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Optimal Global Conformal Surface Parameterization
Miao Jin∗
Computer Science Department SUNY at Stony Brook
Yalin Wang†
Math Department UCLA
Shing-Tung Yau‡
Math Department Harvard University
Xianfeng Gu§
Computer Science Department SUNY at Stony Brook
Figure 1: Uniform global conformal parameterization ((a) and (b)) and region emphasized conformal parameterization ((c) and (d)). (a). Least uniform conformal parameterization with energy: 21. 208 e − 5. (b). Most uniform conformal parameterization with energy: 3. 685 e − 5. (c). Maximizing the parameter area of the left half surface (with percentage: 83 .48%). (d). Maximizing the parameter area of the right half surface (with percentage: 82 .58%.)
ABSTRACT
All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenom- ena, such as heat diffusion and electro-magnetic fields on the sur- face. A good parameterization is crucial for simulation and visual- ization. This paper provides an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algo- rithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implemen- tation is based on a finite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserves an- gular structure, and can play an important role in various applica- tions including texture mapping, remeshing, morphing and simu- lation. The method is demonstrated by visualizing the Riemann surface structure of real surfaces represented as triangle meshes.
CR Categories: I.3.5 [Computational Geometry and Object Mod- eling]: Curve, surface, solid, and object representations—Surface Parameterization
Keywords: Computational geometry and object modeling; Curve, surface, solid, and object representations; Surface parameterization.
1 INTRODUCTION
∗e-mail: [email protected] †e-mail:[email protected] ‡e-mail:[email protected] §e-mail:[email protected]
Surface parameterization is the process of mapping a surface to a planar domain and has many applications in various fields of sci- ence and engineering, including texture mapping, geometric mor- phing, surface matching, surface remeshing, and surface extrapo- lation. For example, texture mapping can be used to enhance the visual quality and generate different visual results. Geometric mor- phing can be used to generate vivid animation results. Essentially, surface parametrization can convert a 3D geometric problem to 2D, thereby improving the efficiency and simplifying the computation. Conformal surface parameterizations have many merits: they preserve angular structure, are intrinsic to geometry, and are stable with respect to different triangulations and small deformations. It has been widely used for many applications, such as non-distorted texture mapping [23], [16],[20], surface remeshing [1], surface fair- ing [22], surface matching [14], brain mapping [2], [13] etc. It is desirable to parameterize surfaces globally without any seams. The existence of global conformal parameterization is a non-trivial fact. This is equivalent to the fact that all orientable sur- faces are Riemann surfaces. The atlas formed by the global confor- mal parameterization is the so-called conformal structure. Confor- mal structure is a fundamental structure between metric structure and topological structure and governs many natural physical phe- nomena. The abstract concept of a Riemann surface can also be visualized by texture mapping special patterns using global confor- mal parameterizations. This is the only means of visually conveying conformal information of surfaces. The early work of global conformal parameterization has been done in [14, 15], where the basis for all possible global conformal parameterizations are computed. Because global conformal param- eterization is non-unique, the problem of finding the optimal one remains open. This paper introduces an explicit method to find the optimal global conformal parameterizations of arbitrary surfaces. First, the metrics for measuring the quality of conformal parameterizations are designed. Second, the major factors affecting the quality of the parameterization are summarized. Then, algorithms are developed to modify the topology, locate the zero points, and determine the cohomology types of the differential forms. The method is based
October 10-15, Austin, Texas, USA
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on finite dimensional optimization and demonstrated by visualizing the Riemann surface structure of real surfaces.
1.1 Contributions
This paper introduces algorithms to optimize global conformal pa- rameterizations. The method is based on Riemann surface theories and differential geometry. Therefore, it is rigorous and general. The optimization algorithms can be generalized to all parameterization methods based on convex combinations [10]. Our main contribu- tions are listed as follows.
