Computing Harmonic Maps and Conformal Maps on Point Clouds
CComputing Harmonic Maps and Conformal Maps onPoint Clouds
Tianqi Wu ∗ Shing-Tung Yau † Abstract
We propose a novel meshless method to compute harmonic maps and conformal mapsfor surfaces embedded in the Euclidean 3-space, using point cloud data only. Given asurface, or a point cloud approximation, we simply use the standard cubic lattice to ap-proximate its (cid:15) -neighborhood. Then the harmonic map of the surface can be approximatedby discrete harmonic maps on lattices. The conformal map, or the surface uniformization,is achieved by minimizing the Dirichlet energy of the harmonic map while deforming thetarget surface of constant curvature. We propose algorithms and numerical examples forclosed surfaces and topological disks.
Roughly speaking, a map between two surfaces is called conformal if it preserves angles,and is called harmonic if it minimizes the stretching energy. Computing harmonic maps andconformal maps has a wide range of applications, such as surface matching, surface param-eterization, shape analysis and so on. See [1][2][3][4][5][6][7][8][9][10][11] for examplesof applications of conformal maps, and [12][13][14][15][16] for examples of applications ofharmonic maps.Existing methods for computing harmonic maps and conformal maps mostly rely on the trian-gle mesh approximation of a surface. However, it is often much easier to get point cloud data,rather than the triangle mesh data. Contemporary 3D scanners can easily provide 3D pointcloud data sampled from the surfaces of solid objects, but sometimes it is inconvenient to gen-erate meshes upon point clouds. Since point clouds data do not contain information about theconnectivity, a lot of algorithms, which were well-established on meshes, cannot be extendedto point clouds.This paper introduces a novel meshless method for computing harmonic maps and conformalmaps for a surface in the Euclidean 3-dimensional space. The basic idea is to use a dense ∗ Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138. Email:[email protected] † Department of Mathematics, Harvard University, Cambridge, MA 02138. Email: [email protected] a r X i v : . [ m a t h . DG ] S e p -dimensional lattice to approximate the (cid:15) -neighborhood of the surface, and then compute thediscrete harmonic map from the lattice to the target surface, by minimizing the Dirichlet energy(i.e., the stretching energy). Conformal maps are computed by minimizing the Dirichlet energyof the harmonic maps, as we deform the target surface of constant Gaussian curvature. In thispaper, we focus on harmonic diffeomorphisms and conformal diffeomorphisms to surfaces ofconstant curvature ± or . These maps are particularly useful for global surface parameteri-zations. More specifically, we propose algorithms and numerical examples for (1) maps to theunit sphere, and (2) maps to flat rectangles, and (3) maps to flat tori, and (4) maps to closedhyperbolic surfaces. Comparing to methods for triangle meshes, there are much fewer works on meshless meth-ods of computing conformal and harmonic maps, especially for higher genus surfaces. Guo etal. [17] computed global conformal parameterizations of surfaces by computing holomorphic1-forms on point clouds. Li et al. [18] computed harmonic volumetric maps by grids dis-cretizations. Meng-Lui [19] developed the theory of computational quasiconformal geometryon point clouds. Using approximations of the differential operators on point clouds, Liang etal. [20] and Choi et al. [21] computed the spherical conformal parameterizations of genus-0closed surfaces, and Meng et al. [22] computed quasiconformal maps on topological disks.Li-Shi-Sun [23] computed quasiconformal maps from surfaces to planar domain, using theso-called point integral method for discretizing integral equations for point clouds.There is an extensive literature on computing conformal maps for triangle meshes. Gu-Yau[24][25] developed the method of computing conformal structures of surfaces by computingthe discrete holomorphic one-forms. Pinkall-Polthier [13] proposed a method of conformalparameterization by computing a pair of conjugate harmonic functions. L´evy et al. [26] andLipman [27] and Lui et al. [28] computed conformal or quasiconformal maps by minimizingor controlling the conformal distortion. There is also a big family of methods based on variousnotions of discrete conformality for triangle meshes, such as circle patterns [29][30][31][32],and inversive distances [33][34], and vertex scalings [35][36][32], and modified vertex scalings[37] [38] [39][40] allowing diagonal switches. Some related convergence results for discreteconformality can be found in [29][41][42][43][44], and other mathematical analysis can befound in [45][46][47][48][49]. Other works on computing conformal maps on triangle meshesinclude [50][51][52][53][26][54][55][56][57][58].For computing harmonic maps on triangle meshes, Gaster-Loustau-Monsaingeon [59][60] givedetailed discussions on the mathematical analysis and algorithms, and produced a computersoftware. Notions of discrete Dirichlet energy have been discussed and used to compute dis-crete harmonic maps, not only in Gaster-Loustau-Monsaingeon’s work [59][60], but also ex-tensively in other works such as [61][13][62][63][64][65][66][67][68].2 .2 Contribution
