s able to guaratee a good rejecto of outlers (pots whose coordates were ot correctly acqured). Aother well-kow approach s the α-shape [53,54], whch assocates a polyhedral shape to a uorgazed set of pots through a parameterzed costructo. Now we propose a classfcato for surface recostructo methods. Our approach ca be compared to prevous works the areas of shape represetato, recostructo, smoothg, ad surface regularzato. The large umber of publshed methods requres a comprehesve classfcato, but t s early mpossble to perform a perfect survey. We descrbe some of the most well-kow approaches, wth a bas towards those more closely related to our regularzato framework. 14

23 . Geometrc methods Oe of the key cocepts these methods s a tragular mesh, because t s obtaed drectly from the data; ths s a smple represetato of the topology of the object, but f other propertes are requred, a better represetato s eeded. I ths cotext the problem of surface recostructo ca be stated as follows: Let be a surface of object O, ad assume that s a smooth twce-dfferetable two dmesoal mafold, embedded a Euclda three-dmesoal space 3. Gve a dscrete set of pots A a a a 3 { 1,,, }, that samples the surface ; fd a surface that approxmates, usg the data set A. The recostructed mesh must be topologcally equvalet to the surface of the orgal object. I order to get ths, a good dea s to take advatage of the dfferet geometrc vewpots uder whch oe ca see a object ad ts surface. Ths s doe studyg propertes that may be local, trsc or global; examples of these are respectvely, taget plaes, Gaussa curvature or Gauss-Boet theorem. All the propertes cosdered should be, sooer or later, dscretzable order to be applcable o pot clouds Geometrc methods rage from the smple but very useful tragular mesh to more sophstcated tools that come from felds such as algebrac topology or Dfferetal geometry. These developmets have coducted to a ew brach of geometry dscrete dfferetal geometry [], whose am s to develop dscrete equvalets of the geometrc otos ad methods of classcal dfferetal geometry, where a surface ca be see as a collecto of pots 3. A popular approach computer vso s to recostruct a tragulated surface usg Delauay tragulatos ad Voroo dagrams. The recostructed surface s usually a subset of the faces of the Delauay tragulatos. A lot of research has bee doe wth ths approach [5,90] ad effcet algorthms have bee desged to compute Delauay tragulatos ad Voroo dagrams. Although ths approach s very versatle ad ca deal wth geeral data sets, the costructed surface s oly pecewse lear ad t s dffcult to hadle o-uform ad osy data. Recetly, mplct surfaces or volumetrc represetatos have bee used most frequetly recet years for surface recostructo from pot clouds. These are based o well establshed algorthms of computer tomography lke marchg cubes ad therefore are easly mplemeted. They produce approxmated surfaces, so that error smoothg s carred out automatcally. The method of Hoppe [90] s able to detect ad model sharp object features. Curless ad Levoy [38] ca hadle mllos of data pots. Levoy et al. [106] used ths recostructo method for ther huge Dgtal chelagelo Project. The tradtoal approach [1,133,183] uses a combato of smooth bass fuctos such as blobs, to fd a scalar fucto such that all data pots are close to a socotour of that scalar fucto. Ths socotour represets the costructed mplct surface. However computato costs are very hgh for large data sets, sce the costructo s global whch 15

24 results solvg a large lear system. The secod approach uses the data set to defe a sged dstace fucto o rectagular grds ad deotes the zero socotour of the sged dstace fucto as the recostructed mplct surface [15,9]. The costructo of the sged dstace fucto uses a dscrete approach ad eeds a estmato of local taget plaes or ormals for the oretato,.e. a dstcto eeds to be made betwee sde ad outsde. Usg the sged dstace represetato, may surface operatos such as Boolea operatos, ray tracg ad offset become qute smple [139, 195]. Effcet algorthms [19,184], are avalable to tur a mplct surface to a tragulated surface. I [41] mplct surfaces are used for amato ad the level set method s used for surface recostructo from rage data [36]. I fact the level set method [01] provdes a geeral framework for the deformato of mplct surfaces accordg to arbtrary physcal ad/or geometrc rules. Tragle meshes are so-called ustructured grds [193] ad therefore t s ot possble to use covetoal modelg or sgal processg methods lke tesor product surfaces or lear flters. Ufortuately, o basc theory exsts for hadlg ustructured data order to estmate surface ormals ad curvature, terpolate curved surfaces, ad subdvde or smooth tragle meshes. A approach for geeral meshes was proposed by Taub [173]. He has geeralzed the dscrete Fourer trasformato by terpretg frequeces as egevectors of a dscrete Laplaca. Defg such a Laplaca for rregular meshes allows to use lear sgal processg tools lke hgh ad low pass flters, data compresso ad multresoluto herarches. However, the traslato of cocepts of lear sgal theory s ot the optmal choce for modelg geometry data. Surfaces of three-dmesoal objects usually cosst of segmets wth low badwdth ad trasets wth hgh frequecy betwee them. They have o reasoable shape, as t s presupposed for lear flters. Optmal flters lke Weer or matched flters usually mmze the root mea square (RS) error. Oscllatos of the sgal are allowed, f they are small. For vsualzato or mllg of surfaces, curvature varatos are much more dsturbg tha small devatos from the deal shape. A smoothg flter for geometrc data should therefore mmze curvature varatos..3 Statstcal methods Statstcal methods expla data D {( a, z ) } 1 from a surface, wth a stochastc model by cosderg that the observatos z, costtute a realzato of a stochastc process. Uder ths pot of vew the surface recostructo problem s solved by fdg the respose f (x) of a ukow fucto f at a ew pot x of a set, from a sample of put-respose pars ( a j, f ( a j)) gve by observato or expermet. Now the respose z j at a j s ot a fxed fucto of a j but rather a realzato of a radom varable Z( a j ). It s assumed that for each x there s a real-valued radom varable Z (x) whch replaces f the determstc approach. The purpose s to get formato about the expectato 16

25 EZ ( ( x)) of (x) Z wth bouded postve varace E[ Z( x) E( Z( x)]. If x s close to y the radom varables Z( x) ad Z ( y) wll be correlated. Ths s descrbed by a covarace kerel cov( x, y) : E( Z( x) Z( y)). The stochastc soluto to ll-posed problems s a straghtforward applcato of Bayesa terpretato of regularzato [99]. I a Bayesa approach [188,18,119], the stablzer correspods to a smoothess pror, ad the error term to a model of the ose the data (usually gaussa ad addtve). Followg Gros et al. [70]; for surface recostructo, t s supposed that D {( a, z ) } 1 are osy data obtaed from samplg the fucto f, thus f ( a ) z, 1,,, (.1) where are radom depedet varables wth a gve dstrbuto. The problem s to recover f or a estmate of t from the set of data D. f s the realzato of a radom feld wth a kow pror probablty dstrbuto. The Bayesa terpretato fd the fucto f whch maxmzes the codtoal probablty of the fucto f gve D P[ f D ] by usg Bayes rule P[ f D] P[ D f ] Pf [ ], (.) where P[ D f ] t s the codtoal probablty of D gve f. Pf [ ] s the a pror probablty of the radom feld f, ad embodes the a pror kowledge of the fucto. Assumg the ose varables equato (.1) are ormally dstrbuted wth varace the P[ D f ] 1 e 1 ( f ( a ) z ) Takg a model kow statstcal physcs t s defed Pf [ ] (.) P[ f D] e 1 ( ( ) ) [ ] f a 1 z H f. [ ] e R f, hece replacg. (.3) Oe estmate of the fucto f from the probablty dstrbuto (.3) s the maxmum a posteror (AP) estmate that maxmzes the a posteror probablty P[ f D ] ad therefore maxmzes the expoet (.3). The AP estmate therefore mmze the fuctoal [ f ] ( f ( a ) z ) R[ f ] 1 whch t s the well-kow regularzato model., (.4) 17

26 I statstcal methods 3D objects ca be represeted by regresso models. Fttg data to a model s a basc statstcal tool that has bee frequetly used computer vso. Objects rage mages are smply represeted by the polyomal surface model. Wth the estmated surface parameters, the rage mage aalyss such as recostructo ad segmetato ca be acheved easly. The model parameters are commoly estmated by the least squares (LS) method that yelds optmum results for the Gaussa ose case. However, t becomes urelable f o-gaussa ose (e.g., mpulse ose) s added. Statstcal approaches to surface parameter estmato clude robust estmators such as the -estmator, least meda of squares (LedS) method, least trmmed squares (LTS) method, ad so o [151,11,105]. Robust estmato techques have bee used to recover the parameters of a surface patch because they are robust to outlers. Outlers have values far from the local tred, resultg a large measuremet error. The local wdow sze employed a robust estmator s a user-specfed parameter. The surface parameters are accurately estmated a large homogeeous rego whereas the estmato error becomes large ear abrupt dscotutes. Thus, t s dffcult to select a sgle optmal wdow sze (resoluto) that yelds relable parameter estmato results over a etre mage. Optmal parameters, therefore, ca be estmated dfferet applcatos by tegratg the surface parameters obtaed at varous resolutos [138]. Rage mage recostructo based o the estmated surface parameters ca be regarded as a meas of ose suppresso. Nose that corrupts the rage mages s modeled by Gaussa ad mpulse ose mxture [101]. Waqar et al. [190] develop a techque based o oparametrc kerel desty estmato to robustly flter a osy pot set whch s scattered over a surface ad cotas outlers. Gve a set 3D scattered pots, a desty fucto of the data s estmated, the ma dea of ths flterg approach cossts o defg a approprate desty estmato to determe those cluster ceters whch delver a accurate ad smooth approxmato of the sampled surface. Recetly, robust statstcs ad statstcal learg techques have gaed popularty Computer Graphcs ad have bee successfully appled to surface recostructo [60,95, 160,169]..4 Regularzato methods Regularzato methods are based varatoal prcples. Varatoal methods have bee employed wth cosderable success computer vso, partcularly for surface recostructo problems. Formulatos of ths type requre the soluto of computatoally complex Euler Lagrage partal dfferetal equatos (PDEs) to obta the desred recostructos. These methods have played a mportat role the aalytc formulato of most early vso problems [9,134,176,178]. Early vso has tradtoally bee cosdered as a array of specal recostructo methods operatg o mages. They clude the recostructo of -D mage testy gradet.ad flow felds, as well as the recostructo of 3-D surface depth, oretato, ad moto felds. A broad rage of vsual recostructo problems may be ufed mathematcally as well-posed varatoal prcples. These ca be characterzed as optmal approxmato problems volvg a class of geeralzed multdmesoal sple fuctoals [6-8,3,175]. 18

27 Jea Ducho, a mathematca at the Uversty Joseph Fourer Greoble, Frace, suggested a varatoal approach mmzg the tegral of f, whch also leads to the th plate sples. Ths work was doe the md 1970s ad s cosdered to be the foudato of the varatoal approach to radal bass fuctos [47-50]. The method troduced by Ducho followed the deas of Attéa [10] ad Lauret [104], for the geeral theory of Hlberta sples ad volved a reproducg kerel. The reproducg kerel of Aroszaj [8] or the Hlberta kerel of Schwartz [163] gves the explct characterzato of the ( ms, ) sples. It s wdely kow that the ( ms, ) -sples belog to a varety of radal bass fuctos whch are codtoally postve defte fuctos [57,6,11,158]. Barrow ad Teebaum [1] were amog the frst to troduce the surface recostructo problem vso. They mplemeted the soluto for the terpolato of approxmately uformly curved surfaces from tal oretato values ad costrats. The soluto apples a relaxato algorthm that volves usg a array of smple parallel processes performg teratve local averagg. Ths techque, although very smple, has may drawbacks. It does ot deal wth local ad global propertes, yelds a terpolated surface that s ot varat to 3-D rgd moto, ad s computatoally effcet. The early work of Grmso [78,79] presets a theory of vsual surface terpolato where rage data s obtaed from a par of stereo mages. The theory deals wth determg a best-ft surface to a sparse set of depth values obtaed from the arr ad Poggo [117] stereo algorthm. Grmso mmzed what he called the quadratc varato E (1.5) of the surface f ( x, y ) E fxx fxy f yy dy dx, (.5) where s a rego the mage plae at whch the depth costrats are specfed. Ths fuctoal also happes to be a partcular case of the semorm we are usg ths work ad represets the eergy of a th plate [47-50]. The mmzato yelds the th plate sples; ths fucto was appled earler by Schumaker [16] for terpolato of scattered 3-D data. Grmso [78] developed a teratve algorthm based upo the bharmoc equato that results from applyg Euler s equatos to mmze E. Hs work does ot apply to the geeral surface recostructo problem sce t dd ot deal wth surface or oretato dscotutes. The recostructed surface s varat to mage plae rotatos ad traslatos. Boult ad Keder [9] dscuss a approach for vsual surface recostructo that s based o semreproducg sple kerels of Ducho [48]. Semreproducg kerel sples are defed terms of the reproducg kerels for the sem-hlbert space [8,10]. The major computatoal compoet of the method s the soluto to a dese lear system of equatos. The algorthm results a true fuctoal form of the surface allowg for symbolc mapulatos (e.g., dfferetato or tegrato). If the orm used the problem s sotropc, the the kerel s rotato, traslato, ad scale varat. Hece, the 19

28 recostructo s varat to such trasformatos appled to the data wth respect to the mage plae. Sce the resultg lear system s dese ad ofte defte, ths lmts the approaches that ca be used ts soluto. Terzopoulos [ ] poeered fte-dfferecg techques to compute approxmate dervatves used mmzg the th-plate eergy fuctoal of a heght-feld. He developed computatoal molecules from the dscrete formulatos of the partal dervatves ad uses a mult-resoluto method to solve the surface. He presets a techque for vsble surface computato from multresoluto mages. Hs prmary cotrbuto was mproved computatoal effcecy ad also the tegrato of surface depth ad oretato dscotuty formato to the surface recostructo problem. He used multgrd methods [178] to speed up hs algorthm by orders of magtude over the sgle grd approach. Hs surface recostructo process treats surfaces of objects as th plate patches bouded by depth dscotutes ad joed by membrae strps alog loc of oretato dscotutes. The descrpto of the surfaces thus obtaed s varat to mage plae trasformatos (rotatos ad traslatos mage plae) but ot to three-dmesoal rgd moto. The cotrolled cotuty stablzer the fuctoal that s mmzed by Terzopoulos s smlar to sple teso. Both Grmso [78] ad Terzopoulos [176,177] used the stadard dscrete bharmoc operator or ther surface recostructo algorthms. Grmso used fte-dfferece methods whle Terzopoulos used more geeral fte-elemet techques. I [14], Poggo ad Torre suggest that the mathematcal theory developed for regularzg ll posed problems leads a atural way to the soluto of early vso problems terms of varatoal problems. They argued that ths s a theoretcal framework for some of the varatoal solutos already obtaed the aalyss of early vso processes. Thus the computatoal ll posed ature of these problems dctates a specfc class of algorthms for solvg them, based o varatoal prcples. They show that these varatoal prcples follow a atural ad rgorous way from the ll posed ature of early vso problems. I a seres of papers [71-73], Gros ad coworkers establshed relatos betwee the problem of fucto approxmato ad regularzato theory. The approach regularzes the ll posed problem of fucto approxmato from sparse data by assumg a approprate pror o the class of approxmatg fuctos whch follows the techque troduced by Tkhoov, detfyg the approxmatg fucto as the mmzer of a cost fuctoal that cludes a error term ad smoothess fuctoal. They show that regularzato prcples lead to approxmato schemes that are equvalet to etworks wth oe hdde layer, called regularzato etworks. I partcular, they descrbed how a certa class of radal stablzers-ad the assocated prors the equvalet Bayesa formulato lead to a subclass of regularzato etworks, the already kow radal bass fuctos etwork. Recetly, Turk [183] used varatoal surface ad th plate sple by specfyg locatos 3D through whch the surface should pass, ad also detfyg locatos that are teror or exteror to the surface. A 3D mplct fucto s created from these costrats usg a varatoal scattered data terpolato approach. They call the so-surface of ths fucto a varatoal mplct surface. 0

29 I [150] t s proposed a varatoal techque for recostructg surfaces from a large set of uorgazed 3D data pots ad ther assocated ormal vectors. The surface s represeted as the zero level set of a mplct volume model whch fts the data pots ad ormal costrats. The resultg model use explct solutos terms of radal bass fuctos. Gokme ad L [76] presets a edge detecto ad surface recostructo algorthm usg regularzato, whch the smoothess s cotrolled spatally over the mage space but they oly use oe regularzato parameter. Beatso et al. [14] apply a method for surface recostructo from scattered rage data by fttg a Radal Bass Fucto (RBF) to the data ad covolvg wth a smoothg kerel (low pass flterg). These sples are a partcular case of our approach. There exst well kow relatos betwee regularzato theory ad other varatoal approaches. I [70] Gros et al. show that regularzato prcples coduct to approxmato schemes called regularzato etworks. I partcular, stadard smoothess fuctoals lead to radal bass fuctos. Addtve sples as well as some tesor product sples ca be obtaed from approprate classes of smoothess fuctoals. I the probablstc terpretato of regularzato, the dfferet classes of bass fuctos correspod to dfferet classes of pror probabltes o the approxmatg fucto spaces, ad therefore to dfferet types of smoothess assumptos. Parallel to the problems of computer vso there has bee a great developmet of varatoal approach ad sples surface recostructo from the pot of vew of approxmato theory. These developmets have coducted to Radal bass fucto theory ad meshfree methods [57,191]. Orgally, the motvato came from applcatos geodesy, geophyscs, mappg, or meteorology. Later, applcatos were foud may other areas such as the umercal soluto of PDEs, computer graphcs, artfcal tellgece, statstcal learg theory, eural etworks, sgal ad mage processg, samplg theory, statstcs (krgg) ad optmzato. Doald Shepard, suggested the use of what are ow called Shepard fuctos the late 1960s [165]. Rollad Hardy, who was a geodesst, troduced the so-called multquadrcs (Qs) the early 1970s. Hardy s work was prmarly cocered wth applcatos geodesy ad mappg [85]. Robert L. Harder ad Robert N. Desmaras [84] who were aerospace egeers at acneal-schwedler Corporato (SC Software), ad NASA s Lagley Research Ceter, troduced the th plate sples 197. Ther work was cocered mostly wth arcraft desg. eguet troduced what he called surface sples the late 1970s. Surface sples ad th plate sples fall uder what are called as polyharmoc sples [1-15]. Rchard Frake [63,64], a mathematca at the Naval Postgraduate School oterey, Calfora, compared varous scattered data terpolato methods, ad cocluded Qs ad TPSs were the best. Frake also cojectured that the terpolato matrx for Qs s vertble. adych ad Nelso proved Frake s cojecture based o a varatoal approach [105]. Charles cchell, also proved Frake s cojecture [16-18]. Hs proofs are rooted the work of Bocher, from 193 [3,4] ad Schoeberg (from 1937) [158,159]; o postve defte ad completely mootoe fuctos. Grace Wahba, a statstca at the 1

30 Uversty of Wscos, studed the use of th plate sples for statstcal purposes the cotext of smoothg osy data ad data o spheres, ad troduced the ANOVA ad cross valdato approaches to the radal bass fucto settg [ ]. Robert Schaback, troduced compactly supported radal bass fuctos (CSRBFs) [154], ad a very popular famly of CSRBFs was preseted by Holger Wedlad hs Ph.D. thess [19] ad [191]. Both of these authors have cotrbuted extesvely to the feld of radal bass fuctos for surface recostructo [155,156].