1 We introduce energy functionals on the space of complex- valued holomorphic mappings.
2 In [10], the author raised the following open question: ”Un- der what boundary condition is a harmonic map between two topo- logical disks conformal?” We answer this question in an algorith- mic way. We compute the double covering of a topological disk (double covering means gluing two copies of the same surface along their boundaries to form a closed symmetric surface; details are described in [15]), and conformally map the double covering to a sphere preserving the symmetry. Thus the disk itself is mapped to a hemisphere. Then a conformal map between two disks is induced by their mappings to the same hemisphere. The boundary condition which makes a harmonic map conformal can be computed using this algorithm directly.
3 The difference between the zero points of a conformal pa- rameterization and singularities of general vector fields is that the zero points cannot be arbitrarily assigned and are determined by the conformal structure. To the best of our knowledge, this state- ment has never been addressed in computer graphics, although it is the major topological obstruction for any surface parameterization method.
4 We propose a way of finding the optimal M¨obius transform that best balances area deformations (note that conformality is in- variant through M¨obius transforms.)
5 This paper explains the following fact: the area stretching factor increases exponentially at the tip of long tubes and it is true for all other parameterization methods. This shows the limits of cur- rent parameterization techniques and justifies topological modifica- tion techniques proposed in this paper. Although some researchers reached the same conclusion by heuristic methods, a rigorous proof is given in this paper.
1.2 Related Work
Surface conformal parameterization algorithms have been thor- oughly studied in the literature. We summarize them according to the topologies of surfaces that they can handle.
Conformal map for topological disks Many researchers propose methods to build a conformal map for topological disks. Pinkall and Polthier derive the discrete Dirichlet energy in [25]. Eck et al. [8] introduce the discrete harmonic map, which approx- imates the continuous harmonic maps by minimizing a metric dis- persion criterion. Duchamp formulates the hierarchical harmonic embedding in [7]. Floater introduces a shape-preserving method in [9], which is very similar to harmonic maps for planar surfaces. Sheffer and de Sturler introduce angle based flattening to compute conformal maps. Desbrun et al. [1, 6] compute the discrete Dirich- let energy and apply conformal parameterization to interactive ge- ometry remeshing. Levy et al.[23] compute a quasi-conformal parameterization by approximating the Cauchy-Riemann equation using the least square method. The above two formulations are
equivalent. Hormann and Greiner propose the MIPS parameteri- zation [18], which roughly attempts to preserve the ratio of singu- lar values over the parameterization. Degener et al. [5] extend the method in [18] and provide a control parameter that allows for me- diation between angle and area distortion.
Conformal map for genus zero closed surfaces Haker et al. [16] introduce a method to compute a global conformal mapping from a genus zero surface to a sphere by representing the Laplace- Beltrami operator as a linear system. Gu et al.[14] introduce a non- linear optimization method to compute global conformal parame- terizations for genus zero surfaces. The optimization is carried out in the tangential spaces of a sphere.
Conformal map for high genus surfaces Few researchers report their work on surfaces with complicated topology. Gu and Yau introduce algorithms to compute conformal structures deter- mined by the metric for general closed surfaces in [14]. The pro- posed method approximates De Rham cohomology by simplicial cohomology and computes a basis of holomorphic 1-forms. Later the method is generalized for surfaces with boundaries in [15]. In [27, 24, 28], the Riemann surface structure is defined for com- binatorial meshes. Because the metric information is ignored in their work, their methods cannot be applied to our problems di- rectly.
2 SKETCH OF MATHEMATICAL THEORIES AND ALGO-
RITHM OVERVIEW
This section introduces the basic concepts in Riemann surface the- ory related to global conformal parameterization and an overview of the optimization algorithms.
2.1 Theoretic Background
The basic concepts of Riemann surface theories are briefly sketched. Further details can be found in [19], [12] and [26].
Conformal Chart Let U be an open set of S ∈ R^3. A pa- rameterization of U is a one to one map z : U → R^2 , which maps U to the (u, v) plane. (U, z) is called a chart of S. In the case of conformal chart, the first fundamental form satisfies: ds^2 = λ (u, v)^2 (du^2 + dv^2 ), where λ (u, v) is called the stretch factor, a function that scales the metric at each point (u, v). The coordinate pair (u, v) is called a conformal parameter of the surface patch U. (U, z) is called a conformal chart of S.