To the best of the authors’ knowledge, our method is the first meshless method in computingharmonic and conformal maps to surfaces of genus greater than 1.Existing meshless methods always use approximating differential operators on point clouds,to compute harmonic or conformal maps. Our method provides a novel type of approach, byjumping out of the point clouds to cubic lattices. It also has the following nice properties. • The idea of lattice approximation is conceptually simple, and could be implemented forany topological types of surfaces. • One can simply tune the density of the lattices to get satisfactory accuracy, within theability of the computing powers. • Numerical experiments indicate that sparse lattices already work well. • The lattice structure should be suitable for parallel computation, and multi-grid methods.We proposed algorithms of computing harmonic maps and conformal maps for any closedsurfaces and topological disks, embedded in R , to constant curvature surfaces. Given a point, or equivalently a vector x in R , denote | x | = (cid:113) x + x + x as the l -norm of x .Given two points x, y ∈ R , denote d ( x, y ) = | x − y | as the distance between x and y .Given a subset A ⊂ R and a point x ∈ R , denote d ( x, A ) = inf y ∈ A d ( x, y ) as the distance from x to A .Given a subset A ⊂ R and r > , denote B ( A, r ) = { x ∈ R : ∃ y ∈ A, s.t. | x − y | < r } as the r -neighborhood of the subset A .Given a closed surface M embedded in R , we say a finite subset P ⊂ R is a δ - point cloud of M if P ⊂ B ( M, δ ) and M ⊂ B ( P, δ ) . G = ( V, E ) is an undirected connected simple graph, and w ∈ R E> is an edge weight, then L = L G,w denotes the discrete Laplacian such that for any f : V → R k , ( Lf )( i ) = (cid:88) j : ij ∈ E w ij ( f ( j ) − f ( i )) . The remaining of the paper is organized as follows. We review the basic mathematical theoryin Section 2, and then introduce the lattice approximation of a surface in Section 3. Thealgorithms, as well as numerical examples, are given in Section 4, 5, 6, 7, for the cases ofspheres, rectangles, flat tori, and hyperbolic surfaces respectively.This work is partially supported by Center of Mathematical Sciences and Applications atHarvard University, and Yau Mathematical Sciences Center at Tsinghua University, and NSF1760471.
We give a formal definition of a conformal map as follows.
Definition 2.1 (Conformal Map)
A diffeomorphsim f between two Riemannian surfaces ( M, g ) and ( N, h ) is called a conformal map if f ∗ h = λ g for some smooth positive function λ , orequivalently, the local angles are preserved under f . In this paper we always assume that a conformal map is a diffeomorphism between two sur-faces. By the celebrated uniformization theorem, it is well known that any closed orientablesurface is conformally equivalent to a surface of constant Gaussian curvature.
Theorem 2.2 (Uniformization for Closed Surfaces)
Given a closed orientable Riemanniansurface ( M, g ) , there exists a Riemannian surface ( N, h ) , with constant Gaussian curvature K , and a conformal map f : M → N such that(a) K = 1 if M has genus and(b) K = 0 if M has genus , and(c) K = − if M has genus > . The Riemannian surface ( N, h ) above is called a uniformization of ( M, g ) and could be viewedas a canonical representative in the conformal equivalence class. A closed surface ( N, h ) of constant curvature − or can be naturally represented as a planar domain, after a fewcuts. 4a) If ( N, h ) has constant curvature 0, then it is isometric to the flat complex plane C moduloa lattice Z ⊕ Z τ where τ ∈ C and Im ( τ ) > . If we properly cut two loops on N , then ( N, h ) can be displayed isometrically as a planar domain in C .(b) if ( N, h ) has constant curvature -1, then it is isometric to the hyperbolic plane H moduloa discrete subgroup Γ of the orientation-preserving isometry group Isom + ( H ) . In thePoincar´e disk model, the hyperbolic plane H is identified as the unit disk { z ∈ C : | z | < } with the complete Riemannian metric | dz | (1 − | z | ) = 4( dx + dy )(1 − x − y ) . Similar to the flat case, if we properly cut a few loops, ( N, h ) can be displayed, isomet-rically in the hyperbolic sense, as a domain in the unit disk {| z | < } .So for genus 0 surfaces, there is essentially a unique conformal equivalence class. For genus1 surfaces, the conformal equivalence classes can be parameterized by a complex number τ with Im ( τ ) > . For a surface M with genus > , the conformal equivalence classes can beparameterized by the generators of the discrete subgroup Γ of Isom + ( H ) . As shown above,a surface of constant curvature ± or can always been identified as a planar domain, bya stereographic projection or cutting loops. So computing diffeomorphisms to such surfacesgives rise to global parameterizations and surface flattening, which have fundamental applica-tions in computational graphics.For nonclosed surfaces, the most important case is the topological disk. Assume ( M, g ) is aRiemannian surface homeomorphic to a closed disk, then again by the uniformization theoremit is conformally equivalent to a closed unit disk. However, for our convenience in utilizing themethod of harmonic maps, we consider conformal maps and harmonic maps to rectangles in-stead of disks. The following uniformization-type theorem for rectangles is well-known. Theorem 2.3