31 Chapter 3 The seve trsc problems of surface recostructo 3.1 Itroducto The recovery ad represetato of 3-D geometrc formato of the real world s oe of the most challegg problems computer vso research ad ts lterature s flled wth a amazgly dverse array of approaches to surface descrpto. The reaso for ths varety s that there s o sgle represetato of surfaces that satsfes the eeds of every problem every applcato area. Apart from beg ll-posed, the problem of surface recostructo from uorgazed pot clouds s challegg because the topology of the real surface ca be very complex, ad the acqured data may be o-uformly sampled or cotamated by ose. oreover, the qualty ad accuracy of the data sets deped upo the methodologes whch have bee employed for acqusto. Furthermore, recostructg surfaces from large datasets ca be prohbtvely expesve terms of computatos. Commoly, recostructo problems correspod to solvg a compact operator equato. But compact operators caot have a bouded verse. Ths meas that the problems we are dealg wth are trscally ll-posed. I ths chapter we study ths ad other dffcultes that arse whe cosderg surface recostructo from scattered data. 3. The problems of surface recostructo Surface recostructo shares a commo structure wth the rest of verse problems whch the put-output relato the operator equato Af z, takes the form of a Fredholm tegral operator of the frst kd wth ( A f)( s) K ( s, t ) f ( t ) dt, where K( s, t ) a s called the kerel of the operator ad represets a smplfed model of the process. The assumpto of learty s very mportat, because spte of the creasg power of computers, oly the case of lear problems t s possble to get a soluto almost real tme for large scale problems. The source of great mathematcal problem s that, o all three couts of Hadamard codtos, Fredholm tegral equatos of the frst kd are ll-posed. These are compact operators, ad the caot have a bouded verse [100,17,13]. 3 b

32 We clam that these lmt stuatos have a trsc ature; therefore we should look for, ot a defte soluto for them but for the best way to deal wth them. Next, we study those whch we cosder the most crtcal trsc problems of surface recostructo from scattered data gvg a way to treat the problem usg the propertes of our regularzato framework Reproducto-geeralzato dlemma I surfaces recostructo from scattered data A { a1, a,, a }, the resultg fucto f that models the surface should be such that t geeralzes well,.e. t should gve practcally useful values of f ( x) to ew pots x A. Furthermore, t should be stable the sese that small chages the trag data do ot chage f too much. However, these goals are coflct wth good reproducto of data. Sometmes we could obta a hghly stable but useless model, whle over fttg occurs f there s too much emphass o data reproducto, leadg to ustable models wth bad geeralzato propertes. Problem: Surface recostructo s subject to the reproducto-geeralzato dlemma ad eed a careful balace betwee geeralzato ad stablty propertes o oe had ad data reproducg qualty o the other. Ths s called the bas-varace dlemma uder certa probablstc hypotheses, but t also occurs determstc settgs. Ths problem ca be llustrated by the ucertaty prcple ts multple forms. For example, t s mpossble for a sgal f to be to be both badlmted ad tme-lmted; that s, t s mpossble for f ad ts Fourer trasform f both to vash outsde a fte terval uless f s detcally zero. Soluto: I our regularzato approach (3.1), we treat the reproducto-geeralzato dlemma o surfaces, usg oe term for reproducto or fdelty to data ad other for geeralzato, wth two fuctoals (local ad global) that hadle dfferet degrees of smoothess by tug parameters 1 ad the followg scheme 1,, z f ( ( ) ) F f a z m [ ] f 1 data reproducto R [ f ] R [ f ]. (3.1) Geeralzato 3.. Restrctos o the choce of spaces Problem: Gve a surface recostructo problem from scattered data, there s ot eough formato to come up wth a useful soluto of the recostructo problem. I partcular, the smple umbers do ot say how to defe f or whch space of fuctos to pck t from. We are facg a ll-posed problem wth plety of dstgushable approxmate solutos. 4

33 Surface recostructo problem requres mmum codtos order to be well posed. Ths s doe by workg a lear or vector space. There are also practcal reasos for ths, because: If the values z f ( a ), from the surface are multpled by a fxed scalar factor, j j the the ew data should be recovered by the fucto f stead of f. If data z j f ( a j) ad z j g( a j) at the same locatos A { a1, a,, a } are recovered by fuctos f ad g, respectvely, the the data f ( a ) g( a ) should be recovered by the fucto f g. But, from a practcal pot of vew, all mathematcal a-pror assumptos o f are useless f we are ot takg the applcato to accout. For example, f we take a lear subspace F of dmeso of a space of multvarate fuctos such a maer that F does ot deped o the set A { a1, a,, a }. There wll always be a degeerated set. It s to say, there exst a fucto g F, dfferet from zero that vashes o. Ths s udesrable because we ca add to the soluto f, all the fuctos of the form g, wthout alterg the reproducto of data. Furthermore, the recovery process wll have a o-uque soluto ad wll be umercally ustable (see arhuber-curts theorem [191]). Soluto: To defe the problem of surface recostructo to fuctoal spaces wth data depedet er products. We do ths chapter 7 (see formula 7.1). Ths s the reaso for obtag solutos the subspace F A, K { j K( x, aj ) : j } The Curse of dmesoalty j1 The curse of the dmesoalty (term coed by Bellma 1961 [16,17]) refers to the followg Problem: the absece of smplfyg assumptos, the sample sze eeded to estmate a fucto of several varables to a gve degree of accuracy grows expoetally wth the umber of varables. Let us say that we wated to do a secod-order polyomal ft. I 1D ths s smply y a0 a1x ax. I D, we have y a0 a1 x a y a3 x y a4 x a5 y, we have goe from three to sx terms. I 3D we eed te terms: y a0 a1x a y a3z a4x y a5x z a6 y z a x a y a z j j 5

34 k It s kow that s the dmeso of k ( ), the as we crease the order of the polyomal, the expoet of the umber of terms requred creases accordgly. Soluto: If our problem s to approxmate the surface f, by the soluto S( x) of the regularzato problems (3.1). The solutos (as we proof chapter 6) ca be wrtte as lear combatos of traslates cotag the orgal data, the form S( x) jk( xa), (3.) j1 d the, gve the approxmato error, wth data A { a1, a,, a }, a d - dmesoal space, we are lookg for approxmatos wth a degree of smoothess s such that f j K( x a ). (3.3) j1 1 s d Usually [69], s such that. It s commo to say that the curse of dmesoalty s the d factor ad the blessg of smoothess s the s factor. Ths meas that fxg the dmeso d of the surface, f we wat to "beat the curse of dmesoalty" we have to crease the smoothess degree s. We do t to our approach, by defg regularzato fuctos spaces for whch the rate of covergece to zero of the error s depedet of ay umber of dmesos (see [69]). Ths avodace of the curse of dmesoalty s due to the fact that these fuctos are more ad more costraed as the dmeso creases. Examples clude spaces of the Sobolev type, whch the umber of weak dervatves ( the sese of Schwartz dstrbutos) s requred to be larger tha the umber of dmesos Nose Problem: Nose s ubqutous measured data. Nose t s a dsturbace that affects a sgal ad that may dstort the formato carred by the sgal. I the case of observatoal data, ose may have may sources cludg quatzato ad ose from mage vdeo systems. Importat object features, such as corer pots or edges are ot drectly recorded; stead, they have to be modelled from data a separate process. I rage sesor for pot clouds, ose may have very hgh frequecy due to measuremet errors. Some recostructo methods teds to smooth ay hgh frequecy feature such as corers ad wrkles that belogs to the geometry of the object creatg accuracy the recostructed object. ay of the errors that are caused by the measurg process (ose, alasg, outlers, etc.) ca be fltered at the level of raw sesor data. A 6

35 specal class of errors (calbrato ad regstrato errors) frst appears after mergg the dfferet vews. Surface mesh models bult usg measuremet data obtaed usg 3D rage scaers ecessarly cotas some type of ose. To remove the ose surface mesh models, may mesh deosg algorthms have bee developed, for example [34,37,41,44,157] ad other refereces there. I evaluatg the effectveess of deosg algorthms both vsual ad umercal comparsos are used [170]. For meshes correspodg to real scaed data, however, we ca most cases oly perform vsual comparsos, because groud truth data requred for evaluato are almost always uavalable. However, to provde a more objectve evaluato ad more through testg, umercal comparsos are requred. Havg sythetc models of real scaer ose would be useful for evaluatg deosg algorthms. Ay sythetc model used for evaluatg deosg algorthms should take a exact model surface ad add ose, whch has to be removed by the algorthms. The ose should have the same characterstcs as ose real measuremet data. Expermetally, measuremet ose s ofte assumed to be Gaussa a wde rage of dscples. Thus, varous mesh deosg algorthms have bee developed based o the Gaussa ose assumpto [39], whle others used sythetc models wth Gaussa whte ose (that s, depedet Gaussa ose per mesh vertex) evaluato of ther algorthms [34,96,170,196,198]. Pot datasets routely geerated by optcal ad photometrc rage fders usually are corrupted by ose. I order to remove these defceces from scaed pot clouds, a large varety of deosg approaches based o low-pass flterg [110], LS fttg [5,43,10] ad partal dfferetal equatos (PDEs) [103] has bee proposed. A very mportat cosequece of beg ll-posed s that mathematcal modellg of the surface recostructo caot uquely cosst establshg the equatos relatg the data to the soluto; t must also clude a model of the ose perturbg the data ad, as far as possble, a model of kow propertes of the soluto. The the data are a lear trasform of a orgal sgal f corrupted by ose, so that we have z Kf ( a ), (3.4) where K s some kow compact lear operator defed o a Hlbert space ad s a whte ose wth ukow varace. Sce K s compact, the equato (3.3) caot be 1 drectly verted sce K s ot a bouded operator. Soluto: The soluto S ( x) to our regularzato schemes (3.1) may be cosdered as the result of applyg a low-pass flter to the data. I frequecy space t ca be show that cotrols the half-power pot of the flter ad m ( m refers to semorm Jm[ f ] (1.3)) the steepess of the roll-off [189]. Further, uder certa codtos the regularzato s fuctoal R[ f ] Jm[ f ] ca be wrtte as R[ f ] f ( ξ) ξ dξ. If ths semorm s defed the Hlbert space assocated wth a kerel K ad ts Fourer trasform m K( ξ) ξ, the semorm reduces to the form 7

36 f ( ξ) R[ f ] dξ, (3.5) K() ξ ths case K () ξ ca be vewed as a low-pass flter [189,70], whch plays a roll of smoothg the sese that the soluto surface f ca be vewed as the covoluto of the observato wth a smoothg flter f ( x) z( x) B( x), where deotes the operato of covoluto product ad B( x) s the verse Fourer trasform of B () ξ gve by K() ξ B() ξ + K() ξ. (3.6) Whe 0 f z, the B( ξ) 1, B( x) ( x), whch correspods to a terpolato problem the oseless stuato [70] Error bouds Error estmates ad error bouds are fudametal ay approxmato process. Remember that the surface recostructo problem preteds to recover the surface f from data D {( a, ) } z 1. From ths formato s desred to compute aother value f ( x). As t s usual approxmato theory, we are lookg for a elemet S a subset or subspace ad the use S ( x) to approxmate f ( x) wth a error, such that f( x) S( x) F. (3.7) It s desrable to obta a a pror measure of the error. But geeral, the formato we have about f s that f s a elemet of a kow lear space of fuctos F ; the the value S ( x) at a fxed pot x s a lear combato F. 1 f ( x) of certa elemets f of Problem: The trsc problem here s that t s mpossble to gve fte bouds for error, terms of f ( a1 ),, f ( a ), f the oly addtoal formato s that f s the elemet of a lear space F. oreover, t ca be demostrated that t s mpossble to obta fte bouds for f the oly addtoal formato s that f s a elemet of F (see [77]). Ths s true o matter how restrcted the fte-dmesoal space F s by codtos of cotuty, dfferetablty, aalytcty, ad so o. Therefore s meagless to speak of the goodess of a lear approxmato wthout referece to some olear costrat F. 8

37 Soluto: The varatoal propertes of regularzato method provde the addtoal formato ecessary for calculatg bouds of the approxmato error. The basc dea s to buld a sem-er product (, ) F, usg data ad the varatoal propertes of regularzato fuctoals (sem-orms), the usg orthogoalty propertes ad Cauchy- Schwarz equalty s possble to obta error bouds the form f S ( f S, f S) F ( f, f S) F f F f S. (3.8) Ths s the most frequet method for obtag error bouds surface recostructo methods (see [77,109,7,8]) The Computatoal complexty A mportat questo about surface recostructo s what the theoretcal lmtatos are o the speed of terpolato ad approxmato algorthms. The computatoal complexty of ay procedure or algorthm ca be cosdered or expressed by meas of a fucto T ( ) that gves ts executo tme terms of the sze of the put. Problem: The tasks assocated wth recostructo problems may have a very hgh computatoal complexty, cludg ssues as amout of data, mplemeted algorthms, or the avalable hardware. I ths thess, we obta terpolators wth the form S ( x) j( x a j ) p( x), (3.9) j1 where the bass fuctos ( x a ) are the Gree s fuctos of the Gram s operator j assocated wth the stablzer. If the stablzer exhbts radal symmetry, the bass fuctos are radally symmetrc as well ad a radal bass fucto s obtaed. s the umber of ceters or pots the cloud. I order to fd, 1,,, we have to solve the lear system A I P z t 0 0, (3.10) P wth A, j ( a a ) j,, j 1,,, P j pj ( a ), 1,,, j 1,, N, z ( z1,, z ), N such that ( 1,, ), β (,, ) 1, ad the polyomals{ p } N form j j 1 a bass for ( ) 1 (see chapters 6, 7 for more detals). m 9

38 The dmeso of the lear system s equal to the umber of cetres, gvg a 3 complexty T( ) O( ). If we cosder that usually a pot cloud may have oe mllo cetres or more, t s easy to mage the huge umber of floatg pot operatos we are dealg wth. The terpolato matrx A s ot sparse ad, except for symmetry, has o obvous structure that ca be exploted solvg the system. Soluto va a symmetrc solver wll 3 requre O ( ) flops. oreover, a sgle drect evaluato of expresso (3.4) 6 requres O ( ) operatos. Soluto: There exsts a dversty of solutos Smplfy the data set reducg the umber of ceters a order to make them computatoally maageable (see [15,191]). The computato of some bass fuctos ( x a ) requre more floatg pot operatos, so we could choose the most coveet. Fast methods for solvg the terpolato matrx A ad the evaluato of S ( x). For example, Beatso ad Powell [14] proposed to use the smoothess of the terpolat ad requred the cardalty propertes be satsfed at a small umber of pots the doma. Ths method was further developed by the groups of Powell ad Beatso for obtag a fast method. The culmato of ths work was a teratve Krylov subspace algorthm [59,80]. j 3..7 Local ad global propertes Vsual surfaces have both mcroscopc ad macroscopc propertes. A pot cloud of a scaed surface may come from objects of dfferet form ad topology. So the shape of the surface vares from smple to a hghly complex oe, wth rapd varatos the local surface oretato. Ths complexty ca be measured wth the dervatves of the surface. A surface that curves cotuously, wthout breaks or creases ad cusps wll be referred to as a regular surface. ore exactly, f f : s a dfferetable fucto a ope set of, the the graph of f, that s the subset of 3 gve by ( x, y, f ( x, y )) for ( xy,, ) s a regular surface. Regularty s measured by the smoothess degree of the surface. Ths s a herarchcal cocept related to the order of cotuty the surface s partal dervatves. Loosely speakg, the more cotuous the dervatves, the smoother the surface. Problem: A surface recostructo approach should be able to reproduce local ad global behavour of the surface. 30

39 Gve a set of scattered data A { a1, a,, a } o a surface, there wll be may possble surfaces cosstet wth these tal pots. How do we dstgush the correct oe? athematcally, we eed to be able to compare two possble surfaces, to determe whch s better. Ths ca be doe by defg a fuctoal Rf [ ] from the space of all possble surfaces to the real umbers, so that comparg surfaces ca be accomplshed by comparg correspodg real umbers, provded R[ f ] R[ g] wheever surface f s better tha surface g. The best surface to ft through the kow pots s the oe that mmzes R[]. Soluto: These fuctoals are chose such a way that permts dfferet smoothess degree. Oe attractve formal characterzato of smoothess s readly related to physcal models, partcular we apply fuctoals comg from elastcty theory to deduce explct expressos that performs local approxmato ad fts a global surface to tal data (see chapter 5). 31

40 Chapter 4 Regularzato ad verse theory surface recostructo 4.1 Itroducto I surface recostructo oe s faced wth the task of dscoverg the ature of objects that produced the pot cloud for the surface receved by the camera. A proper computatoal formulato of these problems recogzes that they volve verse processes that are mathematcally ll-posed. Ths allows oe to derve systematcally, computatoal schemes based upo regularzato theory that esure exstece of soluto ad stablty of verso process. Such approaches ofte suggest partcular types of algorthms for effcet soluto to the problems. The cocepts of regularzato theory gve a comprehesve framework to formulato of the problems vso: at all three levels of problem, algorthm, ad mplemetato. Furthermore, the mathematcal theory of regularzato provdes a useful theory for corporatg pror kowledge, costrats, ad qualty of soluto. 4. Regularzato ad verse problems. I ths secto we show how regularzato theory provdes a framework for formulato of verse problems of computer vso, partcular, surface recostructo. A geeral formulato of verse problems s that oe s gve the sesed data z, whch s produced through the acto of a operator A, actg upo the data, we wsh to recover f from the operator equato A f z. (4.1) I surface recostructo, f s a corrupted verso of z ad A s a tegral operator that models the laser scag process. A problem volvg verso of ths operator s wellposed [8,179,13] f we ca esure exstece, uqueess, ad stablty of solutos. For a varety of reasos, falure of oe or more of these codtos s commo ad thus the problem s ll-posed. For example A may ot be full rak (so that the soluto s ot uque - extra formato may be requred to restrct the soluto space), or A may be vertble but ll-codtoed (thus small chages data lead to large devatos soluto - whch 3

41 ca be dsastrous the presece of ose), or A may be of rak greater tha the umber of degrees of freedom (the system s over-determed ad thus a soluto may ot exst f ay of the measuremets cotas ose). Addtoal costrats ad assumptos restrct the soluto space, but the presece of ose, the problem ca become ll-posed ether of the last two seses. Stadard regularzato theory provdes mathematcal tools that eable oe to tur a ll-posed problem to a well posed problem. 4.3 Ill-posed problems ad regularzato From the pot of vew of moder mathematcs, all problems ca be classfed as beg ether correctly posed or correctly posed. I the laguage of fuctoal aalyss, ths fact ca be stated the followg maer. Let ad be Baach spaces ad the cotuous operator A : (ot ecessarly lear). The equato A f z, represets a correct, correctly posed or well-posed problem the sese of Hadamard f the operator A has a cotuous verse A words (wp1) Exsts a soluto: z there exsts a soluto f. 1 :. I other (wp) The soluto s uque: z there s o more tha oe f such that A f z. (wp3) The soluto f * depeds cotuously o the data: f f 0 whe * z z 0. If oe of these three codtos s ot satsfed, the problem A f z s called ll-posed. Gve that we wat to fd f gve z t s also called ll-posed verse problem. These codtos do ot have the same degree of mportace [94]. If the uqueess codto (wp) s ot satsfed the the problem does ot make sese. However, f (wp1) s ot satsfed, t oly meas that there are ot codtos to guaratee exstece of a soluto. O the other sde, oe may thk (as Hadamard dd) that wthout (wp3) the problem A f zdoes ot have physcal sese ad s computable. Nevertheless, choosg a proper oto of covergece ad the space, t s possble to fx the stuato. For k m, p stace ad may be take from the classcal spaces C ( ) or W ( ). These spaces are a atural settg for mathematcal physcs ad partal dfferetal equatos. They reflect physcal realty ad geerate stable computatoal algorthms. Now we wll deal wth Tkhoov regularzato. The ma dea supportg ths approach [100] s that the soluto of a ll-posed problem ca be obtaed usg a varatoal 33

42 prcple, whch cotas both the data ad pror smoothess formato. For the stable approxmato of a soluto of the operator equato A f z; where t s assumed that oly osy data z of the exact data z are avalable. Tkhoov or varatoal regularzato propose to mmze a fuctoal the geeral form, z [ A f, z ] R[ f ], (4.) oly assumg that s a fuctoal measurg the error betwee A f ad z, 0 ad R [] s a oegatve fuctoal. The real umber s called regularzato parameter. The dea s to fd a soluto for A f z as a elemet mmzg the fuctoal (4.5). Oe may thk that ths elemet s A f z, but t does ot work because s equvalet to (4.4) ad therefore wll also be ll-posed. Aother dea could be to use oore-perose geeralzed verse f A z, defed as the mmum orm soluto of the problem m A z f z. However the operator regularzato s to use the mproved fuctoal,z. 4.4 Surface recostructo as a verse problem A s usually ot bouded. The vewpot of The orgal dea of regularzato s to replace a ll-posed Fredholm tegral equato of the frst kd 1 K( s, t) f ( t) dt z( s), 0 by a earby well-posed Fredholm tegral equato of the secod kd. Here f() t s the ukow fucto ad K( s, t ) s the kerel of the operator. I abstract features ths equato may be wrtte operator form as A f z, where A s a lear tegral operator such that A f ( s) K( s, t) f ( t) dt. a b Let Ax be the tegral operator wth A f f x ( x ). I a pot cloud D {(, ) a z } 1 we have osy data that ca be modeled as b z A a ( f) K ( a, t ) f ( t ) dt a, 1,,. (4.3) Ths s a dscrete (osy) Fredholm tegral equato of the frst kd. I vector form we wll deote z ( z1, z,, z ), such that z A f, where ( ) A f K ( a, t ) f ( t ) dt.(4.4) a b 34

43 Commoly, the errors { } are..d. (depedet ad detcally dstrbuted) radom varables from a ormal dstrbuto N(0, )(see [188]). 4.5 Regularzato of surface recostructo Let ad be two Hlbert spaces ad A : a lear bouded operator. The Tkhoov regularzato scheme [13,14,184] s gve by m f, z f z A f. (4.5) Observe that geeral, the spaces, have dfferet metrcs. Now, we adapt ths framework to the partcular case of surface recostructo. Let be a reproducg kerel Hlbert space [8,188] wth cotuous kerel K :. If x, we let K ( s x ) K ( s, x ) ad by the Resz represetato theorem[100,104,15], defe the bouded lear operator ( A f )( x) ( f, K ) f ( x). (4.6) x Gve the data D {(, ) a z }, from a pot cloud, we obta a dscretzed verso 1 A : of A the followg way x ( A f ) ( f, K ) f ( a ), (4.7) x a where see that, has the er product ( zz, ) 1 zz. The t s straghtforward to 1 Ax f z f a z 1 1 ( ( ) ), (4.8) Usg ths settg, we see surface recostructo as the mmzato of the fuctoal, gve by, z f x f z m [ ] f A R[ f],z where 1 ( f ( a) z) R[ f] 1 m! x. R[ f ] f ( x) d m!, (4.9) 35

44 4.5.1 Iterpolato ad approxmato by regularzato It s a well-kow fact that the classcal methods of terpolato (Lagrage polyomals, for example) have serous problems for fttg data from real problems. Gve a set of kots there exst may ways to terpolate them, so order to choose oe s ecessary to add more formato other tha the kots. Cubc sples were oe of the frst successful aswers to ths problem, addg the requremet of a smoothg fuctoal. Depedg o the org of data ad kd of applcato, t s possble to do terpolato ( S( x) f( x) ) or approxmato ( S( x) f( x) ). For example, some rage scaers data are accurate eough for obtag good results usg terpolato, but f we are gog to terpolate osy data, the advsable way s to make approxmato o the data. We may state both problems terms of regularzato over a fucto space. P1. Iterpolato problem m R[ f], wth { f ( a ) z} 1. (4.10a) f P. Smoothg problem 1, wth { f ( a ) z} 1. (4.10b) m, z ( f ( a) z ) R [ f ] f Proposed model Fally, we exted the approach (4.6) cosderg two regularzato fuctoals R 1 ad R the form m [ f ] 1 ( f ( a ) z ) R [ f ] R [ f ]. (4.11) f 1,, z The dea s to tegrate local ad global features of the surface or pheomeo order to obta models that ft the three requremets for surface recostructo. The frst term s cotrollg fdelty to data ad fuctoals R 1, R are properly combed for hadlg the degree of smoothess or geeralzato by tug parameters ad 1. Ths s oe of the most mportat propertes of regularzato approach, that s, the possblty to add a pror kowledge about the model the form of fuctoals, to a tegrated varatoal approach. 36

