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1 Poltecnco d orno Porto Insttutonal Repostory [Artcle] Study and development of morphologcal analyss gudelnes for pont cloud management: he "decsonal cube" Orgnal Ctaton: Vezzett E. (2011). Study and development of morphologcal analyss gudelnes for pont cloud management: he "decsonal cube". In: COMPUER AIDED DESIGN. - ISSN Avalablty: hs verson s avalable at : snce: March 2011 Publsher: Elsever Publshed verson: DOI: /j.cad erms of use: hs artcle s made avalable under terms and condtons applcable to Open Access Polcy Artcle ("Publc - All rghts reserved"), as descrbed at html Porto, the nsttutonal repostory of the Poltecnco d orno, s provded by the Unversty Lbrary and the I-Servces. he am s to enable open access to all the world. Please share wth us how ths access benefts you. Your story matters. (Artcle begns on next page)

2 NOICE: ths s the author's verson of a work that was accepted for publcaton n "Computer Aded Desgn". Changes resultng from the publshng process, such as peer revew, edtng, correctons, structural formattng, and other qualty control mechansms may not be reflected n ths document. Changes may have been made to ths work snce t was submtted for publcaton. A defntve verson was subsequently publshed n Computer Aded Desgn, Volume 43, Issue 8, August 2011, Pages , DOI: /j.cad

3 SUDY AND DEVELOPMEN OF MORPHOLOGICAL ANALYSIS GUIDELINES FOR POIN CLOUD MANAGEMEN: HE DECISIONAL CUBE Abstract Enrco Vezzett Dpartmento d Sstem d Produzone ed Economa dell Azenda Poltecnco d orno When talkng about reverse engneerng, t s necessary to focus on the management of pont clouds. Generally speakng, every 3D scanner devce codfes smple and complex geometres provdng dfferent pont cloud denstes as an output. Pont cloud densty s usually more correlated wth the techncal specfcatons of the devce employed rather than wth the morphology of the object acqured. hs stuaton s due to the frequent use of structured grds by a large quantty of devces. In order to solve ths problem, we therefore need to ntegrate the classcal structured grd acquston wth a smart selectve one, whch s able to dentfy dfferent pont cloud denstes n accordance wth the morphologcal complexty of the object regons acqured. Currently, we can reach the destnaton n many dfferent ways. Each of them s able to provde dfferent performances dependng on the object morphology and the performances of 3D scanner devces. Unfortunately, there does not yet exst one unversal approach able to be employed n all cases. For ths reason, the present paper ams to propose a frst analyss of the avalable methodologes and parameters, n order to provde fnal users wth some gudelnes for supportng ther decsons accordng to the specfc applcaton they are facng. Moreover, the developed gudelnes have been llustrated and valdated by a seres of case studes of the proposed method. Keywords: Reverse Engneerng, 3D Scanner, Pont Cloud Management, Morphologcal Analyss 1.0 Introducton Reverse engneerng process starts from the usage of a scannng devce that usually provdes a pont cloud, representng a partcular set of ponts descrbng a dscrete sample of a physcal model surface. he Delaunay s approach s only a trangulaton technque whch can be used to generate a polyhedral model of a physcal one, startng from a pont cloud. hese are only two of the many steps of the reverse engneerng process that moves from the pont cloud acquston to the vrtual model reconstructon. hs process s characterzed by many possble settngs and choces whch are sometmes dffcult to defne and may sometmes be the cause of sgnfcant movements of the captured sample regons from the real surface. he effcency of the entre reverse engneerng cycle strongly depends on the ntal pont cloud characterstcs, and n partcular on the number of ponts and on ther placement n the Cartesan space. For ths reason, t s necessary to provde a pont data set, that s a selectve sample, strctly correlated wth the orgnal scanned surface and accurately representng ts morphologcal characterstcs. In order to acheve ths selectve samplng soluton, t s mportant to remember that many 3D Scanners usually acqure the object surface by usng a constant grd, whose dmensons depend on the partcular technology (contact, non-contact) employed [1]. Workng wth a wde number of possble surface morphologes, the use of the constant grd tends to cause two dfferent scenaros: the creaton of too scattered pont clouds whch are not sutable for workng on complex zones, whle scannng planes, cylnders or cone-lke areas wth hgh resolutons performances would be redundant. Consderng that the common assumpton the database qualty mprovng goes always together wth the sampled pont densty s not true, a sample crowded wth too many ponts and obtaned from a relatvely smple surface means wder uncertanty propagaton, due to the measurng tool. It s then necessary to ntroduce a new sentence sayng that the database qualty mprovng s drectly proportonal to the proprety of the ponts of the cloud. Hence, consderng that the acqured pont densty depends on the scannng resoluton, whose value s locally chosen based on the surface morphologcal complexty, the selectve samplng approach can be obtaned by usng an expert operator to locally establsh whch ptch s to be employed. However, n ths case, the whole process would be expected to be extremely tme-consumng because of possble teratons, and the fnal result would probably be sgnfcantly subjectve. 2

