Growing Least Squares for the Analysis of Manifolds in Scale-Space

Transcription

1 Growing Leas Squares for he Analysis of Manifolds in Scale-Space Nicolas Mellado, Gaël Guennebaud, Pascal Barla, Parick Reuer, Chrisophe Schlick To cie his version: Nicolas Mellado, Gaël Guennebaud, Pascal Barla, Parick Reuer, Chrisophe Schlick. Growing Leas Squares for he Analysis of Manifolds in Scale-Space. Compuer Graphics Forum, Wiley, 2012, Proceedings of Symposium on Geomery Processing 2012, 31 (5), pp < /j x>. <hal v2> HAL Id: hal hps://hal.inria.fr/hal v2 Submied on 24 Feb 2015 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 Eurographics Symposium on Geomery Processing 2012 Eian Grinspun and Niloy Mira (Gues Ediors) Volume 31 (2012), Number 5 Growing Leas Squares for he Analysis of Manifolds in Scale-Space Nicolas Mellado, Gaël Guennebaud, Pascal Barla, Parick Reuer, Chrisophe Schlick Inria - Univ. Bordeaux - IOGS - CNRS Absrac We presen a novel approach o he muli-scale analysis of poin-sampled manifolds of co-dimension 1. I is based on a varian of Moving Leas Squares, whereby he evoluion of a geomeric descripor a increasing scales is used o locae perinen locaions in scale-space, hence he name Growing Leas Squares. Compared o exising scale-space analysis mehods, our approach is he firs o provide a coninuous soluion in space and scale dimensions, wihou requiring any paramerizaion, conneciviy or uniform sampling. An imporan implicaion is ha we idenify muliple perinen scales for any poin on a manifold, a propery ha had no ye been demonsraed in he lieraure. In pracice, our approach exhibis an improved robusness o change of inpu, and is easily implemened in a parallel fashion on he GPU. We compare our mehod o sae-of-he-ar scale-space analysis echniques and illusrae is pracical relevance in a few applicaion scenarios. 1. Inroducion The use of muli-scale algorihms is ubiquious in image and geomery processing, and may serve very diverse purposes. In compuer graphics, muli-resoluion represenaions have been widely used. They are based on he decomposiion of a spaial signal ino base and deail layers, which may be used for insance in signal processing (e.g., [BEA83]) or for compression purposes (e.g., [Mal08]). In boh cases, he goal of a muli-resoluion srucure is o help modify a signal in ways ha depend on he argeed applicaion. In his paper, we are raher ineresed in scale-space echniques [Lin94, Rom09], where he goal is alogeher differen: i consiss in analyzing a signal a differen scales o discover is geomeric srucure, independenly of any applicaion. While muliresoluion mehods srive o decompose he signal ino independen layers, scale-space echniques criically rely upon redundan informaion beween scales o characerize he srucure of a spaial signal. A classical approach consiss in racking signal exrema (0-crossings) a increasing scales, using he scale a which hey ge annihilaed o characerize his srucure. Unforunaely, his mehod raises imporan issues. Firs, he consrucion of a scale space requires a dense regular sampling in boh spaial and scale dimensions. Mos imporanly, he racking of exrema provides informaion only for a subse of scale-space (a signal exrema); i is no able o idenify muliple perinen scales per poin; and i is sensiive o noise or nicolas.mellado@inria.fr small changes in he inpu signal, as deailed in Secion 2.1. Moreover, i is difficul o adap scale-space heory o he analysis of poin-sampled manifolds as explained in Secion 2.2. To he bes of our knowledge, here is no mehod ha has proposed a coninuous alernaive o exrema racking, and ha avoids he need of a regular paramerizaion while remaining compuaionally racable for manifolds of co-dimension 1 (e.g., curves in 2D, surfaces in 3D). The key idea of his paper is o perform he scale-space analysis of manifolds by means of coninuous algebraic fis. In our approach, a scale-space is buil hrough leas-square fis of a low-degree algebraic surface ono neighborhoods of coninuously increasing sizes. In some sense, i can be seen as an adapaion of he Moving Leas Squares formalism [Lev98a] o coninuously varying scales, hence he name Growing Leas Squares. The use of an algebraic surface ensures robus fis even a large scales and yields a rich geomeric descripor wih only a few parameers. The coninuiy of he fiing process hrough scales provides for a sable and elegan analysis of geomeric variaions. Our approach exhibis a number of advanages over previous scale-space analysis mehods of manifolds: I provides a fully coninuous alernaive o exrema racking, based on he derivaive of a local geomeric descripor ailored o scale-space analysis; I does no require any conneciviy or paramerizaion, and works from poins equipped wih normals; I is easy o implemen on he GPU, and robus o he scale sampling and o small changes in inpu poins. Compuer Graphics Forum c 2013 The Eurographics Associaion and Blackwell Publishing Ld. Published by Blackwell Publishing, 9600 Garsingon Road, Oxford OX4 2DQ, UK and 350 Main Sree, Malden, MA 02148, USA.