Assume ( M, g ) is a Riemannian surface homeomorphic to a closed disk. Given4 ordered points A , A , A , A on ∂M , there exists unique a ∈ R > and a conformal map f : M → [0 , a − ] × [0 , a ] such that f ( A ) = (0 , , f ( A ) = ( a − , , f ( A ) = ( a − , a ) , f ( A ) = (0 , a ) . Let ( M, g ) and ( N, h ) be two smooth Riemannian surfaces. The Dirichlet energy , i.e., thestretching energy of a smooth map f : M → N is formally defined as E ( f ) = 12 (cid:90) M (cid:107) df (cid:107) dv g where dv g denotes the volume element on ( M, g ) , and (cid:107) df (cid:107) is the norm of the differential of f , with respect to the induced metric on T ∗ M ⊗ f ∗ ( T N ) .5 efinition 2.4 A smooth map f : M → N is called harmonic if it is a critical point of theDirichlet Energy. Smooth harmonic maps have been extensively studied. See [69][70] for examples. Here wefocus on harmonic diffeomorphisms to surfaces of constant curvature ± or , and harmonicdiffeomorphisms to rectangles with given boundary correspondences. Some relevant well-known facts are summarized as the following 2 theorems. Theorem 2.5
Assume
M, N are two closed orientable Riemannian surfaces, and N has con-stant curvature K = ± or 0, and f is a diffeomorphism from M to N .(a) E ( f ) ≥ Area ( N ) , and the equality holds if and only if f is conformal.(b) If f is harmonic, f minimizes the Dirichlet energy in its homotopy class.(c) If f is conformal, then f is harmonic.(d) If f is harmonic and N is a unit sphere, then f is conformal.(e) f is always homotopic to a harmonic diffeomorphism, which is (i) unique up to a M¨obiustransformation if K = 1 , and (ii) unique up to a translation if K = 0 , and (iii) unique if K = − . Theorem 2.6
Assume(i) ( M, g ) is a Riemannian surface homeomorphic to a closed disk, and(ii) A , A , A , A are 4 ordered points on ∂M , and(iii) a ∈ R > , and(iv) F contains all the diffeomorphisms f : M → [0 , a − ] × [0 , a ] such that f ( A ) = (0 , , f ( A ) = ( a − , , f ( A ) = ( a − , a ) , f ( A ) = (0 , a ) . Then(a) For any f ∈ F , E [ f ] ≥ and the equality holds if and only if f is conformal.(b) There exists a unique harmonic map f ∈ F , and it is the unique minimizer of the Dirich-let energy in F .(c) If f ∈ F is conformal, then it is harmonic. By part (b) and (e) of Theorem 2.5, for closed surfaces it makes good sense to compute theharmonic map within a fixed homotopy class. Such a topological constrain also often naturallyarises in applications. Part (b) of Theorem 2.5 and part (b) of Theorem 2.6 somewhat justifythe physical intuition that harmonic maps minimize the stretching energy. Minimizing theDirichlet energy would also be an important idea for computing harmonic maps. Part (c) of6heorem 2.5 and part (c) of Theorem 2.6 connect the notions of conformal maps and harmonicmaps. So conformal maps are really special harmonic maps, and the converse is true if thetarget surface is a unit sphere. Part (a) of Theorem 2.5 and Part (a) of Theorem 2.6 pointout that conformal maps minimize the normalized Dirichlet energy E ( f ) /Area ( N ) amongthe harmonic maps. This optimality provides a method to compute the uniformization usingharmonic maps.(a) If M is a topological sphere, a harmonic diffeomorphism from ( M, g ) to a unit sphere isalready a conformal map.(b) If M is a genus 1 closed orientable surface, one can first compute the harmonic map to aflat torus ( N, h ) = C / Z ⊕ Z τ for some parameter τ , and then minimize the normalizedDirichlet energy, by tuning the parameter τ in the upper half plane.(c) If M is a closed orientable surface with genus > , one can first compute the har-monic map to a hyperbolic surface ( N, h ) = H / Γ for some discrete subgroup Γ of Isom + ( H ) , and then minimize the normalized Dirichlet energy, by continuously de-forming the subgroup Γ .(d) If M is homeomorphic to a closed disk, then one can compute the harmonic map to arectangle [0 , a − ] × [0 , a ] , and then minimize the Dirichlet energy, by tuning the parameter a ∈ R > . Since we will be computing discrete harmonic maps on lattice graphs, here we briefly intro-duce the theory of discrete harmonic maps, which has been well-studied. See [68][71][61] forreferences of the related definitions and theorems. In this subsection, we always assume that G = ( V, E ) is an undirected connected simple graph, and w ∈ R E> denotes the edge weight,and ( N, h ) is a flat rectangle or a closed orientable surface of constant curvature ± or 0.A discrete map f from G to ( N, h ) maps each vertex i ∈ V to a point in N , and each edge ij ∈ E to a geodesic segment in N with endpoints f ( i ) , f ( j ) . Given a discrete map f , wedenote l ij ( f ) as the length of f ( ij ) . Assuming the stretching energy on a single edge ij is w ij l ij , we have the following definition of the discrete Dirichlet energy. Definition 2.7
Given a discrete map f from ( G, w ) to ( N, h ) , the discrete Dirichlet energy isdefined to be E ( f ) = 12 (cid:88) ij ∈ E w ij · l ij ( f ) . The notion of discrete harmonic maps can be defined by the balanced conditions on ver-tices.
