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Discrete Laplace on Meshed Surfaces

[Extended Abstract]

Mikhail Belkin Jian Sun Yusu Wang Dept. of Comp. Sci. & Eng. Dept. of Comp. Sci. Dept. of Comp. Sci. & Eng. The Ohio State University Stanford University The Ohio State University Columbus OH 43210 Palo Alto CA 94305 Columbus OH 43210 [email protected] [email protected] [email protected] state.edu

ABSTRACT by a mesh, which is given by the coordinates of its vertices and In recent years a considerable amount of work in graphics and ge- the connectivity information. Thus, manipulating, deforming and ometric optimization used tools based on the Laplace-Beltrami op- analyzing meshed surfaces is a crucial area within these subjects, erator on a surface. The applications of the Laplacian include mesh with the usual implicit assumption that processing of a mesh corre- editing, surface smoothing, and shape interpolations among others. sponds to analogous processing of the underlying surface. However, it has been shown [13, 24, 26] that the popular cotan- In recent years a class of methods based on discrete differen- gent approximation schemes do not provide convergent point-wise tial geometry of surfaces [1] and the discrete has been used for various tasks of geometric processing. For exam- (or even 2) estimates, while many applications rely on point-wise estimation.L Existence of such schemes has been an open ques- ple, the state-of-the-art report on Laplacian Mesh Processing [21] tion [13]. discusses surface reconstruction, mesh editing, shape representa- In this paper we propose the first algorithm for approximating tion and shape interpolation among other applications of Laplacian- the Laplace operator of a surface from a mesh with point-wise con- based mesh processing methods. Such applications of the Lapla- vergence guarantees applicable to arbitrary meshed surfaces. We cian can be found in [3, 10, 12, 22, 23, 27] among others. show that for a sufficiently fine mesh over an arbitrary surface, our The Laplace-Beltrami operator ( Laplacian) is a funda- mesh Laplacian is close to the Laplace-Beltrami operator on the mental geometric object associated to a and surface at every point of the surface. has many desirable properties. The Laplacian can be used as a Moreover, the proposed algorithm is simple and easily imple- penalty to choose functions varying smoothly along the mentable. Experimental evidence shows that our algorithm exhibits manifold [23] or to smooth the surface itself via the mean convergence empirically and compares favorably with cotangent- flow [10], which is determined by applying the Laplace operator to based methods in providing accurate approximation of the Laplace coordinate functions x, y, z considered as functions on the surface. operator for various meshes. Moreover, of the Laplacian form a natural basis for square integrable functions on the manifold analogous to Fourier harmonics for functions on a circle (i.e. periodic functions). There- Categories and Subject Descriptors fore, computing eigenfunctions of a Laplacian allows one to con- G.2 [ of Computing]: Discrete mathematics struct a basis reflecting the geometry of the surface [12]. Finally, the Laplace operator is intimately related to and the on the surface, and is connected to a large body of classi- General Terms cal mathematics, relating geometry of a manifold to the properties Algorithms, Theory, Experimentation of the heat flow (see, e.g., [20]). Several discretizations of the Laplacian for an arbitrary mesh Keywords have been proposed [9, 10, 14, 19, 23, 25]. Most of the pro- posed methods are variants of the cotangent scheme [19], which Laplace-Beltrami operator, Surface mesh, Approximation algorithm is a form of the finite element method, applied to the Laplace oper- ator on a surface. Suppose we have a surface with a fine mesh. We 1. INTRODUCTION expect that the discretization computed from such a mesh would be an accurate representation of the underlying surface Laplacian. A broad range of topics in geometric modeling and computer However, a detailed theoretical analysis of existing discretizations graphics is concerned with processing two-dimensional surfaces in in [25, 26] shows that while point-wise convergence can be estab- a three-dimensional space. These surfaces are typically represented lished for special classes of meshes, such as certain meshes with valence 6, or for linear functions over a sphere in R3, none of these methods can be expected to converge for surface meshes in gen- Permission to make digital or hard copies of all or part of this work for eral, a finding, which is borne out by experimental results in Sec- personal or classroom use is granted without fee provided that copies are tion 5 and [26]. A notable recent work is [13, 24], where the au- not made or distributed for profit or commercial advantage and that copies thors analyze convergence of various geometric invariants, includ- bear this notice and the full citation on the first page. To copy otherwise, to ing Laplace-Beltrami operators. They apply their analysis to the republish, to post on servers or to redistribute to lists, requires prior specific popular cotangent scheme showing weak convergence (in the dis- permission and/or a fee. tribution sense) for the solution of the Dirichlet’s problem, assum- SCG’08, June 9–11, 2008, College Park, Maryland, USA. Copyright 2008 ACM 978-1-60558-071-5/08/04 ...