3 Mellado & Guennebaud & Barla & Reuer & Schlick / Growing Leas Squares for he Analysis of Manifolds in Scale-Space 2. Background In his secion, we presen previous work relevan o he analysis of manifolds in scale-space. This excludes pyramidal mehods [FMR11], specral processing [LZ09], or opological persisence [ELZ02, CGOS11], since hese are based on formalisms ha do no permi o analyze redundancy Scale-space heory Scale-space heory has firs been applied o 1D signals [Wi83] and gray-level images [Lin94]. Is main idea is o represen he inpu signal as a one-parameer family of smoohed signals. The parameer ha conrols smoohing size is called he scale, and he inuiion behind his approach is o ignore small variaions a increasing scales. The mos common smoohing operaor is he Gaussian kernel, in which case we alk abou linear scale-space. Conrary o muli-resoluion approaches ha mosly rely on pyramidal represenaions, scale-space makes use of sack represenaions: he resoluion is kep unchanged, which creaes a lo of redundancy from one scale o he oher. Such a redundancy is cenral o scale-space analysis, since i permis o sudy accuraely how a signal evolves from scale o scale. The core of analysis echniques relies on differenial invarians (e.g., gradien magniude, curvaure). The common approach consiss in racking 0-crossings of hose invarians along he scale dimension, and o record locaions where pairs of hem annihilae. This analysis exhibis wha has been called he deep srucure (e.g., [Rom09], p.154): 0- crossings are considered more or less perinen depending on he scale a which hey ge annihilaed. Once idenified, he deep srucure may be pu o use in a variey of applicaions: denoising, maching, simplificaion, ec. The main benefi of scale-space heory is ha he signal is analyzed independenly of any poenial applicaion, fully auomaically, wihou requiring he uning of parameers. Unforunaely, heir are some imporan limiaions: he pairing of 0-crossings is no robus o small changes in he inpu (as shown in [Mal00], p.86), and spurious 0-crossings may appear a increasing scales in 2D [Lin94]. Pracical algorihms have limiaions of heir own: he racking of 0- crossings requires a dense sampling in he scale dimension; and he smoohing operaor requires a regular uniform sampling. Alernaive mehods based on he racking of levelcrossings [WS90, DUM 11] have parly addressed heoreical limiaions, bu hey are sill limied by he aforemenioned pracical issues. Mos imporanly, he adapaion of racking-based mehods o he analysis of manifolds is far from sraighforward. As opposed o he image domain which offers a naural paramerizaion, manifolds are (d-1)-dimensional signals in R d, and ofen lack paramerizaion, uniform sampling and even conneciviy. Hence, specific soluions are required o analyze manifolds in scale-space Manifolds in scale-space Differenial geomery. One way o adap he scale-space heory o manifolds is o compue differenial invarians of poin coordinaes. This approach has firs been applied o curves and surfaces by Mokharian e al. [MM86, MKY01]. Is main limiaion is ha i requires a paramerizaion and a regular sampling on exended neighborhoods, which are ofen difficul and/or cosly o obain. A simpler approach employs he Difference of Gaussians (DoG) ha approximaes he Laplacian, firs inroduced for images [MH80] and laer adaped o meshes [ZBVH09]. I works in wo sages: 1) smooh he manifold a wo scales; 2) approximae curvaure a a poin by he disance beween is wo smoohed locaions. This mehod does no require any paramerizaion. The DoG mehod is preferable o mehods based on paramerizaion when dealing wih complex (e.g., scanned) geomery, alhough i sill relies on a conneced mesh in exising implemenaions. Moreover, i only provides local curvaure informaion. Boh approaches suffer from common scale-space issues: need of a dense sampling and lack of robusness of 0-crossings. Even hough spurious 0-crossings could be discarded (e.g., using [ELZ02, CGOS11]), oher pracical issues will remain. Numerical approximaion. An alernaive soluion o avoid compuing a paramerizaion is o perform measuremens of manifold properies ha are relaed o curvaure. One example is direc curvaure space smoohing [ZH97, ZH99], which uses a global spherical space o smooh curvaure-like values a differen scales. Hea kernel smoohing [SOG09] replaces he classic smoohing sep by a geodesic Gaussian diffusion o compue a hea kernel value a each scale. Inegral invarians [HFG 06, PWHY09] compue he volume occupied by he objec in a ball of varying size, which plays he role of scale. Like he DoG, hese mehods do no require any paramerizaion, and have hus been used in applicaions such as maching and regisraion of scanned surfaces (hey also require a mesh reconsrucion sep when applied o poin ses). Alhough hey have proven o be efficien for exracing global characerisics of manifolds, hey are no well adaped o scale-space analysis where we are ineresed in idenifying precise geomeric srucures: he compued values are valid curvaure approximaions only a small scales. Local regression. A hird approach o analyze manifolds in scale-space consiss in firs locally fiing simple geomeric kernels o inpu daa, and hen exracing properies relaed o differenial invarians. Performing regression wih a firs-order kernel inuiively corresponds o he fiing of a plane o inpu poins and normals. Surface anisoropy may hen be exraced [YLHP06, LG05, IT11], as well as a covariance measure [PKG03]. Regression hrough 2nd-order kernels [BSF02,CPG09] provides curvaure informaion, bu requires successive fiing seps (firs fi a plane, hen fi a quadric o he residual heigh daa). This raises issues when he surface folds-over iself. c 2013 The Eurographics Associaion and Blackwell Publishing Ld.