4 On the other hand, the whole process can be automated by employng a morphologcal descrptor parameter, that can be assocated wth the optmal ponts densty (or scannng ptch). At present, the selecton of the most approprate morphologcal descrptor s a complex actvty because dfferent possble solutons have been provded by the techncal lterature. Startng from the approach that aggregates the pont clouds n cluster [2] and evaluates as a morphologcal parameter the normal vector of ts representatve, t s possble to move to a new dfferent methodology whch agan works wth the normal vector but whch evaluates t based on the local Vorono neghbourhoods [3]. As far as the Gaussan Curvature s concerned, ths parameter can be obtaned by workng on a pont cloud and dvdng t nto several elementary regons charactersed by a central node: the Gaussan curvature s then represented by the angular excess of the trangles convergng n the central node [4]. Workng wth the tensor, one soluton s the evaluaton of ths morphologcal parameter by usng the drectonal curvatures and the normal vectors [5] whle another makes use of the prncpal curvatures employng a regon growng approach. All of these methods show dfferent strengths and weaknesses n relaton to the specfc geometry and applcaton nvolved n. Hence, for a new user who needs to understand whch approach s the one to employ, t s not always easy to understand whch drecton to take. Unfortunately, most of exstng studes focus only on one parameter, and evaluate dfferent operatve strateges nstead of gvng any comparson wth the other possble morphologcal descrptors. Moreover, a large part of the techncal lterature only deals wth Gaussan Curvature [6]. In vew of the above dscusson, the present work proposes a structured comparson of the avalable morphologcal parameters, analysng dfferent operatve extracton methodologes, n order to desgn some prelmnary pont cloud management gudelnes. 2.0 Morphologcal descrptor selecton: he evaluaton Method In order to support the selecton of the best morphologcal parameter n accordance wth the specfc acquston scenaro, t s necessary to defne some varables able to descrbe n a consstent way the scenaro tself. 2.1 he evaluaton method: Varables Geometrcal varables and metrologcal performances have been analysed and evaluated by workng wth a survey mplemented on a sample of reverse engneerng users and employng a compatblty dagram approach [7]. Lookng at the results of the study, three varables have been dentfed as key factors for the reverse engneerng users for descrbng n a consstent way the possble workng scenaros. Whle the frst two varables depend on the object geometry, the thrd one s manly correlated wth the 3D scanner devce. Focusng the attenton on geometry, the two man varables are: Shape change amount Sharp edge number On the other hand, as far as the parameter correlated wth the 3D scanner s concerned, t s possble to synthesze the devce performances by usng the: Pont cloud densty Accordng to the value that these three varables may have, t s possble to defne dfferent morphologcal scenaros; each of them could be deal or nadequate for the use of a morphologcal parameter and of a specfc operatve approach for ts extracton. In order to support ths selecton, before startng, t s necessary to make some consderatons. Frst of all, consderng the dfferent dmensons of the acqurable object, t s necessary to defne whch s the dmenson of the area over whch the morphologcal analyss wll be developed (morphologcal detal). hs step s fundamental because morphologcal features have dfferent mportance dependng on the extenson of the object tself. he same feature can be sgnfcant over an object coverng an area of 1 x 1 mm, and neglgble over an object coverng an area of 1000 x 1000 mm. For ths reason, the value of a morphologcal detal cannot be absolute and unversal, but only relatve and, defnable as 10% of the entre object area. (Fg.1). Moreover, consderng that the cted varables are descrbed as an amount, t s necessary to defne ther scale. From the user pont of vew, the scale of all these varables can be consdered as bnary one, because the most frequent scenaros are charactersed by: many or few sharp edges and many or few shape changes. 3

5 From ths pont of vew, t s necessary to splt the varables scales nto two ntervals by usng a threshold. Its value can be defned as the percentage of the morphologcal detal areas composng the object surface charactersed by shape evdences. If ths value s bgger than 50%, the varable descrbes a scenaro wth sgnfcant morphologcal features (hgh complex geometry). On the contrary, f the percentage s equal or lower than 50%, the scenaro wll be characterzed by a qute smooth shape. 10% 100% 100% 10% Fgure 1: Pont cloud elementary regon Startng from ths hypothess, t s possble to formalse the dfferent varables necessary to synthesze the dfferent possble workng scenaros Shape change Amount Startng from the prevous hypothess at ths stage t s possble to ratonally formalse the meanng of surface wth many (few) shape changes. In fact, t wll mean a surface where t s possble to fnd at least (at most) 51% (50%) of the morphologcal detal squares, wth at least a shape change nsde. hs s mportant for objectvely codfyng what an expert eye could subjectvely detect as a shape change and n general as a complex (smooth) surface. he amount of shape changes n an object geometry s dentfed wth the help of the second dervatve. he second dervatve of a functon gves nformaton on the concavty of a curve. A curve varaton always corresponds to a change of concavty. Frst of all, the frst dervatve s analysed n order to study the functon trend and to establsh the exstence of any statonary pont. In order to obtan a hgher precson n the functon study, the second dervatve analyss s also carred out assessng the presence of nflecton ponts (ponts where the second dervatve s equal to 0) and localsng the convexty ntervals. If f ''( x) 0, then f s convex n x, f f ''( x) 0, then f s concave n x, f f ''( x) 0, then x s a pont of nflecton. he nflecton ponts correspond to a change n the curve curvature Sharp edge number It s also possble to formalse the meanng of surface wth many (few) sharp edges. Smlarly to prevous varable (Shape change amount) the threshold s located over the 50%. It n fact wll mean a surface where t s possble to fnd at least (at most) the 51% (50%) of the morphologcal detal squares, wth at least a sharp edge nsde. From a mathematcal pont of vew, t s possble to say that a pont A = (x 0, y 0 ) of a C curve parametersed by the contnuous functon y = f (x) s a cusp (or that the functon f has a cusp at x 0 ) f f s lm f '( x) lm f '( x) xxo xx0 dervable n a deleted neghbourhood and at ths pont, we have. Geometrcally speakng, ths means that at ths pont there are two dfferent tangent lnes, a rght one and a left one. Consderng the surfaces, t s not possble to consder just one tangent because the study must be done n three dmensons. he partal dervatve [8] s a frst generalzaton of the concept of a dervatve of real functons of several varables. If for real functons the dervatve s the slope of the graph of a functon (a 4