$5.00. ing the aspect ratio of mesh elements is bounded. The authors also demonstrate that under the cotangent scheme, convergence in 2 taken to be independent of the point w. However, in general, h can (which is weaker than point-wise convergence) does not generallyL be taken to be a of w, which will allow the algorithm to hold. adapt to the local mesh size. The results presented in our paper provide the first discretization The theoretical results in this paper show that when K is a suffi- h scheme that guarantees convergence at each point (i.e, ∞ and thus ciently fine mesh of a smooth underlying surface S, L is close to L K also 2 convergence). Moreover, by using some recent results [2], the surface Laplacian ∆S (the formal definition will be introduced it canL be shown that convergence of eigenfunctions also follows. shortly). Indeed, our preliminary experimental results demonstrate We note that while ∞ convergence is required in many applica- the converging behavior of this operator, and show that our algo- tions, where various quantities,L e.g., mean curvature, need to be es- rithm outperforms currently available discrete Laplace operators in timated at each node of the mesh, the construction of such a scheme the approximation quality. is not trivial. For example, in [13] (Section 4.3.2) the authors con- In the remainder of this section, after introducing necessary no- jecture that no such discretizations may exist. tations, we give an outline of the theoretical results, which also We summarize the contributions of the paper as follows: explains the derivation of this algorithm. 1. We propose a simple method for approximating over 2.2 Objects and Notations a surface using a mesh and analyze the quality of the result- ing approximation in terms of the parameters of the surface and the mesh. Surface Laplace operator ∆S . In this paper, we consider a smooth compact 2-manifold S without boundary isometrically embedded 2. Combining the approximation results with the idea in some IR3 with geometry induced by the embed- of approximating the heat flow on a mesh, we present and an- ding. (Note that any such surface is necessarily orientable.) The alyze the first algorithm for approximating Laplace-Beltrami corresponding volume form, denoted by ν, determines the area of a operator on a surface with point-wise convergence guaran- surface element. We assume that S is connected — surfaces with tees for arbitrary meshes. multiple components can be handled by applying our results in a 3. The resulting algorithm is simple and straightforward to im- component-wise manner. plement. We provide experimental results showing that it Given a twice continuously differentiable function f C2(S), ∈ outperforms the cotangent scheme in approximation quality let S f denote the vector field of f on S. The Laplace- ∇ and robustness to noise. Beltrami operator ∆S of f is defined as the of the gra- dient; that is, ∆S f = div( S f). For example, if S is a domain in We note that the algorithm for computing the surface Laplacian 2 ∇ ∂2f ∂2f IR , then the Laplacian has the familiar form ∆ 2 f = + . is related to the set of algorithms for computing Laplacian of point IR ∂x2 ∂y2 clouds in data analysis and machine learning [7, 8] by using the h heat equation. We observe that in machine learning, the samples Laplacian FS . To connect the mesh Laplace operator h are usually believed to be drawn independently from a probabil- LK , as defined in Eqn (1), with the surface Laplacian ∆S , we need h ity distribution and the convergence occurs in probability. On the an intermediate object, called the functional Laplace operator FS . other hand, in surface modeling, the nodes of a mesh are typically Given a point w S and a function f : S IR, it is defined as ∈ → not sampled independently from a probability distribution but are follows: x w 2 generated by some deterministic process, e.g. scanning. Hence in h 1 − k − k 4h (2) the case of mesh Laplacian, approximation guarantees need to be FS f(w) = 2 e (f(x) f(w)) dν(x). 4πh Zx S − made for all sufficiently fine meshes, and probabilistic techniques ∈ based on the law of large numbers (usually used in the case of ran- domly sampled point clouds) cannot be applied. Definition of (ε, η)-approximation. We also need a quantitative measure of how well a mesh approximates the underlying surface. 3 2. ALGORITHM AND OVERVIEW OF THE Let the medial axis M of S be the closure of the set of points in IR that have at least two closest points in S. For any w S, the local RESULTS feature size at w, denoted by lfs(w), is the distance∈ from w to the medial axis M. The reach (also known as the condition number) 2.1 Mesh Laplacian Algorithm ρ(S) of S is the infinum of the local feature size at any point in S. We start by describing our algorithm for computing the Laplace At each point p S, np denotes the unit outward of S at p 3 ∈ operator on a meshed surface. Let K be a mesh in IR . We denote and for each face t K, nt is the unit outward normal of the plane the set of vertices of the mesh K by V . Given a face, a mesh, or a passing through t. ∈ surface X, let Area(X) denote the area of X. For a face t K, In the paper, we assume that the vertices of the mesh K lie on the number of vertices in t is denoted by #t, and V (t) is the∈ set of the surface S. Let ρ be the reach of S. We say that K is an (ε, η)- vertices of t. approximation of S, if the following conditions hold: Our algorithm takes a function f : V IR as input and pro- h → h 1. For a face t K, its diameter (maximum distance between duces another function LK f : V IR as output. LK , the mesh ∈ Laplace operator, is computed, for→ any w V , as follows: any two points on t) is at most ερ. ∈ 2. For a face and a vertex , the angle between p w 2 t K p t h 1 Area(t) − k − k ∈ ∈ L f(w) = e 