4 Mellado & Guennebaud & Barla & Reuer & Schlick / Growing Leas Squares for he Analysis of Manifolds in Scale-Space The main benefi of local regression mehods is ha hey work direcly from poin ses, while all oher mehods require a conneced mesh and ignore normals alogeher. As before, he differenial invarians ha may be exraced from fied kernels are limied. Indeed, alhough curvaure is compued wih second order kernels, he use of wo fiing seps grealy complexifies subsequen scale-space analysis: indeed, curvaure values are no direcly relaed when he suppor plane of he fied quadric changes from scale o scale. 3. Our approach The approach aken in his paper is based on local regression. I works direcly wih unorganized poin ses equipped wih normals and hus does no require any conneciviy. Moreover, conrary o previous echniques, i provides a geomeric descripor ha is direcly amenable o coninuous scalespace analysis. Our approach draws inspiraion from Moving Leas Squares (MLS) [Lev98b], whereby local regression is performed in a globally consisen way, resuling in coninuously differeniable surfaces. Our firs main conribuion is o adap MLS o perform scale-space analysis, aking advanage of heir coninuiy along he scale dimension. In a nushell, insead of shifing in space a weigh funcion of consan scale as wih MLS, we sudy he evoluion of he fiing resuls a a consan posiion wih a growing weigh funcion suppor. This leads us o our second key conribuion: insead of considering simple curvaure measures or eigenvalue raios, we propose o exploi and analyze he enire low-degree algebraic surfaces resuling from he fis. 4. Growing leas squares We firs briefly explain our choice of local regression in secion 4.1, along wih a reparamerizaion ha makes i relevan for he analysis of manifolds in scale-space. We hen exend he mehod in secion 4.2 o provide an alernaive o he racking of 0-crossings, based on analyic derivaives along he scale dimension. Third, we show how o adap our approach in secion 4.3 o measure pair-wise geomeric dissimilariies a arbirary scales and posiions Scale-space via local regression The firs sep of our approach is o characerize any poin p of a manifold a any scale by a low-degree algebraic surface ha bes approximaes is neighborhood P. In a discree seing, our manifold is described by a se of poins q i R d, wih d being he dimension of he ambien space, and he neighborhood P consiss in he se of daa poins conained in a ball of radius cenered a p: P = {q i ; q i p }. Inspired by recen work on MLS reconsrucion [GG07], we use algebraic hyper-spheres which have he advanages of being easy o fi in a robus manner, while providing secondorder informaion wih a minimal number of parameers. We assume each poin q i is equipped wih a normal n i R d. In case normals are no provided we esimae hem using covariance analysis, as done in previous work. {x ; su(x)=0} p qi ni Figure 1: Fiing and reparamerisaion. The weigh funcion (in green) around a poin p is defined in a neighborhood of size (dashed green line). Poins q i and normals n i ha belong o his neighborhood are fied by an algebraic hyper-sphere s u; is 0-isosurface is shown in black, and is scalar field wih a color code. This sphere is reparamerized in erms of hree geomeric parameers: he mean curvaure κ, he offse τ, and he gradien direcion η a p. Fiing. An algebraic sphere is implicily defined as he 0- isosurface of he following scalar field (see Figure 1): s u(x) = [1 x T x T x] u, (1) where u R d+2, u = [u c u l u q] T is he vecor of (respecively consan, linear and quadraic) parameers. In order o fi such a sphere ono a se of neighborhood poins P, we employ he fas fiing echnique of Guennebaud e al. [GGG08]. Firs, u l and u q are compued by minimizing i w i () s u(q i ) n i 2, where q i P and w i is a scaledependen weigh funcion: ( ) 2 q w i () = i p (2) Second, he consan coefficien u c is obained by minimizing in a leas square sense he algebraic disance o he samples: i w i () s u(q i ) 2. These wo minimizaions yield closed-form formulas which are recalled in he Appendix. Normalizaion. Conrary o [GG07], our goal is no o reconsruc a surface from a poin se, bu insead o analyze is shape a muliple scales. To his end, we wan o assign a unique and meaningful geomeric descripor for any choice of poin p and scale. A sraighforward soluion would be o use he cener c and radius r of he hyper-sphere. Unforunaely, his leads o degeneraed cases when he surface is locally planar: in paricular, c becomes undefined. We hus raher consider he scalar field iself s u as a geomeric descripor. However, here exiss an infiniy of scalar fields (based on scalar muliples of u) ha correspond o he same hyper-sphere. To solve his issue and consisenly pick a unique soluion, we use Pra s normalizaion [Pra87]: is basic idea is o consrain he scalar field o have a uniary gradien vecor on he 0-isosurface, yielding: û = [û c û l û q] T = u/ u l 2 4u cu q. (3) This choice has he addiional advanage o make algebraic disances near-euclidean for poins close o he 0-isosurface. κ 1 η p τ su c 2013 The Eurographics Associaion and Blackwell Publishing Ld.

