Feature-Preserving Reconstruction of Singular Surfaces

Transcription

1 Eurographics Symposium on Geomery Processing 2012 Eian Grinspun and Niloy Mira (Gues Ediors) Volume 31 (2012), Number 5 Feaure-Preserving Reconsrucion of Singular Surfaces T. K. Dey 1 and X. Ge 1 and Q. Que 1 and I. Safa 1 and L. Wang 1 and Y. Wang 1 1 Compuer Science and Engineering Dep., The Ohio Sae Universiy, U.S.A. Absrac Reconsrucing a surface mesh from a se of discree poin samples is a fundamenal problem in geomeric modeling. I becomes challenging in presence of singulariies such as boundaries, sharp feaures, and non-manifolds. A few of he curren research in reconsrucion have addressed handling some of hese singulariies, bu a unified approach o handle hem all is missing. In his paper we allow he presence of various singulariies by requiring ha he sampled objec is a collecion of smooh surface paches wih boundaries ha can mee or inersec. Our algorihm firs idenifies and reconsrucs he feaures where singulariies occur. Nex, i reconsrucs he surface paches conaining hese feaure curves. The idenificaion and reconsrucion of feaure curves are achieved by a novel combinaion of he Gaussian weighed graph Laplacian and he Reeb graphs. The global reconsrucion is achieved by a mehod akin o he well known Cocone reconsrucion, bu wih weighed Delaunay riangulaion ha allows proecing he feaure samples wih balls. We provide various experimenal resuls o demonsrae he effeciveness of our feaure-preserving singular surface reconsrucion algorihm. Caegories and Subjec Descripors (according o ACM CCS): Generaion Line and curve generaion I.3.3 [Compuer Graphics]: Picure/Image 1. Inroducion Reconsrucing a qualiy surface mesh from a se of discree poin samples is a fundamenal problem in geomeric modeling, wih many applicaions in science and engineering. Numerous algorihms have been proposed for his problem [Dey06, GP07]. Mos of he proposed algorihms focus on reconsrucing smooh surfaces wihou boundaries. Recenly, here have been growing ineress in reconsrucing surfaces while preserving sharp feaures such as nonsmooh creases. However, in pracice, daa can be sampled from more general domains, including non-manifolds where muliple surface paches can mee or inersec. In his paper we consider he so-called singular surfaces which consis of a collecion of smooh surface paches wih boundaries. These surface paches can inersec, or be "glued" along heir common boundaries (i.e, sharp crease lines). For simpliciy, we unify boundaries, inersecions, and sharp creases under he aegis of singulariies, and refer o hem as feaure curves. To dae, an effecive and pracical algorihm o recover a singular surface from poin daa is sill missing Our work In his paper, we propose a simple ye effecive reconsrucion algorihm for singular surfaces ha can handle all hree singulariies menioned above in a unified framework. Our algorihm has wo componens (see Figure 1): (1) feaure curve idenificaion and reconsrucion, and (2) singular surface reconsrucion ha respecs hese feaures. For he firs sep, we employ a novel combinaion of he Gaussian-weighed graph Laplacian [BQWZ12] and he Reeb graph [DW11, GSBW11]. Our approach provides a unified approach o handle all hree ypes of singulariies. I is simple: only he proximiy graph from inpu poins is required, and i does no involve esimaing normals or angen spaces, which could be unreliable around singulariies. Furhermore, higher-order singulariies, such as juncion nodes where muliple feaure curves mee, are auomaically and reliably deeced wihou any special handling. Our approach is robus o noise and can possibly be used in oher applicaions involving poin cloud processing, such as in sylish drawing [PKG03]. For he second sep of feaure-preserving singular surface reconsrucion, we use a varian of he well known Cocone Compuer Graphics Forum c 2012 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.

2 Dey e al. / Feaure-Preserving Reconsrucion of Singular Surfaces (c) (d) (e) (f) (g) (h) Figure 1: Workflow of our algorihm. Original poin clouds. The hidden domain is a sphere inersecing a half-cube wih only hree faces (he hree visible ones). Feaure poins idenified. (c) A zoom-in of feaure poins around inersecions. (d) Coarse feaure curves reconsruced. (e) Refined feaure curves, wih juncion nodes and sharp corners marked. (f) Reconsruced singular surface. Two zoomed-in views of reconsruced model near inersecion in (g) and (h). algorihm [ACDL02]. Inspired by he success of ball proecion idea of [CDR10] in generaing meshes from piecewise smooh complexes, we propose a weighed varian of he Cocone algorihm. The original Cocone algorihm filers riangles from he Delaunay riangulaion of he inpu poins o reconsruc a smooh surface wihou boundary. Afer idenifying he feaure curves and generaing sample poins on hem in he firs sep, we pu a proecing ball cenering each sample poin on hese curves. The balls are urned ino weighed poins. These weighed poins ogeher wih he inpu poins, which remain unweighed, consiue he inpu o he surface reconsrucion algorihm. The Cocone algorihm is run on he weighed Delaunay riangulaion of he resuling poin se. Only he unweighed poins are allowed o choose he cocone riangles. This simple modificaion of he Cocone algorihm allows us o reconsruc singular surfaces quie effecively as our experimens show Relaed work A muliude of algorihms have been proposed in he lieraure ha use Voronoi diagram and Delaunay riangulaion o deermine he ulimae mesh riangles for reconsrucion as we do in his paper. The work of Amena and Bern [AB99] inspired a flurry of aciviies in his area [ACDL02, BC02, ACSTD07]; see he book and survey [Dey06, CG06] for deails. However, none of hese algorihms is concerned wih he reconsrucion of surfaces wih singular feaures wih he only excepion of a recen work by Cazals and Cohen-Seiner [CCS12]. This algorihm reconsrucs compacs provably bu while preserving homoopy. Our goal here is o reconsruc singular surfaces while preserving he full opology and singular feaures. Unlike [CCS12], we canno provide provable guaranee, bu we demonsrae he effeciveness of our mehod by empirical resuls. Recenly, some approaches have been proposed ha aim o reconsruc surfaces from poin daa while preserving sharp feaures. Many of hem are based on implici funcions such as he moving leas-square (MLS) framework and Poisson surfaces [ABCO 01, KBH06]. Earlier work employs anisoropic basis funcions [DTS01, AA06] which