45 Chapter 5 Regularzato fuctoals ad ther physcal terpretatos 5.1 Itroducto At frst glace, t seems to be mpossble to compute the soluto of a problem umercally f the soluto does ot deped cotuously o the data,.e., for the case of ll-posed problems. Uder addtoal a pror formato about the soluto, such as smoothess ad bouds o the dervatves, however, t s possble to restore stablty ad costruct effcet umercal algorthms. The ma purpose of ths chapter s to develop some crtera for choosg smoothg or regularzato fuctoals R[ f ]. I order to do a heurstc dscusso we cosder the behavour of curves ad surfaces uder geometrcal crtera, as for stace, Gaussa ad mea curvature as well as physcal crtera, based o cotuum mechacs. Fg. 5.1: Drac deltas do exst, do t they? (La Ferté pedestra brdge Stuttgart, Germay) [86]. Usg the deas of varatoal calculus we show the strog relatoshp betwee regularzato ad the physcal realtes represeted by classcal partal dfferetal operators of mathematcal physcs. It s also mportat to take to accout more subtle 37

46 mathematcal propertes the fuctoals order to produce solvable mmzato problems ad accordg to the realty of data. 5. Plate ad membrae models Varatoal calculus s the most relable gude for dealg wth the partal dfferetal equatos of mathematcal physcs. The models we wll use as crtera for choosg regularzato or smoothg fuctoals comes orgally from cotuum mechacs, partcular elastcty theory. A elastc body s defed to be a sold whch, oce deformed wll revert to ts orgal shape f the forces causg the deformato are removed. These problems of stable equlbrum, are govered by the varatoal prcple of mmal potetal eergy: A mechacal system wth the potetal eergy U( q1, q,, q ) s equlbrum at a partcular set of values of the coordates q1, q,, q f ad oly f the potetal eergy s statoary for ths set of values. A example oe dmeso s the vbratg strg ad further dmesos the plate ad membrae models. They have show to be very useful for modelg ad recostructo of surfaces. A membrae [86,87,181,176] s a porto of a surface, plae at rest, wth potetal eergy proportoal to chage area; the proportoalty factor s kow as teso. Now, suppose that the membrae at rest covers a rego of the plae ad that the deformato f ( x, y ) s ormal to the equlbrum plae. Suppose ths deformato s small the sese that hgher powers of the partal dervatves f, f of f are eglgble compared wth lower oes. The the expresso the area may be replaced by 1 f f dy dx. ( x y ) 38 1 fx f y dy dx for 1 [1 ( fx f y )] dy dx ad the potetal eergy by For the equlbrum problem of the membrae we suppose that the dsplacemet f ( x, y ) of the membrae have prescrbed values o ts boudary ad that o exteral forces act o the membrae, the the equlbrum posto s characterzed by the followg varatoal problem. The dsplacemet f ( x, y ) the equlbrum posto s the fucto for whch the potetal eergy fuctoal R[ f ] ( fx f y ) dy dx, (5.1) attas the least possble value amog the fuctos whch are cotuous the closed set, take o the prescrbed boudary value, ad possesses cotuous frst ad pecewse cotuous secod dervatves the teror. Observe that the fuctoal (5.1) s the x y

47 1 Ducho semorm Rf [ ] Jm[ f] f m! obta the elastc potetal eergy of a plate. wth m 1. Whe m, we h h A plate s the body ], [ of thckess h, that occupes the rego, oe of ts dmesos, the z drecto, say, s very smaller tha the other two, so that geometrcally the plate s flat. If for the equlbrum problem of the plate we suppose smlar codtos to membrae problem, the the dsplacemet f ( x, y ) the equlbrum posto s the fucto whch mmze the potetal eergy fuctoal f f f J[ f ] ( ) ( ) ( ) dx dy. x xy y The goverg equato for the deflecto of a elastc th plate s where, x x y y D f g, s the bharmoc operator, D s called the bedg 3 stffess; depeds o the materal ad the geometry, ad s defed by D Eh /1(1 ), where E ad are costats kow respectvely as Youg s modulus ad Posso s rato [147]. (a) (b) Fgure 5.: (a) Plate ad (b) beam are classcal models that have spred varatoal theory of sples. The oe-dmesoal verso of plate model s the deflecto of a beam. A beam s a body rectlear shape ad whose legth s cosderable greater tha ts two other dmesos. We wat to fd the -plae deflecto f of the beam (fg.5. ) whle subject to a force of testy g per ut legth. The beam has legth L, wdth b ad depth d ; b ad d are assumed to be much smaller tha ts legth. The goverg equato for deflecto of the beam s 39

48 4 d f EI g( x), 4 dx where E : modulus of elastcty, I : momet of erta of a cross secto about the eutral axs [153,87,153]. The beam model explas the orgal dea of sples. The term sple s derved from the aalogy to a draughtsma s approach to pass a th metal or woode strp through a gve set of costraed pots called ducks (fg 5.3). (a) (b) Fgure 5.3: Physcal terpretato of oe dmesoal cubc sple [153]. We ca mage ay segmet betwee cosecutve ducks (fg. 5.3 (b)) to be a th smply supported beam across whch the bedg momet vares learly. Applyg the learzed Euler-Beroull beam equato for small deformatos d f EI EI Ax B. dx Where EI s the flexural rgdty of the beam, the curvature, A ad B kow costats. Solvg for the deflecto yelds a cubc polyomal A 3 B f x x C1x C 6EI EI. I geeral [35,188], ths cubc sple has the form 3 S( x) ( x a) 0 1x 0, b ad mmze the fuctoal R [ f ] ( f ''). Ths s a approxmato to the total a curvature of the curve, because f the formula f '' k, 3 (1 f ' ) for the curvature of the curve ( x, f ( x )), we cosder f ' small, the k f '' ad R[ f ] b approxmates the total curvature ( x ) dx, of the curve. a k 40

49 5.3 Weak formulato There exst a very close relatoshp betwee dfferetal equatos ad varatoal problems. ay of the PDE of mathematcal physcs ad appled mathematcs are fact the Euler-Lagrage equatos for the mmum of some fuctoal. Laplace's equato f f f 0, xx s the Euler-Lagrage equato for the potetal eergy fuctoal of the membrae, yy, R f f dydx f f dy dx [ ] ( x y ) called the Drchlet tegral. If we cosder the Drchlet boudary value problem for the Posso equato f g, (5.) f 0 o wth gc( ), the desred classcal soluto of ths equato belogs to C ( ) ad vashes o. ultplyg eq. (5.) by a arbtrary fucto wth compact support (.e. 0 o ) we have f g. The, usg tegrato by parts we obta f f f f ds ˆ g C ( ) 0. g, Ths s equvalet to fdg uh 1 ( ) such that usg 0 L - er product 1 (, f) g, H ( ). (5.3) 0 Ths s called the varatoal or week form of equato 5., whch cotrast s called strog form. I the ext chapter we treat ts relato wth Schwartz dstrbuto theory. 5.4 Fudametal solutos Gve that our cetral problem s the recostructo from scattered data, we wll eed physcal terpretatos of operator equatos the form P f g, where P s a partal dfferetal operator, ad g may be Drac s delta. If the plate problem g represets the force actg o the plate the g represets a pot load actg at x 0 ad the correspodg equato s 41

50 f. Hstorcally, depedg of the kd of applcato, the solutos to PD ( ) have take dfferet ames: mpulse respose, Gree s fuctos or fudametal soluto. The pot- force Gree fucto K ( x) for a sple teso must satsfy D K x T K ( ) ( x), where ad are the bharmoc ad Laplace operators, respectvely. The geeral stuato of data costrats f ( a ) z at the locatos A { a1, a,, a }, results (usg dstrbuto theory) the equato wth soluto (see chapter 7) D f x T f x x a 1 ( ) ( ) ( ), f ( x) K( xa). (5.4) Crtera for selecto of regularzato fuctoals I geeral ay fuctoal able to measure the amout of rapd varatos of the surface (that s, the amout of varato the local surface oretato) could be used as stablzer R[ f ]. Ths suggests that ths fuctoal should measure some factor of the secod order dervatve of the surface. Although every requremet restrcts the umber of possble regularzato fuctoals, we would also lke Fuctoals that measure local ad global features. Rotato varat fuctoals. Fuctoals wth geometrcal or physcal terpretato. Fuctoals that ft certa mathematcal requremets. All the requremets are hard to fulfll. There are very complcated fuctoals that yeld surfaces of hgh qualty. But the mmzato procedure s very costly or smple fuctoals that do ot lead to good surfaces. The, thkg geometrcally, oe possblty s to measure the curvature of the surface, whch mplctly reflects varato surface oretato. It may be used mea curvature H or Gaussa curvature K. They both deped o the prcpal curvatures k 1 ad k (see [,45,136]). At ay pot of the surface there exsts a fte umber of ormal sectos that s, plaes that tersect the surface cotag the ormal vector of the surface ad defg curves over the surface. There are two sectos of partcular terest, oe wth maxmum curvature ad other wth mmum curvature. It ca be show that the drectos 4

51 correspodg to these two curves are orthogoal. These drectos are called the prcpal drectos ad the curvatures of the correspodg sectos are the prcpal curvatures K k k, H k1 k. For a surface the form ( x, y, f ( x, y )), these curvatures are gve by fx f y H ( ) ( ), x 1 f f y 1 f f ad x y 1 fxx f yy fxy K (1 f f ) x y. We may defe regularzato fuctoals that takes measures over the whole surface the form R [ f ] H dydx 1 If f x ad f y are assumed to be small, the [ fxx (1 f y ) f yy (1 fx ) fx f y fxy ] dydx. (1 f f ) x 3 x y R [ ] ( ) 1 f fxx f yy dydx Assumg smlar codtos we ca defe other fuctoal y ( f ) dydx. (5.5) R f K dydx [ ] ( fxx f yy fxy ) dydx. (5.6) Note that R1 ad R are both fuctoals cotag secod order dervatves. The relato wth the regularzato fuctoals used ths thess s the followg: Physcally we may thk eergy fuctoals. Th plate eergy fuctoal has a exact ad smple verso E ( k k ) ds, exact 1 E f f f dydx, smple xx xy yy From the physcal pot of vew represets the bedg eergy of a th plate havg the shape of the surface represeted by f. Whe ths surface s ot very complex, E s a good approxmato to E exact. Sce E smple s quadratc, s much easer to mmze tha the hghly o lear fuctoal E exact whch may also be wrtte as smple 43

52 exact ( 1 ) (1 ) 1, E a k k b k k ds where ab, are costats descrbg propertes of the materal (resstace to bedg ad sheerg). 5.6 The Resoluto of the fuctoal I geeral, t s extremely dffcult to fd a fuctoal that measures surface cosstecy ad satsfes the codtos of a er product. Hece f the regularzato fuctoal R[ f ] comes from a sem-er product o a sem Hlbert space of possble surfaces, the the most cosstet surface s uque up to possbly a elemet of the ull space of the fuctoal. The ull space s smply the set of surfaces that caot be dstgushed by the fuctoal from the surface that s zero everywhere. It s to say, we ca cosder a vector space of fuctos wth semorm f ad ull space { u : u 0}, (5.7) that duces a ormed space /, called the quotet space. The duced orm o / s clearly well-defed ad s gve by: f N f. (5.8) Ths meas that we ca study our problems o the space / order to use the propertes of orms. Based o the form of the ull space, we ca determe whether or ot the dffereces mmal surfaces are tutvely dstgushable. The ths crtera that may be used to determe the best fuctoal, usg the sze of the ull space, sce ths determe the resoluto of the fuctoal, that s, the level at whch the fuctoal caot dstgush betwee two dfferet surfaces. For example, t s better to use the quadratc varato m! x, R[ f ] f ( x) d m! because s uable to dstgush betwee two dfferet surfaces oly whe they dffer by a plae, whle the square Laplaca the fuctoal ( f ) caot dstgush betwee two surfaces dfferg by ay harmoc fucto. It s to say, the ull space o the frst fuctoal s smaller tha the secod oe. 44

53 5.7 Addg more fuctoals To llustrate the way these physcal terpretatos may help our uderstadg ad aalyss of surface recostructo, let us cosder the terpolato problem. The dea ths case s to choose a proper codto added to the terpolato costrats f ( a ) z order to obta a well-posed problem. Let us suppose we have a th plate of a flexble materal, plae whe o exteral force s appled o t ad we mpose the codto of crossg the pots represeted by data D {( a, ) } z 1. I equlbrum, the plate teds to form a smooth surface. It s a well-kow fact that the plate mmzes ts potetal eergy, whch s expressed up to certa degree of approxmato mmzg the TPS fuctoal f f f R1 [ f ] [( ) ( ) ( ) ] dy dx. R x xy y The terpolat satsfyg these codtos s the well kow Th Plate Sple. Ths model s very sutable for certa cases, but arse some problems whe we have regos wth rapd chage of gradets the modelled pheomeo. Physcally ths may be terpreted as the resstace of the plate to be stressed beyod ts flexblty. Ths resstace of the plate may be avoded takg a more geeral model mmzg the fuctoal f f R[ f ] R1[ f ] 1 [( ) ( ) ] dy dx, (5.9) R x y where 1 0 s a regularzato coeffcet. Whe 1 0, we aga obta the TPS model. O the other sde, f 1 the resultg sple represets the surface, beg ot a plate but a elastc membrae (rubber sheet) crossg the terpolato pots. Ths membrae overcomes the dffcultes of TPS. The soluto to (5.9) may be terpreted as a th plate wth teso appled o ts boudares ad t s called sple wth teso [7,63]. I ths case the shape wll deped o the amout of teso beg exerted as well as the stffess of the materal. Ths process of extedg the eergy fuctoals to clude more complex codtos the form of adequate fuctoals s the core of ths thess work. We take TPS as tal model to obta explct expressos of sples that may deal wth some of ts lmtatos, especally the recostructo of local ad global propertes of shapes from pot clouds. I geeral, we are terested partal dfferetal operators PD ( ), wth costat coeffcets c P( D) c D, (5.10) m whose propertes maly deped o ts prcpal part order terms [87]. cd m, formed by ts hgher 45

54 Chapter 6 Dstrbuto spaces ad regularzato for surface recostructo 6.1 Dstrbutos ad verse problems The theory of dstrbutos has ts orgs problems of partal dfferetal equatos of mathematcal physcs. However, as usual, after beg rgorously formulated mathematcal terms, the theory has developed very far from the mmedate applcatos ad has become very useful other dscples, partcularly surface recostructo from scattered rage data. Dstrbuto theory was created maly by Sobolev ad Schwartz [163,164,167] to gve aswers to problems of mathematcal physcs. After obtag a sold mathematcal foudato, durg the last decades, t became a depedet dscple. Dstrbuto Theory (or geeralzed fuctos) re-establshes dfferetato as a smple operato of aalyss" [163]. Usg the varatoal propertes of sples wth a abstract approach o dstrbuto spaces, Ducho [48] obtaed expressos for the ow well-kow th plate sple. Usg the Sobolev embeddg theorem, he foud spaces that are cluded to the space of cotuous fuctos C ( ), makg possble the work of terpolato. Durg the last decades, t has bee ecessary to develop very subtle mathematcal models ad algorthms for tacklg the recostructo problem ad there stll rema may others to be solved. Nevertheless, they all have commo features that ca be treated to the framework of verse problems theory [94,19,55,13]. I ths chapter we revew dstrbuto theory ad some dstrbuto spaces, hghlghtg the facts cocerg surface recostructo. The reader should cosult the refereces [51,65,91,163,164] for a better compreheso. We show how dstrbuto spaces are specally suted for dealg wth verse problems of 3D recostructo, provdg a varatoal framework that coducts to the geeralzato from classcal cubc sple to multvarate terpolato ad approxmato. The results of ths approach clude the wellkow th plate sple ad other radal bass fuctos. Although surface recostructo s a ll-posed verse problem, dstrbuto theory gves a settg to costruct spaces where they become well-posed. I ths theory, dscotuous fuctos ca be hadled as easly as cotuous or dfferetable fuctos to a ufed framework, makg t approprate for dealg wth dscrete data. We wll show how ths 46

55 framework may serve as a tool for the double task of modellg these data ad at the same tme to provde solutos for ts recostructo. I ths thess, we use Schwartz s approach to geeralzed fuctos cosderg them as cotuous lear fuctoals o the space of ftely dfferetable fuctos wth compact support. Ths settg s adequate for troducg the cocept of geeralzed dfferetato, whch makes possble the calculus of dstrbutos wth all ts practcal cosequeces. The delta fuctoal ( x), cotradctorly defed by Drac, as ( x) ( x) dx (0) ; s a geeralzed fucto the theory of dstrbutos, establshed rgorously by L. Schwartz. The spaces of dstrbutos ad ts cosequeces for multvarate approxmato ad sple theory have appeared several publcatos [47-49,108,109]. I ths work we wat to remark How dstrbuto spaces ad calculus o dstrbutos provde a approprate framework for solutos of ll-posed problems. The role played by fudametal solutos of dfferetal operators for fdg explct forms of terpolats. 6. The dervatve problem Let us cosder the classcal problem of dfferetato. Gve a fucto f, It s possble to fd ts dervatve by the classcal rules for dfferetato. If the fucto s gve by a formula, a composto of elemetary fuctos or a tegral depedg o a parameter, the ts dervatve s also gve by a formula. However f the problem s to fd the umercal dervatve of a fucto o the bass of expermetal data, the problem becomes ll-posed [55,100]. Our ma task s to fd or costruct spaces where approxmato problems ca be wellposed, ths s, to fulfll propertes (wp1,, 3). Nevertheless, classcal spaces lke C [ a, b ] k may ot be adequate for ths purpose. For example, there may exst a sequece f of fuctos C 1 ( ) such that coverge uformly to f but f C 1 ( ). As we ca see, may of these problems (lke the o vertblty of the order of dfferetato) are o f f dervatves ad t may happe that xy y x. Therefore, t s ecessary a more versatle vewpot for dfferetato of a fucto: dervato the sese of dstrbutos. Eve more; the spaces of dstrbutos solve the exstece problems (wp1) but ths s ot eough for uqueess (wp) ad regularty (wp3). These problems are solved takg subspaces of dstrbutos wth some addtoal propertes. For umercal aalyss the most useful spaces are those as close as possble to Euclda ad Hlbert spaces; these are Sobolev spaces, whch may be troduced usg the 47

56 cocept of fuctoal o a lear (vector) space ad some results from approxmato theory. 6.3 Dstrbutos If X s a fucto space,.e. a space whose elemets are fuctos (e.g. C [ ab, ] ) the a real (complex) fucto T : X o X s called a fuctoal. The set X of all cotuous lear fuctoals o X s called the dual space of X. Especally terestg for us s the so-called space of test fuctos, the set of ftely dfferetable fuctos o wth compact support, symbolzed ( ) C 0, D ( ) or smply by D. The elemets of the lear space D '( ) are called dstrbutos (or geeralzed fuctos) ad are characterzed by ther actos T ( ) or T, ( Dualty bracket ) over the elemets ( ). For stace, (0) s the acto of the delta fuctoal over. I geeral each a determes a lear fuctoal ( ) ( x a) o D ( ) by the expresso, ( ) ( ) a. a Ths s a way to solve the formal cosstecy settled by Drac s samplg property. A mportat thg to ote s that ay locally tegrable fucto f, wll defe a dstrbuto by f, f ( x) ( x) dx ( C ). 0 (6.1) These are called regular dstrbutos. If ths s ot the case, they are called sgular dstrbutos (for example, ). By abuse of otato sgular dstrbutos are also deoted by the symbol f( x ) used for ordary fuctos ad are wrtte as (6.1). The ma goal ow s to exted as may operatos as possble from fuctos to dstrbutos (dervatves, covoluto, Fourer trasform). Oe key dea s that the defto of operatos o dstrbutos should cocde wth the defto for regular dstrbutos. From ( f) f( ), t follows that the product of a dstrbuto T e ( ) ad a fucto C s defed as T, T,. Nevertheless, ths should be doe carefully, because t may arse several lmtatos; for example, t wll ot be possble to multply dstrbutos or defe the Fourer trasform wthout makg extra assumptos. It s a well-kow fact that classcal spaces may ot be sutable order to formulate well-posedess. The dervatve cocept s the core of ths dffculty, so t s ecessary to get a more versatle defto of dervatve. It s kow that classcal dfferetato s a ll-posed problem, because takg dervatves of a fucto a 48

57 amplfes ts oscllatos resultg less smooth fuctos. O the cotrary, tegrato s a smoothg operato. Dstrbutoal dervatves are motvated usg tegrato by parts ad the compact support of ( ), wth the followg defto f f, x f f, x, x x k k k k T the,, T, for T D '( ). As a cosequece, the dervatve of the xk xk delta dstrbuto t s expressed as,, (0). A dstrbuto T has dervatves of all orders. Ths s expressed as T, ( 1) T, ( ). D (6.) Ths meas that the dervatve of a o-dfferetable fucto ca be defed terms of relatos wth smooth fuctos of compact support. A drect cosequece s that a T T dstrbuto T s deftely dfferetable ad. It s also mportat to x x x x j k k j remember that dstrbuto spaces may be defed o a bouded rego ; but for recostructo from scattered data t s coveet to take, such as the above deftos. I ths maer, the boudary codtos are shfted to ad s ot ecessary to solve boudary value problems whch may coduct to ubouded Gree fuctos [188,189]. Aother problem s that some terpolats are ot easy to compute, because ther characterzato volves a kerel gve by a seres, the thgs are much smpler replacg by the whole plae Covoluto A operato wth very useful propertes theory ad applcatos s the covoluto product of two fuctos u ad v u v( x) u( x t) v( t) dt.several codtos o u ad v are ecessary to esure that the tegral exsts [163]. If S ad T are dstrbutos o the ther covoluto product S T s a ew dstrbuto o, defed by S T Sx, Ty, ( x y) ( ). Covoluto product s partcularly useful because of ts regularzg propertes,.e. make a fucto regular or smooth. Covoluto becomes a very powerful ad geeral operato whe cosdered from a dstrbutoal pot of vew. Some kds of dfferetal, dfferece, 49