5 Mellado & Guennebaud & Barla & Reuer & Schlick / Growing Leas Squares for he Analysis of Manifolds in Scale-Space (a) Small- and medium-scale analysis (b) Geomeric parameers (c) Geomeric variaion & finess (d) Dissimilariy profiles Figure 2: 2D analysis. The analysis of a synheic 2D curve composed of wo sinusoids of differen frequencies is illusraed a 3 differen poins. In (a), we show heir geomeric descripor a wo scales, whose parameers are visualized in (b) for all scales, wih one poin per row. In (c), we display geomeric variaions and finess: noe ha he 3rd poin has a more sable srucure a small scales since he magniude of he high-frequency componen is low a is locaion. In (d), we display dissimilariy measures for all pairs of poins. Observe how he use of he finess parameer helps disambiguae beween he wo ypes of inflexion poins a inermediae scales. In all plos, scale sampling is quadraic. Reparamerizaion. Afer normalizaion, we are lef wih a scalar field sû for which a geomeric inerpreaion is far from eviden. Firsly, û c and û l do no correspond o any measurable physical quaniy. Secondly, all d +2 parameers are sill inerdependen, since he normalizaion binds hem ogeher wih: û l 2 4û cû q = 1. We propose an alernaive paramerizaion of he scalar field, which is illusraed in Figure 1. Inuiively, is parameers consis of: he algebraic offse disance τ beween he evaluaion poin p and he 0-isosurface; he uni normal η of he scalar field a p; he signed curvaure κ of he hyper-sphere. When he fiing degeneraes o a plane, τ represens he disance from he origin o he plane, η is normal, and κ vanishes (see Figures 2 and 3 for 2D&3D illusraions, respecively). Formally, he geomeric parameers are given by: τ = sû(p) ; η = s û(p) ; κ = 2ûq. (4) sû(p) Thanks o Pra s normalizaion, he offse τ provides a close approximaion o he Euclidean disance beween p and he 0-isosurface. The normal parameer η gives he direcion o he poin on he hyper-sphere ha is closes o p. The curvaure parameer simply corresponds o he inverse of he hyper-sphere radius r, and has he advanage of behaving coninuously when passing hrough a locally planar surface, while r ends oward infiniy. Noe ha wih hese parameers, he scalar field can no longer be expressed as a linear combinaion of monomials, as shown in he Appendix. Finess. Once reparamerized, he scalar field fied from P yields a univocal geomeric descripor invarian o rigid ransformaions. However, a given geomeric descripor can be associaed o a space of generaor neighborhoods. Considering he sphere as a whole insead of, for insance, local curvaure only, already permis o significanly reduce he size of hese spaces. I is ineresing o remark ha hey can be furher reduced by looking a he finess ϕ ha exhibis how close he q i are o he fied scalar field s u. We define his ϕ = 1 ϕ = 0.91 ϕ = 1 ϕ = 0.86 Figure 4: The finess parameer ϕ helps o disambiguae wo idenical fis, e.g., a smooh versus a bumped neighborhood (lef), or a fla versus a saddle configuraion (righ). addiional parameer by ϕ = i w i () s u(q i ) n i / i w i (), where w i is he same weighing funcion as before. Using he fiing equaions 7, i can be shown ha for his fiing procedure, ϕ boils down o Pra s norm for u: ϕ = u l 2 4u cu q. Hence, ϕ does no have o be measured or sored explicily since i is embedded in he vecor of parameers u. Noe ha by consrucion, ϕ is dimension-less, scale-invarian, and varies in he [0,1] range, wih ϕ = 1 meaning a perfec alignmen beween he fied scalar field and inpu normals. As illusraed in Figure 4, his ypically permis o disambiguae beween surfaces ha locally have he same geomeric descripion, bu differs from a pure algebraic sphere. The chain of operaions described in his secion augmens an arbirary poin p a an arbirary scale wih a geomeric descripor ha characerizes daa poins q i P, as illusraed in Figure 2(a-b) for a 2D curve, and Figure 3(a-c) for a 3D surface. I hus describes an elegan mehod for building a fully coninuous scale-space from sampled manifolds of co-dimension 1, providing meaningful surface informaion in he form of he τ, η κ geomeric parameers, and a finess parameer ϕ ha furher helps disambiguae similar descripors. Anoher handy propery of our geomeric descripor is ha negaing is parameers yields he complemen descripor, which is equivalen o he descripor of he same surface wih an opposie orienaion. As deailed in Secion 5, his is a significan improvemen compared o exising mehods. One may be emped o use i o rack 0-crossings of he curvaure κ for insance, as in previous work. Insead, we show in he nex secion ha scale-space analysis may also be performed in a fully coninuous manner. c 2013 The Eurographics Associaion and Blackwell Publishing Ld.

