Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theory
EEfficient conformal parameterization of multiply-connectedsurfaces using quasi-conformal theory
Gary P. T. Choi
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USAE-mail: [email protected]
Abstract
Conformal mapping, a classical topic in complex analysis and differential geometry, hasbecome a subject of great interest in the area of surface parameterization in recent decadeswith various applications in science and engineering. However, most of the existing conformalparameterization algorithms only focus on simply-connected surfaces and cannot be directlyapplied to surfaces with holes. In this work, we propose two novel algorithms for computing theconformal parameterization of multiply-connected surfaces. We first develop an efficient methodfor conformally parameterizing an open surface with one hole to an annulus on the plane. Basedon this method, we then develop an efficient method for conformally parameterizing an opensurface with k holes onto a unit disk with k circular holes. The conformality and bijectivity ofthe mappings are ensured by quasi-conformal theory. Numerical experiments and applicationsare presented to demonstrate the effectiveness of the proposed methods. The goal of surface parameterization is to map a surface in R onto a simple standardized domain.Over the past few decades, surface parameterization algorithms have been extensively studied [1–3].In general, any parameterization will unavoidably induce angle and/or area distortions. Therefore,it is common to consider conformal parameterizations , which preserve angles and hence the localgeometry of the surfaces. Existing conformal parameterization methods include harmonic energyminimization [4, 5] and its linearizations [6–8], least-square conformal map (LSCM) [9], discretenatural conformal parameterization (DNCP) [10], holomorphic 1-form [11], Yamabe flow [12],angle-based flattening (ABF) [13, 14], circle patterns [15], discrete conformal equivalence [16], Ricciflow [17–20], spectral conformal map [21], curvature prescription [22], zipper algorithms [23, 24],boundary first flattening [25], conformal energy minimization [26] etc. In recent years, quasi-conformal theory has emerged as a useful tool for the development of surface parameterizationmethods [27, 28] with applications to image and video processing [29, 30], geometry processingand graphics [31–35], metamaterial design [36], medical visualization [37, 38] and biological shapeanalysis [39–41]. However, most of the above-mentioned conformal parameterization methods onlywork for simply-connected surfaces, which do not contain any holes.For multiply-connected surfaces with annulus or poly-annulus topology, the computation ofconformal maps is more complicated. Some earlier works have considered mapping a multiply-1 a r X i v : . [ c s . G R ] S e p igure 1: Conformal parameterizations of multiply-connected surfaces achieved by our proposedmethods. (Left) The conformal parameterization of an open surface with one hole onto an annulusby our Annulus Conformal Map (ACM) algorithm. (Right) The conformal parameterization of amultiply-connected open surface onto a unit disk with circular holes by our Poly-Annulus ConformalMap (PACM) algorithm.connected open surface onto a circular domain with concentric circular slits [42, 43]. Also, bythe Koebe’s uniformization theorem, any multiply-connected open surface with k holes can beconformally mapped to a unit disk with k circular holes [44]. Based on this remarkable result, a fewparameterization algorithms have been developed for multiply-connected open surfaces using Ricciflow [17], holomorphic 1-form [45], Laurent series [46], Beltrami energy minimization [47] etc.In this work, we propose two novel algorithms for computing the conformal parameterizationof multiply-connected surfaces using quasi-conformal theory. We first propose an efficient methodfor conformally mapping an open surface with one hole (i.e. a topological annulus) to an annulusdomain with unit outer radius (Fig. 1, left). We then utilize this method to develop another fastalgorithm for conformally mapping a multiply-connected open surface with k holes (i.e. a topologicalpoly-annulus) to a unit disk with k circular holes (Fig. 1, right). With the aid of quasi-conformaltheory, we can effectively achieve the conformality and bijectivity of the parameterizations.The rest of the paper is organized as follows. In Section 2, we review the concepts of conformaland quasi-conformal maps. In Section 3, we describe our proposed methods for the conformalparameterization of multiply-connected surfaces. In Section 4, we demonstrate the effectiveness ofour parameterization methods using numerical experiments. Applications of the proposed methodsare explored in Section 5. We conclude the paper and discuss possible future directions in Section 6. In this section, we review some mathematical concepts related to our work. Readers are referredto [48–50] for more details. 2 .1 Conformal map
Let f : C → C be a map with f ( z ) = f ( x, y ) = u ( x, y ) + iv ( x, y ), where u, v are real-valued functions. f is said to be a conformal map if it satisfies the Cauchy–Riemann equations: ∂u∂x = ∂v∂y ,∂u∂y = − ∂v∂x . (1) M¨obius transformations are a special class of conformal maps on the complex plane. Mathemati-cally, a M¨obius transformation f : C → C is in the form f ( z ) = az + bcz + d , (2)with a, b, c, d ∈ C satisfying ad − bc (cid:54) = 0. Quasi-conformal maps are a generalization of conformal maps. Mathematically, a mapping f : C → C is said to be a quasi-conformal map if it satisfies the Beltrami equation ∂f∂ ¯ z = µ f ( z ) ∂f∂z (3)for some complex-valued function µ f with (cid:107) µ f (cid:107) ∞ <
