Computational Conformal Geometry and Its Applications

Transcription

1 Computational Conformal Geometry and Its Applications Wei Zeng Institute of Computing Technology Chinese Academy of Sciences Thesis Proposal Advisor: Harry Shum Co-advisor: Xianfeng Gu October 11, 2007

2 i Abstract Conformal geometry has deep roots in pure mathematics. It is the intersection of complex analysis, Riemann surface theory, algebraic geometry, differential geometry and algebraic topology. Computational conformal geometry plays an important role in digital geometry processing. Recently, theory of discrete conformal geometry and algorithms of computational conformal geometry have been developed. A series of practical algorithms are presented to compute conformal mapping, which has been broadly applied in a lot of practical fields, including computer graphics, computer vision, medical imaging, visualization, and so on. The thesis focuses on computational conformal geometry and its applications on computer graphics and visualization, including surface conformal spherical parameterization, 3D shape space descriptor, quasiconformal mapping, surface remeshing, and consistent matching. Practical conformal parameterization methods are generated for specified popular applications, like human face expressions matching, and colon flattening. The initial experimental results are very promising.

3 CONTENTS ii Contents 1 Introduction 1 2 Background Conformal Geometry Theory Conformal Structure Riemann Mapping Riemann Uniformization Shape Space General Geometric Structure Computing Conformal Mapping Algorithms Overview Applications Computer Graphics Medical Imaging Computer Vision Geometric Modeling Goal and Solution Conformal Spherical Parameterization Consistent Surface Matching Quasiconformal Mapping Shape Space Descriptor Surface Remeshing Feasibility Analysis 45 5 Innovations 46 6 Resource and Progress Existing Resources Finished Work Ongoing Work Publications Schedule 48 8 Acknowledgement 48

4 1 INTRODUCTION 1 1 Introduction Conformal geometry is in the intersection of many fields in pure mathematics, such as Riemann surface theory, differential geometry, algebraic curves, algebraic topology, partial differential geometry, complex analysis and many other related fields. It has long history in pure mathematics, and is an active field in both modern geometry and modern physics, for example, the conformal fields theory in super string theory. Recently, with the rapid development of three dimensional digital scanning technology, computer aided geometric design, bio-informatics, medical imaging, more and more three dimensional digital models are available. The needs for effective methods to represent, process, and utilize the huge amount 3D surfaces become urgent. Digital geometric processing emerges as an inter-disciplinary field, combining computer graphics, computer vision, visualization and geometry. Computational conformal geometry plays an important role in digital geometry processing. It has been applied in many practical applications already, such as surface repairing, smoothing, denoising, segmentation, feature extraction, registration, remeshing, mesh spline conversion, animation, and texture synthesis. Especially, conformal geometry lays down the theoretic foundation and offers rigorous algorithms for surface parameterizations [Gu and Yau, 2002, 2003b]. Computational conformal geometry is also applied in computer vision for human face tracking [Wang et al., 2005b], recognition [Wang et al., 2006, 2007b], and expression transfer; in medical imaging, for brain mapping [Gu et al., 2003], virtual colonoscopy [Hong et al., 2006], and data fusion. The fundamental reason for conformal geometry to be so useful lies in the following facts: Conformal geometry studies the conformal structure. All surfaces in daily life have a natural conformal structure. Therefore, the conformal geometric algorithms are very general. Conformal structure of a general surface is more flexible than Riemannian metric structure and more rigid than topological structure. It can handles large deformations, which Riemannian geometry can not efficiently handle; it preserves a lot of geometric information during the deformation, whereas, topological methods lose too much information. Conformal maps are easy to control. For example, the conformal maps between two simply connected closed surfaces form a six dimensional space, therefore by fixing three points, the mapping is uniquely determined. This fact makes conformal geometric method very valuable for surface matching and comparison. Conformal maps preserve local shapes, therefore it is convenient for visualization purposes.

5 2 BACKGROUND 2 All surfaces can be classified according to their conformal structures. All the conformal equivalent classes form a finite dimensional manifold. This manifold has rich geometric structures, and can be analyzed and studied. In comparison, the isometric classes of surfaces form an infinite dimensional space. It is really difficult to deal with. Computational conformal geometric algorithms are based on solving elliptic partial differential equations, which are easy to solve and the solving process is stable, namely, the solution is insensitive to the noise of the input surfaces. Therefore, computational conformal geometry method is very practical for real engineering applications. In conformal geometry, all surfaces in daily life can be deformed to three canonical spaces, the sphere, the plane or the disk (the hyperbolic space). In other words, any surface admits one of the three canonical geometries, spherical geometry, Euclidean geometry or the hyperbolic geometry. Most digital geometric processing tasks in three dimensional space can be converted to the task in these two dimensional canonical spaces. It is well known that all orientable surfaces are Riemann surfaces. If two surfaces can be conformally mapped to each other, they share the same conformal structure. Therefore, computing conformal mappings is equivalent to computing conformal structures for surfaces. According to Riemann uniformization theorem, all metric surfaces can be conformally deformed to three canonical surfaces, the sphere, the plane and the hyperbolic disk. Different algorithms are designed to compute the uniformization metrics: (1) for spherical case, harmonic maps are computed by using heat flow method; (2) for Euclidean case, holomorphic 1-forms are computed; (3) for hyperbolic case, discrete Ricci flow method [Hamilton, 1988, Chow and Luo, 2003] is used. 2 Background Riemann surface theory is the intersection field of topology, differential geometry and algebraic geometry, which studies conformal structure of surfaces. Conformal geometry is between topology and geometry, softer than geometry and harder than topology. Conformal geometry plays a fundamental role in nature and the engineering world. This section introduces major concepts and theorems of conformal geometry and a series of computational algorithms and the related applications in geometric modeling, computer graphics, computer vision and medical imaging.