6 Mellado & Guennebaud & Barla & Reuer & Schlick / Growing Leas Squares for he Analysis of Manifolds in Scale-Space (a) Small-scale analysis (b) Medium/large-scale analysis (c) Geomeric parameers (d) Geomeric variaion & finess Figure 3: 3D analysis. The analysis of a 3D golf ball model is illusraed a 3 differen poins: a concaviy, an edge and a juncion. This is shown a a small scale in (a), where we see ha he hree poins (wih heir neighborhoods in blue) have quie dissimilar geomeric descripors (in green). However, a medium and large scales (b), all 3 poins converge o a same global sphere (only one poin is shown for clariy). This is bes observed in (c), where we display heir geomeric parameers (excep for η ), and in (d), where all 3 poins converge ogeher. In all plos, scale sampling is quadraic Coninuous scale-space analysis The main purpose of analyzing a spaial signal in scale-space is o rack is variaions a increasing scales o discover is geomeric srucure. A classical approach consiss in racking invarians in he form of 0-crossings of a spaial derivaive of he signal; hen find locaions in scale-space where hey ge annihilaed. As menioned in Secion 2, his leads o many shorcomings, especially when rying o deal wih manifolds. Mos imporanly, i resrics he analysis o a subse of locaions in scale-space while requiring a paramerizaion. We propose a differen approach o discover he muliscale srucure of a manifold. Our key insigh is o observe ha, in general, a perinen scale for p is one where is geomeric descripor exhibis minimal variaion when he neighborhood size increases. This suggess ha, a such scales, he parameers of our descripor do no crucially depend on scale, bu raher indicae sable geomeric properies of he manifold. In his secion we focus on he derivaion of such a general geomeric variaion; we will show is relevance in Secion 5 and how i can be exploied in Secion 6. One may hink ha he curvaure parameer κ is he one mosly involved in geomeric variaions. Figure 5 shows couner examples where eiher τ or η have significan influence. We hus compue he variaions of all 3 geomeric parameers and combine hem in a naural fashion o yield a geomeric variaion funcion ν(p,) ha describes he scalespace srucure of he inpu manifold. Scale derivaives. The variaion of geomeric descripor parameers we are looking for are simply given by heir parial η/d, derivaives along he scale dimension a (p,): dτ/d, dη and dκ/d. We emphasize ha we are ineresed in geomeric variaions only, hence he finess ϕ does no play a role here. Since we make use of a local regression ha is boh coninuous and given in closed-form, hese derivaives are easily compued analyically, provided he weigh funcions hemselves are differeniable. Their compuaion does no yield any difficuly, and simply involves differeniaing he chain c 2013 The Auhor(s) c 2013 The Eurographics Associaion and Blackwell Publishing Ld. Figure 5: Variaions of τ and η. We show he hyper-spheres fied for he red poin wih various suppor sizes (indicaed by dark dos). On he lef, boh he offse τ and curvaure κ vary, while on he righ only he normal direcion η varies. of equaions presened in Secion 4.1, i.e. from op o boom: weighing (Eq. 2), fiing (Eq. 7), normalizaion (Eq. 3) and reparamerizaion (Eq. 4). Geomeric variaion. The geomeric variaion funcion ν(p,) is obained by a weighed squared sum of hese parial derivaives. In order no o inroduce any bias, a special care has o be aken in he choice of hese weighs. In paricular, we propose he following weighing scheme: 2 2 η 2 dκ dτ dη ν(p,) = + + 2, (5) d d d which has he fundamenal advanages o yield a dimensionless and scale-invarian measure, and o naurally give equal imporance o each parameer. Indeed, le us for insance choose meers m for he uni of lengh. Thanks o our inuiive reparamerizaion of Secion 4.1, we have τ in m, he uni-less η, and κ in m-1. Moreover, by consrucion i is reasonable o expec o have τ o be mosly comprised in ηk = 1. Therefore, a reasonable [,], κ in [ 1, 1 ], while kη η, κ) by (1/, 1,) choice is o scale he parameers (τ,η respecively in order o ge scale-invarian and uni-less quaniies of he same order of magniude. Finally, in order o compensae for he differeniaion over he scale ha is in m, i is naural o muliply by he scale, hus leading o he scaling facors (1,, 2 ) of Equaion 5. The funcion ν(p,) is one of he he key conribuion of his paper. I provides a coninuous descripion of perinen

7 Mellado & Guennebaud & Barla & Reuer & Schlick / Growing Leas Squares for he Analysis of Manifolds in Scale-Space (a) Scale-space for a synheic curve (b) Scale space for he drawn curve on he righ Figure 6: Curvaure scale spaces (from op o boom: DoG, ours and CA), wih orange colors for concaviies and blue colors for convexiies. In (a), we show curvaures for he sinus-based funcion of Figure 2(a); in (b), we show curvaures for he symmeric curve shown on he righ. Compared o DoG, our approach exhibis less noise a small scales and a beer convergence a large scales. CA does no make a difference beween convex and concave regions, which merge ogeher. scales for any poin p, and i is robus o small changes in he inpu. This is o conras wih previous approaches ha rely on he annihilaion of exremal poins (0-crossings) and may lead o alogeher differen srucures when he inpu changes only a lile, as shown in Secion 5. Mos imporanly, our approach is he firs o idenify muliple perinen scales for individual poins on manifolds, as exemplified in Figures 2(c) and 3(d) for 2D and 3D cases, respecively. This opens he door o many new applicaions, for which we skech a few examples in Secion Pairwise dissimilariy in scale-space For a variey of applicaions, i is also ineresing o compare a pair of arbirary scale-space locaions (p a, a) and (p b, b ). In his conex, i is crucial o provide a measure invarian o similariy ransformaions. Invariance o ranslaion is readily available because he geomeric parameers are defined relaive o heir fied poins. Since our descripor is isoropic, invariance o roaion is achieved by aligning he respecive uni normals η a and η b, which amouns o ignore hese parameers. Finally, scale invariance is obained using he weighs derived in he previous paragraph, and we hus define he dimension-less dissimilariy funcion δ a,b by: ( ) 2 δ a,b = a -1 τ a b -1 τ b + (aκa b κ b ) 2 + (ϕ a ϕ b ) 2 (6) Here he fiing errors ϕ a and ϕ b help disambiguae beween similar descripors ha may correspond o differen surfaces. Recall ha i is dimension-less, scale-invarian, and of he same order of magniude han he oher quaniies since i varies beween 0 and 1. The dissimilariy measure is illusraed in Figure 2(d), where we compare hree pairs of poins a muliple scales. Oher examples are presened in Secion 6, where we combine geomeric variaion and dissimilariy measuremens. 