Conformal Atlas All oriented metric surfaces are Riemann surfaces and have a global conformal atlas, or a set of conformal charts. In the following discussion, we treat R^2 as a complex plane, where the point (u, v) is equivalent to z = u + iv, and (u, −v) is equivalent to ¯z = u − iv. In later sections, we use both representa- tions interchangeably. Let S be a surface in R^3 with an atlas {(U α , z α )}, where (U α , z α ) is a chart, and z α : U α → C maps an open set U α ⊂ S to the complex plane C. The atlas is called conformal if (1). each chart (U α , z α ) is a conformal chart. Namely, on each chart, the first fundamental form can be formulated as ds^2 = λ (z α )^2 dz α d z¯ α , (2). the transition maps z β ◦ z− α 1 : z α (U α ∩U β ) → z β (U α ∩U β ) are holomorphic. A chart is compatible with a given conformal atlas if adding it to the atlas again yields a conformal atlas. A conformal structure ( Riemann surface structure ) is obtained by adding all compatible charts to a conformal atlas. A Riemann surface is a surface with a conformal structure.
The computation process for { ωi, i = 1 , 2 , · · · , 2 g} can be sum- marized as computing the homology basis, cohomology basis, har- monic one-form basis and holomorphic one-form basis. Double covering techniques are applied to surfaces with bound- aries to convert them to closed symmetric surfaces. Therefore, in the following discussion, we assume the surfaces are closed.
3.2 Metrics for parameterization
In order to convert the whole mesh to a geometry image or spline surface patches, parameterizations with high uniformity are pre- ferred. It is often desirable to allocate more parameter areas for special regions on the surface in real applications. For example, in surface remeshing, more samples are required for regions with high Gaussian curvature or sharp features. Sometimes, multi-chart ge- ometry images are used to represent the shape. In this case, we can use several global parameterizations, each of which will emphasize a surface region and convert it to one chart in the geometry image. Also in this scenario, the parameterization emphasizing different regions are also desirable. For high genus surfaces, the existence of zero points is unavoidable, and the neighborhoods of zero points will be under sampled in the parameter domain. Therefore, users would like to assign the zero points to positions that have lower curvature or are less visible. In order to allocate zero points at the prescribed positions, we design a special metric to measure the pa- rameter area of the neighborhoods of the given points. If the pa- rameterization of zero points places them at the desired positions, this metric will be close to zero. Suppose Ω ⊂ R^2 is the parameter domain for a surface S and (u, v) are parameters on Ω. Then the functional for measuring uni- formity is
E =
∫
Ω
( λ (u, v)^2 − 1 )^2 dudv, (1)
where λ is the conformal factor, subject to
∫
Ω
dudv =
∫
Ω
λ (u, v)^2 dudv. (2)
Similarly, suppose Ω is divided into two regions Ω 1 and Ω 2 , and we would like to emphasize Ω 1. Then the functional is
E =
∫
Ω 1
λ (u, v)^2 dudv, (3)
subject to ∫
Ω 1 ∪Ω 2
λ (u, v)^2 dudv =
∫
Ω
dudv. (4)
For high genus surfaces, if we want to assign zero points for a global conformal parameterization, different functionals should be formulated to minimize the conformal factor at the desired points. Suppose we want to assign {p 1 , p 2 , · · · , pn} ⊂ S as zero points, where Ui ⊂ Ω is a neighborhood of pi, and ω is a holomorphic 1-form. We define the functional as
E( ω) =
n ∑ i= 1
∫
Ui
ω ∧ ω¯, (5)
∧ represents the wedge product between holomorphic 1-forms. In- tuitively, this functional measures the area of the neighborhoods of zero points on the parameter domain. If there is a holomorphic 1- form ω with zero points at all pi’s, then its E( ω) should be close to zero.