1, where the complex derivatives are given by ∂f∂ ¯ z = f ¯ z = 12 (cid:18) ∂f∂x + i ∂f∂y (cid:19) and ∂f∂z = f z = 12 (cid:18) ∂f∂x − i ∂f∂y (cid:19) . (4)Here, µ f is called the Beltrami coefficient of f . Note that if µ f ≡
0, then Eq. (3) becomes theCauchy–Riemann equations (1) and hence f is conformal.Intuitively, conformal mappings map infinitesimal circles to infinitesimal circles, while quasi-conformal mappings map infinitesimal circles to infinitesimal ellipses with bounded eccentricity. Tosee this, consider the first order approximation of f around a point z ∈ C : f ( z ) ≈ f ( z ) + f z ( z )( z − z ) + f z ( z ) z − z = f ( z ) + f z ( z ) ( z − z + µ f ( z ) z − z ) . (5)This indicates that an infinitesimal circle centered at z is mapped to an infinitesimal ellipse centeredat f ( z ), with the maximum magnification | f z ( z ) | (1 + | µ f ( z ) | ) and the maximum shrinkage | f z ( z ) | (1 − | µ f ( z ) | ) (see Fig. 2 for an illustration). The aspect ratio of the ellipse is then given by | µ f ( z ) | −| µ f ( z ) | . Therefore, the Beltrami coefficient effectively captures the conformal distortion of itsassociated mapping.The Beltrami coefficient is also closely related to the bijectivity of the mapping. Note that if3 𝑧 𝑓 𝑓 𝑧 𝑓 𝑓 Figure 2: An illustration of quasi-conformal maps. Under a quasi-conformal map f , an infinitesimalcircle is mapped to an infinitesimal ellipse with the maximum magnification | f z | (1 + | µ f | ) and themaximum shrinkage | f z | (1 − | µ f | ). f ( x, y ) = u ( x, y ) + iv ( x, y ), then the Jacobian J f of f is given by J f = u x v y − v x u y = 14 (cid:0) ( u x + v y ) + ( v x − u y ) − ( u x − v y ) − ( v x + u y ) (cid:1) = 14 (cid:0) | ( u x + iv x ) − i ( u y + iv y ) | − | ( u x + iv x ) + i ( u y + iv y ) | (cid:1) = | f z | − | f ¯ z | = | f z | (1 − | µ f | ) . (6)Therefore, we have the following result: Theorem 1 . If f is a C map satisfying (cid:107) µ f (cid:107) ∞ <
1, then f is bijective.Besides, the Beltrami coefficient of a composition of two quasi-conformal maps can be expressedexplicitly. Let f : Ω → Ω and g : Ω → Ω be two quasi-conformal maps. The Beltrami coefficientof the composition map g ◦ f is given by µ g ◦ f = µ f + ( f z /f z )( µ g ◦ f )1 + ( f z /f z ) µ f ( µ g ◦ f ) . (7)In particular, if µ f − ≡ µ g , we have µ g ◦ f = µ f − ◦ f = − ( f z /f z ) µ f (8)and hence µ g ◦ f ≡ µ f + ( f z /f z )(( − f z /f z ) µ f )1 + ( f z /f z ) µ f (( − f z /f z ) µ f ) ≡ , (9)which implies that the composition map g ◦ f is conformal. This suggests that one can eliminate theconformal distortion of a quasi-conformal map by composing it with another quasi-conformal map4ith the same Beltrami coefficient, provided that the boundary constraint is admissible. This ideaof quasi-conformal composition [6] will be used in our proposed methods for the computation ofconformal parameterizations.While the above concepts are introduced in terms of mappings on the complex plane, they canbe naturally extended for Riemann surfaces with the aid of local charts. Lui et al. [29] developed a linear method called the
Linear Beltrami Solver (LBS) for computinga quasi-conformal map f ( x, y ) = u ( x, y ) + iv ( x, y ) with a given Beltrami coefficient µ ( x, y ) = ρ ( x, y ) + iη ( x, y ). The idea is to consider the real and imaginary parts in the Beltrami equation (3)separately: ρ ( x, y ) + iη ( x, y ) = µ ( x, y ) = ( u x − v y ) + i ( v x + u y )( u x + v y ) + i ( v x − u y ) , (10)from which we can express v x and v y as linear combinations of u x and u y : (cid:26) v y = α u x + α u y , − v x = α u x + α u y , (11)where α = ( ρ − + η − ρ − η , α = − η − ρ − η , α = 1 + 2 ρ + ρ + η − ρ − η . (12)Similarly, we can express u x and u y as linear combinations of v x and v y : (cid:26) − u y = α v x + α v y ,u x = α v x + α v y . (13)Since ( v y ) x + ( − v x ) y = 0 and ( − u y ) x + ( u x ) y = 0, from Eq. (11) and Eq. (13) we have ∇ · (cid:18) A (cid:18) u x u y (cid:19)(cid:19) = 0 and ∇ · (cid:18) A (cid:18) v x v y (cid:19)(cid:19) = 0 , (14)where A = (cid:18) α α α α (cid:19) . In the discrete case, Eq. (14) can be discretized as two sparse symmetricpositive definite linear systems. Therefore, one can easily obtain u x , u y , v x , v y (and hence the quasi-conformal map f ) for any given µ by solving two linear systems with certain boundary constraints(see [29] for details). We denote the above procedure by f = LBS( µ ). Below, we first develop an efficient algorithm for conformally parameterizing an open surface withone hole onto a planar annulus. We then utilize this algorithm to develop another efficient methodfor conformally parameterizing a multiply-connected open surface with k holes onto a unit disk with k circular holes. 