4h (f(p) f(w)) vectors nt and np, ∠(nt,np), is at most η. K 4πh2 #t − tX∈K p∈XV (t) (1) Intuitively, the first condition ensures that the mesh is sufficiently fine. On the other hand, a very fine mesh can still provide a poor The parameter h is a positive quantity, which intuitively corre- approximation to the underlying surface, as, for example, is seen in sponds to the size of the neighborhood considered at each point. In the Schwarz lantern. Thus the second condition is also necessary many applications and for the theoretical analysis in this paper h is to ensure the closeness between K and S. In many cases, we work with triangular meshes. Note the ap- Theorem 2.4 (See [8]) For a function f C2(S), we have that h ∈ proximation conditions do not require that the triangles in K are limh→0 F f(w) ∆S f(w) = 0. S − ∞ well-shaped, for instance having small aspect ratio. However the ‚ ‚ ‚ ‚ second condition can be implied if we assume that there exists at It is easy to see that Theorem 2.1 follows from the two theorems least one angle whose sin is bounded from below for each trian- above. Theorem 2.4 was shown in [8], and provides an approxi- gle [16], which is a much weaker condition than small aspect ratio. mation to the Laplace operator based on heat diffusion. To make Note that many surface reconstruction algorithms (see [11]) pro- it more transparent we give a brief outline of the ideas involved duce meshes that satisfy these conditions. below in Section 2.4. In the remaining of the paper, we focus on Our technical results use the analysis of the map φ : K S the proof of Theorem 2.3. The proof relies on the following result → defined as follows: For any p K, φ(p) is defined as the closest on approximating integrals over a surface, which is of independent ∈ point to p on the surface S. φ is well-defined when K avoids the interest. Let dS (x, y) denote the geodesic distance between two medial axis of S, which will be the case in our setting. The map points x, y S. φ connects the mesh K with the surface S, and is widely used in ∈ analyzing surface reconstruction algorithms [4, 11], as well as in approximating various quantities for smooth surfaces [13, 16]. Theorem 2.5 Given a Lipschitz function g : S IR, let L = Lip(g) be the Lipschitz constant of function g, i.e., →g(x) g(y) | − | ≤ 2.3 Overview of the Main Results Lip(g)dS (x, y). Set g ∞ = sup g(x) . We can approximate k k x∈S | | Our main result is Theorem 2.1. Intuitively, as the mesh approx- Area(t) S gdν by the discrete sum IKg = t∈K #t v∈V (t) g(v), imating the surface S becomes denser, the mesh Laplace operator soR that the following inequality holds:P P on K converges to the Laplace-Beltrami operator of S. 2 gdν IKg 3 ρLε + g ∞(2ε + η) Area(S). (5) Theorem 2.1 Let the mesh Kε,η be an (ε, η)-approximation of S. ˛ZS − ˛ ≤ k k 1 1 ˛ ˛ ` ´ Put 2.5+α 1+α for an arbitrary fixed positive num- ˛ ˛ h(ε, η) = ε + η Moreover,˛ suppose˛ that g is twice differentiable, with the norm ber α > 0. Then for any f C2(S) ∈ of the Hessian of g bounded by H. Then, for some constant C, h(ε,η) depending only on S, the following inequality holds: lim sup LK f ∆S f = 0 (3) ε,η→0 ε,η Kε,η ‚ − ‚∞ ‚ ‚ 2 2 ‚ ‚ gdν IKg CHε + 3 g ∞(2ε + η) Area(S) (6) where the supremum is taken over all (ε, η) approximations of S. ˛Z − ˛ ≤ k k ˛ S ˛ ` ´ ˛ ˛ It is important to note that for a triangular mesh, we can obtain ˛ ˛ Note that the second part of the above theorem provides a higher- the following corollary which says that the convergence result still order approximation, while the first part only requires the function holds if we replace the second condition in (ε, η)-approximation to be Lipschitz. by requiring the triangles to be well-conditioned, which is an easily g Finally, we remark that by using some very recent results on the verifiable condition. convergence of Laplace spectrum [2], it can be shown that the dis- crete eigenfunctions under this framework also converge to eigen- Corollary 2.2 Let K be a triangular mesh with all the vertices on ε functions of the Laplace-Beltrami operator of S. S and the diameter of each triangle less than ερ. In addition, let s(t) = maxp∈V (t) sin(θp) where θp is the angle of the triangle t 2.4 Functional approximation of the Laplace- at vertex p, and assume that s(K) = mint∈K s(t) is bound from 1 Beltrami operator below by some constant. Put h(ε) = ε 2.5+α for an arbitrary fixed positive number α > 0. Then for any f C2(S) In this section we attempt to demystify Theorem 2.4 by explain- ∈ ing the underlying idea for the functional approximation of the h(ε) Laplacian. The core of the approximation theorem lies in the con- lim sup LK f ∆S f = 0 (4) ε→0 ε Kε ‚ − ‚∞ nection of the Laplace-Beltrami operator to the heat equation. ‚ ‚ where the supremum is taken‚ over all such‚ triangular meshes. The heat equation on the surface S is the partial Corollary 2.2 follows immediately from Theorem 2.1 and Corol- 4 ∂u lary 1 in [16] which says sin ∠(nt,np) < ( + 2)ε for any ∆S u(x, t) = (x, t) s(K) ∂t triangle t K and any p V (t). The proof of Theorem 2.1 it- self relies on∈ the following two∈ results, connecting the mesh Lapla- The heat equation describes the diffusion of the initial heat dis- cian operator to the functional Laplace operator and the functional tribution u(x, 0) at time t. To obtain an integral approximation Laplacian to the Laplace-Beltrami operator, respectively. for the Laplace operator, we use a functional approximation tech- nique from [7]. The approximation is based on the properties of Theorem 2.3 Let K be an (ε, η)-approximation of the surface S, heat propagation with the initial