3.3 Optimal Holomorphic 1-form for High Genus Surface
A global conformal parameterization for a high genus surface can be obtained by integrating a holomorphic one form ω. Suppose { ωi, i = 1 , 2 , · · · , 2 g} is a holomorphic 1-form basis, where an ar- bitrary holomorphic 1-form has the formula ω = (^) ∑^2 i=g 1 λi ωi. The energy for the parameterization is denoted E( ω), which is a func- tion of the linear combination of coefficients λi. The necessary condition for the optimal holomorphic 1-form is straightforward, ∂ E ∂ λi =^0 ,^ i^ =^1 ,^2 ,^ · · ·^ ,^2 g. If the Hessian matrix^ (
∂ 2 E ∂ λi ∂ λ (^) j )^ is positive definite, then E will reach the minimum. If the Hessian matrix is negative definite, E will be maximized. The traditional Newton’s method can be applied for the optimization with the constraint that the total area in the parameter domain is fixed.
3.3.1 Uniform Global Conformal Parameterization
Given any holomorphic one-form ω, ω = ∑ 2 g k= 1 λk^ ωk^ , we require the total parameter area to be equal to the total area of the surface in R^3 ,
∑ [v 0 ,v 1 ,v 2 ]∈K 2
| ω([v 0 , v 1 ]) × ω([v 1 , v 2 ])| = (^) ∑ [v 0 ,v 1 ,v 2 ]∈K 2
S[v 0 ,v 1 ,v 2 ], (6)
where S[v 0 ,v 1 ,v 2 ] is the area of face [v 0 , v 1 , v 2 ] in R^3. The uniformity functional is defined as the sum of the squared area differences of faces,
E( ω) = (^) ∑ [v 0 ,v 1 ,v 2 ]∈K 2
| ω([v 0 , v 1 ]) × ω([v 1 , v 2 ])| − S[v 0 ,v 1 ,v 2 ])^2. (7)
Both the constraint and the energy functional are polynomials with respect to λi’s. For example, the constraint can be reformulated as a quadratic form; if ci, j = ∑[v 0 ,v 1 ,v 2 ]∈K 212 | ωi([v 0 , v 1 ]) × ω (^) j([v 1 , v 2 ])|, then the constraint is ∑ 2 g i, j= 1 ci j^ λi^ λ^ j^ =^ const. We use Newton’s method to optimize the energy with con- straints. Because the energy is of degree 4, the extremal points are not unique. We randomly choose initial values for λi’s, and choose the global optimal solution from local optimal ones. By minimiz- ing the energy, we get the most uniform parameterization, for the purpose of comparison, we get the least uniform parameterization by maximizing the energy. Figures 1 and 2 demonstrate the computation results. In figure 1, three cuts are introduced on the genus 0 bunny surface, two are on its ear tips, one is on the bottom, then the surface is double covered to become a genus 2 surface. In figure 2, the cuts are introduced at horse’ feet and mouth, the double covered surface is of genus 4. The least uniform and the most uniform global parameterization are illustrated by using a checkerboard-texture map. Figure 5 uses the grid pattern to illustrate the computation results.
3.3.2 Emphasized Global Conformal Parameterization
Suppose we subdivide the whole surface into two regions D 0 and D 1. D 0 and D 1 themselves may be disconnected, with complicated topologies, and we want to maximize the parameter areas for D 0. Then, we define the area energy 3 as
E( ω) =
(^2) [v ∑ 0 ,v 1 ,v 2 ]∈D 0
| ω([v 0 , v 1 ]) × ω([v 1 , v 2 ])| (8)
with the same constraint in equation 6.
a. Least Uniform b. Most Uniform
Figure 2: Uniform Global Conformal Parameterization. Least uniform conformal parameterization, energy: 16.983e-5 (a). Most uniform conformal parameterization, energy: 7.878e-5(b).
The functional can be represented as a quadratic form directly. Let ci, j = (^) ∑[v 0 ,v 1 ,v 2 ]∈D 0 | ωi([v 0 , v 2 ]) × ω (^) j ([v 1 , v 2 ])|, then the empha- sized area energy is
E( λ 1 , λ 2 , · · · , λ 2 g) =
2 g ∑ i, j= 1
ci j λi λ (^) j. (9)
By maximizing this functional, we get more samples on D 0 and less samples on D 1 , and vice versa. The critical point is unique in general cases. We use Newton’s method for the optimization with arbitrary initial values for the λi’s. Figure 1 demonstrates the optimization of the emphasized area energy for the bunny surface model. The surface is equally subdi- vided into the left part and the right part. Figure 1 (c) emphasizes the left part, and the parameter area of the left part is 83.48% of the total parameter area. Figure 1 (d) emphasizes the right part, the parameter area is 82.58% of the total parameter area.