5 nput surface Annulus conformal parameterizationRectangular conformal map Exponential map Quasi-conformal composition p qq'p'p q p q qp Figure 3: An illustration of the proposed annulus conformal map (ACM) method for open surfaceswith one hole. We first slice the mesh along a path (highlighted in red) from the inner boundary tothe outer boundary to make the surface open. We then map it onto a rectangle with an optimallength L and unit width. The rectangle is subsequently mapped to an annulus using an exponentialmap. Finally, we compose the map with another quasi-conformal map to achieve a conformalparameterization. Let S be an open surface in R with one hole, i.e. a topological annulus. Denote the surfaceboundary as ∂S = γ − γ , where γ is the outer boundary and γ is the inner boundary. Our goalis to find a conformal parameterization f : S → C that maps S to an annulus on the plane withunit outer radius. The proposed method is outlined in Fig. 3.To begin, we take an arbitrary vertex at the inner boundary γ and find a shortest path from itto the outer boundary γ . By slicing S along the path, we obtain a simply-connected open surface˜ S (see the red curve in Fig. 3, leftmost). Due to the change in the surface topology, it is possible tomap ˜ S onto a planar domain without holes.Now, we consider mapping ˜ S onto a strip conformally (see Fig. 3, second left). Meng et al. [28]developed an efficient rectangular conformal mapping algorithm based on the LBS method [29]. Thealgorithm first computes an initial flattening map of the input surface onto the unit disk. It thenmaps the disk to the unit square using the LBS method. In particular, four boundary vertices arechosen as the four corners of a unit square, and an optimal quasi-conformal map is computed formapping the remaining vertices onto the square domain. Finally, it keeps the length of the squaredomain fixed and optimally rescale the width of it so as to achieve a rectangular conformal map.Here, we follow the approach in [28] with some modifications for obtaining the conformal map ontoa strip.We first compute a disk harmonic map φ : ˜ S → D by solving the following Laplace equation (cid:26) ∆ φ = 0 ,φ ( ∂ ˜ S ) = ∂ D , (15)where the boundary constraint is given by the arc-length parameterization. Here, the Laplacian canbe easily discretized using the cotangent formulation [4]. After flattening the sliced surface onto theunit disk, we compute a quasi-conformal map ψ : D → R = [0 , L ] × [0 ,
1] from the unit disk to arectangular domain with length L and unit width, where L is to be determined. In particular, we6se the LBS method with the target Beltrami coefficient being µ ψ = µ φ − : ψ = LBS ( µ φ − ) , (16)where the four vertices on ∂ ˜ S that correspond to the endpoints of the cut path are set to be thefour corners of the target rectangular domain. More explicitly, denote p, p (cid:48) as the two vertices on ∂ ˜ S that correspond to the endpoint at the inner boundary γ , and q, q (cid:48) as the two vertices on ∂ ˜ S that correspond to the endpoint at the outer boundary γ . The four corners of R are set as follows(see Fig. 3): ψ ( φ ( p )) = (0 , , ψ ( φ ( q )) = ( L, , ψ ( φ ( q (cid:48) )) = ( L, , ψ ( φ ( p (cid:48) )) = (0 , . (17)Note that by Eq. (9), the conformal distortion of the quasi-conformal composition ψ ◦ φ can besignificantly reduced given an appropriate boundary constraint. In the original formulation [28], theboundary vertices are allowed to freely slice along the sides of the rectangular domain to achieveconformality. However, in our case, the top and bottom sides of R correspond to the cut path andare with equal number of corresponding vertices (see the two red curves in Fig. 3, second left). Toenforce their positional consistency, we impose a periodic boundary constraint on the x -coordinatesof the top and bottom boundary vertices. As for the choice of L , we start with an initial guess L = 1and compute the map ψ using Eq. (19). Then, we search for the optimal L which minimizes thenorm of the Beltrami coefficient of ψ ◦ φ to further reduce the conformal distortion.Here we remark that one may look for an extra shear transformation (cid:18) xy (cid:19) (cid:55)→ (cid:18) a (cid:19) (cid:18) xy (cid:19) totransform R into a parallelogram, such that the two bottom corner points do not necessarily havethe same y -coordinates. Theoretically, this can help further reduce the conformal distortion of themapping. However, as the cut path is chosen to be a shortest path, we find that the optimal a isusually very small (with | a | ∼ − ) in our experiments and the improvement in the conformality isnegligible. Therefore, this step can be skipped in practice.After getting the rectangular parameterization ψ ◦ φ , we apply the exponential map η ( z ) = e π ( z − L ) , (18)which maps the rectangular domain [0 , L ] × [0 ,
1] to an annulus with inner radius e − πL and outerradius 1. Because of the periodicity imposed in the computation of the rectangular parameterization,the top and bottom boundaries (i.e. the cut path vertices) are mapped to consistent locations onthe annulus domain. We can then identify every pair of them and obtain a seamless mapping result(see Fig. 3, second right).Finally, we use the quasi-conformal composition to further improve the conformality of theannulus map. Specifically, we compute an automorphism ζ on the annulus ( η ◦ ψ ◦ φ )( S ) with ζ = LBS ( µ ( η ◦ ψ ◦ φ ) − ) , (19)where all boundary vertices are fixed. This results in the final annulus conformal parameterization f = ζ ◦ η ◦ ψ ◦ φ (see Fig. 3, rightmost). We remark that by the quasi-conformal composition, theBeltrami coefficient of the resulting map is with supremum norm less than 1 and hence is bijective.The proposed annulus conformal map (ACM) algorithm is summarized in Algorithm 1.7 lgorithm 1: Annulus conformal map (ACM) for open surfaces with one hole.