6 2 BACKGROUND Conformal Geometry Theory The geometric information of a surface has many layers, as shown in Table 1. Higher structures determine lower structures. Riemannian metric structure determines conformal structure. Lower structures confine higher structure. Topological structure determines the total Gaussian curvature induced by Riemannian metric structure. The inter-relations among structures are profound and subtle. Conformal geometry studies the conformal structure of general surfaces. Conformal structure is a natural structure of all surfaces in real life. Riemannian metric is a structure to measure the lengths of curves on the surface, area of domains on the surface and the intersection angles between curves. Conformal structure is a structure to only measure the intersection angle between two curves on the surface. Topological structure gives the neighborhood information. Roughly speaking, Conformal structure is more rigid than topological structure and more flexible than Riemannian metric. Conformal geometry is between topology and Riemannian geometry. Conformal geometry originated from the study of natural phenomena in classical physics, such as heat diffusion, electromagnetic field, fluid field and elasticity deformation. Mathematically, conformal geometry is the intersection of many mathematical branches. It has rich structures and abundant theoretic tools, such as differential geometry, algebraic topology, complex analysis, algebraic geometry, and complex manifold. Conformal field theory is a quantum field theory, and plays important roles in string theory, statistical mechanics and condensed matter physics. Many engineering applications can not be solved without using conformal geometry. The main work on this field is to convert all surface geometric problems to special problems on three canonical domains, S 2, D 2, and R 2. Here, we will get further understanding of the above statements through further discussions around the following aspects Conformal Structure A conformal structure is a structure assigned to a topological manifold, such that angles can be defined (See Figure 1). It is easy to define angles on the parameter plane. But a manifold can not be covered by a single coordinate system, instead it is covered by many local coordinate systems with overlapping (See Figure 9). In relativity, the physics law is independent of the local coordinate system of the observer. If the transition function from one local coordinates to another is angle preserving, then angle value is independent of the choice of the local chart. Therefore, if the manifold is with a special atlas, such that all transition maps are conformal, then angle can be consistently defined on the manifold.

7 2 BACKGROUND 4 Figure 1: Conformal Structure. [Gu et al., 2003, Gu and Yau, 2003b] Figure 2: Riemann Surface: The manifold is covered by a set of charts (U α, φ α ), where φ α : U α R 2. If two charts (U α, φ α ) and (U β, φ β ) overlap, the transition function φ αβ : R 2 R 2 is defined as φ αβ = φ β φ α 1. If all transition functions are analytic, then the manifold is a Riemann surface. The atlas (U α, φ α ) is a conformal structure.

8 2 BACKGROUND 5 Table 1: Geometric Structure. Geometric Concept Transformations Data Theoretic Structure Structure Tool Topological neighborhood homeomorphism connectivity homology, Structure cohomology, fundamental group Differential differentiability, diffeomorphism / differential Structure smoothness, topology, tangent Morse function Conformal angle, conformal discrete Riemann Structure holo/meromorphic maps holomorphic surface, forms 1-forms complex geometry Riemannian distance, isometry edge Riemannian Structure geodesic, lengths geometry area Euclidean position rigid vertex Euclidean Structure motion position geometry, differential geometry Riemann Mapping A conformal map between two surfaces preserves angles. Riemann mapping theorem states that any simply connected surface with a single boundary (a topological disk) can be conformally mapped to the unit disk. As shown in Figure 3, the frontal part of a human face S is a topological disk and mapped to the unit disk D by a conformal mapping φ : S D. Suppose γ 1,γ 2 are two arbitrary curves on the face surface S, φ maps them to φ(γ 1 ),φ(γ 2 ). If the intersection angle between γ 1,γ 2 is θ, then the intersection angle between φ(γ 1 ) and φ(γ 2 ) is also θ. γ 1 and γ 2 can be chosen arbitrarily. Therefore, we say φ is conformal, meaning angle-preserving. The conformality can be visualized using texture mapping techniques in computer graphics. Figure 4 illustrates the idea. A texture refers to an image on the plane. First a conformal mapping between the face surface (a) to the unit disk (b) is established. Then cover the disk

9 2 BACKGROUND 6 φ θ θ γ 2 γ 1 φ(γ 2 ) φ(γ 1 ) S D Figure 3: A Riemann mapping from a human face to the unit disk, the mapping is anglepreserving. (a)face surface (b) Map to disk (c) Checker texture (d) Circle packing texture Figure 4: Visualization of conformality using texture mapping in computer graphics. by a texture image, and pull back the image onto the surface. In this way, the mapping can be directly visualized. If the texture is a checker board, all the right angles of the corners of the checkers are preserved on the human face as shown in (c). If replacing the texture by circle packing pattern, then planar circles are mapped to circles on the surface, the tangency relation among circles are preserved as shown in (d). Figure 5 shows the conformal mappings of a multi-holed annulus. The planar domains are circular disk with circular holes, which is computed by Ricci flow. The planar domains are determined by the conformal structure of the original surface. A conformal map, also called an angle-preserving map, is a transformation that preserves local angles. Two surfaces are conformal equivalent, if there exists a bijective conformal map between them. Surfaces can be classified by conformal equivalence relation, where each conformal equivalent class is also called a Riemann surface. For example, all genus zero closed

10 2 BACKGROUND 7 γ 0 γ 1 γ 2 γ 2 γ 3 γ 1 γ 1 γ 0 Figure 5: Conformal mappings of a multi-holed annulus. surfaces can be conformally mapped to the unit sphere. Therefore, they are the same Riemann surface. Conformal maps are stronger than harmonic maps. Harmonic maps between two homeomorphic surfaces exist, but conformal maps may not exist. Algorithm (Riemann Mapping): Given a topological disk surface, it can be conformally mapped to the unit disk as follows: 1. Double covering. 2. Conformally map the doubled surface to the unit sphere. 3. Use the sphere Möbius transformation to make the mapping symmetric. 4. Use stereographic projection to map each hemisphere to the unit disk Riemann Uniformization The conformal map between two planar domains is the conventional analytic function, or holomorphic function. From this point of view, conformal mappings are the generalization of holomorphic functions, and Riemann surfaces are the generalization of complex plane. All surfaces in real life are real surfaces. The derivative of a analytic function is called a holomorphic differential. Holomorphic differentials can be defined on surfaces directly. They can be visualized using the same technique as the visualization of conformal mapping. By integrating the holomorphic differentials, the surface can be locally mapped to the plane, the mapping is conformal and visualized by checker board texture mapping (see Figures 6 and 7). Holomorphic, meromorphic functions and holomorphic, meromorphic differentials on the surface form special groups, the group structure is governed by Riemann-Roch theorem, which is a profound theory, connecting geometry, topology and partial differential geometry. Riemann uniformization theorem states that all surfaces in real life can be conformally mapped to three canonical shapes, the unit sphere, the Euclidean plane and the hyperbolic space.

