Estiatig Suface Noals i Noisy Poit Cloud Data Niloy J. Mita Stafod Gaphics Laboatoy Stafod Uivesity CA, 94305 iloy@stafod.edu A Nguye Stafod Gaphics Laboatoy Stafod Uivesity CA, 94305 aguye@cs.stafod.edu ABSTRACT I this pape we descibe ad aalyze a ethod based o local least squae fittig fo estiatig the oals at all saple poits of a poit cloud data (PCD) set, i the pesece of oise. We study the effects of eighbohood size, cuvatue, saplig desity, ad oise o the oal estiatio whe the PCD is sapled fo a sooth cuve i 2 o a sooth suface i 3 ad oise is added. The aalysis allows us to fid the optial eighbohood size usig othe local ifoatio fo the PCD. Expeietal esults ae also povided. Categoies ad Subject Desciptos I.3.5 [ Coputig Methodologies ]: Copute Gaphics Coputatioal Geoety ad Object Modelig [Cuve, suface, solid, ad object epesetatios] Keywods oal estiatio, oisy data, eige aalysis, eighbohood size estiatio. INTRODUCTION Mode age sesig techology eables us to ae detailed scas of coplex objects geeatig poit cloud data (PCD) cosistig of illios of poits. The data acquied is usually distoted by oise aisig out of vaious physical easueet pocesses ad liitatios of the acquisitio techology. The taditioal way to use PCD is to ecostuct the udelyig suface odel epeseted by the PCD, fo exaple as a tiagle esh, ad the apply well ow ethods o that udelyig aifold odel. Howeve, whe the size of the PCD is lage, such ethods ay be expesive. To do suface ecostuctio o a PCD, oe would fist eed to filte out the oise fo the PCD, usually by soe soothig filte [2]. Such a pocess ay eove shap featues, Peissio to ae digital o had copies of all o pat of this wo fo pesoal o classoo use is gated without fee povided that copies ae ot ade o distibuted fo pofit o coecial advatage ad that copies bea this otice ad the full citatio o the fist page. To copy othewise, to epublish, to post o seves o to edistibute to lists, equies pio specific peissio ad/o a fee. SoCG 03, Jue 8 0, 2003, Sa Diego, Califoia, USA. Copyight 2003 ACM -583-663-3/03/0006...$5.00. howeve, which ay be udesiable. A ecostuctio algoith such as those i [2, 4, 8] the coputes a esh that appoxiates the oise fee PCD. Both the soothig ad the suface ecostuctio pocesses ay be coputatioally expesive. Fo cetai applicatios lie edeig o visualizatio, such a coputatio is ofte uecessay ad diect edeig of PCD has bee ivestigated by the gaphics couity [4, 6]. Alexa et al. [] ad Pauly et al. [6] have poposed to use PCD as a ew odelig piitive. Algoiths uig diectly o PCD ofte equie ifoatio about the oal at each of the poits. Fo exaple, oals ae used i edeig PCD, aig visibility coputatio, asweig iside-outside queies, etc. Also soe cuve (o suface) ecostuctio algoiths, as i [6], eed to have the oal estiates as a pat of the iput data. The oal estiatio poble has bee studied by vaious couities such as copute gaphics, iage pocessig, ad atheatics, but ostly i the case of aifold epesetatios of the suface. We would lie to estiate the oal at each poit i a PCD, give to us oly as a ustuctued set of poits sapled fo a sooth cuve i 2 o a sooth suface i 3 ad without ay additioal aifold stuctue. Hoppe et al. [] poposed a algoith whee the oal at each poit is estiated as the oal to the fittig plae obtaied by applyig the total least squae ethod to the eaest eighbos of the poit. This ethod is obust i the pesece of oise due to the iheet low pass filteig. I this algoith, the value of is a paaete ad is chose aually based o visual ispectio of the coputed estiates of the oals, ad diffeet tial values of ay be eeded befoe a