Input : An open surface S with annulus topology. Output : A conformal parameterization f : S → C onto an annulus with unit outer radius. Compute a shortest path from an arbitrary vertex at the inner boundary to the outerboundary. Slice the mesh along the path; Compute the disk harmonic map φ for initialization; Compute the rectangular conformal map ψ with a periodic boundary constraint, where thefour corners correspond to the endpoints of the cut path; Apply the exponential map η to obtain an annulus with unit outer radius; Compose the map with another quasi-conformal map ζ to further improve the conformality; The resulting conformal parameterization is given by f = ζ ◦ η ◦ ψ ◦ φ ; k holes Let S be a multiply-connected open surface in R with k holes, i.e. a topological poly-annulus.Denote the surface boundary as ∂S = γ − γ − γ − · · · − γ k , where γ is the outer boundary and γ , · · · , γ k are the inner boundaries. Our goal is to find a conformal parameterization f : S → D that maps S to the unit disk with k circular holes. The proposed method is outlined in Fig. 4.Analogous to the Koebe’s iteration method [44,45], our method handles the k holes of the surface S one by one. We first fill all but the first holes to get a surface S with annulus topology. In practice,one can simply fill a hole by adding a new vertex at the center of the hole and including its one-ringneighborhood of triangular faces. We can then apply the proposed ACM method (Algorithm 1) toobtain an annulus conformal map g : S → C , with γ and γ mapped to the outer circle and theinner circle respectively. After that, we remove all filled regions to restore the surface topology. Here,we remark that Line 5 in Algorithm 1 can be skipped for now as we will apply the quasi-conformalcomposition later at the last step of the poly-annulus parameterization.By repeating the above process for handling all the remaining holes, we obtain the compositionmap g k ◦ g k − ◦ · · · ◦ g . Note that the k -th hole of S is mapped to the center of the unit disk.Since this may not follow the distribution of the holes on S well, the area distortion of the mapmay be large. To alleviate this issue, we use the M¨obius area correction scheme [24] to reduce thearea distortion of the map while preserving conformality, thereby ensuring that the holes are atappropriate locations on the planar domain. More explicitly, we search for an optimal automorphism τ α on the unit disk in the following form: τ α ( z ) = z − α − αz , (20)where α ∈ C with | α | <
1, such that the composition τ α ◦ g k ◦ g k − ◦ · · · ◦ g minimizes the areadistortion of the parameterization with respect to the input surface.As M¨obius transformations map circles and straight lines to circles and straight lines, this stepdoes not change the circularity of the holes in theory. However, in practice, there may be numericalerrors that affect the circularity of the holes. Therefore, we add a step of enforcing the circularity ofthe holes via projections. More specifically, for each hole ( τ α ◦ g k ◦ g k − ◦ · · · ◦ g )( γ i ), we find themaximum inscribed circle of it and project all boundary vertices onto this circle. After performing8 nput surfacePoly-annulus conformal parameterization Annulus conformal mapfor hole 1 Annulus conformal mapfor hole 2 Annulus conformal mapfor hole 3Optimal Möbius transformationQuasi-conformal composition Figure 4: An illustration of the proposed poly-annulus conformal map (PACM) method for multiply-connected open surfaces. We first repeatedly apply the annulus conformal mapping method(Algorithm 1) with all but one holes filled. After handling all holes, we compute an optimal M¨obiustransformation to adjust the location of the holes. Finally, we compose the map with anotherquasi-conformal map to achieve a conformal parameterization.this operation for all k holes, we obtain a unit disk with k circular holes. We denote the process by ρ : D → D .Finally, we use the quasi-conformal composition to further reduce the conformal distortion. Wecompute an automorphism h on the unit disk with the Beltrami coefficient µ ( ρ ◦ τ α ◦ g k ◦ g k − ◦···◦ g ) − using the LBS method [29]: h = LBS( µ ( ρ ◦ τ α ◦ g k ◦ g k − ◦···◦ g ) − ) , (21)where all boundary vertices are fixed. The composition f = h ◦ ρ ◦ τ α ◦ g k ◦ g k − ◦ · · · ◦ g givesa conformal parameterization of S onto the unit disk with exactly k circular holes. Similar toAlgorithm 1, the quasi-conformal composition here also ensures that the Beltrami coefficient of theresulting map is with supremum norm less than 1 and hence is bijective.The proposed conformal parameterization method for poly-annulus surfaces is summarized inAlgorithm 2. The proposed conformal parameterization algorithms are implemented in MATLAB. The linearsystems are solved using the backslash operator ( \ ) in MATLAB. For the step of mesh slicing, we9 lgorithm 2: Poly-annulus conformal map (PACM) for multiply-connected open surfaces.
Input : A multiply-connected open surface S with k ≥ Output : A conformal parameterization f : S → C onto a unit disk with k circular holes. for i = 1 , . . . , k do Fill all but the i -th holes; Solve for an annulus conformal map g i using Algorithm 1; Remove all filled regions; Search for an optimal M¨obius transformation τ α for reducing the area distortion; Enforce the circularity of the holes by a projection step ρ : D → D ; Compose the map with another quasi-conformal map h to improve the conformality; The resulting parameterization is given by f = h ◦ ρ ◦ τ α ◦ g k ◦ g k − ◦ · · · ◦ g ;use the MATLAB function graphshortestpath to compute a shortest path. For the rectangularconformal map in Algorithm 1, we use the MATLAB function fminbnd to search for the optimallength L of the rectangular domain. For the M¨obius transformation in Eq. (20) in Algorithm 2,we write α = re iθ and use the MATLAB function fmincon to search for the optimal parameters( r, θ ) ∈ [0 , × [0 , π ]. For the step of finding maximum inscribed circles of the holes, we use thefunction find inner circle available in the MATLAB Central FileExchange [51]. All experimentsare performed on a PC with a 4.0 GHz quad core CPU and 16 GB RAM.To assess the conformality of a parameterization f : S → C , we consider the angular distortionof every angle [ v i , v j , v k ] on the surface mesh under the parameterization: d ([ v i , v j , v k ]) = ∠ [ f ( v i ) , f ( v j ) , f ( v k )] − ∠ [ v i , v j , v k ] . (22)For an ideal conformal map, we should have d = 0 for all angles.Fig. 5 shows several surfaces with annulus