11 2 BACKGROUND 8 Figure 6: Visualization of holomorphic differentials on animal surfaces. Figure 7: Visualization of holomorphic differentials on Michelangelo s David. Namely, all surfaces admit one of the three canonical geometries, spherical, Euclidean or hyperbolic geometry. It can also be interpreted as all surfaces admit a canonical Riemannian metric, which is conformal to the original Riemannian metric and induce constant Gaussian curvature, which is +1, 0 or 1. Figure 8 illustrates the uniformization theorem. For closed surfaces without handles as shown in the first column, they can be conformally mapped to the unit sphere. Closed surfaces with one handle can be periodically mapped to the plane. In details, the whole kitten surface in the middle column of the figure can be conformally mapped to a parallelogram on the plane, the repetition of the parallelogram forms a tiling of the whole plane. The third column shows a exotic bottle, it has two handles. The surface has no self-intersection and is embedded in the Euclidean space. It can be conformally periodically mapped to the unit disk,

12 2 BACKGROUND 9 (a) χ > 0 (b) χ = 0 (c) χ < 0 Figure 8: Uniformization Theorem: all surfaces with Riemannian metric can be conformally embedded onto three canonical spaces: sphere, plane and hyperbolic space. [Gu et al., 2003, Gu and Yau, 2003b, Jin et al., 2006a,b] which represents the hyperbolic space. The whole surface is mapped to a hyperbolic octagon, the repetition of the octagon forms a tiling of the whole hyperbolic space Shape Space Surfaces can be classified using conformal geometry. Two surfaces are conformal equivalent, if they can be conformally mapped to each other. It is challenging to verify if two surfaces are conformal equivalent. Roughly speaking, for closed surfaces with one handle, if the shapes of the parallelograms, then they are conformal equivalent. Same result holds for surfaces with more handles. For closed surfaces with two handles, if their hyperbolic octagon are congruent in the hyperbolic space, then they are conformal equivalent. The conformal equivalence classes form a finite dimensional space, which is called the Teichmüller space and the Modular space, which are the space of shapes, therefore, shape space. Each point represents a shape, each curve represents a deformation process. To fully understand the topological and geometric structures of the shape space is the most active research in

13 2 BACKGROUND 10 Figure 9: Riemann Surface: The manifold is covered by a set of charts (U α, φ α ), where φ α : U α R 2. If two charts (U α, φ α ) and (U β, φ β ) overlap, the transition function φ αβ : R 2 R 2 is defined as φ αβ = φ β φ α 1. If all transition functions are analytic, then the manifold is a Riemann surface. The atlas (U α, φ α ) is a conformal structure. mathematics today. Surface classification using conformal structures is described in [Gu and Yau, 2003a]. The methods for computing general geometric structures are in [Jin et al., 2007c] and [Jin et al., 2007b]. Shape space application using Ricci flow is described in [Jin et al., 2007a] General Geometric Structure A conformal structure is a structure assigned to a topological manifold, such that angles can be defined (See Figure 1). It is easy to define angles on the parameter plane. But a manifold can not be covered by a single coordinate system, instead it is covered by many local coordinate systems with overlapping (See Figure 9). In relativity, the physics law is independent of the local coordinate system of the observer. If the transition function from one local coordinates to another is angle preserving, then angle value is independent of the choice of the local chart. Therefore, if the manifold is with a special atlas, such that all transition maps are conformal, then angle can be consistently defined on the manifold. A surface can not be covered by one coordinate system. In general, we can find a collection of open sets to cover the surface and map each open set to the plane, we then use the planar coordinates as the local coordinate system for the corresponding open set. Such kind of local coordinate system is call an atlas of the surface. One point on the surface may be covered by multiple local coordinates, the transformation from one local coordinates system to another is called the transition function or coordinates change (See Figure 9). Suppose X is a topological space, G is the transformation group of X, a (X,G) structure is an atlas, such that the local coordinates are in X, the transition functions are in G. For

14 2 BACKGROUND 11 Figure 10: Visualization of the affine structures of two genus one surfaces. example, a spherical structure is an atlas, where all the local coordinates are on the sphere, all the transition functions are rotations. Figure 8 can be interpreted as the visualization of spherical structure, Euclidean structure and the hyperbolic structure respectively. According to Flix Klein s Erlangen program, different geometries study the invariants under different transformation groups. For example, let X be the Euclidean plane, if G is rigid motion, then the geometry is Euclidean geometry, the invariants are lengths, angles, area etc. If G is affine transformation group, then the geometry is the affine geometry, the major invariant is the ratio for three points on a line, parallelism. If G is the real projective transformation group, then the corresponding geometry is real projective geometry. The major invariant is cross ratio for four points on a line. Figure 11: Visualization of the hyperbolic structure and the real projective structure of a genus two vase model. [Jin et al., 2006b] If a surface admits a (X,G) structure, then the corresponding geometry can be defined on the surface directly. For example, in automobile industry and mechanics engineering fields, surfaces are represented as splines, which are piecewise rational polynomials. Conventional