show higher fideliy in reconsrucing smooh surfaces wih high curvaure feaures. Recenly, Reuer e al. [RJT 05] proposed a projecion operaor for surface reconsrucion, based on he enriched reproducing kernel paricle approximaion, o respec sharp feaures. However, heir algorihm requires poins on sharp feaures o be given as inpu. Lipman e al. [LCOL07] defined a singulariy indicaor field (SIF) which measures, for each poin, is poenial o be on or near a singulariy. They use his SIF o guide he local polynomial fiing in a daa-dependen MLS framework o reconsruc surfaces while preserving high-frequency feaures. By using a coninuous singulariy indicaor field, heir algorihm eliminaes he need for a cuoff hreshold separaing sharp feaures from non-sharp feaures. This allows hem o reconsruc delicae singulariies faihfully as well. However, heir mehod canno handle higher-order singulariies in he hidden surface where muliple feaure curves mee (such as a corner poin). Laer, Özireli e al. [ÖGG09] proposed a robus implici MLS based on local non-linear kernel regression. Avron e al. [ASGCO10] proposed an algorihm c 2012 The Auhor(s) c 2012 The Eurographics Associaion and Blackwell Publishing Ld.

3 o reconsruc sharp poin se surfaces based on an L 1 -sparse mehod. This surface reconsrucion algorihm naurally preserves sharp feaures while a he same ime, is also resilien o noises due o a global opimizaion procedure. There is anoher line of recen work which explicily exracs sharp feaures during he surface reconsrucion process. In paricular, based on he idea ha sharp feaure lines are locaed where he fiing error using a single piece of surface pach is large, Fleishman e al. [FCOS05] developed an ieraive refiing algorihm, called robus MLS, o gradually grow piecewise smooh surface paches. Sharp feaure lines are idenified where disconinuiy of surface occurs. This algorihm iniiaed a new line of aack for handling sharp feaures, and provided reasonable resuls even when he inpu poins have missing daa. Unforunaely, he algorihm is relaively expensive, and requires explici processing for corners where muliple surface paches mee. Also he reconsruced feaure lines may no be smooh. A laer work in [IHOS07] uses a mehod based on his robus MLS algorihm o exrac smooh feaure curves. In [JWS08], Jenke e al. proposed anoher algorihm o classify and reconsruc each piecewise smooh surface pach using he so-called pach-graph, which helps o combine boh local and global informaion o produce more sable classificaion of surface paches. By deecing boundary poins separaely and explicily, his algorihm can also handle surfaces wih boundaries. Anoher relaed work [MW11] can handle non-orienable and self-inersecing surfaces. Specifically, non-manifold regions are idenified by local fiing and a riangulaion is hen consruced by an incremenal (non Delaunay-based) algorihm. This algorihm does no preserve sharp feaures. Our algorihm is mos closely relaed o he work in [SYM10] in erms of he high level wo-sep framework: The algorihm in [SYM10] firs idenifies feaure poins based on he covariance marices of Voronoi cells of daa poins as developed in [MOG09]. I hen fis polylines hrough he feaure poins as feaure curves, and reconsrucs surfaces while preserving hese feaure curves based on a combinaion of he meshing algorihm of [BO05] and he proecion-balls idea of [CDR10]. This algorihm oupus explici sharp feaure curves, in addiion o he feaure-preserving surface reconsrucion. The algorihm shows robus reconsrucion resuls under noise and sparse sampling. To dae, all he previous sharp-feaure preserving reconsrucion algorihms recover a manifold surface (mosly closed surfaces, wih few excepions [FCOS05, JWS08, MW11]). Our algorihm handles no only closed surfaces wih sharp crease lines, bu also non-manifold surfaces wih boundaries, wihin a single framework. Furhermore, by using he Reeb graph o recover he feaure graphs (i.e, feaure curves wih heir connecions), we can recover juncion nodes easily and reliably, while previous approaches require special handling o discover hem. Poins along he exraced feaure curves are fed o he subsequen reconsrucion sep, which employs a weighed version of Cocone algorihm o reliably reconsruc non-manifold singular surfaces wih sharp feaures. 2. Algorihm Overview Suppose we have a collecion of smooh 2-manifolds wih boundary {Ω 1,...,Ω k } isomerically embedded in IR 3. We call each Ω i a surface pach. These surface paches may inersec each oher in heir inerior, giving rise o inersecionfeaure curves. They can also be "glued" along (par of) heir boundaries, producing he sharp-feaure curves. Finally, we call boundary curves ha are no shared by muliple manifolds as boundary-feaure curves. inersecions sharp feaure curve Ω 3 Ω 1 Ω 2 boundary curves See he righ figure and Figure 1 (e) for illusraions of differen ypes of feaure curves we consider. A regular poin refers o a poin no on hese feaure curves. Given a se of poins P sampled from he singular surface Ω, which is he union of Ω i s, our goal is o reconsruc Ω while preserving various ypes of feaure curves. Our approach consiss of wo seps. In he firs sep, we use a combinaion of Gaussian-weighed graph Laplacian wih he Reeb graph o idenify and reconsruc a se of feaure curves. These curves are sampled o generae a poin se F. We nex apply a weighed Cocone algorihm o reconsruc he singular surface, aking he poin cloud P F as inpu. The algorihm is oulined below. The deails of he wo componens are discussed in Secion 4 and 5, respecively. Sep 1 (Feaure Curves Idenificaion and Reconsrucion): (1.a) Idenify feaure poins via graph Laplacian (1.b) Reconsruc feaure curves via Reeb graphs (1.c) Refine feaure curves and generae a poin sample F on hem. Sep 2 (Feaure-aware singular surface reconsrucion): (2.a) Pu a proecing ball cenering each sample poin in F where he weighs are deermined by a parameer used for feaure curve idenificaion (2.b) Delee poins in P inside each proecing ball and le P be he resuling se. Compue a weighed Delaunay riangulaion of he poin se P F (2.c) Compue a cocone surface for each pach using he Cocone algorihm wih he modificaion ha only unweighed poins in P F choose cocone riangles. 3. Gaussian-weighed Graph Laplacian The feaure poins idenificaion algorihm uses he widely used Gaussian-weighed graph Laplace operaor which we describe now. Given a se of poins P = {p 1,..., p n} IR 3, Ω 4 c 2012 The Eurographics Associaion and Blackwell Publishing Ld.