58 ad tegral equatos are all specal cases of covoluto equatos. A great varety of equatos of mathematcal physcs ca be compactly represeted by the smple form u v f. Lear traslato varat systems are modelled by ths kd of covoluto equato. The covoluto of a dstrbuto T D '( ) wth ( ) s a C fucto T o. T s called the regularzato of T. Besdes of beg commutatve ad assocatve, covoluto of dstrbutos has other useful propertes as (a) T T ; (b) ( a) T at; (c) T T; (d) ( a) h( x) h( x a). (6.3) I geeral, f D s a dfferetal operator wth costat coeffcets Thus, f D s the Laplaca operator 6.3. The Schwartz Space x 1, the T T. A very mportat problem whe extedg the Fourer trasform of a fucto f xξ F [ f ] f ( ξ) f ( x) e dx, ξ ( 1,,, )., D T DT. to dstrbutos s that f t s possble that. I order to obta a useful defto of Fourer trasform, Schwartz defed the space of fuctos C such that ad all ts dervatves vash at fty more rapdly that ay power of x. For example x p( x) e belogs to wth p ay polyomal. Wth ths md, s defed the space of tempered dstrbutos as the dual space ( ). As a cosequece mples ad t s possble to defe T, T, T ; preservg ths way, the well-kow ce propertes of Fourer trasform ( 1, for example). Oe mportat property for applcatos o recostructo s that the Fourer trasform of a radal fucto (say f ( x) f0( r ) ) s also radal (say f( ξ) f0( ) ), r x ( ). 1 1/ ( x ) ; ξ 1 1/ Very useful for us wll be the followg results take from Gel fad ad Shlov book [66] 50

59 1 ( ) F [ r ] C, C, ( ) [ l ] F r r C C l. (6.4) 1 C 1, C are also gve terms of the Gamma fucto Fudametal solutos A fudametal soluto of lear dfferetal operator P c wth costat m coeffcets s a dstrbuto K ( xy, ), such that Px ( K( x, y)) ( x-y). If K the K s called a potetal. Here P s appled to K as a fucto of x ad y s a parameter. Fudametal solutos have the remarkable property of beg part of the soluto of homogeeous dfferetal equatos. Reasog formally ad usg propertes (6.3), f PK, the P( K g) ( PK) g g g, therefore K g s a soluto of Pf g. (6.5) Ths s a well-kow fact the theory of dfferetal equatos. I ths ad the ext chapters, t s show ts great utlty for multvarate approxmato. The most remarkable fact that cocer us about fudametal solutos s that the terpolats we are gog to costruct are expressed as a lear combato of traslates of the fudametal soluto for a dfferetal operator. Fudametal solutos are called Gree's fuctos whe they are subjected to boudary codtos. Accordg to algrage-ehrepres theorem [91], every operator P c has a fudametal soluto. Fudametal solutos take dfferet m ames, depedg of the specfc felds where they arse: mpulse respose, Gree s fuctos, fluece fuctos. These multple ames are strogly related wth the hstory of physco-mathematcal sceces [199]. Oe of the most mportat fudametal solutos comes from the terated Laplaca operator m m u. If K s a fudametal soluto of the operator, the m K. Takg m Fourer trasform o both sdes, ( 1) m E 1 ad usg formulas (6.4) are obtaed the fudametal solutos K( x, y) ( x - y ), (6.6) m. m c r l r, eve ( r), d r, odd m, m (6.7) 51

60 c ( 1) m( ) / m1 /, d m / ( m 1)!( m / )! ( / m) ( m 1)! Sobolev ad Beppo Lev spaces The well-kow classcal spaces (e.g. the space of cotuous fuctos) may fal whe dealg wth ll-posed problems. Dstrbuto theory gves a settg to costruct spaces where surface recostructo becomes well posed. The space of dstrbutos ( ) as descrbed above provdes aswers for codtos of exstece ad uqueess. However, ths space s "very large, therefore, the ssues o regularty are treated to some of ts subspaces. It s possble to recostruct the space L ( ) as a Hlbert space of dstrbutos, ufyg ths theory wth the theory of L ( ) spaces; ths leads aturally to Sobolev spaces, very coveet for the pure ad umercal treatmet. The key dea of Sobolev techques t s to assume that the dstrbutos whch solve a partcular problem really come from a fucto f, but wthout makg ay smoothess assumpto about f. The ext step the method, s to take advatage of the most geeral ad operatoal propertes of dstrbutos ad apply them to solve the problem. Oce ths s doe t s possble to use the so called Sobolev embeddg propertes order to determe the smoothess degree of the soluto. If m s a oegatve teger ad m, p p [1, [, The Sobolev space W ( ), s the vector space { D ' }, m, p p W ( ) u ( ): ul ( ), m p provded wth the orm u m, p p W u. L ( ) m m The spaces, m W ( ) ( p ) are symbolzed as H ( ) ad cossts of those fuctos L ( ) that, together wth all ther dstrbutoal dervatves of order m, belog to L ( ) { } m H ( ) u : u L ( ), m. m We cosder real value fuctos oly, ad make H ( ) a er product space wth the Sobolev Ier product ( u, v) ( m u)( v) dx H m u, v H. m Ths er product geerates the Sobolev orm u ( u) dx H. m m Oe mportat fact about these spaces s that H ( ) s a Hlbert space wth respect to the orm m. I geeral, for s, s defed as H 5 m

61 s s / H ( ) { us' ( ):(1 ξ ) u L ( )}, s wth the scalar product ( u, v) ξ u( ξ) v( ξ) dξ. It ca be show [0] that whe s m s m, H ( ) s topologcally equvalet to H ( ). Now, our terest s to fd a proper abstract settg for a atural -dmesoal geeralzato of the mmal orm terpolato problem. Ths s provded by m homogeous Sobolev spaces or Beppo Lev spaces BL ( ) of order m over { u D' }. m ( BL ) : ul ( ); m I words, ths s the vector space of all the (Schwartz) dstrbutos for whch all the partal dervatves ( the dstrbutoal sese) of (total ) order m are square tegrable m Due to ts weaker defto, BL ( ) there s ot a orm but the rotato varat sem-orm defed by (1.3) m! Ru [ ] u( x) dx,! (6.8) m correspodg to the sem-er product m! ( u, v) ( u)( v) dx.! (6.8a) m A sem-er product has early all the propertes of a er product, but ts ull space s dfferet from cero. It ca be show that the ull space of (6.8) s the space ( ) m 1, of k polyomals of total degree o greater tha m 1; the dmeso of k ( ) s ; for stace, {1, x, y, x, y, xy} s a base for ( ). A mportat relato betwee Sobolev ad Beppo-Lev spaces s that the tersecto of all Beppo-Lev spaces k BL ( ) of order k m yelds the Sobolev space W ( ) BL k ( ). m km. Wth the above settg, dstrbuto theory provdes a very effcet tool for treatg wth very complex problems; evertheless, t s very mportat to remember that order to have a realstc applcato ths should be doe wth spaces of cotuous fuctos. Ths s the method used partal dfferetal equatos, where the problems are frst solved the realm of dstrbutos ad the f these dstrbuto solutos are classcal solutos. Sobolev embeddg theorems aswers ths questo. 53

62 By the Sobolev embeddg theorem [], t s well kow that for m, the cluso m H m ( ) C( ) holds, or, to be more precse, that every equvalece class H ( ) m cotas a cotuous represeter. I ths way, H ( ) s terpreted as a set of cotuous m fuctos. The followg theorem shows that BL ( ) C ( ) as well. m Theorem 6.1 ([48,116]). If m the BL ( ) s sem-hlbert space of cotuous fuctos o ad all the evaluato fuctoals wth fte support that ahlate ( ) m1 are bouded. Some of the deas we have metoed up to ow (fuctoals, fudametal solutos, er product spaces) ca be lked usg the deas of Reproducg Kerel Hlbert Spaces (RKHS) [8,171,184,188] Dstrbutos ad ercer codto o kerels I the theory of tegral equatos, the kerel K ( xy, ) s postve defte whe K ( xy, ) satsfes K( x,y ) f ( x ) f ( y ) d x d y ( Kf, f ) 0, (6.9) for ay cotuous fucto f ( x). I the begg of XX-th cetury, ercer [8] dscovered that ths codto s equvalet to the quadratc form, j1 K( a, a ) 0, j j for ay pots a1, a,, a, ad 1,,, \{0}. If K s radal the K( x, y) ( x y) ad (6.9) wll be ( x y) ( x) ( y) x y 0 for all ( C ) 0. To see ths oe oly eed to approxmate ths tegral by a Rema sum. Ths codto ca be wrtte as f ( τ) ( x) ( x τ) dτ dx = f( τ)( )( τ) dτ 0, where s the fucto ( x) x. Ths suggest to defe a dstrbuto T ( ) of postve type f T( ) 0 for all ( ). (6.10) If (,, ) s a Hlbert space of fuctos o a arbtrary set whose er product s wrtte [ ] wll be the space of mappgs from to ad s the vector space of fte 54

63 lear combatos of the Drac, or evaluato fuctoals, [ ] { : a, }. I ths way, 1 a [ ] blear form, : such that fuctoal x( ) exactly oe K [ ] ad L ( f ) ( f ) f ( ) oto x0 ( x0) x0 a 1 1 are sets dualty by the, f f( a ). If the Drac L f s bouded o, by Resz represetato theorem [35], there exsts x such that L ( f ) ( f, K ) f ( x), for all f. x x Kx s kow as the represeter of L x. The becomes a Reproducg Kerel Hlbert Space (RKHS) or Hlbert space of fuctos [35] ad the represeter s a uque postve defte fucto K :, ( x, y) K( x, y), called the reproducg kerel of ; such that f ( x) ( f ( y), K( x, y)) f. (6.11) y The kerels we deal wth ths work are radal, thus they have the form K( x, y) ( x y ), where :[0, [ s a cotuous fucto that depeds of the dstace betwee pots. s called a radal bass fucto. Table 6.1 gves a lst of the most used radal bass fuctos. Useful cases [ ,188] of both postve defte ad radal kerels are K( x, y) ( x-y ) wth Gaussas 1 () r e r, verse multquadrcs () r 1 r 4 ad Wedlad compactly supported fucto ( r) (1 r) (4r 1). A especal case of codtoally postve defte kerel s obtaed wth the th plate sple fucto ( r) r l r,, whch s a partcular case of (6.6). The ext theorem clearly llustrates the two treds that hstorcally have bee followed to study propertes ad applcatos of kerels. I the frst tred, oe s terested prmarly a class of fuctos, ad the correspodg kerel K ( xy, ) s used essetally as a tool the study of fuctos ths space. Those followg the secod tred cosder a gve kerel K ( xy, ) ad study t tself as a tool of research, the space correspodg to K ( xy, ) s troduced a posteror. Theorem 6. (oore-aroszaj [188]). To every RKHS there correspods a uque postve defte fucto (called the reproducg kerel) ad coversely, gve a postvedefte fucto K( xy, ) o t s possble to costruct a uque RKHS of real valued fuctos o wth K( xy, ) as ts reproducg kerel, ths space s called the atve space. 55

64 It ca be show [191] that the atve space for th plate sple kerel m K( x, y) ( x y ) s the Beppo-Lev space BL ( ) ad the atve space for m, K( x, y) ( x y ) wth Wedlad fucto Sobolev space 3 H ( ). s the classcal 4 ( r) (1 r) (4r 1) Name () r Parameters order Lear () r Cubc Gaussa r m 1 3 () r r m () r e r 0 r r m 0 Pol-harmoc () \ 0 m / Th plate sple ( r) r log( r) m / ultquadrc Iverse multquadrc Wedlad fucto / ( r) ( c r ) \ 0 c 0 m / ( r) 1/ c r m 0 m 0 4 ( r) (1 r) (4r 1) Table 6.1: Some well kow Radal Bass Fuctos. 56

65 Chapter 7 Iterpolato of surfaces from pot clouds 7.1 Itroducto Now we are gog to apply the theory developed the former chapter to solve the terpolato problem from a pot cloud ay dmeso D {( a, z ) } 1; ths correspod to the problem (P1) proposed (4.10a) m R[ f], wth { f ( a ) z} 1. (7.1) f The approxmato problem P (4.10b) wll be treated the ext chapter. Based o results from Hlbert space ad dstrbuto theory [35,48,56], we gve a costructve proof to the problem of terpolato from scattered data. As a cosequece of terpolato ad smoothess codtos we obta Th Plate Sple (TPS), whose explct expresso s gve terms of the fudametal soluto of the bharmoc dfferetal operator. Ths sple s also wrtte terms of covoluto wth a fxed fucto. TPS s a geeralzato to of the well-kow cubc sple oe varable [4,161,188]. Cubc sple was developed for solvg terpolato problems arcraft, shpbuldg ad car dustres at the 1950 s. athematcas soo realzed that commo terpolato methods as Lagrage Polyomals were ot sutable for tacklg these problems. It was ecessary to buld more subtle tools. After ths achevemet, there was a great terest to obta the equvalet to cubc sple. Several ways were tred but oly the varatoal approach was successful. I the followg les we descrbe ths approach. 7. m D - Sples or Th Plate Sple m D sples results as the applcato ad geeralzato to varables of the plate model dscussed chapter 5. Ths model was formalzed by Ducho [48] ad eguet [1,13]. To recostruct a fucto (or surface), t s assumed that data D {( a, z ) } 1 comes from the samplg of a fucto f such that f ( a ) z ad t s requred to fd a expresso for f, order to approxmate f ( x ) whe x does 57

66 ot belog to the set of odes or cetres { a1, a,, a }. Ths meas that f should be a cotuous fucto. The dea s to fd a elemet S( x) a proper subset or subspace ad the use S ( x) to calculate f ( x). We suppose cotas a ( ) m1 -usolvet subset { a } N j j1, N dm m 1( ) ad we wat to fd S ( x) such that S( a ) f ( a ) z for 1,, ad mmzes the semorm x, (7.) R [ u] c ( x) u d m wth ull space ( m! ) m 1 ; the c are chose for obtag a rotatoal varat! m semorm. As a cosequece, we beg assumg S BL ( ), the by Sobolev m embeddg theorem, BL ( ) s a lear subspace of C ( ) for m. The key dea s to apply theorems A. ad A.3 o spaces of dstrbutos ad modfy the m! sem er product [ u, v] u( x) v( x) d x o m BL ( ). We the obta a m! er product for buldg a complete space. Oce ths s doe, we are eabled to use all the machery of Hlbert spaces (see Appedx). The er product s defed as ( u, v) [ u, v] u( a ) v( a ). (7.3) a The soluto S( x ) to the mmal orm terpolat s foud usg the projecto operator techque o Hlbert Spaces. A lear trasformato P of a lear space V to tself s sad to be a projecto f P P. The rage R ad the ull space K of P are lear subspaces of V such that V=R K. If R K the I P s also a projecto ad t s wrtte V=R K The deas of the deducto are borrowed from Lght [108] ad eguet [1]. We gve here the detals ecessary for showg the mportace of fudametal solutos of dfferetal operators. Let p1, p,, p N be the Lagrage bass for ( ) 1 wth respect to the pots { a1, a,, a N }; we defe P : wthp f f ( a) p. The ( ) m1 s the rage ofp ad 0 { u : u( a ) 0, a }, ts ull space. By (7.3) t s see that j j ( ) 0 m1, the we have the represetato formula ( = ) 0 m 1, ow t s predctable that S wll be a term 58 m N 1 0 plus a polyomal. The dea s to fd the

67 represeters Ra o m a o BL ( ) the, respectvely. Now we fd ad the use theorem A.(see Appedx). If R h p where a a a the projecto operator( ) 59 Ra s the represeter for p a are the represeters o 0 ad m 1 uder h a ad R usg the dstrbutoal approach. The mage of a I - P s 0, the ( ) P P [ P, hx] [, hx] c ( )( hx ), ( )( x) ( ( ), h x ) m usg dervatve the dstrbutoal sese the last expresso ad by (6.), we have m ( )( x) c, h ( 1) c h, P It s also see that x m m N x m ( 1) m h,. (7.4) x ( P )( x) f ( a ) p ( x) p ( x),, (7.5) ( x) ( a ) 1 1 so by equatos (7.4) ad (7.5) we have show that h x s a soluto of the dstrbutoal dfferetal equato q q x ( x) p x ( a ) 1 m m ( 1) h ( ). (7.6) Usg the fudametal solutos (6.7) of the terated Laplaca ( 1) m m K ; the x ( x ) l 1 Kx p( x ) Ka s a partcular soluto of the dfferetal equato (7.6). We project ths soluto o 0 to obta It ca be show that l x I - P x x 1 { } h ( ) K p ( ) K K p ( x ) K K ( a) p K ( a ) p ( x) p. a x a j j 1 1, 1 a l l l x l p p ( x) p the x 1 R h p ( x) p. akg some calculato a a 1 we fd the terpolat S f, Aas a lear combato of traslates of a fxed fucto plus a polyomal N p( x) jpj( x) where { p } N s a base for ( ) j j 1 m1. Fally, t s obtaed j1 the followg expresso that solve the terpolato problem (7.1) N

68 S ( x) ( x a ) p( x), f, A m, 1 () r d r m, m m c r l r, eve, odd, (7.7) where the costats cdare, kow but are ot ecessary because they are absorbed by the lear combatos S f, A. Ths terpolat s kow as m D sple, polyharmoc sple, Th Plate Sple (TPS) or surface sple. It s worth to say that S f, A s a orthogoal projecto, therefore, s the best approxmato wth respect to the semorm the space m BL ( ),.e. m f S f S S BL ( ). f, A m m Gve the data D {( a, z ) } 1, the terpolat S f, A s completely determed fdg s ad s. The terpolato codtos S( a ) z 1, produce equatos ad the remag N degrees of freedom are absorbed by the codto yeldg the system N c p j ( a ) 0 p j m1 1, (7.8) A t P P 0 z 0, (7.9) are foud by solvg ths lear system, where, A s a terpolato matrx where, (see Appedx), wth A, j m, ( a aj ),, j 1,,, P p ( a ) 1,,, j 1,, N ( 1,, ) t, 1 z ( z,, z ) t. j j It s terestg to ote that S f, Amay be wrtte terms of a covoluto of a dstrbuto wth a fucto the followg way. Let be the dstrbuto ( a ) such that m 1 { : ( p) = 0}, the by 1 codto (7.8), we have that ( p) 0 f p m1. O the other had, defg ( x) ( x ), ad usg propertes (6.3) of the delta fucto, the result s ( a) ( x) ( x a), thus, we ca wrte 60

69 ( x a) ( x). 1 Therefore, takg p m1, m 1ad supp { a1, a,, a }, the terpolat ca be wrtte as Sf, A ( a ) p p. (7.10) 1 Ths suggests that S f, A ca be see as a very geeral form of a low-pass flter. These propertes are completed wth the followg result. Theorem 7.1 ([1]). If m, the m D sple terpolato problem s well-posed: ts soluto exts, s uque, ad depeds cotuously o the data D {( a, z ) } 1 Example 7.1 (Iterpolato wth sem-orm s m D -sples). I the space BL ( ) the correspodg u u u R[ u] ( ) ( ) ( ) x xy y dx dy, that represets the bedg eergy of a ftely exteded plate. Wth ( r) r log r, the terpolat (7.7) s x S( x) ( a ) x y Ths sple terpolato, whether oe or two dmesos, physcally correspods to forcg a th elastc beam or plate to pass through the data costrats. Away from the data pots or cetres the curve (or surface) wll take o the shape that mmzes the stra eergy gve by Ru [ ]. Fgure 7.1 shows the results of terpolatg Frake s test fucto f ( x, y ) (7.11) [64], gve by 1 (9x1) (9 y1) ((9x) (9 y) ) ) f ( x, y) 0.75e 0.75e (9 y3) (9x7) 4 ((9x4) (9 y7) ) , (7.11) e e usg scattered data D {( a, ) } z 1. The lear system to solve s 61

70 A11 A1 A1 1 x1 y1 A1 A A 1 x y A 1 A A 1 x y x1 x x y1 y y z1 z z, (7.1) where the coordates of observato pots are a ( x, y ), A, j ( a aj ). We solve ths system to get the coeffcets or weghts,, 1,,, j 1,,3. j (a) (b) (c) Fg. 7.1: Results terpolatg Frake s test fucto (7.11) wthout ose the terval [0,1] [0,1] (a) Orgal fucto. Iterpolato wth: (b) 40 pots. (c) 100 pots. The terpolato rapdly mproves wth a creasg umber of radom pots. Example 7. (Recostructo of surfaces wth dscotutes). The followg s the mathematcal expresso for Frake s ladslde the terval [0,1] [0,1]. 1 y (1 y) (1 x) y x (7.13) f ( x, y) 1 ( y ) y x 6

71 (a) (b) (c) (d) Fg. 7.: (a) (, ) xy coordates of 100 scattered data o Frake s ladslde surface (7.13) (b) 3D scatter plot of the pots (c) The orgal surface (d) Iterpolato of the scattered data. Ths s a extreme case very useful for testg approxmato models the reproducto of dscotutes. It has smooth zoes as well as vertces, edges ad faces. 63