15 2 BACKGROUND 12 splines are defined on the Euclidean plane and constructed based on affine invariants. The fundamental problem in computer aided geometric design (CAD) field is to construct splines defined on arbitrary surfaces. If the surface admits an affine structure, then splines can be defined on the surface without any difficulty. Unfortunately, very few surfaces admits affine structure. This fact causes intrinsic difficulty for applications in CAD field. Fortunately, all surfaces admits real projective structure. How to construct splines based on real projective geometry is an active research direction in geometric modeling today. Like conformal structure, the shape space of all conformal structures has rich topological and geometric properties. The understanding of the shape space of all (X, G) structures is widely open also. 2.2 Computing Conformal Mapping Figure 12: Face geometries with different expressions of the same person, scanned using real time high speed high resolution scanner. Recently, 3D scanning technology is developing extremely fast. Figure 12 shows several facial surfaces with different expressions of the same person, scanned by a scanner based on phase-shifting method. The scanning speed is as fast as 180 frames per second, each frame has samples. The scanner can capture dynamic facial expressions in real time. It is challenging to process the huge amount geometric data efficiently and robustly. Conformal geometry offers powerful tools to tackle the problem. The main strategy is to use conformal

16 2 BACKGROUND 13 mappings to transform 3D surfaces to canonical 2D domains, and convert 3D geometric problems to 2D ones. In computer graphics and discrete mathematics, much sound research has focused on discrete conformal parameterizations. Here, we briefly overview related works, and refer readers to [Floater and Hormann, 2005, Kraevoy and Sheffer, 2004] for thorough surveys. All parameterizations can be classified according to the type of output produced, which can be a vector valued function, i.e. a mapping, a holomorphic differential form or a flat Riemannian metric. In general, the derivative of a conformal map is a holomorphic 1-form; each holomorphic 1-form induces a flat metric. Therefore, methods which compute metrics are the most general, although they are more expensive to compute. 1. Mappings. First order finite element approximations of the Cauchy-Riemann equations were introduced by Levy et al. [Lévy et al., 2002]. Discrete intrinsic parameterization by minimizing Dirichlet energy was introduced by [Desbrun et al., 2002], which is equivalent to least-squares conformal mapping [Lévy et al., 2002]. Discrete harmonic maps were computed using the cotan-formula in [Pinkall and Polthier, 1993]. Mean value coordinates were introduced in [Floater, 2003]; these generalize the cotan-formula. All these linear methods can easily incorporate free boundary conditions to improve the quality of the parameterization produced, such as the methods in [Desbrun et al., 2002] and [Zayer et al., 2005]. Discrete spherical conformal mappings are used in [Gotsman et al., 2003] and [Gu et al., 2003]. 2. Holomorphic forms. Holomorphic forms are used in [Gu and Yau, 2003b] to compute global conformal surface parameterizations for high genus surfaces. Discrete holomorphy was introduced in [Mercat, 2001] using discrete exterior calculus [Hirani, 2003]. The problem of computing optimal holomorphic 1-forms to reduce area distortion was considered in [Jin et al., 2004]. Gortler et al. [Gortler et al., 2006] generalized 1-forms to the discrete case, using them to parameterize genus one meshes. Recently, Tong et al. [Tong et al., 2006] generalized the 1-form method to incorporate cone singularities. 3. Metrics. There are three major methods for computing edge lengths (or equivalently the angles): angle based flattening, circle packing, and circle patterns. Sheffer and Sturler [Sheffer and de Sturler, 2001] introduced the angle based mesh flattening method. This works by posing a constrained quadratic minimization problem seeking to find corner angles which are close to desired angles in a weighted L2 norm. The efficiency and stability of ABF are improved in [Sheffer et al., 2005] by using advanced numerical algorithm and hierarchical method. The circle packing method was introduced in [Thurston, 1976]. Continuous conformal mappings can be characterized as mapping infinitesimal circles to infinitesimal circles. Circle packings replace infinitesimal circles with finite circles. In

17 2 BACKGROUND 14 the limit of refinement the continuous conformal maps are recovered [Rodin and Sullivan, 1987]. Collins and Stephenson [Collins and Stephenson, 2003] have implemented these ideas in their software CirclePack. The first variational principle for circle packings, was presented in a seminal paper by Colin de Verdiére [de Verdiére, 1991]. Circle patterns based on those in Bobenko and Springborn [Bobenko and Springborn, 2004] have been applied for parameterization in [Kharevych et al., 2006]. Springborn [Springborn, 2003] shows that in theory, circle packing and circle patterns are equivalent. 4. Ricci Flow. Recently, a novel curvature flow method in geometric analysis is introduced to prove the Poincaré conjecture, the Ricci Flow. Ricci flow refers to conformally deform the Riemannian metric of a surface by its Gaussian curvature, such that the curvature evolves according to a heat diffusion process. Ricci flow is a powerful tool to compute the Riemannian metric by the curvature. It can be applied for discrete conformal parameterizations. The connection between circle packing and smooth surface Ricci flow [Hamilton, 1988] was discovered in [Chow and Luo, 2003]. Conventional circle packing only considers combinatorics. The discrete Ricci flow method was introduced in [Jin et al., 2006b, Gu et al., 2007b], which incorporate geometric information and was applied for computing hyperbolic and projective structure and manifold splines Algorithms Overview According to Riemann uniformization theorem, all metric surfaces can be conformally mapped to three canonical shapes, the sphere, the plane and the hyperbolic disk. The mappings are periodic and reflect the intrinsic symmetries of the surfaces. Figure 8 shows three kinds of algorithms for computing conformal structure. For genus zero surface in the first column, the mapping can be computed using spherical harmonic maps, in paper [Gu et al., 2003]. Spherical geometry can be defined on the surface. For genus one surface in the second column, the mapping can be computed using holomorphic 1-forms, in paper [Gu and Yau, 2003b]. Another algorithm is to use Euclidean Ricci flow, in paper [Jin et al., 2006a]. Euclidean geometry can be defined on the surface. For higher genus surface in the third column, the mapping can be computed using hyperbolic Ricci flow, in paper [Jin et al., 2006b]. Hyperbolic geometry can be defined on the surface. The followings are the detailed descriptions for the major algorithms of computing conformal mappings of surfaces with various topologies.