Definition 2.8
A discrete map f is harmonic if for any i ∈ V (cid:88) j : ij ∈ E w ij · l ij ( f ) · (cid:126)e ij = 0 , (1)7 here (cid:126)e ij is the unit vector in T f ( i ) N representing the direction of the geodesic f ( ij ) . This definition consists with the continuous one in the following sense.
Proposition 2.9
A discrete harmonic map f is a critical point of E , if N is not a sphere or l ij ( f ) < π for any ij ∈ E . Here the assumption on N and l ij ( f ) is to guarantee that locally f can be parameterized by [ f ( i )] i ∈ V ⊂ N V . For the cases of non-positive curvatures, we have the following two well-known theorems partially analogue to Theorem 2.5 and 2.6. Theorem 2.10 If ( N, h ) is closed and has constant curvature or − , then any discrete map f from G to N is homotopic to a discrete harmonic map f , which is a minimizer of the discreteDirichlet energy E in the homotopy class. Such discrete harmonic map f is(a) unique up to a translation, if ( N, h ) is a flat torus, and(b) unique if ( N, h ) has constant curvature − . Theorem 2.11
Assume ( N, h ) is a flat rectangle with four ordered edges B , B , B , B , and V , V , V , V are 4 subsets of V such that V ∩ V = ∅ and V ∩ V = ∅ . Then there exists aunique discrete map f from G to N such that f ( V ) ⊂ B , f ( V ) ⊂ B , f ( V ) ⊂ B , f ( V ) ⊂ B , and the balanced condition (1) holds for any i ∈ V − V ∪ V ∪ V ∪ V . Such constraineddiscrete harmonic map minimizes the discrete Dirichlet energy E ( f ) among all the constrainedmaps. Assume that M is a closed smooth surface embedded in R , and P is a δ -point cloud of M .If δ is sufficiently small, comparing to some constant (cid:15) > , then the (cid:15) -neighborhood B ( P, (cid:15) ) of P would be a good approximation of the (cid:15) -neighborhood B ( M, (cid:15) ) of M . If n is sufficientlylarge, comparing to /(cid:15) , then V = B ( P, (cid:15) ) ∩ ( Z /n ) would be a good cubic lattice approximation of B ( P, (cid:15) ) ≈ B ( M, (cid:15) ) (see Figure 1). Connecttwo vertices x, y in V by an edge if d ( x, y ) = 1 /n , and then we obtain a lattice graph G =( V, E ) . Since we are using standard cubic lattice, the weight function is set to be constant w ij = 1 . Such weighted graph ( G, w ) would be our lattice approximation of M .In practice, if the pointcloud data P is given, we need to choose a proper (cid:15) > such that(a) (cid:15) is sufficiently large such that B ( P, (cid:15) ) forms a connected domain in R and has uniform thickness , and 8b) (cid:15) is sufficiently small such that B ( P, (cid:15) ) ≈ B ( M, (cid:15) ) is a good approximation of M .For our second parameter n , if we choose a larger integer n , the lattice graph G should approx-imates the domain B ( P, (cid:15) ) ≈ B ( M, (cid:15) ) better. However, in practice we find that sparse latticesalready work well. Figure 1: Lattice approximation of a cat Suppose G = ( V, E ) is a lattice approximation of P constructed as before, then given a func-tion f : V → R k , we can use the standard trilinear interpolation to extend f to the point cloud P . Assume p = ( p , p , p ) ∈ P lies inside a cubic cell [ x , x ] × [ y , y ] × [ z , z ] of the lattice G . Denote a ijk = f ( x i , y j , z k ) where i, j, k = ∈ { , } , and x (cid:48) = p − x , y (cid:48) = p − y , z (cid:48) = p − z ,x (cid:48) = x − p , y (cid:48) = y − p , z (cid:48) = z − p , then f ( p ) = n (cid:88) i,j,k ∈{ , } a ijk x (cid:48) i y (cid:48) j z (cid:48) k is the trilinear interpolation of f on P . By part (c)(d) of Theorem 2.5, in the spherical case harmonic maps and conformal maps arereally the same. This section solves the following problem of computing harmonic and con-formal maps.