6 curve contaned n the plan), the partal dervatve at a pont n relaton to (for example) the frst varable of a functon of x and y, s the slope of the curve obtaned by ntersectng the graph of f (an area contaned n the 3 space R ) wth a plane passng through the pont and parallel to the plane y = 0. he drectonal dervatve [8] s a tool that generalzes the concept of partal dervatve; t no longer occurs only along drectons parallel to 2 the Cartesan axs, but n any drecton determned by a vector. Let f : R (wth R open set ) be and a pont ( x, y 0 0) ; take a unt vector v ( v 1, v2). hen the drecton dervatve wth respect to unt vector v of f n ( x, y 0 0) s f ( x0 tv1, y0 tv2) f ( x0, y0) Dv f ( x0, y0) lm, f ths lmt exsts fnte[8]. t 0 t Pont cloud densty As far as the pont cloud densty s concerned, t s not necessary to talk about the threshold dscussed before (50%), because of the structured grd employed (unform pont cloud). As a consequence, talkng about very crowded (sparse) pont cloud, means that when workng wth two ponts P 1 and P 2 ther dstance s less (bgger or equal) than 10% of the length of sde of the whole object area : d(p 1,P 2 )<10% (d(p 1,P 2 )>= 10%). 2.2 he evaluaton method: Eucldean Space Scenaros Formalsaton Startng from the consderatons we made before, each varable could then be descrbed on a bnary scale. When workng wth an Eucldean space, each of them could be graphcally represented by employng a three axs framework (Fg. 2). Its orgn descrbes a scenaro where the pont cloud has been obtaned through the acquston of an object charactersed by few shape changes, few sharp edges and low densty. Fgure 2: Eucldean framework varables layout he three varables belong to the nterval 0,1 and, for handness, they wll be ndcated as: x sharp edge, y densty, z shape change. x. hese varables have been located on three orthogonal axes and could be combned each others ndependently. Consderng that every varable moves on a bnary scale, every doman shall be dvded nto two ntervals [0,1/2] and [1/2,1] (ab.1) (Fg.3). and so, y, z 0,1 Intervals 1 1 Varables 0, 2,1 2 x few sharp edges many sharp edges y sparse pont cloud crowded pont cloud z few shape change many shape change able 1: Framework varables ntervals 5

7 Each combnaton wll synthesze a possble scenaro (for greater convenence, scenaro wll also be called case and combnaton n the next lnes), where an optmal morphologcal parameter could be dentfed. In the Eucldean space the possble combnatons wll be located n a specfc place nsde a cube. he entre scenaros set s represented by a cube or sde dmenson 1, called decsonal cube. On the Eucldean space each of these combnatons s represented by a cube of sde length ½ and whch s located nsde of the decsonal cube. Fgure 3: Framework varables ntervals In order to provde a relable analyss about the best morphologcal parameter, all the possble varables values have been combned to analyse as many scenaros as possble (ab.2). Cases Few shape change Many shape change No sharp edges Wth sharp edges Spare Crowed Case 1 X X X Case 2 X X X Case 3 X X X Case 4 X X X Case 5 X X X Case 6 X X X Case 7 X X X 6

8 Case 8 X X X able 2: Eucldean space scenaros representatons 3.0 Morphologcal descrptor selecton: operatve extracton methodologes Once defned the startng hypothess and defned the comparson methodology, and before startng the structured comparson of the avalable morphologcal parameters, the key ponts of the morphologcal parameters extracton methodologes have been analysed by underlnng strengths and weaknesses wth respect to the possble scenaros cted before. 3.1 Morphologcal descrptor: Normal he normal vector, often just called the "normal," to a surface s a vector perpendcular to t. Often the normal unt vector s preferred, sometmes known as the "unt normal". When normals are consdered on closed surfaces, the nward-pontng normal (pontng towards the nteror of the surface) and outwardpontng normal are usually dfferent. An Eucldean vector at a pont of a surface s normal to the surface f t s orthogonal to the tangent plane and, consequently, to every tangent vector to the surface at the pont Normal: Local Vorono Neghbourhood (Method 1) he procedure that nvolves the evaluaton of the normal vector usually ncludes the followng three fundamental steps: 1. Neghbourng ponts dentfcaton where applyng the normal vector estmaton 2. Normal vector evaluaton based on local neghbourng ponts 3. Defnng the nput/output drecton for the normal vector Followng ths method [3], t s possble to dentfy the local Vorono mesh neghbourhoods of a specfc pont startng from the global Vorono neghbours. At the begnnng, the Vorono dagram s created by employng the quckhull [9] algorthm, whch stands out for ts smplcty and ts computatonal effcency. After that, the Vorono mesh neghbours have been made startng from the local trangular mesh growng algorthm as the ball-pvotng approach [3]. Once the neghbourng ponts have been dentfed, t s possble to evaluate the normal vector. hs evaluaton s based on the quadratc curve fttng and allows users to fnd the normal vector of the pont P 0 startng from ts K local Vorono mesh neghbours. he key ponts of ths procedure can be summarsed through the followng lnes (Fg.4): Identfyng the correspondent P j ponts over the local Vorono mesh wth the bggest angle n P 0 Fttng a quadratc curve P(u) through P,P 0 and P j [2] Extractng the drectonal tangent vector from the ftted quadratc curve Evaluatng the normal vector n n the P 0 pont [2]: - he s 2 varance of the n and K drectonal tangent vector dot product s evaluated. - Employng a manpulaton method [9] t s possble to obtan a 3x3 matrx where the column vectors are egenvectors wth 1, 2, 3 as egenvalues. he egenvector correspondent to the mnmal egenvalue s the normal vector that mnmzes the varance. Fgure 4: Normal vector n at P 0 7