72 m D 1 D 1 3 D 3 3 m! u u dx D m m! D D D ( ( )) u x dx ( ( )) u x dx u : [( u u u ) ( ) ( ) ] dxdy x xy y u u u u [( ) 3( ) 3( ) ( ) ] dxdy 3 3 x x y xy y u u u u u u [( ) ( ) ( ) ( ) ( ) ( ) ] dx dydz x y z xy xz y z Table 7.1: Ducho semorm (7.) for some values of m(, m,3 ; 1,,3 ) m represets the degree of dervatves ad the dmeso of data. m p ( x) c c x c c x c x c c x c y c c x c y c x c y c xy c c x c y c z Table 7.: Polyomal terms for m D sples. Now we wll use the same varatoal approach to study other sples whch are m geeralzaros of D sples. The elemets ecessary to defe these sples are a proper fucto space ad sem-er product. Oce ths s doe s possble to fd the explct expressos for them o scattered data, solvg systems of lear equatos wth the structure (7.9) ad (7.1). 7.3 Geeralzatos of m D sples ad semorms There are some mportat sples that ca be see as partcular cases of our proposed framework (4.11), wth m 1,, z [ f ], by extedg the framework used for m D - f sples. For ths, t s coveet to use Fourer aalyss o the sem-er product 64

73 m! ( u, v) ( u)( v) dx!. (7.14) m By Parseval s formula ( u( x))( v( x)) dx ( u( x))( v( x)) dx, so, t s atural to modfy the lke m D sem-er product (7.14 )ad try other er products c ( u( x))( v( x)) w( x) dx, m I ths way t s possble to show that the semorm ξ s ξ ξ, (7.15) m R[ f ] c ( f ( )) d ms, defes a space of cotuous fuctos X uder the assumpto m / s / (see ms, [48,1,13]). The ( ms-, ) sple s the uque fucto X, whch mmzes the semorm R[ f ] (7.15) ad terpolates f o { a1, a,, a },.e. ( a ) f ( a ) z, 1,,. The ( ms, ) sple s gve the form m, s a p, (7.16) 1 ( x) ( x ) ( x) where p s a polyomal ( ) 1, the s satsfy ( ) m1 ad ms, s the radal bass fucto, gve for x m 1 q( x) 0, for ay q, by ms x log( x ), m s ( x) x, m s ms, ms. (7.17) Ths s a geeralzato of m D sple, addg a ew parameter s. 65

74 7.4 Sples teso ( -sples) The th plate sple wth teso (TPST) [63] may be deduced from the model of a th plate subjected to lateral loads ad md-plae forces. Ths equato may be foud Tmosheko s book [180, 181] 1 D f f f x y x y, f q Nx N y N xy where f s the lateral deflecto, q s the lateral load, ad D depeds o the propertes of the plate materal. Settg the former expresso s obtaed a equato of the form teso parameter. I ths case the semorm s N x N y N xy, are the md-plae forces, N x N y ad xy N 0. From f f p. Where s a f f f R[ f ] ( ) ( ) ( ) x xy y f f dx dy ( ) ( ) dxdy. (7.18) x y It ca be proved that there exst a Hlbert space subspace of C ( ) where there s a elemet called sple teso, that mmze the semorm (7.18). s gve the form ( x) ( x a ) p( x), (7.19) where p s a polyomal ( ) 1, the s satsfy 1 ( ) m1 ad ( x) s the radal bass fucto, gve for x m 0 1 q( x) 0, for ay q, by ( x) ( x ) log( x ), (7.0) whe the data have the form D {( a, z ) } 1 wth. It s possble to show [6] that the fucto (7.1) x ( x) Ce ( x ), also gves a soluto to the terpolato problem. 7.5 Coclusos We have deduced a terpolatg sple usg a varatoal approach o Hlbert spaces. The most relevat features of these sples are ther geeralzato ad reproducg 66

75 propertes applcable o scattered data, t s to say, there are ot very restrctve codtos o the spatal dstrbuto of ceters. Followg ths method we show that ( ms, ) sples ad teso sples are partcular cases of our approach. These sples also mmze some semorm wth physcal or geometrcal terpretato. 67

76 Chapter 8 Surface approxmato from osy data 8.1 Itroducto I the former chapter we have solved the mmal orm terpolato problem usg the methods of reproducg kerels ad projecto o Hlbert spaces. Now, we cosder the problem of smoothg of osy data the framework of Schwartz dstrbutos. I other words, we wat to solve the approxmato problem (P) proposed (4.10b) 1, wth { f ( a ) z} 1. (8.1) m, z ( f ( a) z ) R [ f ] f 1 That s, to solve the verse problem of recostructo by Tkhoov regularzato, partcular, the recostructo of surfaces from scattered data. We also study the most geeral results about regularzato of verse problems. I ths case data are cosdered wth or wthout ose. The terestg thg s that these results provde a geeral aswer for several forms of data ad ca be foud by solvg a lear system. The sples we obta here clude all the case studed chapter TPS ad sples teso. It was Hadamard [81], the frst to speak about well-posed problems as havg the propertes of exstece, uqueess ad stablty of the soluto. However, may mportat verse problems scece ad egeerg lead to ll-posed verse problems, though the correspodg drect problems are well posed. Frequetly, exstece ad uqueess ca be forced by elargg or reducg the soluto space as we have doe wth dstrbuto spaces. 8. Soluto of smoothg 3D Whe the soluto of a problem does ot deped cotuously o the data, the computed soluto may be very dfferet from the true soluto. Aga, we have that there s o way to overcome ths dffculty uless addtoal formato about the soluto s avalable. Therefore, a recostructo strategy requres addtoal a pror formato about the soluto. As we have sad before, ths ca be doe by Tkhoov regularzato. The ma dea supportg Tkhoov regularzato theory s that the soluto of a ll-posed problem ca 68

77 be obtaed usg a varatoal prcple, whch cotas both the data ad pror smoothess formato. Now we show the soluto of (8.1) by fdg the TPS for osy pot clouds, D {( a, ) } z 1 ad the smoothg fuctoal R[ f ] from (7.), wth m f f f R[ f ] ( ) ( ) ( ) x xy y dx dy. We the have to mmze ( f ( a) z) ( f ( x ) z ) ( ) x a [ f ] ( f ( a) z) R[ f ] 1 f f f ( ) ( ) ( ) dx dy x xy y f f f ( ) ( ) ( ) dx dy. x xy y Usg the Euler Lagrage equato [ F] u F F F F F x y x xy, for ths fuctoal we obta ad smplfyg u ux uy uxx ux y ( f ( x) z) ( x a) F ( 1) 0, u F yy u yy y y y f f f ( ) 0, 4 4 x x y y ( f ( x ) z) ( x a) f 0, f f f where f s the bharmoc operator. I ths way the dfferetal 4 4 x x y y ( f( ) z ) equato f x ( x a ) s obtaed ad ts soluto s 1 ( f( ) z )) f ( x ) K x ( x a ), 1 where K( x,y), ( x y ) s aga the fudametal soluto of the operator ( f ( a) z), we have f ( x) K ( x a) K( x a) K( x a).. Dog 69

78 As the ull space of (8.) s ( ) we have to add a term 1 p( x ) 1( ) to the fal expresso for the soluto of the optmzato problem m, 1 S ( x) K( x a ) p( x) ( x a ) p( x). Ths s the same expresso obtaed (7.6) but wth a chage the terpolato matrx, whch has the form A I P z t 0 0, (8.) P ad I s the detty matrx of dmeso. These results are vald for data pots o wth {(, ) D a f }. It s also possble to show that the solutos S ( ) 1 x, wth 0 are approxmatos ( S ( a) f ( a) ) that coverge to the terpolato. Ths meas that lm S( x) s the mmum square regresso over ( ) 1 ad lm S( x) s the terpolat of the data D. I other words, the regularzato approach gves a compact soluto appled to both terpolato ad approxmato. Ths s show the followg result for may kds of data. m Extedg Tkhoov regularzato Orgally Tkhoov proposed the mmzato of the fuctoal. wth 1 m [ ] ( f ( a ) z ) R [ f ],, z f f 1. Later, other regularzers were troduced the form R [ f ] f R [ f ] P f, where * s a semorm ad P a dfferetal operator. Now, we cosder our proposed model the form m [ f ] 1 ( f ( a ) z ) R [ f ] R [ f ]. (8.3) f 1,, z Observe that f we add two fuctoals of the form 1 R 1 [ f ] R [ f ], we could wrte ( R [ f ] R [ f ] ). So we wll adopt ths form. I partcular our fuctoals are some staces of Ducho semorm m! R[ f] Jm[ f] f ( x ) dx,! m 70

79 for dfferet values of m. The key dea s to combe fuctoals wth dfferet degree of smoothess. We frst cosder J wth J 1 ( m 1, ), obtag the fuctoal 1 m [ f ] ( f ( a ) z ) ( J [ f ] J [ f ] ) 1,, z 1 1 f 1 1 ( f ( a) z ) 1 ( f xx fxy f yy ) ( f ) x f y. 1 (8.4) Thus, applyg the methods of varatoal calculus o,, z [ f ] gves the PDE 1 ( )K, (8.5) where K s a fudametal soluto. The soluto to ths ad other equatos of the regularzato fuctoals are gve table 8.1. They have bee chose followg the crtera desged chapter fve, about the global or local character of the fuctoals. We try to mx both characterstcs the same expresso. The greater the value of m, the smoother the fuctoal. Here t s mportat to remember that the varatoal dervatve of Jm[ f ] s m u. Due to the physcal terpretato of parameters we have used 1 ad. Fuctoal PDE Kerel J [ f ] K 0 ( r) r log r 1 1( r ) (log 0( )) r K r J f J f [ ] 1[ ] J f J f ( )K 3 [ ] 3[ ] 1 3 ( )K J [ f ] J J [ f ] 3 x ( x) Ce ( x ) 1 r r ( r) ( log r log r 0( )) 4 1 ( )K 3( r) ( log r C 1 0 ( v r) C 0 ( w r)) Tab. 8.1: Sples deduced from varatoal approach, to be appled o scattered data. Each sple ca be obtaed by solvg the Partal Dfeferetal Equato (PDE) that appears the secod colum. Due to ther physcal org we have used 1, (8.3) v, 1 14 w, C 1 w v w, v C, v w 71

80 0 s the modfed Bessel fucto of the secod kd wth Iv( z) Iv( z) v () z, ad s( v ) z z 4 Iv ( z) k v.. For our case 0 k 0 k! ( v k 1) 8.5 The Relato wth Radal bass fuctos The geeralzato of sples to hgher dmesos are called surface sples or radal bass fuctos, they are obtaed by mmzato of orms or fuctoals ad have the rotatoal ad traslato varace propertes. Radal bass approxmators have the form S( x) j( x a j ) p( x), (8.6) j1 where p ( x) s a low degree polyomal. For example, the cubc sple may be expressed 3 ths way wth () r r. If we use the sple S to terpolate the scattered data { a1, a,, a } the S s completely determed by the scalars or weghts s. Ths s doe solvg the lear system K( a, a ) p ( a ) z 1k j k j k k j1 1 j1 N p ( aj ) N, (8.7) such that ( 1,, ) N (,, ) ad the polyomals{ p } N form a j j 1 1 bass for ( ) 1. I ths way the system ca be wrtte as m A t P P 0 z 0, wth A, K ( a, a ) ( a a ), j 1,,, P p ( a ) 1,,, j 1,, N j j 1 z j j j (,, ), ( z1,, z ). Sometmes the sple does ot have polyomal term p ( x), the we have A z. We have see (chapter 6) that uder certa codtos K( xy, ) matrces A, K ( a, a ) ad the the matrx s o-sgular. j j produce postve defte 7

81 Chapter 9 Applcatos, mplemetatos ad umercal expermets 9.1 Itroducto The varatoal framework we have bult ca be appled to multple classes of data. I partcular, ths chapter s devoted to test ad show the performace of our methods the recostructo of data comg from surfaces, gve as pot clouds or Cartesa coordates. We aalyze the sples of table 8.1, whch ca be derved usg the theory of the last three chapters. For evaluatg the performace of these sples we have chose some 3D objects wth dfferet degree of complexty ther shape ad topology. Our models are also aalyzed terms of the key deas of approxmato theory, desty, covergece ad error bouds of the approxmators o scattered data. We also clude some detals about the algorthms developed, ther mplemetato ad codto umber of the terpolato matrx. We obta some results that may be surprsg curret lterature, because some of these sples may have a equal or better performace tha th plate sples. 9. Represetato of surfaces I real world problems we eed a represetato of surfaces a operatve form order to be used to a computer. Ths represetato s gve as dscrete pots. Nevertheless t s mportat to represet a surface as a cotuous set of pots. Explct represetato z f ( x, y) Ths s the usual mathematcal represetato ad correspods to the defto of a fucto f :. It has the property that all the theores of fuctoal aalyss we develop ths work assume ths represetato. Oe dsadvatage (fg.9.1) s that may real pot clouds there are pots ( xy, ) wth more tha oe value for z. 73

82 Fg. 9.1: Ths objects caot be represeted wth a explct form z f ( x, y) m The dea of a fucto f : s the basc tool for ay mathematcal model ad ca be adapted for beg appled to ay kd of object or data. Parametrc represetato r( u, v) ( x( u, v), y( u, v), z( u, v)) Ths represetato s more useful the geometrc methods descrbed chapter. It s the represetato used dfferetal geometry, where ay regular parametrc surface ca be descrbed the form, such that the vectors r u,r v geerate the taget plae at each pot of the surface wth r r ( x y z v,, r x y z ), r v (,, ). u u u u v v v v 9.3 odels ad objects aalyzed Our purpose s to evaluate our regulato framework as well as ts extesos. I order to make comparsos wth the bg amout of classcal ad ew lterature, ths evaluato has bee doe usg some well kow objects gve as pot clouds, the explct form z f ( x, y) or parametrc form. The frst publcatos wth some of these objects appeared the early 80 s. Nowadays there exst a bg repostory ad sources for pot clouds that vary from very smple fgures wth thousads of pots to very complex objects the order of mllos pots. The models we have chose by ther relevace to our work are show below. 1. Frake s data ( ladslde ) 1 y f ( x, y) 1 ( y ) y x, (1 y) (1 x) y x (9.1) 74

83 . Frake s fucto 1 (9x1) (9 y1) ((9x) (9 y) ) ) f ( x, y) 0.75e 0.75e 3. Peaks surface e (9 y3) (9x7) 4 e ((9x4) (9 y7) ) (9.) x 3 5 f ( x, y) 3(1- x) exp(- x - ( y 1) ) - 10( - x - y )exp(- x y ) exp(-( x1) - y ). 3 (9.3) 4. The Tube. u u (, ) 1 3 s 6 s z u v e v e v v y u v e u u 6 v x( u, v) (1 e )cos u cos ( ). u 6 (, ) ( 1 )s cos ( ) (9.4) (a) (b) Fg. 9.: The surface (a) represets the parametrc equato (9.4). I (b) we show a set of 00 sample pots o ths surface. 75

84 (a) (b) (c) (d) Fg. 9.3: The surface (a) correspods to equato (9.1), t s kow as Frake s data. Ths surface shares may features wth the fadsk (b). By ther edges ad vertces, they are both used to test recostructo of dscotutes. The surface (c) s geerated by the fucto (9.3) ad t s well kow to ATLAB users wth the ame peaks. The surface (d) s kow as Frake s fucto ad s the graph of (9.). It has bee wdely used may publcatos ad has become a stadard test for surface recostructo methods. 76

85 (a) (b) Fg. 9.4: The rage scag pot cloud (a) has pots ad we have used t to test our method. I (b) s show a recostructo of these osy data takg 000 pots from (a) ad applyg TPS. 9.4 Iterpolato ad Smoothg wth surface sples The theory developed ths thess produce surface sples represeted as radal bass fuctos. Ths property makes them very sutable for applcatos o scattered data. These sples have the form S ( x) ( x a ) p( x), (9.5) f, A 1 where p ( x) s a low degree polyomal. Although we ca use ay base for these 1 polyomal terms, t s commo to use the base x x x x, thus, { 1, x, y }, 1 {1, x, y, x, y, xy}, {1, x, y, z, x, y, z, xy, xz, yz} are take as bass for ad ( ) ( ). k 3, respectvely. I geeral,{ x }, Z, k 1 ( ), ( ) s take as a bass for We have show (chapter 6, 7, 8) that these terpolats are lear combatos terms of the fudametal solutos of some partal dfferetal equatos. I partcular, we studed sples related wth the PDE s of table 8.1, where 0 s the modfed Bessel fucto of the secod kd (besselk atlab) ad where () r s oe of the fuctos (9.). 77

86 0 ( r) r log r ( TPS) 1 1( r) (log r 0( r)) 1 r r ( r) ( log r log r 0( )) ( r) ( log r C 1 0 ( v r) C 0 ( w r)) x ( x) C( e x ) teso sple (9.6) From ths lst, we have already studed th plate sple ( r) r log r our paper 0 [19]; showg that t has a very good performace for terpolato ad smoothg of scattered data. Our results are smlar to those quoted the lterature. However there exst few reports or o oe about the applcato ad performace of the other fuctos (9.6), so ths chapter we are gog to develop algorthms ad umercal tests to measure the performace of these fuctos ad compare t wth th plate sple, whch s cosdered up to ow oe of the most effcet terpolators. It s possble to show that the fuctos of ths famly are codtoally postve defte [7]. For fdg the explct expresso of the sple S f, A( x) ( x a ) p( x) t s ecessary to determe the coeffcets s the frst term ad c s for the polyomal p( x ) ( ). Ths s doe solvg the ( N) system (8.5) of lear equatos, where m1 N dm ( ). I real cases N s always very small compared to so the complexty m1 3 of solvg the system s O ( ). 1 Other source for creasg the computer tme s represeted by the evaluato of S, ( x ) o a grd. Whe the grd s very large there exst dfferet strateges ad fast algorthms [15]. We have mplemeted umercal algorthms for solvg the system ad evaluatg the sple usg atlab wth 000. It s worth to observe fg.9.5 that a relatvely small umber of pots produce a good vsualzato. Ths fact suggests sples as a good method for mesh ad mage compresso. f A 9.5 How to choose a terpolat There are three very useful crtera to judge effectveess of a terpolat or approxmator: desty, terpolato ad order of covergece. 78

87 Desty A subset U a ormed space s sad to be fudametal f the set of all lear combatos from U s dese. Otherwse, f ad 0 there s a vector ju j wth u j U, such that j1 f j 1 juj. Ths s crucal because our terpolators are costructed o lear combatos of traslates of a fxed fucto. It s kow that f s a compact subset of ad s the subspace of C ( ) defed by spa{ x ( xa ) : x } the s dese C ( ) for fuctos () r of (9.6), amog others. For example, t ca be proved [7,109] that the space 1 ( m ) s dese the space Iterpolato ad error bouds ms, X of ( ms-sples., ) The sples S we have studed are terpolatg, that s S( a ) f ( a ) 1,, or smply S f. Oe mportat observato s that we ca always terpolate uquely wth codtoally postve defte fuctos f we add polyomals to the terpolat ad f the oly polyomal that vashes o our set of data ceters s zero, that s, s ( ) m1 - usolvet. Theorem 9.1 ([191]) Let be a codtoally postve defte fucto of order m o, ad let the data set { a1, a,, a } be ( ) m1 -usolvet. The the system (8.10) s uquely solvable. The accuracy of the terpolat ca be estmated applyg the settg of theorem A. (see appedx) wth V { u : L ( u) 0, 1,..., }. We have Theorem 9.. Let L ( f ) f ( a), 1,,, be cotuous lear fuctoals o such that ux s the represeter for the pot x. Let f have mmal orm terpolat S wth S u. Let P be the orthogoal projecto from oto V, the 1 Covergece Pu f ( x) S( x) [ f, f ]. The covergece of terpolats o scattered data ca be studed terms of the spatal desty of the set of odes { a1, a,, a }. For ths purpose the Hausdorff or fll dstace x 79

88 ha, : sup m y a j, (9.7) y aja of the set wth a eclosg doma s used. ha, gves the radus of the largest ball wthout data stes or data-ste free ball. (a) AX (b) RS (c) ha, Fg. 9.5: Illustrato of the covergece of S f, A. (a) The axmum Error AX ad (b) Root ea Square Error RS decrease whe the Hausdorff measure (c) ha, 0 as a cosequece of creasg. A cetral dea the terpolato problem s that as the set flls, the error betwee the fucto ad ts terpolat should go to zero.; ths s see the values of ha,. If k f C( ) say, ad S f s the terpolat, the oe mght hope to obta f S f ( h ) as h 0, where k s some measure of the smoothess of f. If we take a sequece of observatoal data such that ha, teds to zero, the reproducto error always behaves lke k a power ha,, where k 0 creases wth the smoothess of ( r). If () r s a aalytc fucto as the Gaussa ad ultquadrcs, the error decreases expoetally lke wth c 0 as show by adych ad Nelso [11]. / ce c h But sometmes thgs are ot so easy because ths excellet covergece may have the problem of a ll-codtoed system, whe h 0. I some cases t s ecessary to apply addtoal techques as precodtog. Ducho [48] showed that uder certa codtos lm S f 0, ha, 0 f, A m, s where f deotes the uque elemet of mmal semorm R[ u] J [ u] the set { u X : u f }. ms, m 80