3.4 Optimal M¨obius Transform for Genus Zero Surface
For genus zero surfaces, there are no holomorphic one forms. We conformally map the surface to a unit sphere or a unit disk. Because the parameter domains are fixed, the constraint 6 is unnecessary. We can still use the uniformity energy or the emphasized area en- ergy, but the admissible transformations are changed to the M¨obius transformations.
Topological sphere The M¨obius transformation on the com- plex plane has the formula μ(z) = azcz++bd , ad − bc = 1 , a, b, c, d ∈ C. A sphere can be conformally mapped to the complex plane by a stereographic projection τ : S^2 → C, τ(x, y, z) = (^1) −xz +
y 1 −z. A conformal automorphism φ of the sphere can be formulated as φ = τ−^1 ◦ μ ◦ τ, We first compute a conformal map φ 0 : S → S^2 from the surface to the sphere, all admissible conformal mappings can be represented as φμ = τ−^1 ◦ μ ◦ τ ◦ φ 0. The uniformity functional becomes
E( μ) = (^) ∑ [v 0 ,v 1 ,v 2 ]∈K 2
(| φμ (v 0 ), φμ (v 1 ), φμ (v 2 )| − S[v 0 ,v 1 ,v 2 ])^2 ,
where |p 0 , p 1 , p 2 | represents the area of the triangle formed by p 0 , p 1 , p 2. This is a rational formula with respect to the coeffi- cients of μ. We use Newton’s method to optimize it without any constraints. Similarly, the emphasized area energy is formulated by
E( μ) = (^) ∑ [v 0 ,v 1 ,v 2 ]∈D 0
| φμ (v 0 ), φμ (v 1 ), φμ (v 2 )|. (10)
We use Newton’s method to maximize the energy. Because the optimal solutions are not unique, we randomly choose the initial M¨obius transformation μ 0 , and use φμ 0 as the initial parameteriza- tion. Topological disk For the topological disk case, we use double covering to make it a symmetric topological sphere. However, we restrict the admissible transformations to be in a subgroup of the M¨obius group, which preserves the symmetry; namely μ(z¯) = μ(z). The formula for such a M¨obius transformation can be written as μ(z) = (az + b)/(bz¯ + a¯), a a¯ − bb¯ = 1 , a, b ∈ C. Other steps are similar to those for the case of a topological sphere. Figure 3 illustrates a M¨obius transformation from the disk to itself.
Figure 3: M¨obius transformation from the unit disk to itself.
3.5 Topological Optimization
In this section, we introduce an automatic method to modify the topology of the surface to improve the uniformity of the parameter- ization. For long tube shapes, such as fingers and tails, the area distor- tion is usually very big. We want to show that the problem can- not be solved by linear combination of the holomorphic one-form bases. We have to modify the conformal structure of the surface itself; namely, we either change the Riemannian metric or modify the topology. First, we will demonstrate the fact that the conformal factor will increase exponentially on long tube shapes. Suppose we have a long thin cylinder and we conformally parameterize it. The center of the top is mapped to the origin. If we use polar coordinates ( ρ, θ ), then the conformal factor is a function dependent only on ρ because of symmetry. The Gaussian curvature K of the cylinder is zero, and
K( ρ, θ ) =
λ 2
∆ log λ = 0. (11)
We can deduce λ ( ρ) = exp (a ρ + b), where a, b are constants. No matter what kind of conformal parameterization we choose, the stretching is exponential. We have to change the topology of the surface by introducing a small boundary at the top of the cylinder, and then the conformal factor becomes constant. Based on this observation, we design our greedy topological modification algorithm as follows. First we find the most uniform conformal parameterization for current surface. Second, we locate points with extremely high conformal factors. Third, we introduce a small slice at the neighborhoods of those points. Finally, its con- formal structure is recomputed. We repeat the whole process until the uniformity energy is less than some threshold. Estimating the Conformal Factor Suppose we have ob- tained a global conformal parameterization induced by a holomor- phic one-form ω. The conformal factor for each vertex can be esti- mated by the following formula:
λ (v) =
n (^) [v∑ i,v]∈K 1
|r(vi) − r(v)| | ω([vi, v])|
, vi, v ∈ K 0 , (12)
Mesh Vertices Genus Boundaries Time (s) eight 766 2 0 30 bunny 23996 0 3 150 horse 19994 0 7 250 Max-Planck 23609 0 1 180 Body 40000 0 5 350 David 200000 0 5 1800
Table 1: Performance for global conformal parameterization opti- mization.