distribution u(x, 0) = f(x). It 1 is well known (e.g., [20]) that the solution to the heat equation and ε, η < 0.1. Given a function f C (S), let f ∞ = ∈ k k u(x, t) can be written as u(x, t) = Ht (x, y)f(y) dν(y), where sup f(x) , f ∞ = sup f(x) and ρ denote the S S x∈S | | k∇ k x∈S k∇ k t reach of surface S. We have that for any point w S, HS (x, y) is the of theR surface S, i.e. the measure of ∈ how much heat propagates from x to y in time t. The quantity t S HS (x, y)f(y) dν(y) can be thought of as the sum of all heat h h Area(S) FS f(w) LK (w) comingR to point x from every other point y after time t. ˛ − ˛ ≤ πh2 ˛ ˛ Thus the heat equation can be rewritten as follows: ˛ 2ρ ˛ 2 (1 + )ε + 6εη + 2η f ∞ + ρε f ∞ . „ √h k k k∇ k « ∂ t ` ´ ∆S u(x, t) = HS (x, y)f(y) dν(y). ∂t ZS Taking the as t 0 and recalling that u(x, 0) = f(x) and Furthermore, let u v denote the dot-product between vectors u t → · that HS (x, y) dν(y) = 1, yields and v. Then we have that R ∠ 2 (tx,tp) t tx tp dlx = cos ∠(tx,tp) dlx = (1 2sin )dlx ∆S f(x) = lim HS (x, y)f(y) dν(y) = Z · Z Z − 2 t→0 ZS γ γ γ 2 3 lx l 1 2 dlx = l 2 1 t ≥ Zγ „ − 2ρ « − 6ρ lim HS (x, y)f(y) dν(y) f(x) = t→0 t „ZS − « On the other hand, observe that tx tp dlx measures the length γ · of the (signed) projection of γ alongR the direction tp. That is, 1 t lim HS (x, y) (f(y) f(x)) dν(y) t→0 t ZS − tx tp dlx = (q p) tp. For the case when is N the heat kernel can be written ex- Zγ · − · S IR 2 t 1 kx−yk plicitly HIR2 (x, y) = exp 4t , yielding the func- Hence we have that √(4πt) “− ” tional approximation in Eqn (2). l3 d = p q (q p) tp l For a general manifold, the heat kernel cannot usually be written k − k ≥ − · ≥ − 6ρ2 explicitly. However the heat kernel can be shown to be close to a l3 4d3 Gaussian in the geodesic coordinates (e.g., [20]). By considering l d + d + . the asymptotes of the heat kernel as t 0, the result in Theo- ⇒ ≤ 6ρ2 ≤ 3ρ2 rem 2.4 can still be established (see [8]).→ The last inequality follows from the fact that l 2d. This proves ≤ 3. PROPERTIES OF MESHED SURFACES the lemma. In this section, we present several results on the relations be- tween a smooth surface S and an (ε, η)-approximation K of S. Lemma 3.3 If a mesh K (ε, η)-approximates S with ε, η < 0.1 For simplicity of the exposition, we assume from now on that K then the following conditions hold: is a triangular mesh. However the same analysis works verbatim (i) For any point k K, d(k,S) (ε2 + εη)ρ. for arbitrary -approximation meshes. Let be the reach of , ∈ ≤ (ε, η) ρ S 2 and φ : K S is the closest-point map introduced in Section 2.2. (ii) For any point x S, d(x, K) (ε + εη)ρ. → ∈ ≤ (iii) φ : K S is a homeomorphism. 3.1 Closeness Between S and K → Given any point p IR3, let d(p, X) denote the smallest dis- ∈ 3 ′ n tance from p to any point in the set X IR , and np the unit Proof: To show (i), consider the B p ⊂ normal of S at p. The following result from [6] bounds the surface figure on the right. Suppose the point p k T x normal variation between nearby points. k is contained in the triangle t K p ∈ y and p is any vertex of t. Let Tp de- Lemma 3.1 ([6]) Given two points p, q S with p q ρ/3, note the plane at p. Assume B o ∈ k − kk ≤p−qk that B and B′ are the two balls of the angle between np and nq satisfies that ∠(np,nq) < . ρ−kp−qk radius ρ tangentially touching S at p on each side of Tp, and the centers We will later need the following result bounding the geodesic of B and B′ are o and o′, respectively. distance dS (p, q) between two points p, q S, in terms of the Recall that nt is the normal of the triangle t and np is the unit ∈ ∠ Euclidean distance p q . Note that the difference dS (p, q) surface normal of S at p. Since (np,nt) < η, the angle between k − k − p q is of order 3 in p q , which significantly improves the Tp and the plane passing through t is smaller than η, implying that k − k k − k quadratic bound from [17]. the angle between Tp and the line passing through pk is at most η. Let x denote the projection of k on Tp. We have that Lemma 3.2 Given two points p, q S, let d = p q < ρ/2. x p k p ερ, and ∈ 4d3 k − k k − k ≤ k − k ≤ Then we have that d dS (p, q) d + . ≤ ≤ 3ρ2 k x p k sin η εηρ. Proof: Let γ be the shortest geodesic curve between p and q and k − k ≤ k − k ≤ ′ set l = dS (p, q). By Proposition 6.3 in [17] we have that l 2d. Furthermore, observe that one of the segments xo and xo must ≤ For any x γ, let tx = tx(γ) be the unit tangent vector of γ at x, intersect S. Assume without loss of generality that it is xo. Since B ∈ does not contain any point from S, we have that d(x, S) x y and γ[p, x] the portion of γ from p to x. Set lx = dS (p, x). ≤ k − k Since γ is a geodesic, the unit normal of γ at a point x γ S where y is the intersection between xo and B. This implies that is the same as the unit surface normal n at x, and the curvature∈ ⊆ x d(x, S) ρ2 + x p 2 ρ ε2ρ. of γ is upper bounded by 1/ρ. It then follows from the Frenet ≤ p k − k − ≤ Formulas that dtx/dlx nx/ρ = 1/ρ. Hence we have that Since d(k,S) k x + d(x, S) by the triangle inequality, Part k k ≤ k k (i) then follows.