89 There exst smlar results for terpolatg ad smoothg sples that clude (9.6) [7]. Example 9.1 Testg terpolato ad covergece by sples. I ths example we test the covergece behavor of TPS for terpolatg the peaks surface o dfferet umber of scattered data the rego [ 4,4] [ 4,4]. We have obtaed the model (9.1) for TPS usg a set { a, a,, a} of scattered data, the ths sple s used to approxmate the peaks surface wth explct represetato x 3 5 f ( x, y) 3(1- x) exp(- x - ( y 1) ) - 10( x y )exp(- x y ) 1 exp( ( x 1) y ), 5 3 o a secod fxed set of pots. I the fg. 9.3 the mprovemet of accuracy wth decreasg values for max error ( AX ) (see equato 9.9) ad Root ea Square Error ( RS ) (9.10) as the Hausdorff measure ha, 0 s observed. Ths behavor s also llustrated table 9.1. Test ha, RS AX ,6653 1,0076 6, ,4504 1,4760 4, ,1573 0,146 1, ,1573 0,1769 1, ,9695 0,1044 1, ,7486 0,040 0,4374 Tab. 9.1: Results of umercal expermet for terpolato of peaks surface. Icrease the umber of pots, mples decreasg values ha,, RS ad AX. ost of the exstg results about terpolato, covergece ad error bouds for radal bass fuctos deped o the varatoal propertes ad postve defteess of ( r), so they are applcable to the terpolats studed here, for more detals see [191]. 9.6 Recostructo of data wthout ose Although real pot clouds are commoly osy, the terpolato of exact data t s a mportat crtera for surface recostructo methods ad t s the frst umercal test performed o the sples studed here. I ths case our sples have a very smlar behavor ther reproducg qualty of the tested fuctos but they may have great dffereces the executo tme. 81

90 The table 9. shows the executo tme secods of the sples (9.6). For obtag the data ths table we have terpolated peaks surface, takg 300 scattered pots ( xy, ) the rego [ 3,3] [ 3,3]. The tme cludes all the steps from the geerato of the pots up to the vsualzato of the sple. Obvously the terpolato mproves wth a creasg umber of pots. 0 ( TPS) Tab. 9.: Executo tme ( secods) of terpolato wth sples. Fg. 9.6: Vsual comparso of executo tmes for sples. As we ca see the executo tmes for the terpolato are loger for 1,, 3 due to the evaluato of Bessel fucto these sples. As we ca see table 9., the values of 1 ad are very smlar, so ther graphs are supermposed. A smlar stuato occurs for TPS ad teso sple. 8

91 (a) 00 (b) 00, 0 (TPS) (c) 00, 1, 0.1 (d) 40, 1, 0.1 Fg. 9.7: (a) Three dmesoal scatter plot of = 00 pots o Frake s surface. These pots are used to recostruct the surface. Fgures (b), (c),(d) show a smlar accuracy for TPS ad sple 1 ; evertheless, the performace of 1 ca be mproved by tug the parameter. 83

92 (a) 40,, 0.1 (b) 40 0 (TPS) (c) 40,, , 3, 1, Fg. 9.8 I these fgures, we see that t s oly requred a small umber of pots for obtag a good reproducto wth all the sples lsted (9.6). I geeral, the umercal expermets show a very smlar behavor. As we show later, the great dffereces appears whe dealg wth osy data. 84

93 (a) (b) Fg. 9.9: ay umercal expermets have bee ru for surface terpolato. Fgure (a) shows a typcal scatter plot for pots o Frake s surface ad (b) the exact surface. A mportat test for vsualzato s to recostruct a surface by terpolato from a small umber of scattered data. We have ru may umercal expermets for surface terpolato. I fgures 9.7 ad 9.8 we show some remarkable cases. For ths task, we have used Frake s fucto wth dfferet values for the rego [0,1] [0,1]. 9.7 Codto umbers ad error bouds Assocated wth the soluto of a lear system A x = b there s a umber ( A ) called the codto umber of the matrx A ad s defed as a product of the magtudes of A ad ts verse; that s 1 ( A ) A A, (9.8) where s a matrx orm. If the soluto of A x = b s sestve to small chages the rght-had sde b the small perturbatos b result oly small perturbatos the computed soluto x. I ths case, A s sad to be well codtoed ad correspods to small values of ( A ). If the codto umber s large the A s ll codtoed ad the umercal solutos of A x = b caot be trusted. To uderstad the umercal behavor of the sple approxmatos t s essetal to have bouds o the approxmato error ad o the codto umbers of the terpolato matrx the system A t P P 0 z 0. 85

94 These bouds are usually expressed employg two dfferet geometrc measures. For the approxmato error s crucal to kow how well the data stes { a1, a,, a } fll the rego. Ths ca be also measured by the Hausdorff dstace (9.7), whch gves the radus of the largest data-ste free ball. 0 ( TPS) , , , , , ,0747 0, , , , , ,798 0, , , , , , , , , , , ,640198, Tab. 9.3: Codto umbers for the terpolato matrx of sples. It s remarkable that although these values may be very hgh, the regularzato parameters cotrol the ll-posedess of the terpolato matrx. The codto umber, however, wll obvously oly deped o the data stes ad ot o the rego. oreover, f two data stes ted to coalesce the the correspodg terpolato matrx has two rows whch are almost detcal. Hece, t s reasoable to measure the codto umber terms of the separato dstace 1 qa : m a j ak. jk I the table 9.3 we report the behavor of the codto umber ( A ) of the terpolato matrx for every sple from the lst (9.6). We observe a creasg value of ( A ) wth respect to a creasg umber of scattered data ad comparso wth TPS. I spte of these large values, the soluto of the system ad the recostructo of the surface are possble uder very geeral codtos. 9.8 Evaluato Crtera for the approxmato methods Accuracy For these crtera we use kow surfaces defed the form z f ( x, y), takg samples wth or wthout ose ad approxmatg the whole surface, usg the sple obtaed from the scattered data. We the compare exact ad approxmated values of the fucto. Certaly the usual ad real applcatos of recostructo methods we do ot have a 86

95 represetato of the object the form z f ( x, y), evertheless, f the method makes a fathful approxmato of a varety of surfaces; we expect t to gve reasoable results other staces. I order to test the accuracy of terpolato of exact data wth sples (9.6) We have take a fxed test set B b1, b,, bn ad terpolated the fucto peaks usg scattered data pots A { a1, a,, a } such a way that A ad B are dsjot sets, ts to say they have o commo pot, the A B. We use the dcators axmum Error ( AX ) ( )(, ) max S( b ) f ( b ), (9.9) AX S f 1 ad Root ea Square Error ( RS ) 1 (9.10) N ( RS )( S, f ) N S ( b ) f ( b ). 1 0 ( TPS ) Tab. 9.4: Results for RS terpolatg atlab peaks fucto. A remarkable fact t s the hgh degree of accuracy of ad, better tha TPS. 0 ( TPS) ,611 5,151 3,4896 4,8581 3, ,655,4901 1,1351,554 1, ,5999 0,4793 1,0159,6755 0, ,4003 0,1051 0,16 0,318 0, ,071 0,178 0,0447 0,1151 0,1114 Tab. 9.5: Results for AX terpolatg atlab peaks fucto. 87

96 Fg. 9.10: Ths fgure shows the accuracy of surface recostructo wth our sples. The vertcal axs shows the values of RS for each sple usg data wthout ose. The results ted to be dfferet as creases the umber of pots. As we shall see later, the dffereces are more evdet whe dealg wth osy data. O tables 9.4 ad 9.5 we ca see some of the typcal results obtaed testg the accuracy of sples compared wth TPS. Here s evdet that our sples are sometmes equal or better tha TPS for the reproducg qualty. The performace of Teso sples s aga very remarkable Vsual Aspects As we sad the begg of ths work, huma vsual percepto s a very powerful skll our compreheso of realty. Pot clouds come from laser scag of real objects, so vsualzato s a crtcal task the performace of the models. I geeral we observe that the vsual aspects are smlar for all the sples whe we use exact data. As a example, we ca see (fg 9.11) the excellet terpolato of Frake s fucto by TPS. Nevertheless the dffereces betwee TPS ad the rest of sples may be remarkable whe dealg wth osy data. I fgure 9.11(b) we ca observe that the performace of TPS for terpolatg the osy peaks surface s very poor; but s very good usg the sple correspodg to 3. I geeral ths s the typcal behavor of our famly of sples (9.6). 88

97 (a) (b) (c) (d) Fg. 9.11: I ths umercal expermet we have take a very osy peaks surface (a) ad applyg TPS we obta the surface (b). But tug the parameters of 3, we fally obta the mproved surface (c) that ca be compared wth the exact surface (d). 89

98 (a) (b) (c) (d) Fg. 9.1: We use the pot cloud (a) to test our sples the recostructo of free form objects. I ths example, applyg TPS, wth ad a creasg umber of pots: (b) 100 pots (c) 800 pots (d) 000 pots. 90

99 9.8.3 Sestvty to parameters Tug of parameters permts to ehace or decrease some features of the object. The models based o regularzato have ths remarkable property. Tug of parameters s very useful for fttg dfferet requremets o dfferet applcatos ad t s essetal for osy data. TPS does t have parameters to ts mathematcal expresso, the rest of sples have oe or two parameters we deoted., The ma tred of the curves shows that the optmal, value seems to fall a terval where the accuracy behavor of the terpolat may be ustable. If the same expermet s performed wth a slghtly dfferet data pot resoluto or dfferet, resoluto, the ma tred of the resultg curves remas smlar, but the oscllatory segmets chage. Ths sawtooth stablty has ot yet bee well uderstood although t has bee recogzed the lterature studyg other radal bass fuctos [57]. Nevertheless, our umercal expermets show that t s always possble to tue the values of parameters for obtag good approxmators. I our expermet we have evaluated sples by calculatg RS ad AX o osy data. The values of RS show a smlar performace as the graphs of fgure 9.7. They show a oscllatory behavor ad the grows rapdly. These expermets have show us that the best values for, are the terval [0, 1] ad sometmes a larger terval but ot larger tha [0,10]. (a) (b) Fg : We made a lot of expermets varyg radomly the values for ad the tervals 5 [0,1] ad[1,10 ]. They all have ted to stablze the terval [0,1]. (a) RS. (b) AX. I Fg we show more results wth recostructo (terpolato) for the Frake s data wth 100 scattered pots for dfferet values of ad. 91

100 (a) (b) (c) eps (mache) (d), 1 (e) 1, 1 (f) 0 TPS 0 Fg. 9.14: Ths fgure shows some results from dfferet tests o regularzato parameters. We have used Frake s data for testg a extreme case of dscotutes recostructo. The results show how to preserve dscotutes by tug the values of parameters. 9

101 (a) (b) (c) Fg Fgures (a), (b) shows 100 pots, take radomly o Frake s data (c) Tmg The executo tme of a algorthm s measured by ts computatoal complexty whch, as we sad chapter 3, t s a trsc problem recostructo. Nevertheless a method that s slow ow, may be ot so much a better hardware, or f the method worth the effort, we ca develop faster algorthms. Ths s the case of all our sples; although they may requre a huge umber of floatg pots operatos a computer, the results o recostructo capabltes are very ecouragg. I table 9. we show some typcal performace tmes ( secods) of our sples famly (9.6). It s easly observed, as we may hope, that the sples cotag Bessel fucto are very tme cosumg, specally 0 3. They have the form S ( x) ( x a ) p( x), f, A 1 where s the umber of cetres, so ths evaluato s proportoal to ad may take very log tme Ease of mplemetato. Our purpose has bee to obta explct aalytc expressos for approxmators order to develop better applcatos. From ths pot of vew our sples are very well-suted for mplemetato ay computer laguage. They are lear combatos of a fxed fucto, gve terms of polyomals, expoetals ad logarthms. Nevertheless TPS ad teso sple requre smpler expressos tha 1,, 3 whch cota the Bessel fucto 0. 93

102 9.9 Recostructo from osy data. Cross valdato Other mportat task s the determato of the smoothg parameter. Vsual appearace s a good crtero for vsualzato problems, so these case the values for may be determed by tral ad error. s a very sestve parameter, so small chages ts values may cause large chages the vsualzato of the surface as observed former fgures. s pealzg large values of the fuctoal R[ f ]. The fuctoal cotas a great amout of dervatves such that should assume large values for o smooth fuctos ad small for smooth fuctos. It s possble to fd optmal values ˆ for the smoothg parameter the sese of some crtera. Oe of them s cross valdato. The basc dea of cross valdato for evaluatg the performace of a approxmator of the data D {(, ) a z } 1 s to buld the approxmator over the set { a : k}, that s, wthout cosderg the kot a k ad the repeat ths procedure for k 1,,. Let [ k ] f be the mmzer of 1 H [ f ] ( f ( a) z ) R[ f ], 1 k the t s reasoable to look for the model whch mmzes the ordary cross valdato fucto V ( ), defed as 0 1 [ k ] V0 ( ) ( zk f ( ak) ), k1 The mmzer of V ( ) 0 s kow as the leavg-out-oe estmator of. However, computatoal terms, fdg ths value s very expesve. A alteratve s to use the fluece matrx B, whch may be foud [188] wth f ( a1 ) f ( a ) f ( a ) B( ) z. It ca be show that [ k ( z ( ] k f ( ak) ) zk f ( ak) ), the (1 bkk ( )) 1 ( zk f ( ak) ) V0 ( ). (1 b ( )) k1 kk mzato of the fucto V0 ( ) s a powerful crtero for the choce of a optmal. But there exst a eve easer method called Geeralzed Cross Valdato (GCV), where (1/ ) I B( ) z V ( ), [(1/ ) tr ( I B( ))] 94

103 ad B ( ) s the hat matrx of the rdge regresso [13,188] 1 B( ) X( X T T X I) X. At frst glace, optmzato of V ( ) seems a formdable computatoal problem sce each value of has ts correspodg B ( ). However, [13] gave a method of expressg V ( ) as a easly-calculated ratoal fucto based the sgular value decomposto (SVD) X=UDV T, where U s p wth orthoormal colums, V s p p ad orthogoal, ad D s p p ad dagoal wth dagoal elemets d1 d d p 0, whch are the oegatve square roots of the egevalues of X T X. Fally we obta the expresso V ( ) t j1 t j1 d j dj z j. (9.11) Example 9.: Iterpolato ad Cross valdato. The results for ths example are show Fgure For ths expermet we have take 400 osy pots { a, a,, a } 1 o Frake s fucto [0,1] [0,1] to be approxmated wth the Th Plate Sple. We ca allow the surface to pass close to, but ot ecessarly through, the kow data pots, by settg 0. Whe 0, the fucto terpolates the data pots. As approaches zero, the surface becomes rougher because t s costraed to pass closer to the data pots. At 0, the surface terpolates the data, ad overshoots are much more evdet. The optmum value for s determed mmzg the GCV fucto 0 V ( ) to obta a smooth surface wth fdelty to data. At larger values of ( ), the recostructed model s smoother ad approaches a 0 amorphous bubble (e). It has bee calculated a error (Error = ) for ths expermet wth 1 Error z j f ( a), j 1 where f( a j ) are the exact values o the data pots ad z j obtaed usg GCV. I example 9.4 we study a case whch we apply the other sples to data wth hgher degree of ose. Example 9.3 Testg regularzato parameters. The approxmator TPS, has also bee tested usg k -fold cross valdato. I ths method the data set (Fg.9.11 (a)) s dvded to k subsets whose pots are chose radomly. Each tme, oe of the k subsets s used as the test set ad the other ( k 1) subsets are put together to form a trag set. 95

104 (a) (b) (c) (d) Fg. 9.16: Ths example shows the potetal of sples to smooth a osy surface by tug regularzato parameters by Geeralzed Cross- valdato (GCV). Fgure (a) depcts the osy data ear the surface. I (b) we see the terpolato of the orgal data wth 0, wthout smoothg. (c) The optmum value for the regularzato parameter s foud by mmzg the fucto 0 V ( ) ad the resultg surface s show fgure (d). 96

105 (a) (b) Fg. 9.17: Whe the surface s recostructed by usg values larger tha 0 the surface s too smoothed ad loses fdelty to data as t s the case show fgure (a), wth 1. The data used for k-fold cross valdato are show (b). The fucto approxmator fts a fucto usg the trag set oly. The the fucto approxmator s asked to predct the output values for the data the testg set. I ths case we have take k =5. I fgure 9.11 (f) s show oe case for trag set (red) ad test set (gree) The errors for each test set were very smlar, showg the robustess of the method to dfferet degrees of complexty of the surface or data. The errors were accumulated to gve a mea absolute test set error ( ). Test Error set e e e e e-00 Tab. 9.6: Results for k-fold Cross Valdato. 97

106 (a) (b) (c) (TPS), 0 GCV (d), 1, , 10 4 Fg Fgure (a) shows a osy pot cloud from Frake s surface. (b) terpolato of the former data. I (c) we recostruct the surface wth TPS ad choosg by GCV,but the result 0 stll has ose. I (d) we try the same problem wth but the result s smlar to TPS. 3 Nevertheless, as we show ext (see Fg. 9.19), t s possble to choose adequate values for regularzato parameters to obta hgh accuracy surface recostructo eve hghly osy data. 98

107 (e), , (f), 0.1, (g), 0.1, (h), 0.05, Fg (cot.): Now, we show a successful surface recostructo that remarkably mproves the performace of TPS. Fgure (e) shows that although fals to smooth the ose (d), the 3 parameters ca be tued to obta good results. I (f), we show that ot oly elmates the ose o data, t also gets a hgh accuracy the recostructo of the orgal surface. The other sples have a smlar performace, as see (g) ad (h). 99

108 Example 9.4: Regularzato of a extremely osy data set. Now we take a more osy pot cloud for Frake s fucto fg 9.1(a). I ths case we take 400 hghly osy data o Frake s surface. I fgure 9.1 (b) s depcted the terpolato of the osy surface. The recostructo of the surface s doe usg the regularzato parameter by GCV. We observe (c) ad (d) that TPS ad have a smlar qualty ad t s really remarkable the 3 smoothg doe by the sple teso. The pcture (f) shows that performs much better tha TPS whe treatg very osy data.. Furthermore, (f), (g), (h) show how these sples elmate the ose from data by tug the values of regularzato parameters. Example 9.5: Recostructo wth larger umber of pots. Now we show the performace of regularzato of data comg from free form objects ad take by rage scaers, so geeral, we assume the data are osy. We show some results obtaed by sples teso wth parameter aroud 0.1 ad Further expermets wth the other sples have show smlar behavor. A sample s show Fg Fg. 9.19: Example of recostructo of the face from a large pot cloud wth teso sple. 100

109 9.10 Comparsos wth other methods From the sples tested here oly th plate sples have bee extesvely studed ad although other authors have studed sples teso, ths has bee commoly doe for data wthout ose ad maly for curves [38,153]. After the success of th plate sples at the 1970 s there has bee a great terest mprovg ts performace regularzato problems. Bascally, from the physcal terpretatos chapter 5, we ca uderstad the lmtatos of TPS because the materal aspects of a elastc plate are rather lmtg wth respect to ts physcal realty. That s, oe may abstract from ts thrd dmeso; t s elastc, hece t does t deform, oly bed, ad t s a plate, ot a membrae, whch meas t s stff. Its behavour s rather that of steel tha that of gum. The sples studed ths work deal wth ths problem tegratg local ad global features regularzato model. Oe of the frst to propose surface teso was Frake [63,64], who made extesve study of dfferet sples. Nevertheless, the research of Frake was doe oly for terpolato. Perhaps due to the lmted computatoal resources of the 1980 s, there s ot much research comparg the propertes of the sples we study here. The studes of Frake were doe aroud 198, so we ca say that the avalablty of better computatoal resources we have today allowed us a better aalyss ad vsualzato Coclusos I ths chapter we have performed umercal tests that show how a adequate choce of regularzato fuctoals coduct to stable solutos of severely ll posed verse problems, wth a hgh degree of accuracy. We have show that our sple famly (9.6) s a set of adequate approxmators of surfaces gve as scattered data; cludg free form objects comg from rage scaers. It s possble to fd tervals for adjustg regularzato parameters ad we ca fd ther values by GCV or by tral ad error. Ths last opto does ot mea chaotc choce, because we provde some tervals ad key values that become very useful practcal problems. Comparg wth other works as Frake s, our umercal tests have smlar performace some cases, ad other cases we fd better results. 101

110 Chapter 10 Coclusos ad ope problems 10.1 Geeral coclusos of the thess I the frst part of the thess, we maly focused o buldg a geeral framework for solvg recostructo problems wth dfferet kds of data, that way we obtaed a formulato that jos the classcal verse problems theory wth deas from Schwartz dstrbuto theory. Ths geeral formulato ca be appled to a wde class of verse problems, partcular we dealt wth Lagrage data. Costructg ths formulato we ecoutered ad solved the followg ssues: I order to select the most approprate framework, we have appled verse theory to determe ad classfy the most relevat problems that arse recostructo problems. I ths way we saw that the most crtcal pot s how to clude local ad global features the approach to be appled. We foud that the regularzato framework has eough geeralty for fulfllg these requremets. We foud that t s possble to clude multple fuctoals order to capture local ad global propertes o the data. We foud that the cluso of regularzato fuctoals ca be coducted by physcal ad geometrcal crtera. We have chose as regularzers the famly of Ducho semorms whch have a terpretato elastcty theory. A very mportat dea s the resoluto of the fuctoal, gve by ts ull space. Ths space should ot be too large order to capture the complexty of data. I decdg what kd of fuctoal spaces are adequate for solvg recostructo problems we foud Schwartz dstrbuto theory as a powerful mathematcal tool for desgg sples. By covertg the addto of fuctoals to a optmzato problem, we obta solutos that satsfy local ad global propertes whch ca be tued usg regularzato parameters. Ths method s ot oly for partcular cases, t ca be geeralzed to -dmesoal data ad dfferet kd of problems. We obtaed explct expressos the form of robustly mplemetable radal bass fuctos. 10