zero and it has 5 boundaries after topological modification. The least uniform global conformal parameterization and the most uni- form global conformal parameterization are illustrated in Figure 2 (a) and (b), respectively. The Max Planck head surface in Figure 5 is a topological disk. Figure 5(a) illustrates the result with minimum uniformity energy and Figure 5 (b) illustrates the result with maximum uniformity en- ergy. The human body surface in Figure 7 has 5 boundaries. The dou- ble covering of this surface is of genus 4. We partition the whole surface to the left and right regions equally. The parameterization in Figure 7 (a) emphasizes the right region, which occupies 98.11% of the total parameter area. The parameterization in Figure 7 (b) em- phasizes the left region, which occupies 96.1% of the total parame- ter area. The least uniform and the most uniform parameterization results are shown in Figure 7 (c) and (d) respectively. Figure 6 illustrates the positions of zero points. We can get the desired holomorphic one-forms by minimizing Equation 14. The Michelangelo’s David surface is illustrated in Figure 8. We control the zero points position using the method described in Section 3.6. In Figure 8(a), a zero point is located at the left upper arm near the shoulder. The same global conformal parameterization also has a zero point at his right upper arm near the shoulder as shown in (c). In Figure 8(b), there is a zero point under the left armpit. The same global conformal parameterization also gives a zero point at the right armpit, as shown in Figure 8(d).
5 CONCLUSION AND FUTURE WORK
This work introduces systematic algorithms to optimize global con- formal surface parameterizations. We define uniformity energy to measure the uniformity of the parameterization. We define empha- sized area energy to measure the parameter area of regions of in- terest. We also define special functional to allocate zero points at the desired points. The problem of optimizing global conformal parameterizations is equivalent to searching for a desired M¨obius transformation for genus zero surfaces and a desired holomorphic 1-form for high genus surfaces. We model global parameter op- timizations as finite dimensional optimization problems, and use Newton’s method to solve them. We also introduce algorithms to automatically modify the topology and allocate zero points at the specified positions to improve the quality of the global parameteri- zation. The algorithms developed are efficient, intrinsic, practical, and versatile for different applications. In the future, we will generalize the global conformal parame- terizations to other parameterizations, such as Tuette, Stereo, Alexa parameterizations as in [11], and intrinsic parameterizations as in [6]. The generalization will be based on geometric differential equation theories.
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Figure 6: Two hole torus Model. Locate zero points at different positions
a. Emphasizing Left Part b. Emphasizing Right Part c. Least Uniform d. Most Uniform
Figure 7: Human body Model. (a) Maximizing the parameter areas of left, percentage: 98.11%. (b) Maximizing the parameter areas of right, percentage: 96.01%. (c)Least uniform conformal parameterization, energy: 2.798e-5(c). (d) Most uniform conformal parameterization, energy: 1.501e-5 (d).
a. Zero Point at Left Shoulder b. Zero Point at Left armpit c. Zero Point at Right shoulder d. Zero Point at Right armpit
Figure 8: Zero Point Allocation. Zero point is originally at left shoulder(a). Put zero point at Left armpit(b). Zero point is originally at right shoulder(c). Put zero point at right armpit(d).