Problem 4.1
Assume that M is a genus 0 closed smooth surface embedded in R . Given apoint cloud approximation P of M , how to compute harmonic diffeomorphisms, i.e., conformaldiffeomorphisms from M to the unit sphere? .1 Algorithm Our method is smoothly adapted from Gu-Yau’s method [24] for computing harmonic maps tospheres. First we pick a lattice approximation G = ( V, E ) as in Section 3, and then(1) compute the (approximated) discrete harmonic map f : G → S , and(2) extend f to P by trilinear interpolation, and then do a normalization x (cid:55)→ x/ | x | and getan approximation of a harmonic map from M to the unit sphere.Step (2) is pretty clear and simple, so let us discuss the details of step (1). Recall that a discretemap f from G to S ⊂ R is harmonic if it is a critical point of the energy E ( f ) = 12 (cid:88) ij w ij l ij ( f ) . If the edge length l ij ( f ) is small, l ij ( f ) ≈ | f ( j ) − f ( i ) | , and the discrete energy is approxi-mated by E ( f ) = 12 (cid:88) ij ∈ E w ij | f ( j ) − f ( i ) | . So for simplicity we will minimize E ( f ) instead of E ( f ) to compute the approximation of thediscrete harmonic map. Since ∂ E ∂f ( i ) = − ( Lf )( i ) , a critical point f of E satisfies that for any i ∈ V ( Lf )( i ) ⊥ T f ( i ) S , i.e., ( L (cid:107) f )( i ) = 0 (2)where L (cid:107) f ( i ) denotes the tangential component of Lf ( i ) , i.e., the orthogonal projection of Lf ( i ) to the plane f ( i ) ⊥ . We view a map f satisfying equation (2) as an approximation of adiscrete harmonic map, and wish to compute it by the following discrete heat flow on S dfdt = − L (cid:107) f. We can simply solve the above ODE by the explicit Euler’s method, with a normalization aftereach iteration to keep that all the points are on the unit sphere. More specifically, if the stepsize is δt we have that f t + δt = π ◦ ( f t − δt · L (cid:107) f t ) where π ( x ) = x/ | x | .Two harmonic maps to a sphere can differ by a M¨obius transformation. So a smooth harmonicmap to a sphere is not unique, and would have large distortion if most of the mass is concen-trated near a point of the sphere. To avoid such big distortion, we add a correction mapping m in each iteration. Here the correction mapping m first translate all the points f ( i ) ’s in R to10ake the center of mass be at the origin, and then do a normalization x (cid:55)→ x/ | x | . Now themodified iteration is f t + δt = m ◦ π ◦ ( f t − δt · L (cid:107) f t ) , (3)and we have the following algorithm. Algorithm 1
Harmonic and Conformal Maps to Spheres
Input:
A point cloud P in R Output:
A harmonic map f from P to the unit sphere Translate P such that the origin lies inside of P . Choose parameters (cid:15) and n , and then compute the lattice approximation G = ( V, E ) . Compute f ( x ) = π ( x ) on V as the initial map. Do the iteration by equation (3), until the function f converges. Extend f to P by trilinear interpolation. Return π ◦ ( f | P ) . See the two numerical examples in Figure 2 and Figure 3.Figure 2: Harmonic map from a moai to a unit sphereFigure 3: Harmonic map from retinal to a unit sphere11
Computing Harmonic and Conformal Maps to Rectan-gles
This section solves the following problem on computing harmonic and conformal maps.
Problem 5.1
Assume that M is a smooth surface embedded in R , and is homeomorphic to aclosed disk, and γ , γ , γ , γ are 4 ordered arcs on ∂M divided by 4 points on ∂M . Now weare given the data ( P, P .P , P , P ) such that P is a point cloud approximating M , and P i isa subset of P approximating γ i . Here are two computational problems.(1) Compute the harmonic map f : M → [0 , a − ] × [0 , a ] for a given a > such that f ( γ ) =[0 , a − ] × { } , (4) f ( γ ) = { a − } × [0 , a ] , (5) f ( γ ) =[0 , a − ] × { a } , (6) f ( γ ) = { } × [0 , a ] . (7) (2) Find a positive number a > and a conformal map f : M → [0 , a − ] × [0 , a ] such thatthe above boundary conditions (4)(5)(6)(7) are satisfied. We will first discuss the harmonic maps in Section 5.1, and then discuss the conformal mapsin Section 5.2.