111 I the secod part of the thess we explore applcatos of the regularzato framework to problems of surface recostructo ad vsualzato. I ths feld, ll-posedess appears the form of havg to decde a trade-off betwee a good ft of the data ad some model requremets that comes expressed the degree of localess (or globaless) of regularzato fuctoals. We drew the followg coclusos: The sples famly studed here ufes several sple formulatos ad yelds radal bass fuctos whch produce practcal mplemetatos. These sples perform, some cases, equal results to th plate sples ad better results other cases. The results are smlar for all the sples whe dealg wth data wthout ose ad they are really better recostructo of osy data. The sples studed have two terms. The frst oe cotas a lear combato of traslates of a fxed fucto (parametrc part statstcal laguage) ad a lear polyomal term, that are mplemetable stable form. The optmum regularzato parameters are determed by geeralzed cross valdato ad others are foud to belog to short-legth tervals. The tug of these parameters permt us to obta a trade-off betwee local a global features. The cocepts of regularzato theory gve a comprehesve framework to formulato of the problems vso: at all three levels of problem, algorthm, ad mplemetato. Furthermore, the mathematcal theory of regularzato provdes a useful theory for corporatg pror kowledge, costrats, ad qualty of soluto. The exctg prospect s that the aaloges wth elastcty theory ad physcs wll cotue to provde frutful deas for developmet of regularzato theory ad for uderstadg huma percepto Future research ad ope problems We have gve a wde theoretcal framework for dealg wth may kds of recostructo problems whose data are gve as a set D { L1( f ), L( f ), L ( f )} of bouded lear fuctoals. Applyg ths theory o surface recostructo, very good results are obtaed, that promse smlar performace appled to other kd of data ad o lear fuctoals L ( f ). So, we have a great amout of possbltes for future work the followg subjects. No lear verse problems I recostructo problems the relatoshp betwee object ad mage has the most geeral represetato the form of a o lear tegral equato g( s) h( s, t, f ( t)) dt, where h s a cotuous fucto. Ths equato gves the soluto of the drect problem ad the 103

112 verse problem s obtaed by exchagg the roles of the data ad the soluto: these cases we must fd the object f for a gve mage g. The soluto of ths equato s very dffcult from both the theoretcal ad practcal pots of vew. No geeral theory exsts for such a o lear tegral equato ad each problem requres specfc aalyss, moreover the problem may be ll posed ad o geeral regularzato theory exsts for wde classes of o lear problems. Bomedcal magg I geeral the ew techques of medcal magg are based o the terrogato of the huma body by meas of radato trasmtted, reflected or emtted by the body: the effect of the body o the radato s observed ad a mathematcal model for the body-radato teracto s developed. The veto of computed tomography (CT) by G. H. Housfeld at the begg of the sevetes was a breakthrough medcal magg. The cocepto of CT s based o deas whch opeed ew ad wde perspectves. CT requres a mathematcal model of X-ray absorpto. A specfc feature of medcal magg s that the problems to be solved are ll-posed the sese of Hadamard; ths meas that although the avalable data cotas a bg amout of formato, the fact that the problem s ll posed, combed wth the presece of ose, mples that the extracto of ths formato s ot trval. Ths s a very mportat challege for the future. Fast algorthms Durg ths work we have faced problems regardg the algorthmc complexty of sples, especally for those expressos that cotas the Bessel fucto of the secod kd 0. The applcatos of radal bass fuctos mply two basc problems: 1. The soluto of very large lear systems, ad. The evaluato of very log lear combatos o a very large amout of pots. These two facts could be sometmes dsappotg; evertheless the methods are very promsg, so they deserve the aalyss ad desg of algorthms able to lower ts complexty. Fortuately, ths has bee doe for some radal bass fuctos, but there remas more research about these algorthms, whch become ecessary whe dealg wth thousads or mllos of data. 104

113 Appedx 105

114 The recostructo problem o Hlbert spaces Regularzato problems fte or fte-dmesoal spaces ca be modeled uder the followg framework [35]. Let be a ormed lear space ad D { L1( f ), L( f ), L ( f )} are lear formato fuctoals o. The dea s that for a gve f, the followg formato L ( f) z 1,,, s avalable. From ths formato we wat to compute a value L ( f ) where L s aother lear fuctoal. I partcular, for recostructo of pot clouds the L s are pot evaluato fuctoals of the form L ( f ) f ( a ). From the formato gve by the L s, the goal s to "lear" f (.e. to fd a good approxmato of f ) from radom samples. But wthout ay restrctos o the class of fuctos f, there s o hope of cotrollg L ( f ), smply by kowg the values L1( f ), L( f ), L ( f ). So, we wll look for a approxmato to f o a Hlbert space. At frst stace the oly formato s that the desred elemet s the lear mafold U { u : L ( u) L ( f ), 1,... }. The the problem may be treated as a mmal orm terpolato o the mafold U, where U s a traslate of the subspace V { u : L( u) 0, 1,..., }. If S s the elemet of mmal orm, the S V ad L( S) L ( f ), 1,...,. Theorem A.1 (Resz represetato theorem). If L s a bouded lear fuctoal o a Hlbert space there exsts exactly oe x such that 0 L( x) ( x0, x) x. The elemet x 0 s called the represeter of L. I other words, ths theorem reveals that bouded lear fuctoals o a Hlbert space have a very smple form ad the represeters of the fuctoals L ( f ) ca be used to fd the mmal orm terpolat S. Theorem A.. Let L1( f ), L1( f ),, L ( f ) be cotuous lear fuctoals o, wth represeters u1, u,, u, respectvely. Let f have mmal orm terpolat S o L ( f ), L ( f ), L ( f ). The 1 106

115 S 1 u, where the coeffcets are chose to solve the system of lear equatos ( u, uj ) ( f, u ), 1,...,. j1 Proof. Usg represeters, L ( ) (, ), 1 f f u the V { u : L ( ) 0} u 1 { u : ( f, u ) 0} 1 K spa 1 1 spa 1 { u, u,, u }. O the other had S V the S ( u ) { u1, u,, u }, hece t s possble to wrte S u. 1 The secod part of the theorem comes from L( f) L( S), 1,...,, the ( f, u) L ( f) L ( S) L ( ) j 1 j uj j 1 j j L( u ) ( u, u ), j 1 j j ad fally ( u, uj ) ( f, u ). j1 Let us ow suppose that has a sem-er product [, ] wth ull space N such that dm( N ) N. Now let us choose ths same umber of elemets { L1( f ), L( f ), L N ( f )} from the orgal data L1( f ), L( f ), L ( f ) ad suppose the er product (, ) of ca be expressed as ( u, v) [ u, v] ( u) ( v) N L L. 1 Theorem A.3. The set of represeters { u1, u,, un } of { L1( f ), L( f ), L N ( f )} s a orthoormal base for N, ad N V. Ths ca be see by N ( u, v) [ u, v] L( u) L( v) L( u) L( v) ( u, u )( v, u ) N 1 1 N 1 N ( u, u) u, v, 1 107

116 N the u ( u, u ) u hece { u1, u,, un } geerates N, ad s a bass sce dm( N ) N. 1 O the other had orthoormal. j N u ( u, u ) u, the ( u, u j ) j ad { u1, u,, u N } s j j 1 u V because ( v, u ) L ( v) 0 v V, thus N V. j j Defto A.1. A Hlbert space s called a Hlbert fucto space f the elemets of are complex valued fuctos o a set ad for each x, there exsts a postve costat C such that f ( x) C f for all f. x x It s to say; a Hlbert fucto space the pot evaluato fuctoals L x ( f) f( x) are bouded. The t s possble to apply the Resz represetato theorem. For each x there exsts a uque represeter K such that x L ( f ) ( f, K ) f ( ). x x x Defto A.. The mappg K : such that K( x, y) Kx( y) s called the reproducg kerel (or smply kerel) of. The reproducg kerel s so called because t has the potetal of reproducg each fucto : ( f, Kx) f ( x) for all f ad all x. Some propertes of a kerel are K( xy, ) ( K, K ) xy,. x K( x, y) K( y, x) xy,. K( xx, ) 0 x. y Postve defte fuctos Usually, we have to deal wth approxmato o a arbtrary set { a1, a,, a } of dstct cetres or kots ad a symmetrc kerel K ( xy, ) o. We ca form lear combatos S( x) K( a, x), x. j1 j j Wth such a set we ca form the symmetrc matrx A ( K( a, a )) 1. the terpolato problem S( a ) z 1k k k K( a, a ). j1 j j k j k j k ad pose 108

117 From these terpolato codtos we obta the matrx lear system ( 1,, ), y ( y1,, y ) ad the terpolato matrx A z, wth K( a1, a1 ) K( a1, a ) A. (A.1) K ( a, a1 ) K( a, a ) Defto 4.. We say the fucto K( xy, ) s postve defte f for ay pots a1, a,, a, ad 1 (,,, ) \{0} the quadratc form t A j K( a, a j ) 0, (A.a), j1 ad strctly postve defte f > holds. ore geerally we say that Defto A.3. K( xy, ) s a codtoally postve defte kerel of order m ( m-cpd) o f for ay choce of fte subsets ad all ( 1,,, ) \{0} satsfyg the quadratc form, j1 A a1 a a j1 {,,, } of dfferet pots jpx ( j) 0, ( p ) m1, (A.b) jk( a, a j ) s postve. If m 0, ths defto s reduced to the case of postve defte fuctos ad the codto (A.b) s empty. Defto A.4. The pots A { a1, a,, a } wth N dm m ( ) are called m ( )- usolvet f the zero polyomal s the oly polyomal from m ( ) that vashes all of them. Theorem A.4 ([6]). Let us cosder the space m,, l s X ad a famly 1 compactly supported dstrbutos represetg our data, wth assume: The L s are of order satsfes the codto { L, L,, L } of N dm ( ). We m1 r,.e. cotuous lear fuctoals defed o m r s r C, where r 109

118 The famly s learly depedet D ' ad cotas a subfamly whch s 1 -usolvet. m Ad let Z ( z1, z,, z ) t be a gve vector problem. Cosder the followg terpolato (P1). Fd m,, l s X such that ad the smoothg problem m Ru [ ] u m, l, s m, l, s m,, l s u X L, u z, 1,, (P). Fd m,, l s X that mmze Ru [ ] 1 H [ u] ( L ( u) z ), R [ u] u m.. l s The solutos to problems (P1) ad (P) are gve explctly by a sple of the form L Km,, l s 1 N jp j, (A.3) j1 where { p } N s a bass for j j 1 m1 ad Km,, l s s a fudametal soluto operator m,, l s,.e. m, l, skm, l, s. r C of the I the terpolato problem (P1), the coeffcets 1,,, j j 1,, N satsfyg the terpolatg codtos ad the orthogoal codtos, are solutos of the followg system of lear equatos k m,, l s j k j k 1 j1 N j1 N L, L K L, p z, k 1,, L, p 0, j 1,, N j (A.4) The system ca be wrtte the form 110

119 where K P t P 0 α β Z =, (A.5) 0 K [ L, L K ] s a matrx P s a N matrx j m,, l s 1, j [ L, pj] 1,1 j N ( 1,,, ) t (,,, ) t N Z ( z, z,, z ) t s the data vector. 1 1 I case of smoothg problem (P) we have to solve the system where I s the detty matrx. K + I P α Z t =, (A.6) P 0 β 0 111

120 Refereces [1] Abramowtz,. ad Stegu, I.A. (1970). Hadbook of mathematcal fuctos wth formulas, graphs ad mathematcal tables. Natoal Bureau of Stadards, Appled athematcs Seres. Washgto [] Adams, R. A. (1975). Sobolev Spaces. Academc Press, New York, USA. [3] Ad, B. ad Grevlle, T. (003). Geeralzed verses. Theory ad applcatos. Sprger. [4] Ahlberg, J., Nlso, H.E. ad Walsh, J. L. (1967). The Theory of Sples ad Ther Applcatos. Academc Press, New York. [5] Alexa,., Behr, J., Cohe-Or, D., Fleshma, S. ad Slva, C.T. (001). Pot set surfaces. I Proc. IEEE Vsualzato 001, p.1-8. [6] Ameta, N., Ber,. ad Eppste, D. (1998). The crust ad the b-skeleto: combatoral curve recostructo, Graph. odels Image Process. 60 () pp [7] Arcagél, R., Lopez,.C. ad Torres, J. (004). ultdmesoal mzg Sples. Kluwer Academc Publshers, Dordrecht. [8] Aroszaj, N. (1950). Theory of reproducg kerels. Tras. Amer. ath. Soc., 68, pp [9] Atkso, G. ad Hacock, E. R. (006). Recovery of surface oretato from dffuse polarzato, IEEE Tras. Image. Process., 15 (6), pp [10] Attea,. (199). Hlberta Kerels ad Sple Fuctos. North-Hollad, Elsever Scece, Amsterdam, Hollad. [11] Bajaj, C. Berard, F. ad Xu, G. (1995). Automatc recostructo of surfaces ad scalar felds from 3d scas. SIGGRAPH 95 Proceedgs, pp [1] Barrow, H. G. ad Teebaum, J.. (1981). Computatoal vso, Proc. IEEE, 69 (5), pp [13] Bates, D., Ldstrom,., Wahba, G. ad Yadell, B. (1986). GCVPACK-Routes for Geeralzed Cross Valdato Departmet of Statstcs, Uversty of Wscos. Techcal Report No

121 [14] Beatso R. K. ad Powell,. J. D. (199). Uvarate terpolato o a regular grd by a multquadrc plus a lear polyomal. IA J. of Numercal Aal., 1, pp [15] Beatso, R. K., Carr, J.C., ccallum, B.C., Frght, W.R., clea, T.J. ad tchell, T.J. (003). Smooth surface recostructo from osy rage data. I Proc. Graphte 003, pp [16] Bellma, R. (1961). Adaptve Cotrol Processes: A Guded Tour. Prceto Uversty Press, Prceto. [17] Bellma, R.E. (1957). DyamcProgrammg. Prceto Uversty Press, Prceto. [18] Bertero,., Poggo,T. ad Torre,V. (1986). Ill-Posed Problems Early Vso. Artftal tellgece Laboratory emo, No.94.IT. [19] Bezhaev, A.Y. ad Vasleko, V.A. (001). Varatoal Theory of Sples. Kluwer.New York, USA. [0] Blas, F. (004). Revew of 0 years of rage sesor developmet. Joural of Electroc Imagg 13 (1), pp [1] Bloomethal, J., Bajaj, C., Bl, J., Ca-Gascuel,.P., Rockwood, A., Wyvll, B. ad Wyvll, G. (1997). Itroducto to Implct Surfaces. orga Kaufma, Sa Fracsco, USA. [] Bobeko, A.I., Schröder, P., Sullva, J.., Zegler, G.. (008). Dscrete dfferetal geometry. Brkhäuser. [3] Bocher, S. (193). Vorlesuge uber Fourersche Itegrale, Akademsche Verlagsgesellschaft, Lepzg. [4] Bocher, S. (1933). ootoe Fuktoe, Steltjes Itegrale ud harmosche Aalyse, ath. A., 108 pp [5] Bossoat, J.D. ad Cazals, F. (00). Smooth shape recostructo va atural eghbor terpolato of dstace fuctos. Computatoal Geometry,, (1-3), pp [6] Bouhamd, A. ad Le éhauté, A. (1999). ultvarate terpolatg (m, l, s)- sples. Adv. Comput. ath. 11, pp [7] Bouhamd, A. (005). Weghted th plate sples. Aal. Appl. 3(3), pp

122 [8] Bouhamd, A. (007). Error estmates Sobolev spaces for terpolatg th plate sples uder teso. J. Computatoal ad Appled athematcs, 00, pp [9] Boult, T. E. ad Keder, J. R. (1986). Vsual surface recostructo usg sparse depth data, Proc. IEEE Cof Comp. Vso ad Patter Recogto, pp [30] Brach, J. W. (007). Recostruccó de objetos de forma lbre a partr de mágees de rago empleado ua red de parches NURB. Tess de doctorado. Facultad de as Uversdad Nacoal de Colomba. Sede edellí. Colomba. [31] Bree, D.E., auch, S. ad Whtaker, R.T. (1998). 3D sca coverso of CSG models to dstace. Volume Vsualzato Symposum, pp [3] Buhma,.D. (003). Radal bass fuctos. Cambrdge oographs o Appled ad Computatoal athematcs, Cambrdge Uversty Press, Cambrdge. [33] Carr, J. C., Beatso, R. K.., Cherre, J. B., tchell, T. J., Frght, W. R., ccallum, B. C. ad Evas, T. R. (001). Recostructo ad represetato Of 3D objects wth radal bass fuctos. I proceedgs of AC SIGGRAPH 001, pp [34] Che, C.Y. ad Cheg, H. Y. (005). A sharpess depedet flter for mesh smoothg. Computer Aded Geometrc Desg, (5), pp [35] Cheey, E.W. ad Lght, W.A. (000). A Course Approxmato Theory. Brooks Cole Publshg Compay. Pacfc Grove. [36] Cho J., Youg-seok, H. ad oo, G. K. (008). Hgh Dyamc rage mage recostructo usg multple mages. The 3rd Iteratoal Coferece o Crcuts/Systems, Computers ad Commucatos (ITC-CSCC 008). [37] Clarez, U. Dewald, U. ad Rumpf,. (000). Asotropc geometrc dffuso surface processg. I Proc. Cof. Vsualzato. IEEE Computer Socety, pp [38] Curless, B. ad Levoy,. (1996). A volumetrc method for buldg complex models from rage mages. I H. Rushmeer, edtor, SIGGRAPH 96 Coferece Proceedgs, Addso Wesley, pp [39] Cygaek, B. ad Sebert, J. P. (Eds.). (009). A Itroducto to 3d Computer Vso Techques ad Algorthms. Wley. [40] Dey, J. ad Los, J.L. (1954). Les espaces du type de Beppo-Lev. A. Ist. Fourer 5, pp

123 [41] Desbru,. ad Ca-Gascuel,.P. (1998). Actve Implct Surface for Amato, Graphcs Iterface '98, [4] Desbru,., eyer,., Schroder, P. ad Barr, A. H. (1999). Implct farg of rregular meshes usg dffuso ad curvature flow. I Proc. 6 th Cof. Computer Graphcs ad Iteractve Techques, C Press/Addso-Wesley Publshg Co, pp [43] Dey, T.K. ad Su, J. (005). Adaptve LS surfaces for recostructo wth guaratees. I Eurographcs Symposum o Geometry Processg 005, pp [44] Debel, J. R., Thru, S. ad Brug,. (006). A Bayesa method for probable surface recostructo ad decmato. AC Tras. Graphcs, 5 (1), pp [45] Dh, H. Q., Turk G. ad Slabaugh, G. (00). Recostructg Surfaces By Volumetrc Regularzato Usg Radal Bass Fuctos. IEEE Tras. Patter Aal. ache Itell, pp [46] Dubrov, B.A., Fomeko, A.T. ad Novkov, S.P. (1984). oder Geometry: ethods ad Applcatos. Sprger. [47] Ducho, J. (1976). Iterpolato des foctos de deux varables suvat le prcpe de la exo des plaques mces, Rev. Fracase Automat. Iformat. Rech. Oper., Aal. Numer. 10, pp [48] Ducho, J. (1977). Sples mmzg rotato-varat sem-orms Sobolev spaces, Costructve Theory of Fuctos of Several Varables, Oberwolfach 1976, W. Schempp ad K. Zeller (Eds.), Sprger Lecture Notes ath. 571, Sprger- Verlag, Berl, pp [49] Ducho, J. (1978). Sur l'erreur d'terpolato des foctos de pluseurs varables par les Dm-sples, Rev. Fracase Automat. Iformat. Rech. Oper. Aal. Numer. 1, pp [50] Ducho, J. (1980). Foctos sples homogees a plusers varables. Uverste de Greoble. [51] Duford, N. ad Schwartz, J.T. ( ). Lear Operators., Part 1, Geeral Theory, 1958; Part, Spectral Theory, 1963; Part 3 Spectral Operators. Iterscece Publshers, Ic., New York. [5] Edelsbruer, H. (1998). Shape recostructo wth delauay complex, Proceedgs of LATIN 98. Theoretcal Iformatcs, Lecture Notes Computer Scece, vol. 1380, Sprger- Verlag, pp

124 [53] Edelsbruer, H., Krkpatrck, D. ad Sedel, R. (1983). O the shape of a set of pots the plae, IEEE Tras. If. Theory, 9 (4), pp [54] Edelsbruer, H. ad ücke, E.P. (1994). Three-dmesoal alpha shapes, AC Tras. Graph. 13, pp [55] Egl, H.W., Hake,. ad Neubauer, A. (1996). Regularzato of verse problems. Sprger. [56] Epste, C. (003). Itroducto to the athematcs of edcal Imagg. Pearso. [57] Fasshauer, G. (007). eshfree Approxmato ethods wth atlab. Iterdscplary athematcal Sceces. World Scetfc Publshers. [58] Faugeras, O. D. ad Kerve, R. (1998). Varatoal prcples, surface evoluto, pdes, level set methods, ad the stereo problem, IEEE Tras. Image Process., 7 (3), pp [59] Faul, A.C., Goodsell, G. ad Powell,. J. D. (005). A Krylov subspace algorthm for multquadrc terpolato may dmesos. IA Joural of Numercal Aalyss, 5, pp.1-4. [60] Fe,. ad Stedl, G. (005). Robust local approxmato of scattered data. I Geometrc Propertes from Icomplete Data. R. Klette, R. Kozera, L. Noakes, J. Weckert (Eds.), Sprger, pp [61] Flt, A. ad va de Hegel, D. A. (008). Local 3D structure recogto rage mages IET Comput. Vs., (4), pp [6] Follad, G.B. (199). Fourer aalyss ad ts Applcatos. Amer. athematcal Socety. [63] Frake, R. (1985). Th plate sples wth teso. Comput. Aded Geom. Des., [64] Frake, R. (198). Scattered data terpolato: tests of some methods, ath. Comp., 48, pp [65] Fredlader, F.G. ad Josh,. (008). Itroducto to the Theory of Dstrbutos. Cambrdge Uversty Press. [66] Gel fad, I.. ad Shlov, G.E. (1964, 1968, 1967). Geeralzed Fuctos. Vol Academc Press, New York. 116