9 he method for evaluatng the normal vector n at P 0 s graphcally shown n fgure 4. Durng the followng steps, the normal vectors of both nternal and external drectons are defned n order to obtan a consstent global orentaton for the normal vectors n the sampled ponts. For ths purpose the local mesh s employed for dentfyng the two poles of the Vorono cells (ab.3) (Fg.5). Strengths hs approach s accurate and consstent; n fact, whle ncreasng the sample dmenson, ts probablty dstrbuton converges on the estmated parameter. he normal vector evaluaton method features a wde number of geometrc analyss algorthms for pont cloud. Moreover, the Vorono dagram s ndpendent from pont cloud densty and, more mportant, the Vorono nebourng ponts form a set that relably represents local surface geometry. he Delaunay trangular mesh s a global structure that allows users to dentfy the shape mnmum volume 3D representaton n any case. hs feature allows a better shape approxmaton proportonal to the pont cloud densty. Generally speakng, expermental results comng from the techncal lterature descrbe ths method as robust and able to evaluate the normal vector wth consstency. Weaknesses he normal vector s a local geometrc property. As a consequence of ths, t s necessary to make a careful evaluaton of the neghbourng ponts. In fact, by ntroducng too many ponts the local characterstc of the estmated normal vector could degrade, addng msleadng nformaton nto the pont cloud. However, f the pont cloud s too scattered the choce of the optmal neghbourng pont s strongly lmted, because t s not able to provde a consstent representaton of the surface geometry. able 3: Local Vorono neghbourhood method strengths and weaknesses Fgure 5: Local Vorono neghbourhood flowchart Normal: global clusterng approach (Method 2) hs method starts from a dense pont cloud formed by M ponts belongng to a contnuous surface S. he goal of ths methodology s to fnd a subset P N P wth a specfed number of ponts N M whch mnmzes the geometrc devaton between the surfaces represented from P N and P [2]. hs approach s dvded nto three fundamental steps: the objectve functon defnton, the objectve functon evaluaton and the objectve functon mnmzaton. he defnton s based on clusterng and pont- 8

10 to-surface dstance approxmaton concepts. N ponts belongng to P are consdered and named r. hey are defned as representatves. he set of r s P N (Fg.6). Fg. 6 Confguraton change: representatve r 1 replaced by r 1 After that, a cluster s created wth a representatve r, belongng to P, and every pont N p P / P, N so that the dstance between p and r s mnmum. An objectve functon s defned and used to evaluate an optmal confguraton for P N, but t s not suffcent to obtan a contnuous surface representaton startng from the pont cloud. he geometrc shape of the surface S, represented by a smplfed data set P, s approxmated by the normal vector, whch s N evaluated for each pont belongng to P N. he normal vector neghbourng ponts n P N [3]. Another problem that occurs frequently s that, when the N nr n r P N s evaluated by usng the r P N confguraton changes (Fg.6), the cluster representaton and the correspondent normal vector also change. In ths approach, these changes are controlled n a predctable way and they can therefore modfy the objectve functon. he objectve functon mnmzaton process conssts of two correlated steps n whch the layout cluster s mproved and hence the representatves choce s refned (ab.4)(fg.6). Strengths In ths approach the geometrc devaton accuracy s made n a cluster-by-cluster way. Few clusters at each teraton are taken nto consderaton and the evaluaton can be carred out n a rgorous way. It s possble to obtan a data seres by varyng the pont number n P N. hs extracted data set wll represent the nput data s orgnal geometry at dfferent detal levels. However, the estmate requres much fewer neghbourng ponts (mnmum three) compared to the quadratc surface fttng method that requres at least nne neghbourng ponts. Snce the Vorono dagram of P N usually supples at least three neghbourng ponts for every representatve r PN, ths method can proft from the 3D Vorono dagram structure, whch well constructed for the desred neghbourng ponts choce. Weaknesses he whole method s based on the assumpton of a dense pont cloud. It s better not to take a very small N, because the object functon s stablty s based on the pont-to-surface dstance approxmaton whch, n such a case, mght not be effcent enough. he problem of the relatonshp between pont densty and the approxmaton resultng error remans. able 4: Global clusterng approach method strengths and weaknesses 9

11 Fgure 7: Global clusterng approach neghbourhood flowchart 3.2 Morphologcal descrptor: Gaussan Curvature When workng wth a generc geometry t s possble to evaluate ts morphologcal complexty by employng two perpendcular planes cuttng the surface n one pont p 0. Lookng at these planes, t s possble to work on two curves to analyse ther geometrcal behavour. hs operaton provdes the prncpal curvatures k 1 and k 2 of the two curves and on the same pont. he product of these values provdes the Gaussan Curvature morphologcal descrptor Gaussan Curvature: percentles (Method 3) In the methodology [10], there s a process used for parametrc surfaces and another smlar one for dscrete surfaces. When dealng wth pont clouds, t s necessary to use the dscrete one and therefore to work on a trangular mesh M composed by a vertex set V={v } R 3, a edges set E={e j =v j1 v j2 } whch connect the vertexes and a trangles set ={t k = v k1 v k2 v k3 }. he ncdent angles n v are defned as { 1, 2,, d} where d s the vertex degree v (Fg.8). Fgure 8 : Vertex and related varables As a consequence of ths, the Gaussan curvature K [10], used n the dfferentable surfaces, can be approxmated n the dscrete ones as the sum of the K n each vertex n a specfc regon. Hence, n a specfc vertex v the curvature ntegral can be approxmated from [12,4]: Consderng that for dscrete surfaces the Gaussan curvature K K S Kds jd 2. j1 s only an approxmated value, the presence of nose durng the data acquston process and of geometrc and topologc features embedded n the mesh can lead up to msleadng nformaton. he cover surface boundary [10] and the assocated topologcal features often result n edges not shared wth the other two trangles (Fg.9). j 10