18 2 BACKGROUND 15 Harmonic Maps for Topological Disks The harmonic map between simply connected surfaces with a single boundary and a convex planar domain can be computed by solving Dirichlelt problem. First fix the boundary on a convex planar curve, and compute the interior by minimizing harmonic energy. The Euler-Lagrange equation of the critical point of the harmonic energy is the Laplace equation. By using finite element method, the Laplace equation is formulated as a symmetric positive definite linear system. The problem is a linear problem. The energy optimization can be performed using conjugate gradient method efficiently. Figure 13 shows a harmonic map between a human face to a rectangle on the plane. Figure 13: Harmonic maps for topological disks. Harmonic Maps for Topological Spheres The harmonic map between a topological sphere and the canonical unit sphere is automatically conformal. The computational algorithm is based on non-linear heat diffusion process. First construct a degree one map, such as the Gauss map, then compute the Laplacian of the map, and update the map along the negative direction along the tangential component of the Laplacian. Because of the projection to the tangential space, the heat diffusion process becomes non-linear. Different solutions differ by Möbius transformations. Therefore, normalization conditions are necessary. Figure 14 shows one example of conformal mapping of a topological sphere. Riemann Mappings of Topological Disks Harmonic maps between a topological disk to the unit disk may not necessarily be conformal. First compute the double covering of the topological disk, which is a topological sphere, then compute a conformal map between the doubled surface and the unit sphere, such that, each copy of the topological disk is mapped to a hemisphere. Then use stereo-graphic to project the unit sphere onto the whole plane, the lower hemisphere is mapped to the unit disk. This induces the mapping from the surface to the unit

19 2 BACKGROUND 16 Figure 14: Harmonic maps for topological spheres. disk, and the map is conformal. Figure 15 shows a Riemann mapping from a human face to a disk on the plane. Figure 15: Riemann mapping. Conformal Mappings with Free boundaries Conformal mappings with free boundaries can be achieved by discrete approximation of Beltrami equation, a special case is the Riemann- Cauchy equation. The advantage of this method is that it is linear and efficient. The disadvantage is the less control of the boundaries. It mainly handle genus zero surfaces. The mapping results may have self-overlapping. Extra constraints can be added to enhance the mapping result, such as feature point constraints. Figure 16 shows one example of conformal mapping with free boundary from a human face to the plane.

20 2 BACKGROUND 17 Figure 16: Solving Beltrami equation using free boundary condition. Holomorphic 1-forms All metric surfaces are Riemann surfaces, which admit special complex differential forms, holomorphic 1-forms. The group of the holomorphic 1-forms has special structure, the generators can be explicitly calculated. A holomorphic 1-form has zero points, the number of zero points equals to the absolution value of the Euler number. In the neighborhood of normal points, holomorphic 1-form induces conformal maps between the neighborhood to the complex domains. Iso-parametric curves through zero points can be used to segment the surface. Figure 17 illustrates a conformal texture mapping induced by a holomorphic 1-form. Figure 17: Conformal Mapping induced by a holomorphic 1-form. [Gu and Yau, 2003b] Holomorphic 1-forms for Affine structure General geometric structures on the surfaces refer to the atlases, such that all the local coordinates changes belong to the special transformation groups. Most popular spline schemes are constructed based on affine invariants, therefore can be generalized to be defined on the surfaces with affine geometric structure. Unfortunately, the existence of affine atlas depends on the topology of the surface. If the

21 2 BACKGROUND 18 Figure 18: Affine structure induced by holomorphic 1-forms. [Gu and Yau, 2003b] surface is with boundaries, or the surface is a closed torus, then it admits an affine atlas. In general cases, extraordinary points have to be introduced. Conventional subdivision surfaces are splines defined on the surfaces with extraordinary points. Holomorphic 1-forms naturally induce affine structures on the surfaces. The extraordinary points are the zero points of the holomorphic 1-form, as show in the Figure 18, the centered octagons are zero points. The number of zero points equals to the Euler number of the surface. This pave the way to defining various planar splines on general surface domains. Conformal Mapping of Multi-Holed Annuli to Annulus with Concentric Circular Arcs Special holomorphic 1-forms can be constructed on a multi-holed annulus, such that the whole surface is mapped to an annulus with concentric circular arcs. Two boundaries are mapped to the inner and the outer boundaries of the annulus, the other boundaries are mapped to the slits. Computing such holomorphic 1-forms is a linear problem, and the most difficult part is to find harmonic 1-forms. Chapter Holomorphic Forms algorithms explains the details. Figure 19 shows the conformal mapping of a three-holed face surface, (the mouth is cut open), the target domain is a unit disk with two concentric circular arcs. The exterior boundary of the face is mapped to the outer circle of the annulus, the mouth boundary is mapped to the inner circle. The boundaries of eyes are mapped to the two circular slits. Then conformally mapping the annulus to the rectangle, the outer and inner circles are mapped to parallel lines and the boundaries of eyes are mapped to horizontal slits. Euclidean Ricci Flow for Genus One Surface Euclidean Ricci flow method computes special metrics of the surface conformal to the original metric with prescribed target curvature. For genus one closed surface, set the target Gaussian curvature to be zero everywhere, and compute the flat metric, which is conformal to the original induced Euclidean metric. The universal cov-

22 2 BACKGROUND 19 Figure 19: Conformal Mapping between a multi-holed annulus to an annulus with concentric circular arcs and a rectangle with slits. ering space of the surface can be isometrically embedded on the plane. Figure 20 shows one example. The kitten surface is of genus one, the universal covering space is embedded on the plane. The rectangle is a fundamental polygon. Figure 20: Conformal flat metric of a genus one surface, computed using Euclidean Ricci flow. [Jin et al., 2006a] Hyperbolic Ricci Flow for High genus Surface For high genus surfaces, there exists a unique Riemannian metric, which is conformal to the original Riemannian metric, and induces constant Gaussian curvature everywhere, the constant is 1. Such kind of metric can be computed using hyperbolic Ricci flow. The universal covering space of the surface can be isometrically embedded on the hyperbolic space. Figure 21 demonstrates the embedding of the universal covering space of a genus two surface on the Poincaré model of hyperbolic space.