5. Comparisons wih previous work In his secion, we demonsrae he benefis of our scale space consrucion and analysis echniques and compare hem o sandard approaches. We only consider mehods ha, like ours, avoid fold-over issues, do no require any paramerizaion and provide valid approximaions of curvaure a all scales: DoG curvaure scale space [ZBVH09] and covariance analysis (CA) [PKG03]. Neverheless, his secion mosly makes use of paramerized 2D curves o 1) visualize he differen scale-spaces in a comprehensive manner, and 2) permi comparisons wih DoG and 0-crossings. Curvaure scale space. Le us firs focus on he scalarvalued scale-space obained using our curvaure measure κ. All our 2D visualizaions use he arclengh l and scale parameers for he horizonal and verical axis, respecively. Figure 6(a) compares our appraoch o DoG and CA for he smooh sinus-like curve of Figure 2. On his example, all mehods manage o idenify he wo signals, hough CA canno disinguish beween concaviies and convexiies, and DoG fails o capure imporan variaions a curve borders. On a more complex example, as shown in Figure 6(b), he differen mehods sar o exhibi more differen behaviors. In paricular, DoG does no reach convergence a large scales: i keeps on inroducing meaningless srucures. In conras, our curvaure κ produces a much smooher and coheren scale space, even hough i does no rely on inpu poin conneciviy. Sabiliy comparison. In Figure 7, we illusrae an imporan limiaion of racking-based scale-space analysis. When he inpu curve is modified so ha one of is bumps is slighly more prominen, hen he corresponding se of 0-crossings changes abruply, suggesing ha he inrinsic srucure of he curve has changed in similar respecs. In conras, our geomeric variaion scale-space evolves coninuously o reflec he more suble change of srucure implied by his sligh modificaion. In paricular, inermediae scales around he bump region are no longer considered persisen once he more prominen bump is se in place. Indeed, increasing he ampliude of he bump has he effec of breaking he srucure of he slope on he lef side of he curve. Relaionship o 0-crossings. Figure 8 shows a layering of 0-crossings on op of our geomeric variaion scale-space. c 2013 The Eurographics Associaion and Blackwell Publishing Ld.

8 Mellado & Guennebaud & Barla & Reuer & Schlick / Growing Leas Squares for he Analysis of Manifolds in Scale-Space Figure 7: Sabiliy comparison. Top row: a convex bump is added o he lef side of a simple open curve. Middle row: he srucure of 0-crossings changes abruply even for small changes in he inpu (red and green lines show he 0-crossings of κ and of is curvilinear derivaive κ, respecively). Boom row: he geomeric variaion ν changes coninuously o reflec he suble change in he inpu curve (whie means low variaion). Noe how he sable region a inermediae scales, which corresponds o he slope on he lef side, is progressively filled wih a more complex srucure. comparison wih dedicaed echniques in each of hese applicaion domains is ou of he scope of his paper. Insead, we focus on he novelies brough by our scale-space analysis, essenially is coninuiy and is abiliy o deec muliple perinen scales per poin. Some implemenaion deails are given a he end of his secion. Figure 8: Relaionship o 0-crossings. The geomeric variaion ν for he second drawing in Figure 7 is displayed wih 0-crossings of κ (in red) and κ (in green) layed on op. Two ypes of differenial invarians are shown: 0-crossings of κ and is spaial derivaive κ. Le s wrie he corresponding scale-space poins C and C for convenience. Firs, noe ha all annihilaion evens for boh C and C occur in regions of low geomeric variaion, bu here are oher such regions which are never reached by eiher C or C. This suggess ha he racking of exrema acually leads o a subse of perinen scale-space locaions according o ν. I is also ineresing o noe ha poins in C (resp. C ) seem o be araced oward regions of high (resp. low) variaion, a endency we plan o sudy in fuure work. Our second observaion concerns he larges scale: by consrucion, only a few poins from C (and none from C) reach he op of scale-space, and hey are relaed hrough racking o a single poin on he original manifold. However, here is no reason o idenify a precise locaion as a represenaive of such a large scale, since i corresponds o an exremely smooh shape. Our approach reflecs his convergence since mos poins have a low geomeric variaion in his case, and correspond o a global geomeric descripor. 6. Applicaion scenarios We have shown ha our geomeric variaion scale-space provides a robus and coninuous characerizaion of he geomeric srucure of a manifold. We now give examples of is usefulness hrough hree applicaion scenarios relaed o denoising, feaure deecion and maching/regisraion. A full Adapive Bandwidh. When a poin-sampled manifold is corruped wih spaially varying noise as shown in Figure 9, i is no appropriae o reconsruc he signal a a single global scale. Insead, we mus find a spaially-varying scale locally adaped o he ampliude of noise, also called adapive bandwidh [WSS09]. However, care mus be aken no o over-esimae his minimum scale, oherwise perinen manifold srucures may be damaged. Thanks o our geomeric variaion scale-space, i is possible o adapively esimae a proper bandwidh: inuiively, we only have o find he smalles perinen scale for each poin p. As a proof of concep, we propose a simple op-down heurisic ha works well for smooh objecs. We deec he noise in a coarse-o-fine fashion: for each poin we se a bandwidh a scales where dν/d is greaer han a given hreshold (we use 0.01 in his example). We hen regularize he resul wih a spaial smoohing across daa poins. We will sudy he exension of his mehod o more complex objecs in fuure work. The resuling adapive bandwidh is used as a variable suppor size o reconsruc a smooh curve, shown in orange in Figure 9 for differen kind of noise. Coninuous feaures. To deal wih complex manifolds, many geomery processing applicaions rely on a preliminary feaure exracion sep, which idenifies a subse of salien poins o consider for furher processing. Probably he mos famous echnique in his field is SIFT (e.g., [IT11]), which idenifies local feaures ha are relaed o locaions where 0-crossings of κ annihilae. Because such 0-crossings are sensiive o noise and small changes in he inpu, SIFT poins may be filered ou in a number of ways o keep only he mos salien locaions. c 2013 The Eurographics Associaion and Blackwell Publishing Ld.