12 Fgure 9 : Degenerate vertex due to a specfc feature (hole) In ths case, the smooth surfaces curvature s not well defned. For the vertces n ths case, the K values, computed by the ntegral, are drectly lnked to the boundary curve s curvature nstead than only to the surface curvature. hese vertces, whose computed K value cannot represent the surface shape, are defned as degenerate vertces. In fact, once defned the threshold value, K e = 2π / 3, f the vertces satsfy K <K e they are erased. On the bass of the computed values, a 2D sphercal map s desgned. Wth t s possble to evaluate the curvature assocated wth x, y and wth r and the dstance from the center. he vertces are then grouped together on the bass of the percentle and dvded nto n concentrc regons. For each regon j, the ntegral of the Gaussan curvature K j s calculated as the summaton of all the K, namely K K. For each nput t s possble to obtan a vector K=[ K 1, K2,.., Kn ] that represents j I j the curvature ntegral of every concentrc regon. he smlar meshes are then compared by means of some correlaton coeffcents. he values of these coeffcents wll correspond to a hgh or low smlarty level (ab.5)(fg.10). K Strengths One of the man advantages provded by the method s the effcent computatonal behavour. Because of the use of a two meshes comparson, when analysng the dfferent morphologcal behavour of the two geometres, the calculaton of the curvature ntegral s reduced to the comparson of the K values n par; from a computatonal pont of vew, ths process s equvalent to comparng two vectors. hs methodology ncludes both contnuous and dscrete models approaches. Weaknesses he proposed methodology s suggested for freeform surfaces. In fact, t s not applcable to the study of prsmatc geometres or complcated topologes wth flat geometres because there are many degenerate vertces. When workng wth dscrete surfaces, the Gaussan curvature value provded s just an approxmaton whch hghly depends on the mesh qualty. In fact, bg mesh sze and layout varatons could lead to ncorrect conclusons. Accordng to ths methodology, another lmtaton n the use of ths parameter s the densty of the pont cloud. he approxmaton of the Gaussan curvature ntegral through the formula mentoned above, leads to satsfyng results f the sample s suffcently large. able 5: Percentles method strengths and weaknesses 11

13 Fgure 10: Percentles method flowchart Gaussan Curvature: angle defct n sphercal mage (Method 4) he curvature of a surface can be approxmated wth another method that has been developed startng from the Rodrguez heorem [13]. he surface s approxmated by a polyhedron wth trangular faces whose vertces are the ponts of the surface. For example, pont 0 s surrounded by the trangular faces P OP 1 (Fg.11). he sphercal mage of the polyhedron s a set of ponts of the unt sphere (the head of the unt vectors parallel to n, 1 ). hese ponts are lnked by arches and a sphercal polygon s created on the unt sphere. Fgure 11: Angle defct method 2 1 he area of the sphercal polygon s the angle defct of the polyhedron [11],,. he area of each trangular face of the polyhedron can be subdvded nto three equal parts, one for each vertex, so 1 that the area relatve to pont O on the polyhedron s S hs value s consdered as an approxmaton of the area of the curve on the surface around O, although the curve has not been specfed. Hence an approxmaton of the curve n O s 2, 1. K 1 S, 1 3 hs formula for calculatng K s for example used by [6,14,15,11] (ab.6)(fg.12). Strengths In order to obtan the best curvature approxmaton wth ths method, t s necessary to have a very hgh pont cloud densty sample. Furthermore, from Weaknesses he curvature approxmaton wth the angle defct method has proved to be not too accurate at tmes. For unform and not unform data, the angle defct 12

14 a computatonal pont of vew, ths methodology s hghly ndcated for free-form surfaces. method s able to approxmate the curvature wth precson O(1). It has been shown that for hgh pont cloud sample densty other known methods lead to a hgher approxmaton error, because the method s ndependent by the ponts dstance. able 6: Angle defct method strengths and weaknesses Fgure 12: Angle defct method flowchart Gaussan Curvature: quadratc surface fttng (Method 5) hs method provdes a curvature approxmaton startng from a quadratc nterpolaton surface. he normal and the curvature of a quadratc surface can, n fact, approxmate the normal and the curvature of a surface. A quadratc surface that passes through the orgn s gven by z=a 10 x+a 01 y+ A 20 2 x2 +A 11 xy+ A 02 2 y2. (1) It s requred to pass through other fve ponts as well. Consderng the ponts ( X, Y, Z), 1,,5 t s possble to express ths condton through the followng system: 2 2 A10 Z1 X 1 Y1 X1 Y1 X1Y1 A01 Z2 2 2 A20 Z3 (2) 2 2 X 5 Y5 11 Z4 X 5 Y A 5 X 5Y5 2 2 A02 Z5 If system (2) has a unque soluton, t s possble to fnd the curvature through ths method. he curvature of the quadratc equaton (1) wll be gven by: K A A A L , where A A01 L. 13

15 2 2 he form of the quadratc formula can be extended ncludng the terms n x y, xy and 2 y 2 [4]. In ths case eght ponts wll be necessary and a system of 8x8 lnear equatons wll be the result (ab.7)(fg.13). x Strengths hs method works for both unform and not unform data gvng an approxmaton O(h) of the curvature. hs means that the method has a lnear dependence on the pont dstance nsde of the pont clouds. Besdes, expermental results comng from the techncal lterature show that the dstance between ponts, and also the densty of the pont cloud, do not excessvely affect the surface approxmaton. Whenever the surface can be classfed as a free-form surface, the approxmaton provded by ths method supples very good results thanks to the absence of sgnfcant rregulartes. Weaknesses Workng wth a fttng quadratc equaton (1), the method nvolves matrces representaton (2). For ths reason, f these are badly condtoned the method could provde poor accuracy results. If ths method s used for not free-form surface, the process of fttng the quadratc equaton could supply msleadng results and, as a consequence, the curvature approxmaton s lkely to have a level of naccuracy proportonal to the rregulartes presence. In fact, the use of ths methodology s not recommended when the geometry s charactersed by a lot of sharp edges. able 7: Angle defct method strengths and weaknesses Fgure 13: Angle defct method flowchart 3.3 Morphologcal descrptor : curvature tensor he curvature tensor of a surface S s the map p k p that assgns to each pont p of S a functon measurng the drectonal curvature k p ( ) of S n p n the drecton of the unt vector, tangent to S n p. he curvature tensor s representable through a 3x3 symmetrcal matrx M. Its egenvalues are k 1, k 2, 0 and the correspondent egenvectors are k 1, k 2, N. k 1 and k 2 represent the prncple curvature; k 1 and k 2 the correspondent prncple drectons; and N the normal to the area. M s nterpretable as the normal vector varaton n small neghbourhoods. Hence, the necessary nformaton s avalable for buldng M as M=PDP -1 wth k1 0 0 P=( k1, k2, ) e D=. 0 k