good selectio of is foud. Futheoe, the sae value of is used fo oal estiatio at all poits i the PCD. We ote that the accuacy of the oal estiatio usig a total least squae ethod depeds o () the oise i the PCD, (2) the cuvatue of the udelyig aifold, (3) the desity ad the distibutio of the saples, ad (4) the eighbohood size used i the estiatio pocess. I this pape, we ae pecise such depedecies ad study the cotibutio of each of these factos o the oal estiatio pocess. This aalysis allows us to fid the optial eighbohood size to be used i the ethod. The eighbohood size ca be coputed adaptively at each poit based o its local ifoatio, give soe estiates about the oise, the local saplig desity, ad bouds o the local cuvatue. The coputatioal coplexity of estiatig all 322
oals of a PCD with poits is oly O( log ).. Related Wo I this sectio, we suaize soe of the pevious wos that ae elated to the coputatio of the oal vectos of a PCD. May cuet suface ecostuctio algoiths [2, 4, 8] ca eithe copute the oal as pat of the ecostuctio, o the oal ca be tivially coputed oce the suface has bee ecostucted. As the algoiths equie that the iput is oise fee, a aw PCD with oise eeds to go though a soothig pocess befoe these algoiths ca be applied. The wo of Hoppe et al. [] fo suface ecostuctio suggests a ethod to copute the oals fo the PCD. The oal estiate at each poit is doe by fittig a least squae plae to its eaest eighbos. The value of is selected expeietally. The sae appoach has also bee adopted by Pauly et al. [6] fo local suface estiatio. Highe ode sufaces have bee used by Welch et al. [5] fo local paaeteizatio. Howeve, as poited out by Aeta et al. [3] such algoiths ca fail eve i cases with abitaily dese set of saples. This poble ca be esolved by assuig uifoly distibuted saples which pevets eos esultig fo biased fits. As oted befoe, all these algoiths wo well eve i pesece of oise because of the iheet filteig effect. The success of these algoiths depeds lagely o selectig a suitable value fo, but usually little guidace is give o the selectio of this cucial paaete..2 Pape Oveview I sectio 2, we study the oal estiatio fo PCD which ae sapligs of cuves i 2, ad the effects of diffeet paaetes o the eo of that estiatio pocess. I sectio 3, we deive siila esults fo PCD which coe fo a suface i 3. I sectio 4, we povide soe siulatios to illustate the esults obtaied i sectios 2 ad 3. We also show how to use ou theoetical esult o pactical data. We coclude i sectio 5. 2. NORMAL ESTIMATION IN 2 I this sectio, we coside the poble of appoxiatig the oals to a poit cloud i 2. Give a set of poits, which ae oisy saples of a sooth cuve i 2, we ca use the followig ethod to estiate the oal to the cuve at each of the saple poits. Fo each poit O, we fid all the poits of the PCD iside a cicle of adius ceteed at O, the copute the total least squae lie fittig those poits. The oal to the fittig lie gives us a appoxiatio to the udiected oal of the cuve at O. Note that the oietatio of the oals is a global popety of the PCD ad thus caot be coputed locally. Oce all the udiected oals ae coputed, a stadad beadth fist seach algoith [] ca be applied to obtai all the oal diectios i a cosistet way. Though out this pape, we oly coside the coputatio of the udiected oals. We aalyze the eo of the appoxiatio whe the oise is sall ad the saplig desity is high eough aoud O. Ude these assuptios, which we will ae pecise late, the coputed oal appoxiates well the tue oal. We obseve that if is lage, the eighbohood of the poit caot be well appoxiated by a lie i the pesece of cuvatue i the data ad we ay icu lage eo. O the othe had, if is sall, the oise i the data ca esult i sigificat estiatio eo. We ai fo the optial that sties a balace betwee the eos caused by the oise ad the local cuvatue. 