topology and the annulus conformal parameterizationsachieved by our proposed ACM method. Note that our method is capable of handling surfaces witha highly non-convex hole (see the top example) as well as surfaces with a highly tubular geometry(see the bottom example). From the histograms of the angular distortion d , it can be observedthat the parameterizations are highly conformal. For comparison, we consider the Ricci flow (RF)method [17] with implementation available in the RiemannMapper toolbox [52]. Table 1 records thecomputation time and the conformal distortion of our proposed ACM method and the RF method,from which it can be observed that our method outperforms the RF method in both the conformalityand efficiency. The improvement in the conformality can be explained by that the RF method isbased on the circle packing metric, which is highly dependent on the quality of the triangulations.For surface meshes with coarse or irregular triangulations, the approximation of the metric maybe inaccurate, thereby yielding a large conformal distortion in the parameterization results. Bycontrast, our method effectively reduces the conformal distortion using the idea of quasi-conformalcomposition. As for the improvement in the efficiency, note that the RF method uses a gradientdescent approach for minimizing the Ricci energy and hence is time-consuming, while our methodonly involves solving a few linear systems and a one-dimensional optimization problem.Fig. 6 shows the poly-annulus conformal parameterizations of several multiply-connected opensurfaces achieved by our proposed PACM method. It can be observed that our method works wellfor surfaces with different size, shape, and number of holes. For a more quantitative analysis, weagain compare our method with the RF method in terms of the computation time and the angular10
150 -100 -50 0 50 100 150
Angle difference (degree) N u m be r o f ang l e s Angular Distortion
Figure 5: Examples of the annulus conformal parameterization achieved by our proposed ACMmethod (Algorithm 1). Left: The input open surfaces with annulus topology. Middle: The annulusconformal parameterization results. Right: The histograms of the angular distortion d .distortion. As shown in Table 2, our method achieves a significant improvement in both the efficiencyand conformality when compared to the RF method. Therefore, our method is more advantageousfor the computation of the poly-annulus conformal parameterization.11
150 -100 -50 0 50 100 150
Angle difference (degree) N u m be r o f ang l e s Angular Distortion -150 -100 -50 0 50 100 150
Angle difference (degree) N u m be r o f ang l e s Angular Distortion -150 -100 -50 0 50 100 150
Angle difference (degree) N u m be r o f ang l e s Angular Distortion
Figure 6: Examples of the poly-annulus conformal parameterization achieved by our proposed PACMmethod (Algorithm 2). Left: The input multiply-connected open surfaces with k holes. Middle: Thepoly-annulus conformal parameterization results. Right: The histograms of the angular distortion d . The proposed conformal parameterization methods can be effectively applied to texture mapping.Using our methods, any multiply-connected open surface in R can be conformally mapped onto aunit disk with circular holes. Textures can then be designed on the plane and mapped back onto12urface | d | ) Time (s) Mean( | d | )Amoeba1 (Fig. 3) 7K 0.3 1.1 4.7 21.5Amoeba2 (Fig. 5) 7K 0.3 4.0 4.5 21.6Niccol`o (Fig. 5) 10K 0.3 1.5 5.9 18.6Sophie (Fig. 1) 21K 1.0 0.6 15.3 9.5Lion vase (Fig. 5) 25K 1.1 3.2 17.0 26.3Table 1: Performance of the proposed ACM method (Algorithm 1) and the Ricci flow (RF)method [17] for the annulus conformal parameterization of open surfaces with annulus topology.Here, the angular distortion d is evaluated using Eq. (22).Surface | d | ) Time (s) Mean( | d | )David (Fig. 7) 2 25K 2.6 0.8 16.6 13.8Alex (Fig. 1) 3 14K 2.0 1.3 10.5 13.1Face (Fig. 8) 3 1K 0.2 4.4 0.8 14.1Lion (Fig. 6) 5 17K 3.5 6.8 12.4 10.2Peaks (Fig. 6) 7 2K 0.4 5.1 1.4 15.8Twisted hemisphere(Fig. 7) 8 25K 8.8 7.6 22.0 9.1Amoeba (Fig. 6) 10 7K 2.0 4.3 5.8 21.7Table 2: Performance of the proposed PACM method (Algorithm 2) and the Ricci flow (RF)method [17] for the poly-annulus conformal parameterization of multiply-connected open surfaces.Here, the angular distortion d is evaluated using Eq. (22).the surface easily. As our methods are angle-preserving, the local geometry of the designed textureswill be well-preserved. Also, as our methods produce global parameterizations of the surfaces, thetexture mapping results will be seamless. Fig. 7 shows two texture mapping results produced usingour parameterization methods. The orthogonality of the checkerboard patterns on the surfacesindicates that our parameterizations are highly conformal. The proposed conformal parameterization methods can also be applied to surface remeshing. Supposewe would like to improve the mesh quality of a given multiply-connected surface. A simple way is tomap it onto the unit disk with circular holes using our parameterization methods, and then performthe remeshing process on the plane.In particular, DistMesh [53] is a powerful toolbox for generating triangular meshes, which usessigned distance functions to specify the geometry of the domain and control the mesh quality. Inour case, the disk domain with circular holes can be easily expressed as a difference between signeddistance functions for several circles using the ddiff and dcircle functions in DistMesh. Thetarget mesh quality is set using a scaled edge length function in DistMesh, which can also be easily13igure 7: Texture mapping for multiply-connected surfaces achieved by our proposed conformalparameterization methods. We first compute the conformal parameterization of an input multiply-connected surface onto a planar domain using our proposed methods. Then, we can design textures onthe planar domain and map them back onto the input surface with the local geometry well-preserved.controlled using the signed distance functions for circles. Therefore, our parameterization methodscan be naturally combined with DistMesh for the remeshing task. Once a new planar mesh isgenerated, we can map it back to the surface via the parameterization. Fig. 8 shows two examplesof remeshing a multiply-connected human face surface, from which it can be observed that differentremeshing effects can be easily achieved. As the parameterization is angle-preserving, the regularityof the triangles in the new planar meshes is well-preserved in the final remeshed surfaces.