23 2 BACKGROUND 20 Figure 21: Conformal hyperbolic mapping of a genus two surface. [Jin et al., 2006b] Hyperbolic Ricci Flow for Real Projective Structure All surfaces admit a special atlas, such that all chart transitions are real projective transformations. Such kind of the projective atlas can be computed using hyperbolic Ricci flow method. Figure 22 demonstrates the computing result. First compute the conformal hyperbolic metric of the surface, then embed its universal covering space on the Poincaré model of the hyperbolic space, finally transform the Poincaré model to the Klein model, where all the rigid motions are real projective transformations. This embedding induces a real projective atlas of the surface. Figure 22: Real projective structure of a genus two surface. [Jin et al., 2006b] Conformal Metric Designed by the Prescribed Curvature The conformal metrics and the curvatures of a surface are essentially of one-to-one correspondence. The conformal metric can be computed using the prescribed curvature on the surface using Euclidean Ricci flow method. Figure 23 shows one example. The input surface is a topological disk. It is mapped to the planar domains specified by curvature on the boundaries. The curvature of interior points are zero

24 2 BACKGROUND 21 Figure 23: Conformal flat metrics are designed by the target curvature. everywhere. The conformal mapping induced by the metric is fully controlled by the prescribed curvatures. It is also possible to concentrate all the curvature of a surface with arbitrary topology to a single point. 2.3 Applications Computational conformal geometric methods are valuable for a broad range application in geometric modeling, computer graphics, computer vision, visualization medical imaging and scientific computing and many other engineering fields. In the following, we briefly browse some most direct applications of conformal geometric methods Computer Graphics Conformal geometry has numerous applications in computer graphics, including surface parameterization, mesh repairing, texture mapping and synthesis, surface re-meshing, mesh matching, mesh-spline conversion, geometric morphing, efficient rendering, animation and many other applications. Global Conformal Parameterization In computer graphics, surface Parameterizations refer to the process to map the surface onto 2D planar domains. To avoid the problems associated

25 2 BACKGROUND 22 with discontinuous boundaries, global conformal parametrization, which preserves conformality everywhere (except for a few points), is highly desirable. (a) Texture mapped bunny (b) Front view (c) Back view (d)planar image Figure 24: Conformal parameterization of Stanford bunny model. [Gu and Yau, 2003b] Gu et. al. [Gu and Yau, 2003b] solved the problem of computing global conformal parameterizations for general surfaces, with nontrivial topologies, with or without boundaries, using the structure of the cohomology group of holomorphic 1-forms. Jin et. al. [Jin et al., 2004] provided an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. Figure 24 illustrates the whole process. As a powerful geometric tool to compute the uniformization metric, Ricci flow has been introduced to compute global conformal parameterization, which appears in the work [Gu et al., 2005b]. Jin et. al. [Jin et al., 2006a] parameterized surfaces with different topological structures in an unified way using Euclidean Ricci flow. Jin et. al. [Jin et al., 2006b] introduced discrete variational Ricci flow to compute the hyperbolic structure and real projective structure for general surfaces with negative Euler characteristic numbers. The method is efficient and robust in practice. Texture Mapping In computer graphics, surfaces are approximated by triangular meshes (polygonal surfaces, each face is a triangle), which can be supported by graphics hardware directly. The rendering efficiency of the hardware depends on the resolution of the mesh. For real time applications, time is critical, therefore low resolution meshes are highly preferred. Small geometric details, and material properties are modeled as texture images. The parameterization process map each vertex to the planar domain, and obtain its 2D coordinates, which is called the texture coordinates. Then graphics hardware will glue the texture to the meshes using the texture coordinates. Figure 25 illustrates the process.

26 2 BACKGROUND 23 (a) Geometry (b) Parameterization (c) Texture mapping Figure 25: Texture mapping using hyperbolic parameterization. [Jin et al., 2006b] Remeshing and Geometry Images Surfaces are represented as meshes in computer graphics. In order to convert to spline format in geometric modeling, it is highly desirable to retessellate the triangular meshes to quad meshes. Because most popular spline schemes are based on tensor-product. First parameterize the triangular mesh onto the planar region, and use regular grids to tessellate the planar image of the surface, this induces the tessellation of the original surface and convert it to a quad mesh. Figure 26 demonstrates one example for remeshing a triangular mesh to a quad-mesh. (a) Original mesh (b) Quad mesh Figure 26: Surface remeshing using conformal parameterization. General meshes have both connectivity information of the triangulations and geometric information represented as the coordinates of the vertices. After remeshing, the quadmesh connectivity is regular, it is unnecessary to encode the connectivity any more and just record the coordinates of vertices. Color encoding the coordinates, the surface is represented as an image, which is called geometry image. There are two pipelines in graphics hardware, one handles

27 2 BACKGROUND 24 meshes, the other one handles texture images. Geometry image unifies both geometry and texture, which has the potential to simplify the graphics hardware. Geometry images can be applied for efficient rendering. The details of geometry image is presented in [Gu et al., 2002] Medical Imaging With the rapid development of medical imaging technologies, vast medical imaging data are available today. In order to fuse medical images acquired from different modalities, extract surfaces or volumes, register, fuse and compare different geometric data sets, conformal geometric algorithms have been developed and proven to be valuable for real applications. Conformal Brain Mapping Conformal mapping has its natural and intrinsic characteristics, and is involved for cortical surface flattening. Under the Riemann mapping theorem, no other extraneous cuts are required. (a) Brain cortex surface (b) Conformal spherical brain mapping Figure 27: Conformal Brain Mapping. [Gu et al., 2003, 2004] Brain imaging technology has accelerated the collection and databasing of brain maps. Computational problems arise when integrating and comparing brain data. The cortex surface of a brain is highly convoluted and anatomical structures varies from person to person. One way to analyze and compare brain data is to map them into a canonical space while retaining geometric information on the original cortex surface as far as possible. Cortical surfaces are of genus zero, therefore, they can be conformally mapped onto the unit sphere. All such conformal mappings differ by Möbius transformations of the sphere, which form a 6 dimensional group. For genus zero closed surfaces, harmonic maps are also conformal. A conformal mapping can be obtained by optimizing the harmonic energy. Further constraints are added to ensure that the conformal map is unique. Empirical tests on magnetic resonance imaging (MRI) data show that the mappings preserve angular relationships, are stable in MRIs