2. Modelig Without lost of geeality, we assue that O is the oigi, ad the y-axis is alog the oal to the cuve at O. We assue that the poits of the PCD i a dis of adius aoud O coe fo a seget of the cuve (a -D topological dis). Ude this assuptio, the seget of the cuve ea O is locally a gaph of a sooth fuctio y = g(x) defied ove soe iteval R cotaiig the iteval [, ]. We assue that the cuve has a bouded cuvatue i R, ad thus thee is a costat κ > 0 such that g (x) < κ x R. Let p i = (, y i) fo i be the poits of the PCD that lie iside a cicle of adius ceteed at O. We assue the followig pobabilistic odel fo the poits p i. Assue that s ae istaces of a ado vaiable X taig values withi [, ], ad y i = g() + i, whee the oise tes i ae idepedet istaces of a ado vaiable N. X ad N ae assued to be idepedet. We assue that the oise N has zeo ea ad stadad deviatio σ, ad taes values i [, ]. Usig Taylo seies, thee ae ubes ψ i, i such that g() = g (ψ i)x 2 i /2 with ψ i. Let γ i = g (ψ i), the γ i κ. Note that if κ is lage, eve whe thee is o oise i the PCD, the oal to the best fit lie ay ot be a good appoxiatio to the taget as show i Figue. Siilaly, if σ / is lage ad the oise is biased, this oal ay ot be a good appoxiatio eve if the oigial cuve is a staight lie, see Figue 2. I ode to eep the oal appoxiatio eo low we assue a pioi that κ ad σ / ae sufficietly sall. κ 2 Figue : Cuvatue causes eo i the estiated oal Figue 2: Noise causes eo i the estiated oal We assue that the data is evely distibuted; thee is a adius 0 > 0 (possibly depedet o O) so that ay eighbohood of size 0 i R cotais at least 2 poits of the s but o oe tha soe sall costat ube of the. We obseve that the ube of poits iside ay dis of adius is bouded fo above by Θ()ρ, ad also is bouded fo below by aothe Θ()ρ, whee ρ is the saplig desity of the poit cloud. Hee we use Θ() to deote soe sall positive costat, ad fo otatioal siplicity, diffeet appeaaces of Θ() ay deote diffeet costats. We ote that distibutios satisfyig the (ɛ, δ) saplig coditio poposed by Dey et. al. [7] ae evely distibuted. 323
Ude the above assuptios, we would lie to boud the oal estiatio eo ad study the effects of diffeet paaetes. The aalysis ivolves pobabilistic aguets to accout fo the ado atue of the oise. 2.2 Total Least Squae Lie I this sectio, we biefly descibe the well-ow total least squae ethod. Give a set of poits p i, i, we would lie to fid the lie a T x = c, with a T a = such that the su of squae distaces fo the poits p i s to i the lie is iiized. Let f(a, c) = 2 at 2 pipt a We would lie to fid a ad c iiizig f(a, c) ude the costait that a T a =. To solve this quadatic optiizatio poble, we eed to solve the followig syste of equatios: i f(a, c) = λa p ip T a c p = λa, a (at p i c) 2 = c p T a + 2 c2 whee p = pi. c f(a, c) = 0 pt a + c = 0, T whee λ is a Lagagia ultiplie. It follows that c = p T a, pipt i p p a = λa, ad f(a, c) = λ. Thus 2 λ is a eigevalue of M = pipt i p p T with a as the coespodig eigevecto. It is clea that to iiize f(a, c), λ has to be the iiu eigevalue of M. The coespodig eigevecto a is the oal to the total least squae lie ad is ou oal estiate. Note that this appoach ca be geealized to highe diesioal space. The oal to the total least squae fittig plae (o hypeplae) of a set of poits p i, i i d fo d 2 ca be obtaied by coputig the eigevecto coespodig to the sallest eigevalue of M = pipt i p p T. We obseve that M ca be witte as M = (pi p)(pi p)t ad thus it is always syetic positive sei-defiite, ad has o-egative eigevalues ad o-egative diagoal. 