Another possible application of the proposed conformal parameterization methods is multiply-connected surface registration [54]. As shown in Fig. 9, given two multiply-connected open surfaces(denoted as S and S ) with the same topology, we can first compute the poly-annulus conformalparameterizations (denoted as f and f ) of them using our proposed methods. Then, we cancompute a quasi-conformal map g between the two planar domains f ( S ) and f ( S ) with allcorresponding holes exactly matched using the LBS method. The composition f − ◦ g ◦ f thengives a registration mapping between the surfaces S and S . From the final registration result inFig. 9, it can be observed that the two multiply-connected surfaces are matched very well. With the advancement in computer technology, there has been a surge of interest in the developmentof conformal parameterization algorithms for science and engineering applications in recent decades.However, most of the existing methods only work for simply-connected surfaces. In this work, we haveproposed two novel algorithms for the conformal parameterization of multiply-connected surfacesonto either an annulus or a unit disk with circular holes using quasi-conformal theory. As thereare a vast number of analytical and numerical conformal mapping methods for multiply-connectedplanar domains [55–59], the proposed parameterization algorithms pave the way for applying thesemethods to multiply-connected surfaces.Besides the applications discussed in this work, it is natural to explore the use of the proposed14igure 8: Remeshing a multiply-connected open surface via our proposed conformal parameterizationmethods. Given a multiply-connected open surface (top left), we first compute the poly-annulusconformal parameterization using our proposed methods (bottom left). We can then generatetriangular meshes on the poly-annulus domain using DistMesh [53] with different desired effects,such as having finer triangulations at the central part (bottom middle) or around one of the holes(bottom right). The new planar meshes can then be mapped back onto the given surface via theparameterization (top middle and top right).conformal parameterization methods for shape analysis [60,61], greedy routing in sensor networks [62]etc. Another possible future direction is to combine the proposed methods with the optimal masstransport (OMT) [63–65] or the density-equaling map (DEM) [66, 67] for efficiently computingarea-preserving parameterizations of multiply-connected surfaces.
References [1] M. S. Floater and K. Hormann, “Surface parameterization: A tutorial and survey,” in
Advancesin multiresolution for geometric modelling , pp. 157–186, Springer, 2005.[2] A. Sheffer, E. Praun, and K. Rose, “Mesh parameterization methods and their applications,”
Found. Trends Comput. Graph. Vis. , vol. 2, no. 2, pp. 105–171, 2006.15 urface 1 Poly-annulus conformal parameterization f Surface 2 Quasi-conformal map g Poly-annulus conformal parameterization f Inverse mapping f -1 Figure 9: Registration of multiply-connected surfaces via the proposed conformal parameterizationmethods. Given two multiply-connected surfaces with the same topology, we can first conformallyparameterize them onto two unit disk domains with the same number of circular holes. We canthen compute a quasi-conformal map between the two planar domains, with all corresponding holesexactly matched. Finally, we can map the planar mapping result back onto the target surface toobtain the final registration result.[3] K. Hormann, B. L´evy, and A. Sheffer, “Mesh parameterization: Theory and practice,”
ACMSIGGRAPH 2007 Courses , 2007.[4] U. Pinkall and K. Polthier, “Computing discrete minimal surfaces and their conjugates,”
Exp.Math. , vol. 2, no. 1, pp. 15–36, 1993.[5] X. Gu, Y. Wang, T. F. Chan, P. M. Thompson, and S.-T. Yau, “Genus zero surface conformalmapping and its application to brain surface mapping,”
IEEE Trans. Med. Imaging , vol. 23,no. 8, pp. 949–958, 2004.[6] P. T. Choi, K. C. Lam, and L. M. Lui, “FLASH: Fast landmark aligned spherical harmonicparameterization for genus-0 closed brain surfaces,”
SIAM J. Imaging Sci. , vol. 8, no. 1,pp. 67–94, 2015.[7] G. P.-T. Choi, K. T. Ho, and L. M. Lui, “Spherical conformal parameterization of genus-0 pointclouds for meshing,”
SIAM J. Imaging Sci. , vol. 9, no. 4, pp. 1582–1618, 2016.[8] G. P.-T. Choi and L. M. Lui, “A linear formulation for disk conformal parameterization ofsimply-connected open surfaces,”
Adv. Comput. Math. , vol. 44, no. 1, pp. 87–114, 2018.[9] B. L´evy, S. Petitjean, N. Ray, and J. Maillot, “Least squares conformal maps for automatictexture atlas generation,”