4 he (un-normalized) Gaussian-weighed graph Laplace operaor L is an n n marix where 1 pi p 2 j e L [i][ j] = n 2, if i j; 1 p 2 (1) k p i n 2 n k=1,k i e, if i == j. For any fixed funcion f : P IR and a poin p i P, i is easy o verify ha L applied o his funcion f is: L f (p i ) = 1 n 2 n j=1 e p i p j 2 [ f (p i ) f (p j )]. I was shown in [BN03] ha if poins in P are uniformly randomly sampled from a smooh d-manifold M, hen in he limi as n goes o infiniy and ends o 0 a an appropriae rae, L f (p) converges o f (p) where is he Laplace- Belrami operaor for M. The connecion o M is made via he so-called funcional Laplacian [BN03]: L f (p) = 1 d 2 +1 M e x p 2 ( f (p) f (x))dx. Inuiively, he Gaussian-weighed graph Laplacian L is simply he discreizaion of he inegral operaor L a poins in P. I has been shown ha for sufficienly small, L f (p) = f (p) + o(1). (2) Now suppose ha he underlying domain where daa are sampled from is no a manifold, bu a singular d-manifold Ω. Ineresingly, i urns ou ha he funcional Laplacian L behaves differenly around various singulariies (i.e, feaure curves in 2D) from around a regular poin (see [BQWZ12] for deails). In paricular, for a poin p lying on a boundaryfeaure curve of some Ω i, we have ha L f (p) = 1 π n f (p) + o( 1 ), (3) where n is he uni ouward normal o he boundary-feaure curve a p (ha is, n is in he angen space of Ω i a p and normal o he boundary-feaure curve); and n f (p) is he direcional derivaive of f in direcion n. We remark ha he erm f (p) as well as erms depending on inrinsic curvaures a p are now hidden in he lower-order o( 1 ) erm which does no dominae for small. Comparing wih Eqn (2), we see ha while L f (p) reflecs he manifold Laplacian (a second-order differenial quaniy) a a regular poin, i reflecs a firs-order parial derivaive a a boundary poin. More imporanly, he scale dependence on is differen: O(1) for a regular poin versus O( 1 ) for a singular poin. For small (which is usually he case in pracice), 1 is large, implying ha L f (p) is significanly larger a a boundary poin han a a regular poin (assuming n f (p) is bounded from below by a consan). The dominance of he O( 1 ) erm in fac affecs a "band" of poins O( wihin Θ( ) disance away from 1 ) he boundary-feaure curves. Bu i follows a Gaussian disribuion cenered a a boundary poin wih 0 variance ; hus his effec wears off rapidly. See he adjacen figure for an illusraion of he firs erm of Eqn (3) (he y-axis in he figure) as he disance of poin x o he boundary-feaure curve (he x-axis in he figure) increases. The cases for poins on or around he oher wo ypes of feaures curves are similar, alhough more complicaed. We briefly summarize he behavior of L f around hose feaure curves in Appendix A. As menioned earlier, L is simply a discreizaion of L and hus shares he same behavior as L. Our algorihm leverages he differen scaling behavior of L around feaure curves o help idenify feaure poins. Non-uniform sampling. Suppose he inpu poins P are sampled from a non-uniform densiy funcion ρ : Ω IR which is smooh on each surface pach Ω i. I urns ou ha under mild assumpions on ρ, he differen scaling behavior of L sill holds. In paricular, he O( 1 ) erm will now simply be muliplied by a facor of ρ(x). Hence our feaure idenificaion algorihm is reasonably robus agains nonuniform sampling of he inpu domain. 4. Feaure Curves Idenificaion and Reconsrucion 4.1. Poenial feaure poins idenificaion Le he inpu poin cloud be P = {p 1,..., p n}. We apply he graph Laplacian L o he coordinae funcions X,Y,Z : IR 3 IR, where X(p) = p.x, Y(p) = p.y and Z(p) = p.z. Le P = [X Y Z] denoe he coordinae funcions resriced o he inpu poins P, which is a 3 n marix and can be considered as n 3-dimensional vecors. Applying L o P gives rise o anoher lis of 3-dimensional vecors V = L P, where he ih row v i := V [i] is a 3-dimensional vecor associaed wih poin p i P. Le V be he n-dimensional vecor (funcion) where V [i] := v i is he norm of vecor v i. I is known (see e.g, [DMSB99]) ha P (p) = H p n p for a regular poin p from a smooh surface, where H p is he mean-curvaure a p and n p is he uni surface normal a p. Hence v i = L P (pi ) H pi n pi, a a regular poin p i. However, if p i is on or near he hree-ypes of feaure curves we consider, v i reflecs fundamenally differen informaion. We noe ha his is no rue a a regular poin: for poins sampled from a densiy funcion ρ, L f (x) converges o ρ(x) ρ 2 f (x) a a regular poin, whereas he weighed Laplacian ρ 2 f (x) equals 1 ρ 2 div[ρ 2 grad f (x)]; see e.g, [Gri06] for deails. c 2012 The Eurographics Associaion and Blackwell Publishing Ld.