2.3 Eige-aalysis of M We ca wite the 2 2 syetic atix M, as defied i the pevious sectio, as 2 Note that i absece of oise ad cuvatue, 2 = 22 = 0 which eas 2 22. 0 is the sallest eigevalue of M with [0 ] T as the coespodig eigevecto. Ude ou assuptio that the oise ad the cuvatue ae sall, y i s ae sall, ad thus 2 ad 22 ae sall. Let α = ( 2 + 22)/. We would lie to estiate the sallest eigevalue of M ad its coespodig eigevecto whe α is sall. Usig the Geshgoi Cicle Theoe [9], thee is a eigevalue λ such that λ 2, ad a eigevalue λ 2 such that 22 λ 2 2. Whe α /2, we have that λ 2 22 + 2 λ 2. It follows that the two eigevalues ae distict, ad λ 2 is the sallest eigevalue of M. Let [v ] T be the eigevecto coespodig to λ 2, the Thus 2 2 22 λ 2 2 v = v = λ 2 v, 2 22 λ 2. v = ( λ2)2 + 2(22 λ2), ( λ 2) 2 + 2 2 () v 2 ( λ2 + 2 ), ( λ 2) 2 α( + α) ( α) 2. Thus, the estiatio eo, which is the agle betwee the estiated oal ad the tue oal (which is [0 ] T i this case), is less tha ta (α(+α)/( α) 2 ) α, fo sall α. Note that we could wite the eo explicitly i closed fo, the boud it. Ou appoach is oe coplicated, though as we will show late, it ca be exteded to obtai the eo boud fo the 3D case. To boud the estiatio eo, we eed to estiate α. 2.4 Estiatig Eties of M The assuptio that the saple poits ae evely distibuted i the iteval [, ] iplies that, give ay ube that iteval, the ube of poits p i s satisfyig x /4 is at least Θ(). It follows easily that = (xi x)2 Θ() 2. The costat Θ() depeds oly o the distibutio of the ado vaiable X. Fo the eties 2 ad 22, we use ad y i κ 2 /2 + to obtai the followig tivial boud: Thus, 2 = 22 y i 2 2(κ 2 /2 + ), y 2 i 2((κ 2 /2) 2 + 2 ). α Θ() κ + 2 + κ2 2 + 2 Θ() κ +. (2) This boud illustates the effects of, κ ad o the eo. Fo lage values of, the eo caused by the cuvatue κ doiates, while fo a sall eighbohood the te / is doiatig. Nevetheless, the expessio depeds o the absolute boud of the oise N. This boud ca be uecessaily lage o ubouded fo ay distibutio odels of N. We would lie to use ou assuptio o the distibutio of the oise N to ipove ou boud o α futhe. y i 324
ɛρ M Note that 2 = + 2 y i 2 i i) y (γ ix 3 i /2 + i) (γ ix 2 i i /2 + i Θ()κ 3 + +Θ() κ 2 +. Futheoe, ude the assuptio that X ad N ae idepedet, we have E[ i] = E[]E[ i] = 0 sice E[ i] = 0 ad Va( i) = Θ() 2 σ 2 sice Va( i) = σ. 2 Let ɛ > 0 be soe sall costat. Usig the Chebyshev Iequality [3], we ca show that the followig boud o 2 holds with pobability at least ɛ: 2 Θ()κ 3 + Θ() 2 σ 2 ɛ + Θ() σ 2 ɛ = Θ()κ 3 + Θ() 2 σ 2 ɛρ + Θ() σ 2 ɛρ Θ()κ 3 + Θ()σ ɛρ. (3) Fo easoable oise odels, we also have that 22 2(γ 2 i x 4 i /4 + 2 i ) Θ()κ 2 4 + Θ()σ 2. 2.5 Eo Boud fo the Estiated Noal Fo the estiatios of the eties of M, we obtai the followig boud o α, with pobability at least ɛ: α Θ()κ + Θ() σ + Θ() σ2 3. (4) 2 Note that this boud depeds o the stadad deviatio σ of the oise N athe tha its agitude boud. Fo a give set of paaetes κ, σ, ρ, ad ɛ, we ca fid the optial that iiizes the ight had side of iequality 4. As this optial value of is ot easily expessed i closed fo, let us coside a few extee cases. Whe thee is o cuvatue (κ = 0) we ca ae the boud o α abitaily sall by iceasig. Fo sufficietly lage, the boud is liea i σ ad it deceases as 3/2. Whe thee is o oise, we ca ae the eo boud sall by choosig as sall as possible, say = 0. Whe both oise ad cuvatue ae peset, the eo boud caot