First we construct a lattice approximation G = ( V, E ) of P as in Section 3, and let V i = ∪ x ∈ P i V x be the lattice approximation of γ i , where V x consists of the 8 vertices of the cubic cellof G containing x ∈ P . Then we would like to(1) compute the (constrained) discrete harmonic map f : G → [0 , a − ] × [0 , a ] , such that f ( V ) ⊂ [0 , a − ] × { } ,f ( V ) ⊂{ a − } × [0 , a ] ,f ( V ) ⊂ [0 , a − ] × { a } ,f ( V ) ⊂{ } × [0 , a ] , and(2) extend f to P by trilinear interpolation, and then f | P is our approximation of the desiredharmonic map.Step (2) is clear and simple. Computing f = ( f , f ) in step (1) amounts to solve the following2 systems of linear equations f ( i ) = 0 if i ∈ V f ( i ) = a − if i ∈ V Lf ( i ) = 0 if i ∈ V − V − V , f ( i ) = 0 if i ∈ V f ( i ) = a if i ∈ V Lf ( i ) = 0 if i ∈ V − V − V . (8)12hese two equations can be solved efficiently by the preconditioned conjugate gradient method.In a summary we have the following Algorithm 2 . Algorithm 2
Harmonic Maps to Rectangles
Input:
A point cloud P in R as an approximation of a surface M , and 4 subsets P , P , P , P as approximations of 4 boundary arcs of M , and a parameter a ∈ R > . Output:
A harmonic map f from P to the rectangle [0 , a − ] × [0 , a ] . Choose parameters (cid:15) and n , and then compute the lattice approximation ( G, V , V , V , V ) of ( P, P , P , P , P ) . Solve the two systems of linear equations (8). Extend f to P by trilinear interpolation Return f | P . Assume f a is the discrete harmonic map from the lattice approximation G = ( V, E ) to therectangle [0 , a − ] × [0 , a ] . To compute conformal maps, we only need to find a proper edgelength a such that the computed discrete harmonic map f a approximates a conformal mapping.Inspired by part (a) of Theorem 2.6, we compute the conformal map by minimizing E ( f a ) over a ∈ R > . If ¯ a is the minimizer, then f ¯ a : V → [0 , ¯ a − ] × [0 , ¯ a ] would approximate a conformalmap.Our method is summarized as Algorithm 3 . Algorithm 3
Conformal Maps to Rectangles
Input:
A point cloud P in R as an approximation of a surface M , and 4 subsets P , P , P , P as approximations of 4 boundary arcs of M . Output:
A parameter a , and a conformal map f from P to the rectangle [0 , a − ] × [0 , a ] . Choose parameters (cid:15) and n , and then compute the lattice approximation ( G, V , V , V , V ) of ( P, P , P , P , P ) . For a given a ∈ R > , solve the two systems of linear equations (8) and get a discreteharmonic map f a . Compute the discrete Dirichlet energy E ( f a ) . Compute the minimizer ¯ a of E ( f a ) , and f ¯ a . Extend f ¯ a to P by trilinear interpolation. Return ¯ a and f ¯ a | P . Here are two numerical examples for harmonic maps and conformal maps respectively.13igure 4: Harmonic maps to rectanglesFigure 5: Conformal maps to rectangles
This section solves the following problem on computing harmonic and conformal maps.
Problem 6.1
Assume that M is a genus 1 closed smooth surface embedded in R , and we aregiven a point cloud approximation P of M .(1) How to compute a harmonic map f : M → C / Z ⊕ Z τ for a given τ with Im ( τ ) > ?(2) How to find the complex number τ with Im ( τ ) > and the map f : M → C / Z ⊕ Z τ such that f is conformal? We will first discuss the harmonic maps in Section 6.1, and then discuss the conformal mapsin Section 6.2.