16 3.3.1 Curvature tensor: based on Normal Cycle (Method 6) hs s an effcent algorthm [16] to decompose a trangulated mesh. It s based on the curvature tensor feld and t conssts of two complementary steps: 1) a regon based segmentaton, whch s an mprovement of what was already computed by Lavoue and others [17] and that decomposes the object nto several patches wth a constant curvature; 2)a boundary rectfcaton based on curvature tensor drectons, that corrects the boundares and elmnates ther artefacts or dscontnutes. An orgnal method of segmentaton s presented to decompose an orgnal 3D mesh nto patches characterzed by a unform curvature and clear boundares. he smple and effcent classfcaton dentfes any curvature transton; hence, t allows to segment the object nto confnng regons wth constant curvature wthout cuttng the rght object n ts sharp edges. In the frst step, curvature based regon segmentaton, a pre-processng step dentfes sharp edges and vertces; the curvature tensor s then calculated for each vertex accordng to the work of Coree- Stener et al. [18], based on the Normal Cycle. Vertces are classfed nto clusters, accordng to the prncpal curvature values Kmn and Kmax. Wth the help of a regon-growng algorthm, the trangles are assembled nto connected labelled regons accordng to the vertex clusters. Fnally, a regon adjacency graph (RAG) s processed and reduced n order to merge smlar regons accordng to several crtera (curvature smlarty, sze and common permeter). Durng the second step, boundary rectfcaton, boundary edges are extracted from the prevous regon segmentaton step. hen, for each of them, a boundary score s calculated whch notfes a degree of correctness. Accordng to ths score, estmated correct boundary edges are marked and used n a contour trackng algorthm to complete the fnal boundares of the object (ab.8)(fg.14) Strengths he procedure used to estmate the tensor provdes satsfactory results even for not well tasselled objects. It s ndependent from acqured pont cloud densty and offers the possblty to flter nosed objects. he fact of workng wth the orentaton of the curvature tensor allows to elmnate "artefacts". he densty of the cloud does not nfluence the algorthm effcency because, n the presence of a too crowded pont cloud, the method tself carres out a selecton of the optmal neghbourng ponts, excludng those that are consdered superfluous for the surface segmentaton. If the cloud s sparse, the problem s solved through the regon growng process. Weaknesses hs methodology nvolves a sgnfcant computatonal cost due to ts several passages, but t does not provde sgnfcant weaknesses. he presence of many steps could cause loss of nformaton or error dsperson and for ths reason, when workng wth free-from surfaces, there are other more effcent solutons. able 8: ensor method strengths and weaknesses 15

17 Fgure 14: Normal cycle method flowchart. (a) constant curvature regon segmentaton, (b) Boundary rectfcaton Curvature tensor: approxmaton matrx for tensor evaluaton (Method 7) o analyse the curvature tensor, the matrx M s defned wth an ntegral. hs matrx has the same p egenvectors as K [5], and ts egenvalues are connected to a few lnear homogeneous transformatons. p he computaton of the curvatures and of the prncple drectons of S n p derves from the dagonalzaton of the matrx M, whch can be obtaned n closed form. A scheme of fnte dfferences s then used to p approxmate the drectonal curvatures [5]. If q s another pont belongng to the surface S, close to p, and t s the normalzed projecton on the tangent plane N of the vector q p, the drectonal curvature can be t approxmated as follows 2N q p k p (3). 2 q p In ths method a polyhedron s consdered as an approxmaton of the unknown surface. Only trangulated areas are consdered, both closed and lmted, but supposedly orentated and consstent [20]. he frst goal s to evaluate the normal vectors. As explaned n [19], the faces of the surface are planar, every face f k has a normal unt vector N. he normal n the vertex f k v s defned as the normalzed weghted sum of the normals ncdent nto the faces, wth a weght proportonal to the surface face areas [19]. he second am s to calculate the matrx M v, approxmatng t wth a summaton of weghed sums on the t neghborhoods of V : M ~ w k. vector v j v j V For each neghbourng pont v v on the tangent plane j j j j v j of v, t s possble to defne j as the normalzed projecton of the v N [19]. It s now possble to approxmate the drectonal curvature k v ( j ) usng the formula of the equaton (3). he weghts are chosen proportonally to the sum of the areas of all the rectangles of the surface that are ncdent both n the vertex v j and n the vertex v. herefore w 1. v j V j 16