be abitaily educed. Whe the desity ρ of the PCD is sufficietly high, α Θ()κ + Θ()σ/ 2 2. The eo boud is iiized whe = Θ()σ 2/3 κ /3, i which case α Θ()κ 2/3 σ 2/3. The sufficietly high desity coditio o ρ ca be show to be ρ > Θ()ɛ σ 4/3 κ /3. / Whe thee ae both oise ad cuvatue, ad the desity ρ is sufficietly low, α Θ()κ +Θ()σ ɛρ 3. The boud is sallest whe = Θ()(σ/(ɛρκ 2 2 )) /5, i which case, α Θ()(κ 3 σ/(ɛρ)) 2 /5. The sufficietly low coditio o ρ ca be expessed oe specifically as ρ < Θ()ɛ σ 4/3 κ /3. We would lie to poit out that the costat hidde i the Θ() otatio i the sufficietly low coditio is 3/4 of that i the sufficietly high coditio. 3. NORMAL ESTIMATION IN 3 We ca exted the esults obtaied fo cuves i 2 to sufaces i 3. Give a poit cloud obtaied fo a sooth 2-aifold i 3 ad a poit O o the suface, we ca estiate the oal to the suface at O as follows: fid all the poits of the PCD iside a sphee of adius ceteed at O, the copute the total least squae plae fittig those poits. The oal vecto to the fittig plae is ou estiate of the udiected oal at O. Give a set of poits p i, i, let M = p p T, whee p = pi. As poited out i subsectio 2.2, the oal to the total least squae plae fo this set of poits is the eigevecto coespodig to the iiu eigevalue of the M. We would lie to boud the agle betwee this eigevecto ad the tue oal to the suface. 3. Modelig pipt i We odel the PCD i a siila fashio as i the 2 case. We assue that O is the oigi, the z-axis is the oal to the suface at O, ad that the poits of the PCD i the sphee of adius aoud O ae saples of a topological dis o the suface. Ude these assuptios, we ca epeset the suface as the gaph of a fuctio z = g(x) whee x = [x, y] T. Usig Taylo Theoe, we ca wite g(x) = 2 xt Hx whee H is the Hessia of f at soe poit ψ such that ψ x. We assue that the suface has bouded cuvatue i soe eighbohood aoud O so that thee is a κ > 0 such that the Hessia H of g satisfies H 2 κ i that eighbohood. Wite the poits p i as p i = (, y i, z i) = (, z i). We assue that z i = g( ) + i, whee the i s ae idepedet istaces of soe ado vaiable N with zeo ea ad stadad deviatio σ. We siilaly assue that the poits ae evely distibuted i the xy-plae o a dis D of adius ceteed at O, i.e. thee is a adius 0 such that ay dis of size 0 iside D cotais at least 3 poits aog the s but o oe tha soe sall costat ube of the. We also assue that the oise ad the suface cuvatue ae both sall. 3.2 Eige-aalysis i 3 = We wite the aalogous atix M = M 3 M3 T As poited out i subsectio 2.2, M is sy 33. 2 3 2 22 23 3 23 33 325
etic ad positive sei-defiite. Ude the assuptios that the oise ad the cuvatue ae sall, ad that the poits ae evely distibuted, ad 22 ae the two doiat eties i M. We assue, without lost of geeality, that 22. Let α = ( 3 + 23 + 33)/( 2 ). As i the 2 case, we would lie to boud the agle betwee the coputed oal ad the tue oal to the poit cloud i te of α. Deote by λ λ 2 the eigevalues of the 2 2 syetic atix M. Usig agai the Geshgoi Cicle Theoe, it is easy to see that 2 λ, λ 2 22 + 2. Let λ be the sallest eigevalue of M. Fo the Geshgoi Cicle Theoe we have λ 3 + 23 + 33 = α( 2) αλ. Let [v T ] T be the eigevecto of M coespodig with λ. The, as with Equatio, we have that: v = (M λi) 2 + M 3M T 3 ((M λi)m 3 + M 3( 33 λ)) = (M λi) 2 I + (M λi) 2 M 3M T 3 ((M λi)m 3 + M 3( 33 λ)), v 2 (M λi) 2 2 Note that Thus It follows that v 2 I + (M λi) 2 M 3M3 2 T ( (M λi) 2 M 3 2 + M 3 2 33 λ ). (M λi) 2 M 3M T 3 2 (M λi) 2 2 M 3 2 M T 3 2 (λ λ) 2 ( 2 3 + 2 23) ( α) 2 α 2. I + (M λi) 2 M 3M3 2 T ( α)2 ( α) 2 α2 2α. ( α) 2 λ 2 α( + α) 2α ( α) 