≤ k − k 1 lx tx tp = dty dly We now show claim (iii), and claim (ii) follows easily from (i) k − k k Zγ[p,x] k ≤ Zγ[p,x] ρ ≤ ρ and (iii). Note that by (i), K does not intersect the medial axis of ∠(tp, tx) lx S. Hence the map φ is well-defined and actually continuous. Since sin . ⇒ 2 ≤ 2ρ K is compact, to show (iii), it suffices to show that φ is bijective. nx First, assume that φ is not injective, and x Lemma 3.5 Let K be a triangular mesh that (ε, η)-approximates ′ S it maps two points k, k K to the same with . For any point in the interior of an arbitrary ∈ ′ S ε, η < 0.1 x point, say x S. In this case, both k and k t′ k′ 2 ∈ triangle t K, we have that J(x) 1 2(2ε+η) . In particular, are on the line l passing through x in direc- this implies∈ that the areas of |S and−K|satisfies: ≤ tion nx. Without loss of generality, assume t k ′ that and are two consecutive intersec- l Area(S) 2 k k 1 2(2ε + η) . tion points between l and K. Let t and t′ be the triangles that ˛ Area(K) − ˛ ≤ ˛ ˛ contain k and k′, respectively. See the right figure for an example. ˛ ˛ ˛ ˛ By (i), we have that both k x and k′ x are upper-bounded Proof: Let P be the plane where the triangle t lies, and Tφ(x) by (ε2 + εη)ρ. k − k k − k be the tangent plane of S at φ(x). Set α denote the angle between Consider any vertex p of t. We have that x p x k + the planes P and Tφ(x). Note that α < 2ε + η by Eqn (7). Easy to k p (ε + ε2 + εη)ρ, by trianglek inequality.− k ≤ Sincek − bothk see that there exists an orthogonal (u, v) on P k − k ≤ such that ∠ and ∠ . Let and x and p are in S, we have that ∠(nx,np) 2ε by Lemma 3.1. It (u, Tφ(x)) = α (v, Tφ(x)) = 0 φu(x) then follows that ≤ φv(x) be the partial of φ at a point x = (u0, v0) in the interior of t, respectively. The Jacobian J(x) = J(u0, v0) of φ at 2 ∠(nx,nt) and ∠(nx,nt′ ) 2ε + η . (7) x then equals √EG F , where E = φu φu, G = φv φv and ≤ − · · F = φu φv. The above equation implies that t = t′, as otherwise, l lies in · n n 6 ∠ To bound φu, consider the point y = (u + x y the plane passing through t and as such (nx,nt) = π/2, which 0 b ′ δ, v ), where δ is a real number. Note that φ(y) is not possible. Since k and k are two consecutive intersection 0 φ(x) z by the choice of the direction u, the angle points between L and K, nx must point to the inside of K at l ∠ θ y one of these two points, say . If lies in the interior of , then ((y x), Tφ(x)) between y x and the k k t − − ∠(nx,nt) π/2. Otherwise, k is on the boundary of t, in which plane Tφ(x) is α. Let z be the projection ≥ y case ∠(nx,nt) π/2 2η because it follows from the definition point of y onto Tφ(x) and b be the projection x ≥ − ∠ of (ε, η)-approximation that the neighboring triangle of t forms an of φ(y) onto the line yz. See the right figure. Set θ = (ˆnx, nˆy); δ 2 angle of at most with . Either case contradicts Eqn (7) since note that θ by Lemma 3.4. Since φ(y) y (ε + 2η t ≤ (1−ε)ρ k − k ≤ ε, η < 0.1 by hypothesis. Hence we cannot have two such points k εη)ρ by Lemma 3.3, we have that and k′ and the map φ must be injective. ε2 + εη Finally, note that the image of K under the map φ, φ(K) S, φ(y) b = φ(y) y cos θ < δ ⊆ k − k k − k 1 ε | | is a compact surface as well. Since S is connected, by Theorem − (7.6) [18], φ(K) = S. It then follows that φ : K S is a ′ homeomorphism. → Now let B and B be the two balls of ra- B b dius ρ that tangentially touch S at φ(x). By φ(x) φ(y) the definition of the reach ρ, the surface S z 3.2 Approximating Integrals on the Surface S x y Lemma 3.3 implies that the mesh K is close to the underlying has to lie outside these two balls. B′ First we show z b is of order δ2. Let surface S both geometrically and topologically. Intuitively, quan- ′ k − k tities defined on S are closely related to their analogs defined on z be the projection of φ(y) onto Tφ(x); z z′ = φ(y) b is of order δ. Hencek by− triangle inequality, K (i.e, the pre-images under φ). Indeed, consider an arbitrary but k k − k fixed triangle t K. Note that although the map φ is not differen- φ(x) z′ φ(x) z + z z′ tiable on the entire∈ domain K, its restriction to the interior of t is k − k ≤ k − k k − k differentiable. We now present a result (Lemma 3.5) which states is of order δ. Furthermore, since φ(y) has to lie in between B and ′ ′ 2 that J(x), the Jacobian of the map φ at an interior point x t, is B , we can show that φ(y) z , thus z b , is of order δ . ∈ k − k k − k close to 1. First, for any point x t, let nˆx be the surface nor- Hence we have that: ∈ mal at φ(x); that is, nˆx = nφ(x). We need the following lemma z φ(y) z b + b φ(y) from [6], saying that the rate that changes as moves around in lim k − k lim k − k k − k nˆx x δ→0 δ ≤ δ→0 δ t is bounded. | | | | (8) ε2 + εη ≤ 1 ε Lemma 3.4 Let K be a triangular mesh that (ε, η)-approximates − kφ(x)−φ(y)k S with ε, η < 0.1. Given any triangle t K, for any x in the By definition, √E = limδ→0 . Consider the triangle ∈ |δ| interior of t, and for any unit vector v in the plane passing through consisting of φ(x), φ(y) and z. Since t, we have that φ(x) z = y x = δ cos α, k − k k − k ∠(ˆnx+δv, nˆx) 1 lim , it follows from the triangle inequality and Eqn (8) that ˛δ→0 δ ˛ ≤ (1 ε)ρ ˛ ˛ − ˛ ˛ ε2 + εη ˛ ˛ √E 1 (1 cos α) + . | − | ≤ − 1 ε We now present our result on bounding the Jacobian of the map − Furthermore, let αu be the angle between the vectors z φ(x) and φ. Note that by a nice differential geometric argument, Morvan and − φu (i.e, φ(y) φ(x)). We have for small enough δ that Thibert [16] and Hildebrandt et. al. [13] analyze the Jacobian of φ − when the meshed surface K is close to S under the Hausdorff dis- z φ(y) ε2 + εη tance and normal variation. A bound similar to that in Lemma 3.5 sin αu k − k . ≤ z φ(x) ≤ (1 ε) cos α can be derived by combining Lemma 3.3 with their results. For k − k − ′ completeness of the exposition we include an elementary geomet- Similarly, let w = (u0, v0 + δ) and z the projection of w onto the ′ ric proof here. plane T . Let αv = ∠((z φ(x)), φv). Similar argument can φ(x) − show that where L = Lip(g) is the Lipschitz constant of g. This implies that ε2 + εη √G 1 and g(φ(u, v)) g(p) dudv 2ρLεArea(t). | − | ≤ 1 ε ˛Z − ˛ ≤ ˛ t ˛ − 2 ˛ ˛ z φ(y) ε + εη Combining it with Eqn (9) and noticing that p is an arbitrary vertex sin αv k − k . ˛ ˛ ≤ z φ(x) ≤ (1 ε) of t, we have that k − k − Finally, observe that by the triangle inequal- 2 gdν IKg 2ρLεArea(K) + 2(2ε + η) Area(K) g ∞ ity, ˛Z − ˛ ≤ k k ˛ S ˛ ˛ ˛ ∠(φu, φv) π/2 (αu + αv). Since˛ by Lemma˛ 3.5, we have that Area(K) 1.5Area(S). This | − | ≤ thus proves the first part of Theorem 2.5. ≤ Since F = √EG cos ∠(φu, φv) , it follows that | | | | Second part of Theorem 2.5. In the case when we have a triangu- √EG (1 cos2 ∠(φ , φ )) EG F 2 √EG u v lar mesh and the function g is twice differentiable, this analysis can − ≤ p − ≤ and the lemma follows from the bounds derived for E,G,au, av. be improved to show higher-order convergence. Below we provide a sketch of the proof. Consider a twice-differentiable function h : t IR on a triangle 3.3 Proof of Theorem 2.5 t K, and let p, q, r be the vertices of t. As above,→ let x = (u, v) ∈ We now prove one of our main results on approximating inte- be an orthonormal coordinate system on the plane of t. Consider gral on a meshed surface. Let g : S IR be a function de- the unique l(x) defined on the plane of the triangle, fined on S. We wish to approximate the→ integral g(x) dν(x) s.t. l(p) = h(p),l(q) = h(q),l(r) = h(r). The first observation S is that 1 using the values g(v) at vertices v of the mesh KR. Specifically, we construct a discrete version of the integral gdν by setting 1 S l(u, v)dudv = (h(p) + h(q) + h(r))Area(t). Area(t) R Z 3 IKg = t∈K #t v∈V (t) g(v), where #t is the number of t verticesP in the face t (3P for a triangulation). Each face of the mesh Now without loss of generality assume that p is the origin O of contributes the amount equal to its area multiplied by the average the coordinate system and that h(p) = 0 (additive constants will value of its vertices. It can be shown that, for a triangular mesh, the not change the computation). Let q = (u1, v1), r = (u2, v2). above discretization of the integral is the same as the one obtained Since l is a linear function, straightforward calculations show that by intepolating a function g linearly on each triangle. u v −1 h(q) l(u, v) = (u v) 1 1 . (10) First part of Theorem 2.5. We prove Theorem 2.5 via the map „u2 v2« „h(r)« φ. Since φ is a homeomorphism, the set of surface mesh faces Writing the Taylor expansion for h with a quadratic remainder φ(t),t K partitions the surface S. Therefore from the change term, and recalling that h(0) = 0, we have that of{ variable∈ formula} we have 2 2 h(u, v) hu(0)u hv(0)v < H(u + v ), | − − | g(x) dν(x) = g(x) dν(x) where H is maximum of the norm of the Hessian of h on the trian- Z Z gle, and h (resp. h ) is the partial of h along u (resp. S tX∈K φ(t) u v v). Now based on the observation that the function value of a linear = g(φ(u, v))J(u, v) dudv , function inside a triangle is bounded by the sum of its (absolute) Z tX∈K t values on the vertices, and that the length of each side of t is at most ερ, it can be shown that where (u, v) is any orthogonal coordinate system on the plane pass- 2 ing through the face t. It then follows from Lemma 3.5 that l(u, v) hu(0)u hv(0)v 4H(ερ) . | − − | ≤ Combining this with Eqn (10) leads to that h(u, v) l(u, v) < 5H(ερ)2. ˛ g(x) dν(x) g(φ(u, v)) dudv˛ | − | ˛ZS − Zt ˛ ˛ tX∈K ˛ Finally, set h(x) = g(φ(u, v)). Note that φ is infinitely differen- ˛ ˛ (9) tiable outside of the medial axis. Thus by a standard compactness ˛ 2 ˛ ˛ g(φ(u, v))2(2ε + η) dudv˛ argument, the of φ, and thus of g(φ(x, y)) (from ≤ ˛ Zt ˛ ˛tX∈K ˛ the ), is bounded. Hence ˛ 2 ˛ 2(2˛ ε + η) Area(K) g ∞ ˛ 2 ≤ k k g(u, v) l(u, v) < CHε , | − | On the other hand, for any vertex p of t and any point x = where C is some constant depending on the geometry of S. Inte- (u, v) t, we have that p x ερ by definition of (ε, η)- grating, and using Eqn (9) we obtain approximation.∈ It then followsk − fromk Lemma ≤ 3.3 (i) and the triangle inequality that 2 2 gdν IKg CHε Area(K) + 2(2ε + η) Area(K) g ∞ ˛Z − ˛ ≤ k k 2 ˛ S ˛ p φ(x) (ε + ε + εη)ρ. ˛1 ˛ k − k ≤ ˛ We remark that˛ this implies that, for a triangular mesh, we have Since ε, η < 0.1, we have that dS (p, φ(x)) 2ερ by Lemma 3.2. IKg = gˆ, where the integral is taken over the mesh (considered as Therefore ≤ a piece-wiseR linear manifold) and gˆ is a (unique) piece-wise func- tion obtained by linearly interpolating the values of g on vertices of g(φ(x)) g(p) 2ρLε, each triangle t to the interior of t. | − | ≤ Since Area(K) 1.5Area(S), the second part of Theorem 2.5 5. EXPERIMENTS ≤ then follows. In this section, we apply our algorithm of computing the mesh Laplacian to different surfaces and show its convergence as the 4. DISCRETE LAPLACE OPERATOR mesh becomes finer. One currently widely used mesh Laplace We now show our main result (Theorem 2.1) which states that the operator is the cotangent (COT)n scheme, originally proposed by mesh Laplacian approximates the surface Laplacian well. We first Pinkall and Polthier [19]. We compare our algorithm with its modi- prove Theorem 2.3; that is, given a mesh K (ε, η)-approximating fied version [15, 25], which produces better results than the original a surface S, for any point w S, the difference between the method. Our experiments show that the COT scheme does not pro- ∈ h h duce convergence for functions other than linear functions, while mesh and the functional Laplace operators FS f(w) LK (w) is bounded, and converges to zero as ε and η|go to 0.− Theorem 2.1| our method, in addition to exhibiting convergence behavior, is also then follows from this theorem and Theorem 2.4. much more robust with respect to noisy data.