9 Mellado & Guennebaud & Barla & Reuer & Schlick / Growing Leas Squares for he Analysis of Manifolds in Scale-Space A B D C (a) Noisy 2D poin se A B C D (b) Exraced bandwidh Figure 9: Adapive bandwidh: a 2D poin se (a) is corruped wih noise on: normals and posiions (A), posiions (B) and normals (C) ; D is wihou noise. The reconsruced curve (shown in orange) is obained using an adapive bandwidh (black curve) exraced from our coninuous scalespace represenaion (b). Insead of relying on individual poins, we propose he coninuous feaure funcion f (p) = ν(p,)d, where ν is a smooh remapping of ν ha disregards oo srong geomeric variaions (we use a anh funcion o his end). The inuiion behind his formula is ha poins for which f is small are subjec o nearly no geomeric variaions across scales (e.g., f = 0 on a sphere); whereas poins wih a high f indicae ha our geomeric descripor is perinen only a a few scales. This is shown on complex curves and surfaces in Figure 10 wih a color gradien. Observe how he mos pronounced feaures (in red) may correspond o widely differen concave and/or convex regions; his is desirable because here is ofen no a priori on he sign of curvaure of feaures. Also noe ha regions wih repeiive small deails are no considered as imporan, whereas isolaed or more prominen shape deails emerge as key feaures of he model. Muli-scale dissimilariy. The abiliy o evaluae local surface similariies is he cornersone of many maching or regisraion echniques. However, he noion of muli-scale similariy hides an imporan quesion: which scales should acually be aken ino accoun? One may hink ha aking all scales a once is a naural soluion. However, as shown in Figure 11(a), his migh no always be he case, a leas for some applicaions. The quesion becomes delicae as soon as muliple scales are nesed in a same objec: one may wan o find similariies a he smalles of hese scales, or perhaps in oher siuaions a he larges scale. Our approach permis o make his choice in a coninuous manner, and we illusrae i wih a simple picking ool: he user selecs a poin on a surface, and he sysem finds all similar poins on he same objec, given a rough scale prior. The basic idea is o combine he global prior wih he local geomeric variaion o compue a per-scale dissimilariy, which is hen inegraed over he scales i o yield a muli-scale dissimilariy p,q = i δ p,q( i )h( i ), where δ p,q() is he dissimilariy measure from Equaion 6, and h is a normalized weighing funcion over scales ha defines he global prior. In our example, we use a simple box filer for h. Figure 11(b-c) visualizes p,q as a funcion of q (in blue) for a given poin p (in red) locaed boh on a small ridge and on he S of he SGP acronym. By varying he global prior, our mehod idenifies unambiguously eiher one deail layer or he oher. Alhough our approach is based on an isoropic regression, his resul demonsraes ha anisoropic feaures a perinen scales are properly exraced, even juncions and corners. Moreover, his is done irrespecive of he shape of he base surface, since he leers SGP are exraced similarly on differen locaions of he orus. Implemenaion deails. In our sysem we use a kd-ree for he neighbor search, and he fiing a scale is performed by collecing all he neighbors wihin he disance of he curren poin. This makes he complexiy of our algorihm quadraic wih respec o for a 3D surface. We implemened our approach boh on he CPU (on all 8 cores of an Inel I7 3.40Ghz) and on he GPU (using CUDA on a GTX 580). As expeced, he GPU implemenaion ouperforms he CPU version; for he example of Figure 10(c), he analysis akes only 6.3s versus 221s. All repored imings correspond o our CUDA implemenaion. We believe a huge speed-up could be achieved by ieraively smoohing and simplifying he inpu poin cloud o compue he larger scales [PKG06]. 7. Discussion & Fuure work We have presened a novel approach o scale-space analysis based on local regression, ha auomaically and robusly characerizes he sable srucure of a manifold of codimension 1 in a coninuous manner. A srengh of his approach is ha i is enirely independen of any argeed applicaion. Indeed, our coninuous geomeric variaion scalespace may be inerpreed and processed differenly by differen ypes of applicaions, as demonsraed in Secion 6. Our soluion considerably improves previous work: i is fully coninuous in boh space and scale, i is robus o noise (Figures 9 and??), i does no require any paramerizaion or conneciviy, and i naurally deals wih manifold borders. The DoG mehod could be adaped o handle poin ses and manifold borders using local regression for smoohing. However, i would sill have o be applied o daa poins wih geodesic neighborhoods, which are ofen difficul o obain, and i would provide only curvaure measuremens. Our approach idenifies a more complee geomeric descripor while only requiring an Euclidean neighborhood wih a reasonable number of poin samples (a leas abou 5 in 2D and 12 in 3D). Quadric fiing is anoher alernaive o compue curvaure, bu i is no adaped o scale-space analysis since i requires a local planar paramerizaion which is no globally coheren and no robus o fold-overs. In conras, our mehod avoids hese shorcomings by employing a oalleas square fiing procedure. A limiaion of our approach comes from he choice of an isoropic regression: surface anisoropy is no explicily idenified. However, as shown in Figure 11, our mehod is able o deec indirecly complex anisoropic srucures. As shown in Figure??, our scale-space analysis even permis o idenify saddle-like shapes. The case of a perfec saddle c 2013 The Eurographics Associaion and Blackwell Publishing Ld.