ACM Trans. Graph. , vol. 21, no. 3, pp. 362–371, 2002.1610] M. Desbrun, M. Meyer, and P. Alliez, “Intrinsic parameterizations of surface meshes,”
Comput.Graph. Forum , vol. 21, no. 3, pp. 209–218, 2002.[11] X. Gu and S.-T. Yau, “Global conformal surface parameterization,” in
Proceedings of the 2003Eurographics/ACM SIGGRAPH Symposium on Geometry Processing , pp. 127–137, 2003.[12] F. Luo, “Combinatorial Yamabe flow on surfaces,”
Commun. Contemp. Math. , vol. 6, no. 05,pp. 765–780, 2004.[13] A. Sheffer and E. de Sturler, “Parameterization of faceted surfaces for meshing using angle-basedflattening,”
Eng. Comput. , vol. 17, no. 3, pp. 326–337, 2001.[14] A. Sheffer, B. L´evy, M. Mogilnitsky, and A. Bogomyakov, “ABF++: Fast and robust anglebased flattening,”
ACM Trans. Graph. , vol. 24, no. 2, pp. 311–330, 2005.[15] L. Kharevych, B. Springborn, and P. Schr¨oder, “Discrete conformal mappings via circle patterns,”
ACM Trans. Graph. , vol. 25, no. 2, pp. 412–438, 2006.[16] B. Springborn, P. Schr¨oder, and U. Pinkall, “Conformal equivalence of triangle meshes,”
ACMTrans. Graph. , vol. 27, no. 3, pp. 1–11, 2008.[17] M. Jin, J. Kim, F. Luo, and X. Gu, “Discrete surface Ricci flow,”
IEEE Trans. Vis. Comput.Graph. , vol. 14, no. 5, pp. 1030–1043, 2008.[18] Y.-L. Yang, R. Guo, F. Luo, S.-M. Hu, and X. Gu, “Generalized discrete Ricci flow,”
Comput.Graph. Forum , vol. 28, no. 7, pp. 2005–2014, 2009.[19] Y.-L. Yang, J. Kim, F. Luo, S.-M. Hu, and X. Gu, “Optimal surface parameterization usinginverse curvature map,”
IEEE Trans. Vis. Comput. Graph. , vol. 14, no. 5, pp. 1054–1066, 2008.[20] M. Zhang, W. Zeng, R. Guo, F. Luo, and X. D. Gu, “Survey on discrete surface Ricci flow,”
J.Comput. Sci. Technol. , vol. 30, no. 3, pp. 598–613, 2015.[21] P. Mullen, Y. Tong, P. Alliez, and M. Desbrun, “Spectral conformal parameterization,”
Comput.Graph. Forum , vol. 27, no. 5, pp. 1487–1494, 2008.[22] M. Ben-Chen, C. Gotsman, and G. Bunin, “Conformal flattening by curvature prescription andmetric scaling,”
Comput. Graph. Forum , vol. 27, no. 2, pp. 449–458, 2008.[23] D. E. Marshall and S. Rohde, “Convergence of a variant of the zipper algorithm for conformalmapping,”
SIAM J. Numer. Anal. , vol. 45, no. 6, pp. 2577–2609, 2007.[24] G. P. T. Choi, Y. Leung-Liu, X. Gu, and L. M. Lui, “Parallelizable global conformal param-eterization of simply-connected surfaces via partial welding,”
SIAM J. Imaging Sci. , vol. 13,no. 3, pp. 1049–1083, 2020.[25] R. Sawhney and K. Crane, “Boundary first flattening,”
ACM Trans. Graph. , vol. 37, no. 1,pp. 1–14, 2017.[26] M.-H. Yueh, W.-W. Lin, C.-T. Wu, and S.-T. Yau, “An efficient energy minimization forconformal parameterizations,”
J. Sci. Comput. , vol. 73, no. 1, pp. 203–227, 2017.1727] P. T. Choi and L. M. Lui, “Fast disk conformal parameterization of simply-connected opensurfaces,”
J. Sci. Comput. , vol. 65, no. 3, pp. 1065–1090, 2015.[28] T. W. Meng, G. P.-T. Choi, and L. M. Lui, “TEMPO: Feature-endowed Teich ¨muller extremalmappings of point clouds,”
SIAM J. Imaging Sci. , vol. 9, no. 4, pp. 1922–1962, 2016.[29] L. M. Lui, K. C. Lam, T. W. Wong, and X. Gu, “Texture map and video compression usingBeltrami representation,”
SIAM J. Imaging Sci. , vol. 6, no. 4, pp. 1880–1902, 2013.[30] C. P. Yung, G. P. T. Choi, K. Chen, and L. M. Lui, “Efficient feature-based image registrationby mapping sparsified surfaces,”
J. Vis. Commun. Image Represent. , vol. 55, pp. 561–571, 2018.[31] Y. Lipman, “Bounded distortion mapping spaces for triangular meshes,”
ACM Trans. Graph. ,vol. 31, no. 4, pp. 1–13, 2012.[32] O. Weber, A. Myles, and D. Zorin, “Computing extremal quasiconformal maps,”
Comput.Graph. Forum , vol. 31, no. 5, pp. 1679–1689, 2012.[33] T. W. Wong and H.-k. Zhao, “Computation of quasi-conformal surface maps using discreteBeltrami flow,”
SIAM J. Imaging Sci. , vol. 7, no. 4, pp. 2675–2699, 2014.[34] L. M. Lui, K. C. Lam, S.-T. Yau, and X. Gu, “Teichm¨uller mapping (T-map) and its applicationsto landmark matching registration,”
SIAM J. Imaging Sci. , vol. 7, no. 1, pp. 391–426, 2014.[35] G. P.-T. Choi, M. H.-Y. Man, and L. M. Lui, “Fast spherical quasiconformal parameterizationof genus-0 closed surfaces with application to adaptive remeshing,”
Geom. Imaging Comput. ,vol. 3, no. 1, pp. 1–29, 2016.[36] G. P. T. Choi, L. H. Dudte, and L. Mahadevan, “Programming shape using kirigami tessellations,”
Nat. Mater. , vol. 18, no. 9, pp. 999–1004, 2019.[37] W. Zeng, J. Marino, K. C. Gurijala, X. Gu, and A. Kaufman, “Supine and prone colonregistration using quasi-conformal mapping,”
IEEE Trans. Vis. Comput. Graph. , vol. 16, no. 6,pp. 1348–1357, 2010.[38] G. P. T. Choi, Y. Chen, L. M. Lui, and B. Chiu, “Conformal mapping of carotid vessel walland plaque thickness measured from 3D ultrasound images,”