5 Specifically, consider a poin p on he boundary of a surface pach Ω j. Le n = [ n.x n.y n.z] T denoe he uni ouward normal vecor o he boundary curve a p (noe, n is no he surface normal a p, bu he uni vecor in he angen plane a p normal o he boundary curve). By Eqn (3), seing C = π 1/2 2 we have ha L X(p) C X n = C [1,0,0] T, n = C n.x, where [1,0,0] T is he uni vecor in he x-direcion, and, sands for he inner produc. By using, we omi lower order erms o( 1 ). Similarly, we have ha L Y(p) C n.y and L Z(p) C n.z. Puing hem ogeher, we have ha L P (p) L P (p) C [ n.x, n.y, n.z] T = 1 π1/2 2 n. Noe ha L P (p) is independen of he choice of he coordinae sysem. The magniude of L P (p) is simply π 1/2 2. As a poin moves away from he boundary, he magniude of L P (p) decreases rapidly from π 1/2 2 o he mean curvaure (i.e, he case for regular poins), following a Gaussian disribuion wih variance. The direcion of L P (p) also changes from he ouward normal o he boundary curve o he surface normal for a regular poin. Poins on and around oher ypes of feaure curves share similar behavior. In oher words, V [i] is he mean curvaure (and hus of order O(1)) for a regular poin p i. Bu V [i] is of order Θ( 1 ) for a poin on and near feaure curves. By "near", we mean poins wihin O( ) disance away from feaure curves, where consan hidden in he big-o noaion depends on he specific singulariy we have. Hence our algorihm idenifies poenial feaure curve poins as hose whose V value is above a given hreshold τ. Examples. In Figure 1 and (c), we show he poenial feaure poins idenified by our algorihm. Noe ha as we lower he hreshold τ, poins wih local high mean-curvaure will also sar o appear as feaure poins. We remark ha as consisen wih he heoreical analysis in [BQWZ12], poins on inersecion-feaure curves have low L P magniude, bu poins around hem have high values and are capured as feaure poins. Hence around an inersecion-feaure curve, here are wo small bands of feaure poins (see Figure 1 (c)). We will see laer ha our feaure curve reconsrucion algorihm is able o close he gap beween hese wo bands. Implemenaion. Given a parameer, we firs compue he proximiy graph from he inpu where every poin p P is conneced o all neighbors wihin 5 disance o p. Le G = (P, E) denoe he resuling proximiy graph. We hen build he marix L as inroduced in Eqn (1) bu only resriced o edges in G: ha is, if here is no edge beween p i and p j, hen L [i][ j] = 0. This resuls ino a sparse marix L. In our curren implemenaion, we use a kd-ree srucure o help compue he proximiy graph G and hence L Coarse feaure curves reconsrucion Once we compue a se of poenial feaure poins Q from inpu poin clouds P, we aim o produce feaure curves (graphs) from hem. For he subsequen surface reconsrucion sep we do no need he feaure curves per se. However, we sill need o generae sample poins on he feaure curves and idenify he branching poins which necessiaes reconsrucing hem from Q. We achieve his in wo seps. Firs, we obain an iniial reconsrucion of he feaure graph Π. This coarse feaure graph capures he correc opology of he feaure graph ha we wish o reconsruc. However, i may no have saisfacory geomery (such as smooh feaure lines aligned wih boundaries or sharp feaures). To his end, we furher refine he feaure graph. The feaure poins we obain are around he feaure curves ha we wish o reconsruc. The work in [DW11] suggess ha he hidden graph srucure can be exraced from a cerain Rips complex from hese sampled poins. Hence we firs build a simplicial complex K from Q o connec discree poins, where K is Rips complex of Q using parameer ; ha is, wo feaure poins are conneced if heir disance is smaller han, and when he hree edges spanned by p,q,u Q are in K, we also add he riangle pqu o K. Since all he poenial feaure poins are roughly wihin a band of widh O( ) around he feaure curves, we choose as he radius parameer o compue he Rips complex. The resuling simplicial complex K "hugs" he feaure curves (including closing he gap around inersecion-feaure curves as seen in Figure 1 (c) and (d)). Nex, he algorihm of [GSBW11] compues he Reeb graph of some specific funcion defined on K o capure he skeleon of he he underlying space of K. The oupu is a graph where each branch is a polygonal curve wih verices being inpu feaure poins in Q. The branching nodes where muliple feaure curves mee are obained naurally as nodes in he Reeb graph. Previously, special handling is ofen required o deec and compue such nodes (see e.g, [IHOS07, SYM10]). Furhermore, we can easily simplify he resuling feaure graphs by he simplificaion of before simp. afer simp. he Reeb graphs o remove noise and less imporan loops or branches. See he above figure for an example where several small spurious loops in he Reeb graph are removed. c 2012 The Eurographics Associaion and Blackwell Publishing Ld.