2 2α λ 2 λ. (λ2αλ + αλαλ) Whe α is sall, the ight had side is appoxiately (λ 2/λ )α, ad thus the agle betwee the coputed oal ad the tue oal, ta v 2, is appoxiately bouded by (λ 2/λ )α (( 22 + 2 )/( 2 ))α, 3.3 Estiatio of the eties of M As i the 2 case, fo the assuptio that the saples ae evely distibuted, we ca show that Θ() 2, 22 2. We ca also show that 33 Θ()κ 2 4 + Θ()σ. 2 Let ρ be the saplig desity of the PCD at O, the = Θ()ρ 2. Agai, let ɛ > 0 be soe sall positive ube. Usig the Chebyshev iequality, we ca show that 3, 23 Θ()κ 3 + Θ()σ / ɛ Θ()κ 3 + Θ()σ / ɛρ with pobability at least ɛ. Fo the te 2, we ote that E[y i] = 0 ad V a(y i) = Θ() 4, ad so, by the Chebyshev iequality, 2 Θ()/ ɛρ with pobability at least ɛ. 3.4 Eo Boud fo the Estiated Noal Let β = 2/. We estict ou aalysis to the cases whe β is sufficietly less tha, say β < /2. This estictio siply eas that the poits s ae ot degeeate, i.e. ot all of the poits s ae lyig o o ea ay give lie o the xy-plae. With this estictio, it is clea that (λ 2/λ )α ( 22/ )(( + β)/( β))α = Θ()α. Fo the estiatios of the eties of M, we obtai the followig boud with pobability at least ɛ: λ 2 σ α Θ()κ + Θ() λ 2 ɛρ Θ()κ 2 2 + Θ() σ2 2 σ Θ()κ + Θ() 2 ɛρ + Θ() σ2 2 This is a appoxiate boud o the agle betwee the estiated oal ad the tue oal. To iiize this eo boud, it is = clea that we should pic σ + c 2σ κ c 2, (5) ɛρ /3 fo soe costats c, c 2. The costats c ad c 2 ae sall ad they deped oly o the distibutio of the PCD. We otice that fo the above esult, whe thee is o oise, we should pic the adius to be as sall as possible, say = 0. Whe thee is o cuvatue, the adius should be as lage as possible. Whe the saplig desity is high, the optial value of that iiizes the eo boud is appoxiately = Θ()(σ/κ) 2 /3. This esult is siila to that fo cuves i 2, ad it is ot at all ituitive. 4. EXPERIMENTS I this sectio, we discuss soe siulatios to validate ou theoetical esults. We the show how to use the esults i obtaiig a good eighbohood size fo the oal coputatio with the least squae ethod. 4. Validatio We cosideed a PCD whose poits wee oisy saples of the cuves (x, κ sg(x) x 2 /2), fo x [, ] fo diffeet values of κ. We estiated the oals to the cuves at the oigi by applyig the least squae ethod o thei coespodig PCD. As the y-axis is ow to be the tue oal to the cuves, the agles betwee the coputed oals ad the y-axis gives the estiatio eos. To obtai the PCD i ou expeiets, we let the saplig desity ρ be 00 poits pe uit legth, ad let x be uifoly distibuted i the iteval [, ]. The y- copoets of the data wee polluted with uifoly ado oise i the iteval [, ], fo soe value. The stadad deviatio σ of this oise is / 3. Figue 3 shows the eo as a fuctio of the eighbohood size whe = 0.05 fo 3 diffeet values of κ, κ =,, ad.2. As pedicted by Equatio 4 fo lage value of, the eo iceases as iceases. I the expeiets, it ca be see that the eo is popotioal to κ fo > 0.2. Note 326
that the PCD we chose geeates the wost case behavio of the eo. 0.9 0.07 0.033 0.05.2.2 0.7 0.6 Eo Agle 0.5 Eo Agle 0.6 0.3 0.2 0. 0.2 0 0.02 0.04 0.06 0.08 0. 0.2 Radius 0 0 0. 0.2 0.3 0.5 0.6 0.7 0.9 Radius Figue 5: The aveage eo ove 50 us exhibits a clea tedecy to decease as iceases fo sall. Figue 3: The oal estiatio eo iceases as iceases fo > 0.2. Figue 4 shows the estiatio eo as a fuctio of the eighbohood size fo sall whe κ =.2 fo 3 diffeet values of, = 0.07, 0.033, ad 0.05. We obseve that the eo teds to decease as iceases fo < 0.08. This is expected as fo Equatio 4, the boud o the eo is a deceasig fuctio of whe is sall. Figue 6: Taget plaes o the oigial buy Eo Agle.8.6.4.2 0.6 0.2 0.07 0.033 0.05 0 0.02 0.04 0.06 0.08 0. 