kx−wk2 1 − Experimental setup. To analyze the convergence behavior, we Proof of Theorem 2.3. First, set g(x) = e 4h (f(x) 4πh2 need the “ground truth”, that is, we need to know the Laplace- f(w)). Comparing Eqn (2) and Eqn (1), we obtain that − Beltrami operator for the underlying surface approximated by in- put meshes. This somewhat limits the type of surfaces that we can h h (11) LK f(w) FS f(w) = gdν IKg experiment with. In this paper, we consider two types of surfaces: ˛ − ˛ ˛ZS − ˛ ˛ ˛ ˛ ˛ planar surfaces and spherical surfaces. By (the first˛ part of) Theorem 2.5,˛ to bound˛ the above˛ quantity, it ˛ ˛ For the planar surface, we uniformly sample the region [ 1, 1] suffices to bound g and the Lipschitz constant Lip(g) of g. ∞ [ 1, 1] with different density, and construct the Delaunay− triangu-× Easy to verify thatk kg 1 f . For Lip(g), since f, ∞ 2πh2 ∞ lation− for the resulting sample points. For the spherical surface, 1 k k ≤ k k ′ and thus g, is C -continuous, it is upper bounded by g ∞ = we sample a , either uniformly or non-uniformly, with ′ ′ k k supx∈S g (x) . Notice that g (x) is a map from the tangent space different density, and then use the COCONE software [5] to gen- k k ′ Tx to IR. By definition, g (x)(v) = S g(x) v for any vector erate the triangular meshes. The goal of the non-uniform sampling ′ ∇ · v Tx and hence g (x) = S g(x) . On the other hand, it is ∈ k k k∇ k is to create “bad” meshes, such as those with skinny triangles and easy to verify that triangles of different sizes. We also produce noisy data from the x w 2 1 − k − k uniform sampling of the sphere, by perturbing each sample point S g(x) (2 f ∞ S e 4h + S f(x) ). k∇ k ≤ 4πh2 k k · k∇ k k∇ k randomly by a small shift. Some examples of these three types of spherical meshes are shown in Figure 1. Since S is a surface embedded in IR3 with the induced Riemannian metric, it holds that

x w 2 x w 2 − k − k − k − k S e 4h 3 e 4h k∇ k ≤ k∇IR k x w 2 − k − k 1 = e 4h x w /(2h) , k − k ≤ √h 2 where the last inequality holds as y/ey 1 for any real number y. It then follows that ≤ uniform ′ 1 2 f ∞ ′ g ∞ = S g(x) k k + f ∞ . k k k∇ k ≤ 4πh2 √ k k ` h ´ Combining with Theorem 2.5, and putting everything together, we conclude with Theorem 2.3.

Proof of Theorem 2.1. Now let Kε,η denote an (ε, η)-approximation of S with ε, η < 0.1, and h(ε, η) an appropriate constant depend- ing on ε and η. We have that for any point w S, non-uniform ∈ h(ε,η) h(ε,η) h(ε,η) LK f(w) ∆S f(w) LK f(w) FS f(w) ˛ ε,η − ˛ ≤ ˛ ε,η − ˛ ˛ ˛ ˛ h(ε,η) ˛ ˛ ˛ + ˛FS f(w) ∆S f(w) . ˛ ˛ − ˛ 1 ˛ 1 ˛ By choosing h(ε, η) = ε 2.5+α +˛ η 1+α , where α > 0 is˛ an arbi- trary fixed positive number, it follows from Theorem 2.3 that

h ε,η α α ( ) 2.5+α 1+α LK f(w) ∆S f(w) O(ε ) + O(η ) uniform with noise ˛ ε,η − ˛ ≤ ˛ ˛ h(ε,η) ˛ ˛ + FS f(w) ∆S f(w) Figure 1: Meshes for spherical surface with 500 sample points | − | and 8000 sample points. The big-O notation above hides terms linear in the area Area(S), ′ the reach ρ(S), f ∞ and f ∞. On the other hand, the limit h(ε,η) k k k k limt→0 F f(w) ∆S f(w) = 0 by Theorem 2.4( [8]). Hence | S − | Error measure. Given a mesh K approximating a surface S and by taking the limit, we see that h(ε,η) , limε,η→0 LKε,η f(w) = ∆S f(w) an input function f : S IR, we evaluate the surface Laplacian which proves Theorem 2.1. and the mesh Laplacian at→ each of the n vertices of K, and obtain two n dimensional vectors U and U, respectively. To measure the Method 500 2000 8000 16000 error between the mesh Laplacian and the surface Laplacian, we COT 0.058 0.030 0.015 0.011 b OUR 0.606 0.142 0.034 0.017 b kU−Uk2 consider the commonly used normalized L2 error E2 = . kUk2 f = x We remark that our theoretical result is that our mesh Laplacian COT 0.171 0.157 0.158 0.155 converges under the L∞ norm (i.e, point-wise convergence), which OUR 0.488 0.115 0.013 0.005 2 is a stronger result than the L2-convergence (as L∞-convergence f = x OT implies L2-convergence for compact spaces). Such convergence C 0.124 0.101 0.102 0.099 is indeed observed for all types of meshes that we have conducted OUR 0.613 0.140 0.028 0.015 experiments on. However, since the COT scheme does not show f = exp(x) any convergence under the L error metric, we will mainly show ∞ Table 2: Normalized L error for spherical meshes with uni- results under the error metric from now on — our method com- 2 L2 form sampling. pares even more favorably under the L∞ error. We will show re- sults under the L error for one data set (Table 3), and the perfor- ∞ Method 500 2000 8000 16000 mances over other data sets are similar. COT 0.229 0.185 0.083 0.062 UR 2 O 2.097 0.492 0.081 0.037 Results for planar surface. Given a function f : IR IR on the f = x →∂2f ∂2f 2 COT 1.147 1.577 1.478 1.375 plane, its surface Laplacian is simply: ∆IR f(x, y) = ∂x2 + ∂y2 . OUR 2.915 0.838 0.062 0.025 Specifically, ∆ 2 