6 4.3. Feaure curve refinemen The feaure graphs Π reconsruced above reflec he srucures of he hidden feaure graphs ha we wish o capure. However, he feaure curves are no smooh, and may no be aligned wih real feaures. In his sep, we wish o improve he qualiy of he geomery of feaure graphs. There are wo componens involved Smoohing of feaure graphs Firs, we wan o smooh each branch in he feaure graph, which is represened as a polygonal curve. A sandard curve smoohing algorihm, such as he Laplacian smoohing, ends o push he curve o he cener of he "band" of feaure poins involved. This causes shrinkage for boundary curves and sharp feaure curves as hey are smoohed; see Figure 2. To align wih real feaures while smoohing, we use an acive-conour based mehod o deform he feaure curve, similar o [PKG03]. inersecion-ype of singulariy, poins wih high V values are around he inersecion lines bu no on hem (recall Figure 1 (c) and Figure 8 (c) in Appendix A). However, due o symmery, i is sill beneficial o pu q k in he middle of hese band of high V values, which is on he inersecion curves. While we canno prove ha his energy funcion achieves minimum on he singulariies, we found ha i is effecive in pracice in handling all hree ypes of singulariies wihin a single simple framework. Since we have he explici form for E ex(q), we can compue is gradien direcly in closed form, which helps o reduce he compuaional cos Locaions of sharp feaure corners To use he acive conour approach described above for refining each branch in he feaure graph, we wish o fix heir endpoins, which correspond o he graph nodes. They can represen eiher a node where muliple feaure curve pieces mee (graph nodes of degree 3 or more) or a ip of feaure curves (degree-1 graph nodes). We also wan o preserve sharp corners wihin a single feaure curve: see Figure 3 and ; such degree-2 corners are no available in he Reeb graph and have o be idenified separaely. Figure 2: Laplacian smoohing causes feaure curves shrink inside he model. Our acive conour approach alleviaes his problem. In paricular, we design he following energy funcion: n E snake = [E in (q k ) + E ex(q k )], k=1 where q k is he k-h verex of curren piece of feaure curve we are smoohing. The firs erm E in (q k ) aims o ensure he smoohness of he feaure curve a q k, and is he same as he one inroduced in he original acive-conour work [KWT88] based on approximaed derivaives using finie differences. The second erm E ex aims o align curves wih real feaures, and is defined as n E ex(q k ) = e p i q k 2 σ 2 V (p i ) i=1 where σ is aken as 2 in experimens. Noe ha in pracice, we only consider poins wihin 2σ = 4 disance from q k o compue E ex(q k ). Inuiively, o minimize E ex(q k ), we wan o maximize 2 he sum of e p i q k σ 2 V (p i ) for poins p i s wih high V values. Hence we wan o relocae q k close o poins wih high V values o maximize his sum. For boundary and sharp feaure curves, poins wih high V value lie along hese singulariies, and his energy erm pushes q k owards hem. For (c) Figure 3: The sharp degree-2 corner is smoohed ou afer acive conour. Our algorihm firs idenifies degree-2 sharp corners and preserves hem during he acive conour. (c) Corner poins (red dos) compued by our algorihm: hey are firs idenified as nodes of degree 3 in he Reeb graph, and hen relocaed o align wih geomeric corners. To idenify degree-2 sharp corners, we simply compue he local maxima of he funcion V (he magniude of L P ) in he Rips complex K ha we consruced o compue he Reeb graph. I urns ou ha he V values are no effecive a idenifying very sharp corners, due o he small number of poins available around such corners. So we also compue he normalized version of he weighed graph Laplacian, L, and also ake he local maxima of he magniude of L P. This does no incur any exra ime complexiy: when our algorihm compues he sandard graph Laplacian L P, we also compue he normalized version and sore boh values a each poin. Corners where 3 or more feaure curves mee appear auomaically as nodes in he Reeb graph. For each Reeb graph node p i, le NN(p i ) denoe is neighbors wihin he Rips c 2012 The Eurographics Associaion and Blackwell Publishing Ld.