Med. Biol. Eng. Comput. , vol. 55,no. 12, pp. 2183–2195, 2017.[39] G. P. T. Choi and L. Mahadevan, “Planar morphometrics using Teichm¨uller maps,”
Proc. R.Soc. A , vol. 474, no. 2217, p. 20170905, 2018.[40] G. P. T. Choi, H. L. Chan, R. Yong, S. Ranjitkar, A. Brook, G. Townsend, K. Chen, and L. M.Lui, “Tooth morphometry using quasi-conformal theory,”
Pattern Recognit. , vol. 99, p. 107064,2020.[41] G. P. T. Choi, D. Qiu, and L. M. Lui, “Shape analysis via inconsistent surface registration,”
Proc. R. Soc. A , to appear.[42] X. Yin, J. Dai, S.-T. Yau, and X. Gu, “Slit map: Conformal parameterization for multiplyconnected surfaces,” in
International Conference on Geometric Modeling and Processing , pp. 410–422, Springer, 2008. 1843] Y. Wang, X. Gu, T. F. Chan, P. M. Thompson, and S.-T. Yau, “Conformal slit mapping andits applications to brain surface parameterization,” in
International Conference on MedicalImage Computing and Computer-Assisted Intervention , pp. 585–593, Springer, 2008.[44] P. Koebe, “ ¨Uber die konforme abbildung mehrfach zusammenh¨angender bereiche.,”
Jahresberichtder Deutschen Mathematiker-Vereinigung , vol. 19, pp. 339–348, 1910.[45] W. Zeng, X. Yin, M. Zhang, F. Luo, and X. Gu, “Generalized Koebe’s method for conformalmapping multiply connected domains,” in , pp. 89–100, 2009.[46] E. Kropf, X. Yin, S.-T. Yau, and X. D. Gu, “Conformal parameterization for multiply connecteddomains: Combining finite elements and complex analysis,”
Eng. Comput. , vol. 30, no. 4,pp. 441–455, 2014.[47] K. T. Ho and L. M. Lui, “QCMC: Quasi-conformal parameterizations for multiply-connecteddomains,”
Adv. Comput. Math. , vol. 42, no. 2, pp. 279–312, 2016.[48] O. Lehto,
Quasiconformal mappings in the plane , vol. 126. Springer-Verlag Berlin Heidelberg,1973.[49] F. P. Gardiner and N. Lakic,
Quasiconformal Teichm¨uller theory , vol. 76. American Mathemat-ical Society, 2000.[50] L. V. Ahlfors,
Lectures on quasiconformal mappings , vol. 38. American Mathematical Society,2006.[51] T. Birdal, “Maximum inscribed circle using Voronoi dia-gram.” .[52] X. Gu, “RiemannMapper: A mesh parameterization toolkit.” .[53] P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,”
SIAM Rev. , vol. 46,no. 2, pp. 329–345, 2004.[54] T. C. Ng, X. Gu, and L. M. Lui, “Teichm¨uller extremal map of multiply-connected domainsusing Beltrami holomorphic flow,”
J. Sci. Comput. , vol. 60, no. 2, pp. 249–275, 2014.[55] D. Crowdy, “The Schwarz–Christoffel mapping to bounded multiply connected polygonaldomains,”
Proc. R. Soc. A , vol. 461, no. 2061, pp. 2653–2678, 2005.[56] D. Crowdy and J. Marshall, “Conformal mappings between canonical multiply connecteddomains,”
Comput. Meth. Funct. Th. , vol. 6, no. 1, pp. 59–76, 2006.[57] D. Crowdy, “Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions,”
Math. Proc. Camb. Phil. Soc. , vol. 142, no. 2, p. 319, 2007.[58] M. M. S. Nasser, “Numerical conformal mapping via a boundary integral equation with thegeneralized Neumann kernel,”
SIAM J. Sci. Comput. , vol. 31, no. 3, pp. 1695–1715, 2009.1959] M. M. S. Nasser, “PlgCirMap: A MATLAB toolbox for computing conformal mappings frompolygonal multiply connected domains onto circular domains,”
SoftwareX , vol. 11, p. 100464,2020.[60] W. Zeng, L. M. Lui, X. Gu, and S.-T. Yau, “Shape analysis by conformal modules,”
MethodsAppl. Anal. , vol. 15, no. 4, pp. 539–556, 2008.[61] J. Zhao, X. Qi, C. Wen, N. Lei, and X. Gu, “Automatic and robust skull registration based ondiscrete uniformization,” in
Proceedings of the IEEE International Conference on ComputerVision , pp. 431–440, 2019.[62] S. Li, W. Zeng, D. Zhou, X. Gu, and J. Gao, “Compact conformal map for greedy routing inwireless mobile sensor networks,”
IEEE Trans. Mobile Comput. , vol. 15, no. 7, pp. 1632–1646,2015.[63] X. Zhao, Z. Su, X. D. Gu, A. Kaufman, J. Sun, J. Gao, and F. Luo, “Area-preservationmapping using optimal mass transport,”
IEEE Trans. Vis. Comput. Graph. , vol. 19, no. 12,pp. 2838–2847, 2013.[64] K. Su, L. Cui, K. Qian, N. Lei, J. Zhang, M. Zhang, and X. D. Gu, “Area-preserving meshparameterization for poly-annulus surfaces based on optimal mass transportation,”
Comput.Aided Geom. Des. , vol. 46, pp. 76–91, 2016.[65] A. Pumarola, J. Sanchez-Riera, G. P. T. Choi, A. Sanfeliu, and F. Moreno-Noguer, “3DPeople:Modeling the geometry of dressed humans,” in
Proceedings of the IEEE International Conferenceon Computer Vision , pp. 2242–2251, 2019.[66] G. P. T. Choi and C. H. Rycroft, “Density-equalizing maps for simply connected open surfaces,”
SIAM J. Imaging Sci. , vol. 11, no. 2, pp. 1134–1178, 2018.[67] G. P. T. Choi, B. Chiu, and C. H. Rycroft, “Area-preserving mapping of 3D carotid ultrasoundimages using density-equalizing reference map,”