18 he normal vector N v s an egenvector of the matrx M ~ v assocated to the egenvalue 0. he M ~ v, by usng the prncpal curvatures are drectly obtaned from the two correspondent egenvalues of approprate formula [4]. o compute the two remanng egenvectors and egenvalues t's necessary to restrct the matrx M ~ v to the tangent plane N usng Householder transformaton [20]; after that, the resultng 2x2 matrx v must be dagonalzed n closed form wth a Gvens rotaton [20]. In ths way, the computed prncpal drectons have to be orthogonal to the normal vector N v, even f one of the values of M ~ v s zero, or close to zero. ~ ~ It s possble to obtan an angle so that the vectors 1 cos( ) 1 sn( ) 2 and ~ ~ sn( ) cos( ) are the remanng egenvectors of v. he prncpal curvatures are obtaned from the two correspondng values of M ~ v, or the prncpal drectons of the surface n M ~ v, usng the defned equatons [19]. Fnally, a pre-processng smoothng step s requred for surfaces wth noses, due to measurng errors or systematc problems. (ab. 9) (Fg.15) Strengths he algorthm complexty s lnear, both n tme and space, as a functon of the number of vertces and faces of the polyhedral surface. All calculatons are smple and straghtforward. Expensve numercal teratve algorthms are not necessary, even for the calculaton of egenvectors and egenvalues of the matrces nvolved. he experments performed [11] show that the accuracy of ths algorthm s not worse than that of other algorthms avalable, n some cases t s nstead even better. Weaknesses As any other method workng wth the estmaton of prncpal drectons, even ths approach could provde a not relable behavour. In fact, f the two remanng egenvalues of M ~ are equal, the prncpal v drectons cannot not unquely determned. able 9: Matrx tensor method strengths and weaknesses Fgure 15: Matrx tensor method flowchart 17

19 4.0 he evaluaton Method: Implementaton hs paragraph analyses the dfferent combnatons presented above (ab.2). In each of these cases, one or more of the methods prevously descrbed can be used. Some methods are usable n many stuatons, whle other methodologes are not hghly recommended n any of these combnatons of varables. o get a more complete pcture of the method applcablty t s not suffcent to pont out whch s the best method n a gven case, but rather to hghlght what are the cases where a partcular method cannot be used. 4.1 Implementaton: Occuped postons Dfferent coloured lttle cubes have been used to hghlght the cases n whch a parameter can be the best morphology descrptor (Gaussan Curvature: Yellow ensor: Lght Blue Normal: Volet)(ab.10). CASE SCENARIO DESCRIPION POSIION ON EUCLIDEAN SPACE 1 hs combnaton smulates an deal free-form surface. In fact t has only few shape changes and few sharp edges. Moreover, here the acquston devce has provded a qute sparse pont cloud. he methods employng the Gaussan curvature are able to provde the most effcent and relable results, because they do not requre dense pont cloud n order to provde good performances. Besdes, a surface wthout sharp edges represents the deal applcablty scenaro because the method s not able to manage ths knd of surface features. 2 hs scenaro shows a geometry charactersed by sharp edges, few shape changes and low ponts densty. Snce we are here workng wth geometres charactersed by sharp edges, the best morphologcal parameter to use s the tensor. Besdes, a specal attenton must be pad to the trangular growng method because t s able to remove relably spkes comng from anomalous pont cloud management, and to provde good performances both wth few or many ponts and both for few or many shape changes. 3 hs scenaro represents the deal stuaton where there are few sharp edges, many ponts and few shape changes. he surface representaton wll be the most accurate one and for ths reason all of the cted morphologcal descrptor are able to provde relable nformaton. 4 Here the pont cloud s crowded, there are many sharp edges and few shape changes. he presence of many sharp edges mposes the use of almost all the morphologcal descrptors rather than the tensor. It provdes the most relable performances and t s ndependent by the ponts densty and shape change amount. 18

20 5 he presence of few sharp edges and the many shape changes suggest the use of the Gaussan curvature wth specfc attenton to the percentles method. hs method can n fact provde the best performances when workng wth smooth surfaces by collectng ponts accordng to ther curvature, and wthout lmtng ts applcablty wth respect to the shape change amount. 6 he presence of many sharp edges mpose the use of almost all of the morphologcal descrptors rather than the use of just the tensor. It provdes the most relable performances and t s ndependent by the ponts densty and shape change amount. 7 hs case s characterzed by many shape changes, crowded pont cloud and few sharp edges; hence, the best soluton s here provded by the use of the normal vector. In ths specfc scenaro, and generally speakng when workng wth many ponts, the method that s able to guarantee the best performances s the cluster approach. hs method s employable ndependently from the sharp edges amount. 8 Here, we are dealng wth a hgh ponts densty, many sharp edges and many shape changes: t s therefore possble to employ the normal vector whle mplementng the cluster method. In ths scenaro ths method provdes optmal results. ab.10 Eucldean space parameters postons. 4.2 Implementaton: Covered areas By locatng all the cubes of the same colour n one sngle cube, we obtan the decsonal cube. hs cube descrbes all the scenaros where a sngle parameter can be optmal and where t doesn t assure and optmal behavour but t s anyway applcable because provdng acceptable results. Only n case 3 (ab.10), whch can be consdered as the deal scenaro, wth few shape changes, not many of sharp edges and a dense pont cloud, all of the already descrbed parameters and methods can provde optmal results. For ths reason, n the next lnes case 3 wll be consdered and attenton wll be focused only on the other more complex cases Gaussan Curvature he surface descrbed n case 1 (ab.10) s characterzed by few shape changes, absence of sharp edges and a low densty pont cloud. In such a case the percentle method that uses the Gaussan curvature seems to be the most approprate one. In fact, part of the methodology used n ths case requres the elmnaton of degenerate vertces from the study, whch n other words means that those vertces havng a Gaussan curvature hgher than a certan predetermned level must not be taken nto account. Hence, a surface presentng sharp edges cannot be analysed wth ths type of approach, because many ponts would be automatcally excluded from the analyss losng a sgnfcant porton of the surface geometrcal behavor. 19