28 2 BACKGROUND 25 acquired at different times, and are robust to differences in data triangulation, and resolution. Figure shows the conformal brain mapping of a real human cortical surface. Hurdal et.al. [Hurdal et al., 2001, 1999] have adapted a method that uses circle packing to compute an approximation to the conformal map of a cortical surface. For genus zero surfaces, a unique mapping between any two genus zero manifolds can be found by minimizing the harmonic energy of the map. Conformal brain mapping using nonlinear heat diffusion is introduced in [Gu et al., 2003] and [Gu et al., 2004]. Spherical conformal mapping is presented in [Gotsman et al., 2003]. Conformal brain mapping based on Riemann surface structure is explained in [Wang et al., 2007c]. Computational and visualization tools are needed to interact with these conformal flat maps to gain information about spatial and functional relationships that might not be apparent. Such information can contribute to earlier diagnostic tools for diseases and improved treatment. Conformal Colon Flattening Colon cancer is one of the leading causes of cancer deaths in the United States. Computed tomographic (CT) colonography is a technique used in the detection of colonic polyps, the precursors of colorectal carcinoma, with CT of the cleansed and air-distended colon. Virtual colonoscopy has been successfully demonstrated to be more convenient and efficient than the real optical colonoscopy. However, because of the length of the colon, inspecting the entire colon wall is time consuming, and prone to errors. Moreover, polyps behind folds may be hidden, which results in incomplete examinations. (a) Colon surface (b) Conformal colon (c) Conformal reconstructed from CT flattening images parameterization Figure 28: Conformal colon flattening. [Hong et al., 2006]. Virtual dissection is an efficient visualization technique for polyp detection, in which the entire inner surface of the colon is displayed as a single 2D image. The straightforward methods [Balogh et al., 2002, Wang and Vannier, 1995] extract the isosurface by straightening the central path and unfolding the cross sections. Paik et al. [Paik et al., 2000] use cartographic projections

29 2 BACKGROUND 26 Figure 29: Genus Two Models. [Jin et al., 2007d] Figure 30: Genus Three Models. [Jin et al., 2007d] to project the whole solid angle of the camera onto a cylinder which is mapped finally to the image. Haker et al. [Haker et al., 2000a] propose a method based on the discretization of the Laplace-Beltrami operator to flatten the colon surface onto the plane in a manner which preserves angles. Hong et al. [Hong et al., 2006] conformally map the 3D colon surface to a 2D rectangle using conformal structure, which is general and can handle high genus surfaces (see Figure 28) Computer Vision Conformal geometry has been applied in computer vision for surface matching, shape comparison, shape classification, geometric analysis and tracking. Shape Classification Recognition, retrieval, and classification are the common applications in computer vision field. With graphics hardware getting faster and 3D scanning hardware getting cheaper, the number of 3D geometric models (see Figures 29 and 30) in online repositories is growing dramatically, and the demand for effective retrieval of models is continuously increasing. The primary challenge in building a shape-based classification and retrieval system is to find a computational representation of shape descriptors for which an index can be built, and similarity queries can be answered efficiently.

30 2 BACKGROUND 27 Survey papers to shape descriptor literature have been provided by [Tangelder and Veltkamp, 2004] and [Iyer et al., 2005], taking into account the applicability to surface models as well as to volume models. The corresponding shape searching methods are evaluated with respect to several requirements of content-based 3D shape retrieval, such as: (1) shape representation requirements, (2) properties of dissimilarity measures, (3) efficiency, (4) discrimination abilities, (5) ability to perform partial matching, (6) robustness, and (7) necessity of pose normalization. For 3D Surfaces, they can be classified by different transformation groups. Traditional classification methods mainly use topological transformation groups and Euclidean transformation groups. In recent years, conformal geometry has been studied and applied for shape classification analysis. For many shape classification problems based on geometric features, conformal invariants can offer sufficient information to differentiate the different shapes. [Gu and Yau, 2003a] introduces a novel method to classify surfaces by conformal transformation groups. Conformal equivalent class is refiner than topological equivalent class and coarser than isometric equivalent class. Also, conformal invariants are concise and efficient to compute, and can be used as search keys conveniently. Hence conformal classification is more suitable for practical surface classification problems, such as human face surface matching [Wang et al., 2006], and human brain surface matching [Gu and Vemuri, 2004], which is the first paper to classifies surfaces with arbitrary topologies by global conformal invariants. Surface Matching Surface matching is a fundamental task for computer vision, graphics and medical imaging. Figure 31 shows the basic idea of using conformal parameterizations to convert 3D matching problems to 2D ones. Suppose S 1 and S 2 are two surfaces in R 3. π 1 : S 1 D and φ 2 : S 2 D are conformal mappings to map surfaces to the canonical planar domain. φ : D D is a a map from D to itself, this is a 2D matching process. Then φ = π 1 2 φ π 1,S 1 S 2, is the desired 3D matching. If S 1 and S 2 are similar to each other in terms of their geometries, then their conformal structures are close to each other. Under some appropriate boundary conditions, the images of corresponding feature points will be close to each other on the planar images. In Figure 31, we can see the images of the nose tips of two surfaces are very close to each other on the 2D plane. The images of the eye corners are close also. Figure 32 demonstrates the fact that isometric deformations preserves conformal structures. The original surface in (a) is a plastic mask, which can only be bent and hardly stretched. It is deformed to get another surface shown in (c). Their shapes are acquired using 3D scanner to acquire their shapes, denoted as S 1 and S 2. Then use conformal map φ 1 : S 1 R 2 and φ 2 : S 2 R 2 with the only constraints that the images of all the boundaries are circles. The