0.2 Radius Figue 4: The oal estiatio eo deceases as deceases ad iceases fo < 0.08. The depedecy of the eo o fo sall values of ca be see oe easily i Figue 5, which shows the aveage of the estiatio eos ove 50 us fo each. 4.2 Estiatig Neighbohood Size fo the Noal Coputatio I this pat, we used the esults obtaied i Sectio 3 to estiate the oals of a PCD. The data poits i the PCD wee assued to be oisy saples of a sooth suface i 3. This is the case, fo exaple, fo PCD obtaied by age scaes. To obtai the eighbohood size fo the oal coputatio usig the least squae ethod, we would lie to use Equatio 5. We assued that the stadad deviatio σ of the oise was give to us as pat of the iput. We estiated the othe local paaetes i Equatio 5, the coputed. Note that this value of iiizes the boud of the oal coputatio eo, ad thee is o guaatee that this would iiize the eo itself. The costats c ad c 2 deped o the saplig distibutio of the PCD. While we could attept to copute the exact values of c ad c 2, we siply guessed the value c ad c 2. The value of ɛ was fixed at 0.. Give a PCD, we estiated the local saplig desity as follows. Fo a give poit p i the PCD, we used the appoxiate eaest eighbo libay ANN [5] to fid the distace s fo p to its -th eaest eighbo fo soe sall ube, = 5 i ou expeiets. The local saplig desity at p was the appoxiated as ρ = /(πs 2 ) saples pe uit aea. To estiate the local cuvatue, we used the ethod poposed by Guhold et al. [0]. Let p j, j be the eaest saple poits aoud p, ad let µ be the aveage distace fo p to all the poits p j. We coputed the best fit least squae plae fo those poits, ad let d be the distace fo p to that best fit plae. The local cuvatue at p ca the be estiated as κ = 2d/µ 2. Oce all the paaetes wee obtaied, we coputed the eighbohood size usig Equatio 5. Note that the estiated value of could be used to obtai a good value fo, which ca to be used to e-estiate the local desity ad the local cuvatue. This suggests a iteative schee i which we epeatedly estiate the local desity, the local cuvatue, ad the eighbohood size. I ou expeiets, we foud that 3 iteatios wee eough to obtai good values fo all the quatities. We still have pobles with obtaiig good estiates fo the costats c ad c 2. Fotuately, we oly have to estiate the costats oce fo a give PCD, ad we ca use the sae costats fo ay PCD with a siila poit distibutio. I ou expeiets, we used the sae value fo both c ad c 2. This value was chose so that the coputed oals o a sall egio of the PCD wee visually satisfactoy. Figue 6 shows the coputed taget plaes fo the oig- 327
We also wat to tha the ueous efeees of the pevious vesios of this pape fo thei exteely useful suggestios. Figue 7: Noal estiatio eos fo the buy PCD with oise added. The subfigues show the poits of the PCD usig the pi colo wheeve the eos ae above 0, 8, ad 5 espectively. ial Stafod buy. The plaes ae daw as sall fixed size squae patches. We oted that ou coputed oals ae siila to those obtaied usig the cocoe ethod by Aeta et al. [4]. Noisy PCD used i ou expeiets wee obtaied by addig oise to the oigial buy. The x, y, ad z copoets of the oise wee chose idepedetly ad uifoly ado i the age [ 0.0005, 0.0005]. The aplitude of this oise is copaable to the aveage distace betwee the saple poits ad thei eaest saples. We coputed the oals of the oisy PCD, ad used the agles betwee those oals ad the oals of the oigial PCD as estiates of the oal coputatio eos. I Figue 7, we colo coded the estiatio eos usig a covetio i which