10 Mellado & Guennebaud & Barla & Reuer & Schlick / Growing Leas Squares for he Analysis of Manifolds in Scale-Space f =1 f =0 (a) (b) 2D symmeric curve: 5k ps, 1k scales, 3s (c) 3D Armadillo models: 173k ps, 20 scales, 6.3s Figure 10: Coninuous feaures are displayed using a color gradien (a), on a symmeric 2D closed curve (b), and on 3 varians of he Armadillo 3D model. Observe how feaures of varying shapes and sizes (in red) are properly seleced in all cases. (a) All scales (b) Fine scales (c) Coarse scales Figure 11: Muli-scale similariy. Top row: Poins similar o he picked poin (in red) are seleced (in blue) via our dissimilariy measure. Boom row: he ype of seleced feaure depends on a user-conrolled global prior (blue box), which is locally refined by our geomeric variaion. In (a), all scales are seleced. In (b), only he fine displacemen paern emerges. In (c), he largescale SGP leers are properly segmened (SGP orus: 500k ps, 20 scales, 42sec). We use a log remapping on he scale axis. pach requires he ake finess ino accoun, as shown in Figure 4. In fuure work, we also plan o sudy spaial variaions of our geomeric descripor, in a way similar o wha is classically done wih MLS. I would permi o explicily idenify direcion fields on manifolds and he scales a which hese are perinen. In his paper, we have only considered shape geomery and ignored addiional informaion such as color aribues. In fuure work, we plan o exend our weigh funcions o rejec neighbor poins of dissimilar aribues, yielding a non-linear version of our coninuous scale-space analysis. We have also observed oscillaions in ν for inpu manifolds wih regular srucures (see Figures 2(c) and 3(d)). For some applicaions, hey migh lead o false posiives, which could be deeced hrough a frequenial analysis of ν. Our versaile and coninuous geomeric scale-space analysis has a grea poenial for a large variey of applicaions ha we wan o explore in fuure work. For example, in he domain of Culural Heriage, where i is common o conc 2013 The Auhor(s) c 2013 The Eurographics Associaion and Blackwell Publishing Ld. fron differen hypoheses, our coninuous scale-space could be used o design semi-auomaic mehods ha guide archaeologiss in shape maching asks. Muli-resoluion represenaions could also benefi from our analysis as a pre-process, for insance for he idenificaion of he mos meaningful scales a which deails should be compued. This could be paricularly ineresing in he conex of base/relief surface decomposiion. Finally, he geomeric descripors ha correspond o perinen scales could be used for curve and surface absracion purposes. Acknowledgmen This work has been suppored by he ANR SeARCH projec (ANR-09-CORD-019), and he European Communiy s Sevenh Framework Program [FP7-2007/2013] under he Gran Agreemen (V-mus.ne). The model of figure 3 is provided couresy of S. Rusinkiewicz, and figure 10 couresy of AIM@SHAPE Shape Reposiory.

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P.: Surface feaure deecion and descripion wih applicaions o mesh maching. In CVPR (2009). 2, 6 [ZH97] ZHANG D., HEBERT M.: Muli-scale classificaion of 3D objecs. In CVPR (1997), pp [ZH99] ZHANG D., HEBERT M.: Harmonic maps and heir applicaions in surface maching. In CVPR (1999), pp Appendix Sphere fiing. For ease of implemenaion we recall he closed-form formula for algebraic sphere fiing [GGG08]: u q = 1 w i q T i n i w i q T i w i n i 2 w i q T i q i w i q T i w i q i u l = w i n i 2u 4 w i q i (7) u 0 = u T l w i q i u 4 w i q T i q i where w i is he normalized weigh of he sample q i : w i = w i / j w j. The differeniaion of hese formulas is sraighforward and yield o he differeniaion of w i () (Equaion 2) over he scale : dw i d () = 4 p i p 2 3 ( ) p i p Reparamerizaion. Afer our reparamerizaion (Equaion 4), he scalar field sû can be rerieved by: (8) sû(x) = s τ,η,κ(x) = τ + (1 + 2τκ) 1 2 η (x p) + κ 2 (x p)2 (9) c 2013 The Eurographics Associaion and Blackwell Publishing Ld.