125 [67] Ghae,., Sattara,., Samadzadega, F. ad Azzb, A. (008). A revew of recet rage mage recostructo algorthms. The Iteratoal Archves of the Photogrammetry, Remote Sesg ad Spatal Iformato Sceces. [68] Gbso, J.J. (1979). The ecologcal approach to vsual percepto. Houghto ffl. [69] Gros, F. ad Azellott, G. (1995). Covergece Rates of Approxmato by Traslates.A.I. emo No. 188-assachusetts Isttute of Techology Artfcal Itellgece Laboratory. [70] Gros, F., Joes,. ad Poggo, T. (1995). Regularzato Theory ad Neural Networks Archtectures, Neural Computato, 7 (), pp [71] Gros, F. ad Poggo, T. (1990). Networks ad the best approxmato property. Bol. Cyberet. 63, [7] Gros, F. (1993). Regularzato theory, radal bass fuctos ad etworks. I From Statstcs to Neural Networks. Theory ad Patter Recogto Applcatos, V. herkassky, J. H. Fredma, ad H. Wechsler (Eds.). Sprger-Verlag, Berl. [73] Gros, F., Joes,. ad Poggo, T. (1993). Prors, Stablzers ad Bass Fuctos: From Regularzato to Radal, Tesor ad Addtve Sples, A.I. emo No. 1430, C.B.C.L. Paper No.75, assachusetts Isttute of Techology Art cal Itellgece Lab. [74] Gleerster, A. (00) Computatoal theores of vso. Curret Bology, 1 (0), pp [75] Gleerster, A. (007). arr's vso: Twety-fve years o. Curret Bology, 17 (11) pp [76] Gokme,. ad L, C. C. (1993). Edge Detecto ad Surface Recostructo Usg Refed Regularzato. IEEE Tras. Patter Aalyss ad ache Itellgece, 15 (5), pp [77] Golomb,. ad Weberger, H.F. (1959). Optmal approxmato ad error bouds. I Numercal Approxmato. Lager, R. E. Ed., Uversty of Wscos Press, adso, pp [78] Grmso, W.E.L. (1983). A mplemetato of computatoal theory for vsual surface terpolato, Comput. Vw, Graphcs, Image Processg, 6 (), pp [79] Grmso, W.E.L. (1981). From mages to surfaces: A Computatoal study of the huma early vso system, IT Press, Cambrdge, A. 117

126 [80] Gumerov, N. A. ad Duraswam, R. (007). Fast radal bass fucto terpolato va precodtoed Krylov terato, SIA Joural o Scetfc Computg, 9 (5), pp [81] Hadamard, H. (193). Lectures o the Cauchy problem lear partal dfferetal equatos. Yale Uversty Press. New Have. [8] Hase, P. C. (1998). Rak-defcet ad dscrete ll-posed problems: umercal aspects of lear verso. Socety for Idustral athematcs. [83] Hase, P. C. (008). Regularzato Tools. A atlab Package for Aalyss ad Soluto of Dscrete Ill-Posed Problems. Verso 4.1 for atlab 7.3. [84] Harder, J. R., Desmaras, R. N. (197). Iterpolato usg surface sples; Joural Arcraft; 9, pp [85] Hardy, R. L., (1971), ultquadrc equatos of topography ad other rregular surfaces, J. Geophys. Res., 76, pp [86] Hartma, F. ad Katz, C. (007). Structural aalyss wth fte elemets. Sprger [87] Hlbert, D. ad Courat, R. (1989). ethods of athematcal Physcs. Wley. [88] Hlto, A. Stoddart, A.J., Illgworth, J. ad Wdeatt, T. (1998). Implct surface - based geometrc fuso. Comput. Vso ad mage Uderstadg, 69, pp [89] Hestroza, D., uro, A. ad Zha, S. (1999). Regularzato techques for olear problems. Computers ad athematcs wth Applcatos, 37, pp [90] Hoppe, H. (1994). Surface recostructo from uorgazed pots, Ph.D. Thess Computer Scece ad Egeerg, Uversty of Washgto. [91] Hormader, L. (1963). Lear Partal Dfferetal Operators. Sprger-Verlag, Berl. [9] Hor, B. K. P. ad Brooks,. J. (1986). The varatoal approach to shape from shadg, Comp. vso, Graphcs, ad Image Processg, 33 (), pp [93] Hwag, W.L., Lu, C.S. ad Chug, P.C. (1998). Shape from texture: Estmato of plaar surface oretato through the rdge surfaces of cotuous wavelet trasform, IEEE Tras. Image Process., 7 (5), pp [94] Isakov, V. (006). Iverse Problems for partal Dfferetal Equatos. Sprger. 118

127 [95] Ivrssmtzs, I., Lee, Y., Lee, S., Jeog, W.K. ad Sedel, H.P. (004). Neural mesh esembles. I d Iteratoal Symposum o 3D Data Processg, Vsualzato, ad Trasmsso, Y. Alomoos, G. Taub (Eds.), pp [96] Joes, T. R., Durad, F. ad Desbru,. (003). No-teratve, feature-preservg mesh smoothg. AC Tras. Graphcs, (3), pp [97] Jue, H. ad Chelberg, D.. (1995). Dscotuty-Preservg ad Vewpot Ivarat Recostructo of Vsble Surfaces Usg a Frst Order Regularzato IEEE Tras. Patter Aal. ache Itell., 17 (6), pp [98] Kelly, A. R. ad E. Hacock. (004). A graph-spectral approach to shape-fromshadg, IEEE Tras. Image Process., 13 (7), pp [99] Kere, D. ad Werma,. (1999). A Full Bayesa Approach to Curve ad Surface Recostructo, Joural of athematcal Imagg ad Vso, 11, pp [100] Krsch, A. (1996). A Itroducto to the athematcal Theory of Iverse Problems. Sprger. [101] Kovue, V. (1995). A robust olear flter for mage restorato. IEEE Tras. Image Process. 4, pp [10] Kress, R. (1999). Lear Itegral Equatos. Sprger. [103] Lage, C. ad Polther, K. (005). Asotropc farg of pot sets. Comput. Aded Geom. Des. (7), pp [104] Lauret, P.J. (197). Approxmato et optmsato. Herma, Pars. [105] Lee, K.., eer, P. ad Park, R.H. (1998). Robust adaptve segmetato of rage mages. IEEE Tras. Patter Aal. ache Itell. 0, pp [106] Levoy,., Pull, K., Curless, B., Ruskewcz, S., Koller, D., Perera, L., Gzto,., Aderso, S., Davs, J. Gsberg, J. Shade, J. ad Fulk, D. (000). The dgtal mchelagelo project: 3D scag of large statues. Proceedgs of SIGGRAPH 000, pp [107] Lght, W. A. ad Waye, H. (1998). O power fuctos ad error estmates for radal bass fucto terpolato, J. Approx. Theory 9, pp [108] Lght, W. (1998). Varatoal ethods for terpolato, Partcularly by Radal Bass Fuctos. Uversty of Lecester. Techcal Report. [109] Lght, W. ad Waye, H. (1997). Spaces of Dstrbutos ad Iterpolato by Traslates of a Bass Fucto. Uversty of Lecester. Techcal Report. 119

128 [110] Lse, L. (001). Pot cloud represetato. Tech. Rep , Fakultät für Iformatk, Uverstät Karlsruhe. [111] adych, W. R. ad Nelso, S. A. (1983). ultvarate terpolato: a varatoal theory, mauscrpt. [11] adych, W. ad Nelso, S. (1988). ultvarate terpolato ad codtoally postve defte fuctos, Approx. Theory Appl. 4, pp [113] arr, D. (198). Vso: A Computatoal Ivestgato to the Huma Represetato ad Processg of Vsual Iformato. W. H. Freema ad Compay New York. [114] arr, D. ad Poggo, T. (1976). Cooperatve computato of stereo dsparty. Scece, 194, pp [115] arr, D. ad Poggo, T. (1977). From uderstadg computato to uderstadg eural crcutry. Neurosceces Res. Prog. Bull., 15, pp [116] arr, D. ad Poggo, T. (1979). A computatoal theory of huma stereo vso. Proceedgs of the Royal Socety of Lodo B, 04, pp [117] arr, D. ad T. Poggo, (1979). A theory of huma stereo vso, Proc. Roy, Soc. Lodo B, 04, pp [118] arr, D. ad Poggo, T. (1976) Cooperatve computato of stereo dsparty. Scece, 194, pp [119] arroqu, J., tter, S. ad Poggo, T. (1987). Probablstc Soluto of Ill-posed Problems Computatoal Vso, Joural of Amerca Statstcal Assocato, 8 (397), pp [10] ederos, B., Velho, L. ad de Fgueredo, L.H. (004). Smooth surface recostructo from osy clouds. J. Brazla Comput. Soc. [11] eer, P., tz, D., Rosefeld, A. ad Km, D.Y. (1991). Robust regresso methods for computer vso. Arevew. Iterat. J. Comput. Vso 6, pp [1] eguet, J. (1979). ultvarate terpolato at arbtrary pots made smple, Z. Agew. ath. Phys., 30, pp [13] eguet, J. (1979). A trsc approach to multvarate sple terpolato at arbtrary pots, Polyomal ad Sple Approxmatos, N. B. Sahey (ed.), Redel, Dordrecht, pp

129 [14] eguet, J. (1979). Basc mathematcal aspects of surface sple terpolato, Numersche Itegrato, G. Hammerl (ed.), Brkhauser, Basel, pp [15] eguet, J. (1984). Surface sple terpolato: basc theory ad computatoal aspects, Approxmato Theory ad Sple Fuctos, S. P. Sgh, J. H. W. Burry, ad B. Watso (Eds.), Redel, Dordrecht, pp [16] cchell, C. A. (1986). Iterpolato of scattered data: dstace matrces ad codtoally postve defte fuctos, Costr. Approx.,, pp [17] cchell, C. A. ad Rvl, T. J (1977). A survey of optmal recovery, Optmal estmato approxmato theory, C. A. cchell ad T. J. Rvl (Eds.), Pleum Press, New York, pp [18] cchell, C. A ad Rvl, T. J. (1980). Optmal recovery of best approxmatos, Resultate ath. 3 (1), pp [19] otegraaro, H. ad Esposa, J. (007). Regularzato Approach for Surface Recostructo from Pot Clouds. Appled athematcs ad Computato, 188, pp [130] otegraaro, H. (008). A ufed framework for 3D recostructo. PA - Proc. Appl. ath. ech. 7, [131] otegraaro, H. ad Esposa, J. (009). A geeral framework for regularzato of -dmesoal verse problems. VII Cogreso Colombao de odelameto Numerco. Bogota. Colomba. [13] orozov, V. A. (1993). Regularzato ethods for Ill-Posed Problems. CRC Press; [133] urak, S. (1991). Volumetrc shape descrpto of rage data usg blobby model. I Computer Graphcs. Proc. SIGGRAPH 91, 5, pp [134] Narayaa, K. A., O Leary, D. P. ad Rosefeld, A. (198). Image smoothg ad segmetato by cost mmzato, IEEE Tras. Syst. a, Cyberetcs, vol. SC- 1, pp [135] Ng, P. ad Bloomethal, J. (1993). A evaluato of mplct surface tlers. IEEE Computer Graphcs ad Applcatos, 13(6), pp [136] O Nell, B. (1966). Elemetary Dfferetal Geometry. Academc Press. [137] Osher, S. ad Setha, J. (1988) Frots propagatg wth curvature depedet speed, algorthms based o a Hamlto-Jacob formulato. J. Comp. Phys., 79, pp

130 [138] Park, D.J., Nam, K.. ad Park, R.H. (1995). ultresoluto edge detecto techques. Patter Recogto 8, [139] Pasko, A., Adzhev, V. Sour, A. ad Savcheko, V. (1995). Fucto represetato geometrc modelg: cocepts, mplemetato ad applcatos. The Vsual Computer, 11 (8), pp [140] Pegel, L. ad Tller, W. (1996). The NURBS Book, Sprger-Verlag. [141] Poggo, T. ad Gros, F. (1990) Regularzato Algorthms for Learg that are Equvalet to ultlayer Networks, Scece, 47, [14] Poggo, T. ad Torre, V. (1984). Ill-posed problems ad regularzato aalyss early vso. IT Artftal Itellgece Lab.A.I. emo 773. [143] Poggo, T., Voorhees, H. ad Yulle, A. (1988). A Regularzed Soluto to Edge Detecto, Joural of Complexty, 4, pp [144] Poggo, T., Torre, V. ad Koch, C. (1985). Computatoal Vso ad Regularzato Theory, Nature, 317, pp [145] Poggo T. ad Torre, V. (1984). Ill-Posed Problems ad Regularzato Aalyss Early Vso. AI emo 773/CBIP Paper 1, assachusetts Isttute of Techology, Cambrdge, A. [146] Raja, V. ad Ferades, K. J. (Eds.). (008). Reverse Egeerg. A Idustral Perspectve. Sprger. [147] Reddy, J.N. (1984). Eergy ad varatoal methods appled mechacs. Joh Wley ad Sos. [148] Rglg, B. D. ad oses, R. L. (005). Three-dmesoal surface recostructo from multstatc SAR mages, IEEE Tras. Image Process., 14 (8), pp [149] Rogers, D.F. (000). A Itroducto to NURBS, orga Kaufma. [150] Rogjag, P., eg, X. ad Whagbo, T. (009). Hermte varatoal mplct surface recostructo. Sc Cha Ser F-If Sc., 5 (), pp [151] Rousseeuw, P.J. ad Leroy, A.. (1987). Robust Regresso ad Outler Detecto. Wley, New York, USA. [15] Rud, W. (1973). Fuctoal Aalyss. cgraw-hll, New York. [153] Saxea, A. ad Sahay, B. (005). Computer Aded Egeerg Desg. Sprger. 1

131 [154] Schaback, R. (1995). Creatg Surfaces from Scattered Data usg Radal Bass Fuctos, athematcal ethods for Curve ad Surfaces,. Dæhle, T. Lyche, ad L. Schumaker (Eds.), Vaderblt Uversty Press, Nashvlle, pp [155] Schaback, R. (1999). Natve Hlbert spaces for radal bass fuctos I, New Developmets Approxmato Theory,. W. uller,. D. Buhma, D. H. ache ad. Felte (Eds.), Brkhauser, Basel, pp [156] Schaback, R. (000). A ufed theory of radal bass fuctos. Natve Hlbert spaces for radal bass fuctos II, J. Comput. Appl. ath. 11, pp [157] Schall, O., Belyaev, A. ad Sedel, H.P. (007). Feature-preservg o-local deosg of statc ad tme-varyg rage data. I Proc. AC Symp. Sold ad Physcal odelg, AC Press, pp [158] Schoeberg, I. J. (1938). etrc spaces ad completely mootoe fuctos, A. of ath., 39, pp [159] Schoeberg, I. J. (1964). Sple fuctos ad the problem of graduato, Proc. Nat. Acad. Sc. 5, pp [160] Scholkopf, B., Gese, J. ad Spalger, S. (005). Kerel methods for mplct surface modelg. I Advaces Neural Iformato Processg Systems 17, pp [161] Schultz,. H. (1973). Sple Aalyss. Pretce Hall, Eglewood Clffs, N. J. [16] Schumaker, L. L. (1976). Fttg surfaces to scattered data, Approxmato Theory II, G. G. Loretz, C. K. Chu, ad L. L. Schumaker, (Eds.). Academc Press, pp [163] Schwartz, L. (1966). Théore des dstrbutos. Herma, Pars. [164] Schwartz, L. (1966). athematcs for the Physcal Sceces. Addso-Wesley. [165] Shepard, D. (1968) A two dmesoal terpolato fucto for rregularly spaced data, Proc. 3rd Nat. Cof. AC, pp [166] Sh, D. ad Tjahjad, T. (008). Local hull-based surface costructo of volumetrc data from slhouettes, IEEE Tras. Image. Process., 17 (8), pp [167] Sobolev, S.L. (1936). éthode ouvelle à résoudre le problème de Cauchy pour les équatos léares hyperbolques ormales. atematcheskĭ Sbork. 1 (43) (1), pp

132 [168] Speckma, P. ad Su, D. (003). Fully Bayesa sple smoothg ad trsc autoregressve prors, Bometrka, 90 (), pp [169] Steke, F., Schölkopf, B. ad Blaz, V. (005). Support vector maches for 3D shape processg. Comput. Graph. Forum 4(3), pp [170] Su, X., Ros, P. L., art, R. R. ad Lagbe, F. C. (007). Fast ad effectve feature-preservg mesh deosg. IEEE Tras. Vsualzato ad Computer Graphcs, 13(5), pp [171] Suykes, J. A. K., Gestel, T. V., Brabater, J. D., oor, B. D. ad Vadewalle, J. (003). Least Squares Support Vector aches. World Scetfc Publshg Compay. [17] Taratola, A. (005). Iverse Problem Theory ad odel Parameter Estmato, SIA. [173] Taub, G. (1995). A sgal processg approach to far surface desg. I R. Cook, edtor, SIGGRAPH 95 Coferece Proceedgs, Addso Wesley, pp [174] Terzopoulos, D. (1988). The Computato of Vsble- surface represetato.ieee Trasactos o Patter Aalyss ad ache Itellgece. 10 (4), pp [175] Terzopoulos, D. (1983). ultlevel computatoal processes for vsble surface recostructo. Comput. Vso, Graphcs ad Image Processg, 4, pp [176] Terzopoulos, D. (1984).ultresoluto computato of vsble-surface represetatos. Ph.D. dssertato, assachusetts Isttute of Techology, Dept. of Electrcal Egeerg ad Computer Scece, Cambrdge. [177] Terzopoulos, D. Wtk, D. A. ad Kass,. (1988) Costrats o deformable models: Recoverg 3D shape ad orgd moto, Artfcal Itell., 36, pp [178] Terzopoulos, D. (1986). Regularzato of Iverse Vsual Problems Ivolvg Dscotutes, IEEE Tras. Patter Aalyss ad ache Itellgece, 8(4), pp [179] Tkhoov, N. ad Arse, V.Y. (1977). Solutos of ll-posed problems. W. H. Wsto, Washgto D.C. [180] Tmosheko, S. ad Woowsky, S. (1981). Theory of Plates ad Shells. cgraw- Hll. [181] Tmosheko, S P. ad Gere, J.. (195). Theory of Elastc Stablty. Wley ad sos. 14

133 [18] Turk, G. ad O Bre, J.F. (00). odellg wth Implct Surfaces that Iterpolate. AC Trasactos o Graphcs, 1 (4), pp [183] Turk, G. ad O`Bre, J.F (1999). Shape trasformato usg varatoal mplct fuctos. SIGGRAPH99, pp [184] Vapk, V. N. (1998). Statstcal Learg Theory. Wley, New York. [185] Vesh, R. ad Ferádez, K. J. (Eds.). (008). Reverse Egeerg. A Idustral Perspectve. Sprger. [186] Wahba, G. ad Luo, Z. (1997). Smoothg sple ANOVA ts for very large, early regular data sets, wth applcato to hstorcal global clmate data, The Hertage of P. L. Chebyshev: a Festschrft hoor of the 70th brthday of T. J. Rvl, A. Numer. ath. 4 (1-4), pp [187] Wahba, G. (1979). Covergece rate of "th plate" smoothg sples whe the data are osy, Sprger Lecture Notes ath., 757, pp [188] Wahba, G. (1990). Sple odels for Observatoal Data. CBS-NSF, Regoal Coferece Seres Appled athematcs, SIA, Phladelpha. [189] Wahba, G. ad Wedelberger, J. (1980). Some ew mathematcal methods for varatoal objectve aalyss usg sples ad cross valdato; o.wea.rev., 108, pp [190] Waqar, S., Schall, O., Pata, G., Belyaev, A. ad Sedel, H.P. (007). O stochastc methods for surface recostructo. Vsual Comput., 3, pp [191] Wedlad, H. (010). Scattered Data Approxmato. Cambrdge Uversty Press. [19] Wedlad, H. (1994). E Betrag zur Iterpolato mt radale Bassfuktoe, Dplomarbet. Uverstat, Gottge. [193] Westerma. R. (000). Volume models. I Prcples of 3D Image Aalyss ad Sythess, B. Grod, G. Greer, ad H. Nema, (Eds). Kluwer Academc Publshers, Bosto-Dordrecht-Lodo, pp [194] Wyvll, B., Guy, A. ad Gal, E. (1999). Extedg the CSG tree, warpg, bledg ad boolea operatos a mplct surface modelg system. Computer Graphcs Forum, 18(), pp [195] Wyvll, B. cpheeters, C. ad Wyvll, G. (1986). Data structure for soft objects. The Vsual Computer, (4), pp

134 [196] Yoshzawa, S., Belyaev, A. ad Sedel, H.P. (006). Smoothg by example: esh deosg by averagg wth smlarty-based weghts. I Proce. Shape odelg ad Applcatos, IEEE Computer Socety, pp. 9. [197] Yosda, K. (1995). Fuctoal Aalyss, Sprger-Verlag, Berl. [198] Yu, Y., Zhou, K., Xu, D., Sh, X., Bao, H., Guo, B. ad Shum, H. (004). esh edtg wth Posso-based gradet feld mapulato. AC Tras. Graphcs, 3 (3), pp [199] Zedler, E. (1984). Nolear Fuctoal Aalyss ad Its Applcatos: Part II ad III. Sprger. [00] Zexao, X., Jaguo, W. ad Qume, Z. (005). Complete 3D measuremet reverse egeerg usg a mult-probe system. Iteratoal Joural of ache Tools & aufacture, 45, pp [01] Zhao, H.K., Osher, S. ad Fedk, R. (001). Fast Surface Recostructo Usg the Level Set ethod. Proceedgs. IEEE Workshop o Varatoal ad Level Set ethods Computer Vso, pp [0] Zuly, C. (1988). Problems Dstrbutos ad Partal Dfferetal Equatos. Elsever Scece. 16