the colo of the squae patch at a poit of the PCD showed the eo at that poit. The colo of a patch is blue whe thee is o eo, ad it gets dae as the eo iceases. Whe the eo is lage tha a cetai theshold, the patch becoes pi. Figue 7 shows the taget plaes whee the thesholds ae 0, 8, ad 5 espectively. We a the least squae oal estiatio algoith o the buy with diffeet aouts of oise added to it ad obseved that the algoith woed well. We also oted that the oal estiatio ethod based o cocoe pefoed pooly i the pesece of oise. 5. CONCLUSIONS We have aalyzed the ethod of least squae i estiatig the oals to a poit cloud data deived eithe fo a sooth cuve i 2 o a sooth suface i 3, with oise added. I both cases, we povided theoetical boud o the axiu agle betwee the estiated oal ad the tue oal of the udelyig aifold. This theoetical study allowed us to fid a optial eighbohood size to be used i the least squae ethod. 6. ACKNOWLEDGMENTS We would lie to tha Leoidas Guibas fo his suggestios, coets ad ecouageet. We ae gateful to Taal K. Dey fo useful discussios ad also fo povidig the softwae fo evaluatig oals usig cocoe. We also tha Mac Levoy, Ro Fediw fo helpful discussios. We acowledge the geeous suppot of the Stafod Gaduate Fellowship poga ad of NSF CARGO gat 038456. The sall holes i the buy ae obsevable due to the fact that the patches do ot cove the buy etiely. 7. REFERENCES [] M. Alexa, J. Beh, D. Cohe-O, S. Fleisha, D. Levi, ad C. T. Silva. Poit set sufaces. IEEE Visualizatio 200, pages 2 28, Octobe 200. ISBN 0-7803-7200-x. [2] N. Aeta, M. Be, ad M. Kavysselis. A ew Voooi-based suface ecostuctio algoith. Copute Gaphics, 32(Aual Cofeece Seies):45 42, 998. [3] N. Aeta ad M. W. Be. Suface ecostuctio by voooi filteig. I Syposiu o Coputatioal Geoety, pages 39 48, 998. [4] N. Aeta, S. Choi, T. K. Dey, ad N. Leeha. A siple algoith fo hoeoophic suface ecostuctio. Iteatioal Joual of Coputatioal Geoety ad Applicatios, 2(-2):25 4, 2002. [5] S. Aya, D. M. Mout, N. S. Netayahu, R. Silvea, ad A. Y. Wu. A optial algoith fo appoxiate eaest eighbo seachig fixed diesios. Joual of the ACM, 45(6):89 923, 998. [6] J.-D. Boissoat ad F. Cazals. Sooth suface ecostuctio via atual eighbou itepolatio of distace fuctios. I Syposiu o Coputatioal Geoety, pages 223 232, 2000. [7] T. K. Dey, J. Giese, S. Goswai, ad W. Zhao. Shape diesio ad appoxiatio fo saples. I Poc. 3 th ACM-SIAM Sypos, Discete Algoiths, pages 772 780, 2002. [8] S. Fue ad E. Raos. Sooth-suface ecostuctio i ea-liea tie, 2002. [9] G. Golub ad C. V. Loa. Matix Coputatios. The Joh Hopis Uivesity Pess, Baltioe, 996. [0] S. Guhold, X. Wag, ad R. MacLeod. Featue extactio fo poit clouds. I 0 th Iteatioal Meshig Roudtable, Sadia Natioal Laboatoies, pages 293 305, Octobe 200. [] H. Hoppe, T. DeRose, T. Duchap, J. McDoald, ad W. Stuetzle. Suface ecostuctio fo uogaized poits. Copute Gaphics, 26(2):7 78, 992. [2] I. Lee. Cuve ecostuctio fo uogaized poits. Copute Aided Geoetic Desig, 7:6 77, 2000. [3] A. Leo-Gacia. Pobability ad Rado Pocesses fo Electical Egieeig. Addiso Wesley, 994. [4] S. Rusiiewicz ad M. Levoy. QSplat: A ultiesolutio poit edeig syste fo lage eshes. I K. Aeley, edito, Siggaph 2000, Copute Gaphics Poceedigs, pages 343 352. ACM Pess / ACM SIGGRAPH / Addiso Wesley Loga, 2000. [5] W. Welch ad A. Witi. Fee-fo shape desig usig tiagulated sufaces. Copute Gaphics, 28(Aual Cofeece Seies):247 256, 994. [6] M. Zwice, M. Pauly, O. Koll, ad M. Goss. Poitshop 3d: A iteactive syste fo poit-based suface editig. I Poc. ACM SIGGRAPH 02, Copute Gaphics Poceedigs, Aual Cofeece Seies, 2002. 328