7 complex K. Our goal is o find a good locaion for p i o align wih he sharp geomeric corner (0-dimensional singulariies) of he hidden domain. Simply aking he cenroid of poins in NN(p i ) does no serve as a good choice as i ends o push p i off he domain and away from he sharp corners. So we again ake he poin wih larges V value in NN(p i ) as he new posiion for he corner node p i. See Figure 3 for an example of he degree-3 nodes ha our algorihm compues. 5. Feaure-Aware Singular Surface Reconsrucion 5.1. Weighed-Cocone reconsrucion The acive conour algorihm described in he previous secion deforms feaure curves. Le F denoe he sample poins on he deformed feaure curves, wih sharp nodes (of degree- 2 or more) marked. We firs ake a sparse subse of feaure poins in F so ha no wo of hem are wihin 0.8r disance o each oher, where r is aken as he radius of proecion balls around hese poins. Le F sill denoe he resuling sparsified feaure curve samples. The radii r of he balls are aken as o reflec he fac ha anomalies due o he presence of feaures are assumed o have an influence radius of roughly while idenifying hem. For he corner poins, we make an excepion ha heir proecing balls are made larger by aking heir radii 3. All poins from P ha are conained wihin he proecion balls are removed. Le he remaining se of poins be sill denoed as P. The poins in P do no have any ball, or equivalenly, a ball of radius zero. A sample poin p wih a ball of radius r p 0 cenering i is urned ino a weighed poin ˆp = (p,r p). The disance beween wo weighed poins ˆp, ˆq is measured as he weighed disance p q 2 r 2 p r 2 q. Noice ha any or boh of r p and r q could be zero. Wih his weighed disance, one can define he weighed Voronoi diagram whose dual is he Weighed Delaunay riangulaion of (P F). We apply he Cocone algorihm on his weighed Delaunay riangulaion. The original Cocone algorihm for a unweighed se of poins P works as follows. For each poin p P, he algorihm compues a pole vecor from he Voronoi cell of p which is known o approximae he normal vecor a p on he sampled surface. Then, a double cone wih apex a p and an angle of π/8 wih he pole vecor is considered. All Voronoi edges in he Voronoi cell of p inersecing he complemen of his double cone, also called he co-cone of p, are deermined and heir dual Delaunay riangles are chosen as he firs approximaion of he sampled surface. A subsequen manifold exracion sep furher prunes his iniial approximaion o compue he oupu surface mesh. We follow he same procedure o exrac he mesh approximaing he sampled singular surface wih he modificaion ha he enire compuaion is performed on he weighed Delaunay riangulaion ha accouns for he weighed poins on he feaure curves. The weighed poins do no paricipae in filering he riangles; ha is, he riangles dual o he Voronoi edges ha are inerseced by he co-cones of he un-weighed poins are seleced. The raionale behind his choice is ha normal esimaion a and near he feaure curves is generally unreliable. The balls around hese poins ac as a guard so ha un-weighed poins are sufficienly away from hese feaure curves. The riangles adjoining weighed poins in he final reconsrucion are chosen by adjacen un-weighed poins. The assumpion is ha, because of he dense sampling, each riangle adjoining a weighed poin in he final reconsrucion is chosen by an un-weighed poin. Our experimens show ha his assumpion is no unreasonable Reconsrucion from noisy daa Our feaure curve reconsrucion sep is reasonably robus agains noise. However, he inpu for weighed Cocone algorihm should be noise-free. When inpu daa conains noise, we perform he following o reconsruc our singular surfaces. Firs, we exrac feaure curves using he mehod described in Secion 4. Nex, we perform a small number of ieraions of he version of he mean-shif algorihm proposed in [WCPn10] o denoise he inpu daa. Noe ha similar o mos denoising algorihms, mean-shif algorihm ends o smooh ou sharp feaures: his is one reason why we exrac feaure curves before he denoising sep. In his way, even hough sharp feaures are smoohed afer mean-shif, hey are sill recorded as feaure curves, and hus are reconsruced by our weighed Cocone algorihm which operaes on smoohed daa poins. See Figure 4, where we obain a noisy sample of OcaFlower by perurbing each poin randomly wihin 1% of he diagonal of he bounding box. In (c) we show he reconsruced surface by our algorihm, and in (d) we zoom-in o show he deail of one par. In (e) we show he surface reconsruced by he original Cocone algorihm from he smoohed (denoised) daa se, where we can see ha sharp feaures are no well-reconsruced due o missing riangles and smoohing. However, by preserving he feaure curves (shown in ), which we compue before denoising, we can reconsruc hese sharp feaures. More examples on reconsrucion from noisy daa are shown in Figure Experimenal Resuls In his secion, we firs provide several examples o show ha our algorihm can reconsruc singular manifold while preserving various singulariies faihfully. See Figure 5. We also show more examples of reconsrucion from noisy daa in Figure 6, including he reconsrucion of he non-manifold SphereCube model. In boh cases, each poin is perurbed randomly wihin 1% of he size of he diagonal of he bounding box of he model. To exhibi he behavior of our algorihm wih respec o he sparsificaion of he inpu, we show in Figure 7 he reconsrucion of he block model from poin cloud of differen sizes. We sar wih an inpu of 74K poins, and down-sample c 2012 The Eurographics Associaion and Blackwell Publishing Ld.

8 Dey e al. / Feaure-Preserving Reconsrucion of Singular Surfaces (c) (d) (e) Figure 4: Noisy poins and feaure curves reconsruced from noisy poin cloud, boh overlayed on op of he clean surface model for visualizaion purpose. (c) and (d): Reconsruced surface by our algorihm. (e) Reconsrucion using he original Cocone algorihm from he denoised (smoohed) poins. (c) (d) Figure 5: Reconsrucion for various models. i o 47K, 21K and 8.5K poins. Noe ha small misakes (including dens along sharp feaure lines and small missing riangles near he feaure curves) sar o appear in he model of size 21K which worsens o more global misakes (large riangles connecing differen pars of he model) as he inpu sparsifies o 8.5K poins. We observed hough if he deeced feaure curves were sampled wih more densiy in 8.5K poin model, he reconsrucion would no incur global misakes. Parameers. The main parameer involved in our algorihm is, and we se as hree imes he average disance from a poin o is 5h neares neighbor. The defaul values of oher parameers depend on, excep for τ, he hreshold o decide poenial feaure poins, which we choose as hree imes he average V of all poins. We use as he parameer o build he Rips complex. We use as he radius of proecion ball for feaure poins in he singular manifold reconsrucion sep: we observe ha corner poins in he reconsrucion sep requires a bigger proecion ball radius, so ha no riangles are formed beween poins from differen feaure lines. Hence we choose 3 o be he proecion ball size for corner poins. For he noisy poins daa, we have o increase he size of so ha is larger han he noise level. Timing. We have implemened a proo-ype of our algorihm. The code is no ye opimized. Our curren implemenaion uses a kd-ree o speed up compuaion of proximiy graph. In Table 1 we show he iming for our experimens. Figure 6: Reconsrucion from noisy models for block model and SphereCube model. The experimens were carried ou on a mac-lapop wih Inel Core i7 2.2 Ghz, and 4 GB RAM. The iming is broken ino 5 sages: kd-ree consrucion; compuaion of graph Laplacian and feaure poins deecion; feaure curve consrucion; feaure curve refinemen; surface reconsrucion. Models SphCube (65K) Fandisk(114K) Block(47K) Flower(107K) Beele(63K) Sg Sg Sg Sg Sg Table 1: Timing (in seconds) for each sage of our algorihm. c 2012 The Auhor(s) c 2012 The Eurographics Associaion and Blackwell Publishing Ld.