21 Even n case 5 (ab.10), where the consdered surface s characterzed by many shape changes, absence of sharp edges and a low densty cloud, t s possble to use the percentle approach (Fg16a). By consderng also the cases where the Gaussan curvature s applcable but doesn t assure the best results, t s possble to conclude that ths parameters s adoptable where the geometry doesn t nclude sharp edges. However, there are no problems for those geometres charactersed by dfferent level of shape changes. Pont densty does not narrow the use of ths parameter (Fg.16b) ensor Curvature a) b) Fgure 16 : Gaussan curvature. a) optmal area, b) total area When workng wth sharp edges, the most sutable parameter seems to be the tensor wth specfc attenton to the based on normal cycle method [16]. In fact, ths method allows to segment the pont cloud accordng to the real boundares of the surface and then to subdvde t nto several patches on the bass of the curvature. he use of ths approach allows not to underestmate or neglect the presence of sharp edges leadng to a not optmal fnal approxmaton of the surface. hs stuaton occurs because when smulatng the manual segmentaton the method s able to handle the sngularty ponts due to the presence of sharp edges. In the he cases 2,4 and 6 (ab.10) the represented geometres are characterzed by many sharp edges. For ths reason, n these cases the parameter that better locates the dfferent morphologcal complexty areas s the tensor. Moreover the use of ths parameter s ndpendent by the ponts number n the cloud and by the presence of shape changes (Fg.17a). From the developed analyss t appears that the tensor s a parameter that can be used n every case. It hasn t partcular weaknesses or, anyhow, ts use never leads to erroneous conclusons f used n the rght way(fg.17b) Normal a) b) Fgure 17 : ensor curvature: a) optmal area, b) total area Wth hgh precson devces, t s possble to obtan a hgh densty pont cloud. In these cases the use of the normal provdes optmal results adoptng the cluster segmentaton approach. hs method reles on the mnmzaton of a functon dependent on a subset of ponts belongng to the acqured pont cloud. It's necessary that the number of startng ponts s not too low, otherwse a subset adequate to our goal would result excessvely small and there would be the rsk of obtanng a non optmal result, or a result too far from realty. For ths reason, t s necessary to use ths approach for cases 7 and 8, charactersed by the presence of a very hgh densty pont cloud. hs methodology s able to well approxmate sgnfcant shape changes. Furthermore, thanks to ts stablty, t works very well both n the presence of sharp edges and n the absence of these features (Fg.18a). he use of the normal vector s not recommended n stuatons n whch the pont cloud acquston nstrument cannot provde a hgh amount of ponts related to the surface analyzed (low densty), but does not 20

22 present any problem regardng the surface geometry. Sgnfcant or neglgble shape changes and the presence or absence of sharp edges do not nfluence the use of ths method (Fg.18b). a) b) Fgure 18 : Normal: a) optmal area, b) total area 4.3 Expermental valdaton: case studes In order to verfy the effcency of the methodology prevously ntroduced, four case studes have been analyzed. For each of the objects represented n Fg.20,21,22,23 the deal geometres and ther pont clouds, comng from the same 3D scanner (Roland Pcza)[21], have been adopted for the expermental valdaton. hrough the Gaussan curvature functon [22] t has been possble to subdvde every surface nto areas wth dfferent morphologcal complextes. Once obtaned the orgnal parametrc surfaces, t has been possble to compute the Gaussan curvature, whose value vares accordng to the ponts p belongng to the surface. By analysng ths functon, the dfferent morphologcal zones, charactersng the curvature map of the selected geometres have been dentfed wth a specfc set of colours (Fg. 19). Fgure 19: Fandsk and ts curvature areas map he obtaned surface curvature can be classfed nto one of the followng three groups: postve, negatve or null value. Geometrcally speakng, snce the Gaussan curvature can also be nterpreted as the product of the two prncpal curvatures, t depends on the maxmum and mnmum value of the normal curvatures. he surface zones wth dfferent Gaussan curvatures are therefore dentfable collectng those ponts wth the same curvature n confnng areas, whose borders are gven by the ponts n whch the curvature s null. In fact, snce the curvature vares wth contnuty on the surface, when passng from negatve values to postve values, borderng ponts, n whch the curvature s null, have to exst. hese are the ponts that defne the areas to dentfy. A clear example of ths stuaton s the orus (Fg.20) n whch two curvature types are dentfable: the external one wth a postve value, and therefore defned as ellptcal, and the nternal one wth a negatve value, defned as hyperbolcal. he ponts unfyng these two zones are the ponts n whch the curvature s null and are ndcated n red n fgure

23 Fgure 20: orus: boundary ponts example Subsequently the methodologes descrbed n the prevous sectons have been appled to the pont clouds obtaned from the objects n fgures 22,23,24,25, n order to dentfy, through the parameter used n each one of them, the varous morphologcal dfferences nsde of the surface tself. he zones wth a dfferent morphologcal complexty have been evdenced to dstngush them by usng dfferent colors. It s possble to see an example of ths applcaton n fgure 21, where the dfferent approaches provde a dfferent subdvson of the surface nto these zones. Once obtaned the curvature maps over the deal parametrc surfaces, these have to be compared wth those comng from the avalable dscrete methods, descrbed n the prevous paragraph, n order to dentfy the best matchng and, as a consequence, the most effcent method for the specfc pont cloud and scenaro (shape changes and sharp edges amount and densty). a b c d e f g Fgure 21 : Expermental results: a) method 1, b) method 2, c) method 3, d) method 4, e) method 5, f) method 6, g) method 7 he results comng from the expermental valdatons have been analyzed by employng the followng parameters: Correspondence: t dentfes the correspondence percentage between the zones dentfed on the deal geometry and those on the pont clouds by the valdated methods Non-exstent zones: t dentfes the non-exstent zones parentage, anomales comng from the use of the valdated methods, dfferng from the Gauss map. Sometmes the methods were able to clearly dentfy the areas characterzed by dfferent complextes. hese zones not dentfable by the curvature functon were due to an erroneous nterpretaton of the acqured data (ab.11,12,13,14). 22