f(x, y) = 0 if f is a linear function. It turns IR 2 out that the COT scheme produces exactly 0 for a linear function f = x on a planar mesh. Hence we compare the mesh Laplacians for two COT 0.798 0.887 0.928 0.849 OUR 4.000 0.873 0.112 0.054 non-linear functions: f(x, y) = x2 and f(x, y) = ex+y. The f = exp(x) results under the normalized L2 error metric are shown in Table 1. For both functions, for different values of the parameter h, our Table 3: Normalized L∞ error for spherical meshes with uni- method always presents convergence behavior; while finer mesh form sampling. does not lead to lower approximation error by the COT scheme. Note that we obtain better approximation even when the number of sample points is small (500). Since different values of h show constructed from points non-uniformly sampled from the unit sphere. similar behavior, from now on, we fix the parameter at 4h = 0.04. The error obtained for both methods are fairly similar to that on the regular mesh (as reported in Tables 2 and 3). Method Param 500 2000 8000 16000 Finally, in Table 4, we show how our method and the COT scheme COT non 0.220 0.173 0.197 0.207 4h=0.01 0.450 0.146 0.040 0.022 perform for noisy data. Note that even though the COT scheme has OUR 4h=0.04 0.126 0.038 0.010 0.005 been proven to converge for linear functions on a sphere, once the 4h=0.09 0.069 0.017 0.004 0.002 mesh becomes noisy, such convergence behavior is lost; while the f = x2 approximation error by our algorithm stays almost the same as be- COT non 0.198 0.188 0.190 0.202 fore. This is not too surprising as our scheme considers a much 4h=0.01 0.875 0.128 0.055 0.027 larger neighborhood than COT scheme when evaluating the mesh OUR 4h=0.04 0.189 0.037 0.022 0.016 Laplacian at a point, which has the “smoothing” effect. However, it 4h=0.09 0.099 0.033 0.027 0.025 is not clear how to extend COT scheme to incorporate larger neigh- f = exp(x + y) borhood than its current one-ring neighborhood.

Table 1: Normalized L2 error for planar mesh. We show the results of our method with there different h values. Method 500 2000 8000 16000 COT 0.398 1.532 3.015 0.936 OUR 0.599 0.155 0.051 0.022 Spherical surfaces. Given a function f : S2 IR defined on f = x COT 0.267 0.914 1.840 0.545 the unit sphere S2, where f is parametrized by the→ spherical coor- OUR 0.484 0.128 0.028 0.006 dinates θ and φ, its spherical surface Laplacian is: f = x2 1 ∂ ∂f 1 ∂2f COT 0.308 1.271 2.631 0.817 ∆S2 f(θ, φ) = sin θ + . OUR 0.612 0.153 0.043 0.018 sin θ ∂θ ∂θ sin2 θ ∂φ2 ` ´ f = exp(x) Xu [25] proved that his modification of the COT scheme can pro- duce convergent result for linear function defined on spheres. In- Table 4: Normalized L2 error for the experiments on the sphere deed, this is observed in our experimental results as shown in Table with noise. 2, where we also compare the mesh Laplacians for two non-linear functions f(x, y, z) = x2 and f(x, y, z) = ex for a uniformly sampled spherical mesh. However, while our method converges for Remark. Finally, we remark that we have also compared our mesh all the three functions, the COT scheme only converges under the Laplacian with the graph Laplacian. Graph Laplacian has been linear function. In Table 3, we also show the approximation error shown to converge for points randomly sampled from the uniform of the two mesh Laplacians under the L∞ error metric. Note that distribution on the underlying manifold [8], which is also observed compared to the case of normalized L2 error, the numerical values in our experiments when points are randomly sampled. However, of L∞ error can be big, as it is the absolute error and not normal- our mesh Laplacian produces consistently smaller errors. For non- ized. uniformly sampled meshes, graph Laplacian is not expected to con- We also test the COT scheme and our mesh Laplacian for meshes verge to the surface Laplacian. 6. CONCLUSIONS remeshing. In SIGGRAPH ’02: Proceedings of the 29th In this paper, we have developed the first algorithm for approx- annual conference on Computer graphics and interactive imating the Laplace operator on a meshed surface with point-wise techniques, pages 347–354, New York, NY, USA, 2002. convergence guarantee. Such convergence is required in many ap- ACM Press. plications, where quantities, such as mean curvature, need to be [4] N. Amenta and M. Bern. Surface reconstruction by voronoi estimated at each node of the mesh. The convergence result does filtering. Discr. Comput. Geom., 22:481–504, 1999. not require the aspect ratio of mesh elements to be bounded. Ex- [5] N. Amenta, S. Choi, T. K. Dey, and N. Leekha. A simple perimental results show that our algorithm indeed exhibits conver- algorithm for homeomorphic surface reconstruction. gence empirically, and outperforms current popular methods both Internat. J. Comput. 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