9 (c) Figure 7: The block is reconsruced from a poin cloud of size 74K, 47K, (c) 21K and (d) 8.5K, respecively. In (d) we noe ha feaure curves (blue curves) are sill reasonably reconsruced, alhough no well sampled (orange dos). 7. Limiaions and Conclusion Our curren algorihm can idenify he approximae locaion of corner poins where muliple feaure curves mee. However, we find i difficul o align hese poins wih he real corner poin of he hidden domain when hey represen concave corners (convex corners usually pose no difficuly). We plan o invesigae furher o idenify he posiion of hese concave corners using only local informaion. We will also explore how our algorihm perform on real daa, such as range scans daa, which can conain missing daa and more general ypes of noise. Missing daa can be challenging as he boundary of he sampling holes may no be well sampled. I will be ineresing o invesigae how o make our algorihm more robus for real daa. Finally, we do no ye have a heoreical guaranee for our reconsrucion mehod. We believe ha i migh be possible o provide a opological guaranee for he oupu along he line of [CDR10], who applied weighed Delaunay refinemen for meshing wih heoreical guaranees. Acknowledgemen The auhors hank he anonymous reviewers for heir helpful commens. Mos of he models used in his paper are couresy of AIM@SHAPE Shape Reposiory. The auhors acknowledge he suppor of NSF under grans CCF , CCF and CCF (d) References [AA06] ADAMSON A., ALEXA M.: Anisoropic poin se surfaces. In AfriGraph 06 (2006), pp [AB99] AMENTA N., BERN M.: Surface reconsrucion by Voronoi filering. Discree & Compuaional Geomery 22, 4 (1999), [ABCO 01] ALEXA M., BEHR J., COHEN-OR D., FLEISHMAN S., LEVIN D., SILVA C. T.: Poin se surfaces. In IEEE Visualizaion 01 (2001), pp [ACDL02] AMENTA N., CHOI S., DEY T. K., LEEKHA N.: A simple algorihm for homeomorphic surface reconsrucion. Inern. J. Compu. Geom. Appl. 12, 1 2 (2002), [ACSTD07] ALLIEZ P., COHEN-STEINER D., TONG Y., DES- BRUN M.: Voronoi-based variaional reconsrucion of unoriened poin ses. In Sympos. Geomery Processing (2007), pp [ASGCO10] AVRON H., SHARF A., GREIF C., COHEN-OR D.: L 1 -Sparse reconsrucion of sharp poin se surfaces. ACM Trans. Graph. 29, 5 (2010), 135:1 135:12. 2 [BC02] BOISSONNAT J.-D., CAZALS F.: Smooh surface reconsrucion via naural neighbor inerpolaion of disance funcions. Compu. Geom.: Theory Appl. (2002), [BN03] BELKIN M., NIYOGI P.: Laplacian Eigenmaps for dimensionaliy reducion and daa represenaion. Neural Comp 15, 6 (2003), [BO05] BOISSONNAT J., OUDOT S.: Provably good sampling and meshing of surfaces. Graphical Models 67, 5 (2005), [BQWZ12] BELKIN M., QUE Q., WANG Y., ZHOU X.: Toward undersanding complex daa: graph Laplacians on manifolds wih singulariies and boundaries. In Conference on Learning Theory (COLT), o appear (2012). 1, 4, 5, 10 [CCS12] CAZALS F., COHEN-STEINER D.: Reconsrucing 3d compac ses. Compu. Geom.: Theory and Appl. 45 (2012), [CDR10] CHENG S.-W., DEY T. K., RAMOS E. A.: Delaunay refinemen for piecewise smooh complexes. Discree & Compuaional Geomery 43, 1 (2010), , 3, 9 [CG06] CAZALS F., GIESEN J.: Delaunay riangulaion based surface reconsrucion. In Effecive Compuaional Geomery for Curves and Surfaces, J. Boissanna and M. Teillaud (eds.) (2006), Spirnger Verlag, Mah. and Visualizaion, pp [Dey06] DEY T. K.: Curve and surface reconsrucion: Algorihms wih mahemaical analysis. Cambridge Universiy Press, New York, , 2 [DMSB99] DESBRUN M., MEYER M., SCHRÖDER P., BARR A. H.: Implici fairing of irregular meshes using diffusion and curvaure flow. Compuer Graphics 33, Annual Conference Series (1999), [DTS01] DINH H. Q., TURK G., SLABAUGH G.: Reconsrucing surfaces using anisoropic basis funcions. In Inern. Conf. Compu. Vision (ICCV) (2001), pp [DW11] DEY T. K., WANG Y.: Reeb graphs: Approximaion and persisence. In Proc. 27h Sympos. Compu. Geom. (2011), pp , 5 [FCOS05] FLEISHMAN S., COHEN-OR D., SILVA C. T.: Robus moving leas-squares fiing wih sharp feaures. ACM Trans. Graph. 24 (July 2005), [GP07] GROSS M., PFISTER H.: Poin based graphics. Morgan Kauffman, Massachuses, c 2012 The Eurographics Associaion and Blackwell Publishing Ld.