First we construct a lattice approximation G = ( V, E ) as in Section 3, and then need to141) compute the discrete harmonic map f from G to C / Z ⊕ Z τ , or equivalently, to a funda-mental domain in C , and(2) extend f to P by trilinear interpolation, and then f | P represents an approximation of thedesired harmonic map.Step (2) is kind of clear and simple, so let us discuss the details of step (1). First we ”cut” thelattice G along two ”loops” γ , γ , so that we can represent a discrete map from G to a torus asa map from V to a planar domain in C . Secondly we fix the homotopy class of discrete maps f : G → C / Z ⊕ Z τ , by requiring that γ corresponds to the translation z (cid:55)→ z + 1 , and γ corresponds to the translation z (cid:55)→ z + τ . Now computing the discrete harmonic map in thefixed homotopy class amounts to solve a system of complex linear equations. For a vertex i that is away from the loops γ and γ , the balanced condition (1) gives us that Lf ( i ) = 0 . Things become a bit subtle when the vertex i is near a loop. Assume the edge ij pass throughthe loop γ , and l ij ( f ) is small, and all the other edges adjacent to vertex i do not pass through γ or γ . Then f ( j ) ≈ f ( i ) ± τ, where the sign depends on the orientation of γ and the relative position between γ and i . Forsuch a vertex i , the corrected balanced condition should be Lf ( i ) = ± τ. Similar corrections should be made for all the edges passing through γ or γ . After makingall the corrections, we arrived at a system of complex linear equations of the form Lf ( i ) = b ( i ) , ∀ i (9)where b ( i ) is computed by adding up all the correction terms. The equation (9) can be solvedefficiently by the preconditioned conjugate gradient method.In a summary we have the following Algorithm 4 . Algorithm 4
Harmonic Maps to Flat Tori
Input:
A point cloud P in R as an approximation of a surface M , and a parameter τ ∈ C with Im ( τ ) > . Output:
A harmonic map f from P to the flat torus C / Z ⊕ Z τ . Choose parameters (cid:15) and n , and then compute the lattice approximation G = ( V, E ) . Choose two ”loops” γ , γ in G . Compute the total correction term b ( i ) ’s. Solve the linear systems (9). Extend f to P by trilinear interpolation. Return f | P . 15 .2 Algorithm for Conformal Maps Assume that f τ is the discrete harmonic map from the lattice approximation G = ( V, E ) to theflat torus C / Z ⊕ Z τ . Inspired by part (a) of Theorem 2.5, we compute the conformal map byminimizing E ( f τ ) Area ( C / Z ⊕ Z ¯ τ ) = E ( f τ ) Im ( τ ) over the parameter τ . If ¯ τ is the minimizer, then f ¯ τ : V → C / Z ⊕ Z ¯ τ would approximate aconformal map.Our method is summarized as Algorithm 5 . Algorithm 5
Conformal Maps to Flat Tori
Input:
A point cloud P in R as an approximation of a surface M . Output:
A parameter τ ∈ C with Im ( τ ) > , and a conformal map f from P to the flat torus C / Z ⊕ Z τ . Choose parameters (cid:15) and n , and then compute the lattice approximation G = ( V, E ) . Choose two ”loops” γ , γ in G . Compute the total correction term b ( i ) ’s. Solve the linear systems (9). Compute the normalized discrete Dirichlet energy E ( f τ ) /Im ( τ ) . Compute the minimizer ¯ τ of E ( f τ ) /Im ( τ ) , and f ¯ τ . Extend f ¯ τ to P by trilinear interpolation. Return ¯ τ and f ¯ τ | P . Here are two numerical examples for harmonic maps and conformal maps respectively.Figure 6: Harmonic maps to flat tori16igure 7: Conformal maps to flat tori
This section solves the following problem on computing harmonic and conformal maps.
Problem 7.1
Assume that M is a genus 1 closed smooth surface embedded in R , and we aregiven a point cloud approximation P of M .(1) How to compute a harmonic map f : M → H / Γ for a given closed hyperbolic surface H / Γ homeomorphic to M ?(2) How to find the discrete subgroup Γ of Isom + ( H ) and the map f : M → H / Γ such that f is conformal? We will first discuss the harmonic maps in Section 7.1, and then discuss the conformal mapsin Section 7.2.
First we construct a lattice approximation G = ( V, E ) as in Section 3, and then need to(1) compute the discrete harmonic map f from G to H / Γ , or equivalently, to a fundamentaldomain in the unit disk representation H = {| z | < } , and(2) extend f to P by trilinear interpolation, and then f | P represents an approximation of thedesired harmonic map.Step (2) is kind of clear and simple, so let us discuss the details of step (1). First we need to”cut” the lattice G along · genus ( M ) ”loops” γ i , so that we can represent a discrete mapfrom G to a hyperbolic surface as a map from V to a planar domain in the unit disk. Secondly17e fix the homotopy class of discrete maps f : G → H / Γ , by requiring that each loop γ i corresponds to a certain deck transformation α i in Γ .Now computing the discrete harmonic map in the fixed homotopy class amounts to solve asystem of nonlinear equations. For a vertex i that is away from any loop γ i , the balancedcondition (1) gives us that (cid:88) j : j ∼ i d H ( f ( i ) , f ( j )) · (cid:126)e ( f ( i ) , f ( j )) = 0 where (cid:126)e ( x, y ) denotes the unit tangent vector in T x H indicating the direction to point y . Thingsbecome subtle when the vertex i is near some loop γ i . Assume the edge ij pass through theloop γ k , and l ij ( f ) is small. Then f ( i ) ≈ α ij ( f ( j )) , where α ij is a deck transformation in Γ , which is determined by(1) the choice of the loops γ i , and(2) the choice of α i ’s, and(3) the relative position between i and γ k .Properly choosing α i ’s and computing α ij ’s are involved, particularly for higher genus sur-faces. One need to use polygonal representations for higher genus surfaces, and parameteriza-tions of the deformation spaces of Γ . See [72][73] for references. Once we computed all the α ij ’s, it remains to solve the following system of nonlinear equations. (cid:88) j : j ∼ i d H ( f ( i ) , α ij ( f ( j ))) · (cid:126)e ( f ( i ) , α ij ( f ( j ))) = 0 ∀ i. (10)The existence and uniqueness of the solution is guaranteed by Theorem 2.10. Notice thatequation (10) indicates that f ( i ) is the hyperbolic center of mass of α ij ( f ( j )) ’s where j goesover all the neighbors of i . So one can use the so-called center of mass method to solve equation(10). This is an iterating method where in each iteration f ( i ) is replaced by the mass-center ofits neighbors α ij ( f ( j )) , i.e., the ( n + 1) -th function f n +1 ( i ) is determined by (cid:88) j : j ∼ i d H ( f n +1 ( i ) , α ij ( f n ( j ))) · (cid:126)e ( f n +1 ( i ) , α ij ( f n ( j ))) = 0 ∀ i. It has been proved by Gaster-Loustau-Monsaingeon [59] that the center of mass method willconverge to the unique discrete harmonic map. But the disadvantage is that there is no directway to compute a hyperbolic mass-center and this iteration is slow. In our numerical exper-iments, we use the so-called cosh-center of mass method introduced also in Gaster-Loustau-Monsaingeon’s paper [59]. Instead of minimizing the exact discrete Dirichlet energy, we min-imize E [ f ] = (cid:88) ij ∈ E (cosh l ij ( f ) − ≈ (cid:88) ij ∈ E l ij ( f ) l ij aresmall. The balanced condition for critical points of the new energy E is (cid:88) j : j ∼ i sinh[ d H ( f ( i ) , α ij ( f ( j )))] · (cid:126)e ( f ( i ) , α ij ( f ( j ))) = 0 ∀ i, and the induced iterating formula is (cid:88) j : j ∼ i sinh[ d H ( f n +1 ( i ) , α ij ( f n ( j )))] · (cid:126)e ( f n +1 ( i ) , α ij ( f n ( j ))) = 0 ∀ i. (11)The miracle is that such cosh -center of mass defined as in equation (11) can be computeddirectly. Using the hyperboloid model H h = { ( x, y, z ) ∈ R : z − x − y = 1 } , the cosh -center of points p , ..., p n ∈ H h is shown, in [59], to be π H (cid:18) p + ... + p n n (cid:19) where ( p + ... + p n ) /n is computed as vectors in R , and π H is the radial projection to H h .In a summary we have the following Algorithm 6 . Algorithm 6
Harmonic Maps to Hyperbolic Surfaces
Input:
A point cloud P in R as an approximation of a surface M , and a hyperbolic surface H / Γ . Output:
A harmonic map f from P to the hyperbolic surface H / Γ . Choose parameters (cid:15) and n , and then compute the lattice approximation G = ( V, E ) . Choose · genus ( M ) ”loops” in G for the cut. Assign a proper deck transformation α i ∈ Γ to each loop γ i . Compute the correction transformations α ij . Do the cosh -center of mass iterations (11), with the trivial initial map f ≡ . Extend f to P by trilinear interpolation in the unit disk. Return f | P . Assume that f Γ is the discrete harmonic map from the lattice approximation G = ( V, E ) to thehyperbolic surface H / Γ . Inspired by part (a) of Theorem 2.5, we compute the conformal mapby minimizing E ( f Γ ) Area ( H / Γ) = E ( f Γ ) − πχ ( H / Γ) = E ( f Γ ) − πχ ( M ) Γ . If ¯Γ is the minimizer, then f ¯Γ : V → H / ¯Γ would approxi-mate a conformal map. Our numerical experiments used Maskit’s nice parameterization of thedeformation space of Γ for genus 2 surfaces [72]. Maskit also proposed methods of parame-terizing for any genus hyperbolic surfaces using Fenchel-Nielsen coordinates [73].Our method is summarized as Algorithm 7 . Algorithm 7
Conformal Maps to Hyperbolic Surfaces
Input:
A point cloud P in R as an approximation of a surface M . Output:
A hyperbolic surface H / Γ and a conformal map f from P to the hyperbolic surface H / Γ . Choose parameters (cid:15) and n , and then compute the lattice approximation G = ( V, E ) . Choose · genus ( M ) ”loops” in G for the cut. For a given hyperbolic surface H / Γ homeomorphic to M , assign a proper deck transfor-mation α i ∈ Γ to each loop γ i . Compute the correction transformations α ij . Do the cosh -center of mass iterations (11), with the trivial initial map f ≡ . Compute the discrete Dirichlet energy E ( f Γ ) . Compute the minimizer ¯Γ of E ( f Γ ) , and f ¯Γ . Extend f ¯Γ to P by trilinear interpolation in the unit disk. Return f ¯Γ | P . Here are numerical examples for harmonic maps and conformal maps on genus 2 surfaces.Figure 8: Harmonic maps to hyperbolic surfaces20igure 9: Conformal maps to hyperbolic surfaces
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