10 [Gri06] GRIGOR YAN A.: Hea kernels on weighed manifolds and applicaions. Con. Mah. 398 (2006), [GSBW11] GE X., SAFA I., BELKIN M., WANG Y.: Daa skeleonizaion via reeb graphs. In Proc. 25h Annu. Conf. Neural Informaion Processing Sysems (NIPS) (2011). 1, 5 [IHOS07] II J. D., HA L. K., OCHOTTA T., SILVA C. T.: Robus smooh feaure exracion from poin clouds. In Shape Modeling Inernaional (2007), pp , 5 [JWS08] JENKE P., WAND M., STRAβER W.: Pach-graph reconsrucion for piecewise smooh surfaces. In Proceedings Vision, Modeling and Visualizaion (VMV) (2008). 3 [KBH06] KAZHDAN M., BOLITHO M., HOPPE H.: Poisson surface reconsrucion. In Sympos. Geom. Processing (2006), pp [KWT88] KASS M., WITKIN A., TERZOPOULOS D.: Snakes: Acive conour models. Inernaional Journal of Compuer Vision 1 (1988), [LCOL07] LIPMAN Y., COHEN-OR D., LEVIN D.: Daadependen mls for faihful surface approximaion. In Sympos. Geom. Processing (2007), pp [MOG09] MÉRIGOT Q., OVSJANIKOV M., GUIBAS L.: Robus voronoi-based curvaure and feaure esimaion. In 2009 SIAM/ACM Join Conference on Geomeric and Physical Modeling (2009), SPM 09, pp [MW11] MARTIN S., WATSON J.-P.: Non-manifold surface reconsrucion from high-dimensional poin cloud daa. Compu. Geom. Theory Appl. 44, 8 (Oc. 2011), [ÖGG09] ÖZTIRELI A. C., GUENNEBAUD G., GROSS M. H.: Feaure preserving poin se surfaces based on non-linear kernel regression. Compu. Graph. Forum 28, 2 (2009), [PKG03] PAULY M., KEISER R., GROSS M. H.: Muli-scale feaure exracion on poin-sampled surfaces. Compu. Graph. Forum 22, 3 (2003), , 6 [RJT 05] REUTER P., JOYOT P., TRUNZLER J., BOUBEKEUR T., SCHLICK C.: Surface reconsrucion wih enriched reproducing kernel paricle approximaion. In Sympos. Poin-Based Graphics (2005), pp [SYM10] SALMAN N., YVINEC M., MÉRIGOT Q.: Feaure preserving mesh generaion from 3d poin clouds. Compu. Graph. Forum 29, 5 (2010), , 5 [WCPn10] WANG W., CARREIRA-PERPINÁN M. A.: Manifold blurring mean shif algorihms for manifold denoising. In Compuer Vision and Paern Recogniion (2010), pp Appendix A: Graph Laplacian around Oher Singulariies We now give a brief descripion of he behavior of he funcional Laplacian on a singular surface near he hree ypes of singulariies: he boundary-feaure curves, he sharp-feaure curves and he inersecion-feaure curves. These are resuls of [BQWZ12], and we include a brief synopsis here for compleeness and o help o provide inuiion for our approaches. Le x be a poin near he singulariy, and le x 0 be is neares neighbor in he feaure curves (of one of hree ypes). In general, we have ha L f (x) = 1 D(x,x 0,) + o( 1 ), (4) where D(x,x 0,) is a quaniy dominaed by a Gaussian (or a O( 1 ) 0 O( 1 ) O( ) (c) Figure 8: The plos for he value of he erm 1 D(x,x 0,) in Eqn (4) (y-axis in he graphs) for poins on and around boundary, inersecion and (c) sharp corner singulariy. In all hree graphs, 0 is where he singulariy locaes, x-axis indicaes he disance beween a poin o he singulariy. combinaion of Gaussians) whose variance is c. In paricular, seing r = x x0, we have: (1) boundary-feaure curves: See Figure 8. 0 O( ) L f (x) = C 1 e r2 + o( 1 ). (5) (2) inersecion-feaure curves: See Figure 8. L f (x) = C 2 r e r2 sin 2 θ + o(( 1 ), (6) where θ is he angle beween he angen spaces of he wo inersecing surface paches a x 0. (3) sharp-feaure curves: See Figure 8 (c). L f (x) = C 3 e r2 + C 4 r e r2 sin 2 θ + o( 1 ) (7) Inuiively, a poin around a sharp feaure will boh see he boundary effec and parial effec from he oher surface pach. In each case, C 1, C 2, C 3 and C 4 are proporional o a cerain firs order derivaive of f a x 0, and is independen of. The illusraions in Figure 8 assume ha (he absolue values of) hese erms are bounded from below. For cerain funcions, heir parial derivaives may vanish and hese erms C i s may be zero. However, for he specific funcions (he coordinae funcions) ha we will use in our algorihm, hese C i s are always bounded by a consan from below. I is worh poining ou ha for a poin x on he inersecion-feaure curves (i.e, x = x 0 and r = 0), he O( 1 ) erm vanishes and Eqn (6) evenually leads o ha L f (x) = S1 f (x) + S2 f (x) + o(1), which is simply he addiion of he wo manifold-laplacian S1 f (x) and S2 f (x) for he wo inersecing surface paches a his poin x. See also Figure 8 where a he origin 0 (i.e, on he inersecion singulariy), he firs erms becomes zero. This is differen from he oher wo ypes of singulariies, where he value of L f (x) is of order Θ( 1 ) a singulariies; recall Figure 8 and (c). The poins wih high L f values are concenraed around wo bands which are abou O( ) disance around he inersecion singulariy (i.e, he origin 0 in he picure). O( 1 ) 0 O( ) c 2012 The Eurographics Associaion and Blackwell Publishing Ld.