31 2 BACKGROUND 28 f φ 1 φ 2 f Figure 31: Surface matching using conformal mapping. [Wang et al., 2006] (a) Original surface (b) Planar image of (a) (c) Deformed surface (d) Planar image of (c) Figure 32: Isometric deformations from (a) to (c) preserves conformal structures, their planar images of conformal mappings are consistent shown in (b) and (d). [Wang et al., 2005a] centers and radii of the images of the boundaries do not been specified and they are calculated automatically by the conformal geometric algorithms. Their planar images are shown in (b) and (d), which are identical. Then the 2D map φ is the identity of the two hole annulus, the 3D map φ = π2 1 π 1, which is exactly the isometric deformation. This example shows that conformal geometric methods can recover isometric maps automatically. Therefore, for the purpose of surface matching, conformal geometric methods reduce the dimensionality and recover isometric maps, furthermore, they can handle surfaces with arbitrary topologies. In Figure 33, two genus two surfaces and two genus six surfaces are matched using hyperbolic Ricci flow method. The matching is visualized by transferring the textures from the domain surfaces to the range surfaces. Figure 34 shows the surface matching between two genus two surface by a geometric morphing.

32 2 BACKGROUND 29 Figure 33: Surface matching between two high genus surfaces using conformal geometric methods. Figure 34: Visualization of surface matching by surface morphing. Surface Stitching Surface matching with exact feature alignment is in [Carner et al., 2005]. 3D surface matching and recognition and stitching using conformal geometry is described in [Wang et al., 2006] and [Wang et al., 2007b]. Figure 35 demonstrates the alignment and stitching of two 3D surfaces undergoing non-rigid deformations. 3D faces are captured using 3D scanner. Each face has approximately 80K 3D points with both shape and texture information available. The subjects were not asked to keep their head and facial expression still during the 3D face scanning. An important property of conformal mappings is that they can map a 3D shape to a 2D domain in a continuous manner with minimized local angle distortion. This implies that conformal mappings are not sensitive to surface deformations, e.g., if there is not too much stretching between two faces with different expressions, they will induce similar planar images. Therefore, matching on the planar images of conformal mappings are more reliable and accurate than direct matching in 3D. For partial surface matching, extra constraints are needed, such as feature points, feature lines, or area distortion factors.

33 2 BACKGROUND 30 (a) (c) (e) (g) (b) (d) (f) (h) Figure 35: An example of surface alignment and stitching: (a,b) Two original 3D faces with texture in different poses and deformations. (c,d) Original 3D faces without texture. (e,f) The conformal mapping images of the faces. (g) The aligned planar images of the two faces. (h) The resulting 3D face by stitching a part of (c) into (d). [Wang et al., 2007b] Geometric Modeling The surfaces obtained by 3D scanners are represented as point clouds. After geometric processing means, triangular meshes are constructed. In geometric modeling fields, surfaces are usually represented as piecewise polynomials or rational polynomials with higher order continuity, called splines. In order to convert meshes to splines, conformal geometric method is a most useful tool. For the purpose of generalizing splines from planar domain to manifold domain, special atlas needs to be constructed using conformal geometric methods. Manifold Splines Conventional splines are defined on planar domains. Manifold splines define polar forms directly on manifolds. Constructing splines, whose parametric domain is an arbitrary manifold, and effectively computing such splines in real-world applications are of fundamental importance in solid and shape modeling, geometric design, graphics, etc. [Gu et al., 2005a] shows the existence of manifold splines is equivalent to the existence of affine structure of the manifold, which is obstructed by topology. Figure 36 demonstrates the key components of manifold spline. Practical methods to compute affine atlas for general surfaces and generalize

34 2 BACKGROUND 31 F cafa cbfb fab faua fa fb fbub Z ua ub M Figure 36: Framework of manifold spline. various planar splines to surfaces are explained. (a) Holomorphic 1-form (b) Domain surface (c) Spline surface (d) control net Figure 37: Manifold Spline for a genus three surface. [Gu et al., 2006]. The key is to construct an affine atlas for the domain surface. Any surface with boundaries admits an affine structure and only genus one closed surfaces admit affine structure. In the figure, two holes are punched on the genus two surface, and then an affine atlas using holomorphic differential forms and manifold splines are constructed. Holomorphic differential forms induce affine atlas covering the whole surface except several zero points, the affine atlas can be used to construct the manifold splines. Figure 37 shows a manifold triangular B-Spline defined on a

35 3 GOAL AND SOLUTION 32 genus three surface. The theoretic framework of manifold splines is established in [Gu et al., 2005a] and [Gu et al., 2006]. Then the theory is applied to generalized many spline schemes on manifolds, such as manifold T-Splines [He et al., 2006b], triangular B-Spline [He et al., 2006a], polycube splines [Wang et al., 2007a]. Especially, manifold splines with single singularity [Gu et al., 2007a] is constructed, which reaches the theoretic limit. 3 Goal and Solution In this section, we demonstrate our research goals on conformal geometry theory and its applications on computer graphics and visualization, as follows: 1. Conformal spherical parameterization; 2. Consistent surface matching. 3. Shape space descriptor; 4. Quasiconformal mapping; 5. Surface remeshing; For each goal, we will give the detailed description, solution, analysis and initial experimental results. 3.1 Conformal Spherical Parameterization Abstract Surface parameterization establishes bijective maps from a surface onto a topologically equivalent standard domain. It is well known that the spherical parameterization is limited to genus-zero surfaces. In this work, we design a new parameter domain, two-layered sphere, and present a framework for mapping high genus surfaces onto sphere. This setup allows us to transfer the existing applications based on general spherical parameterization to the field of high genus surfaces, such as remeshing, consistent parameterization, shape analysis, and so on. Our method is based on Riemann surface theory. We construct meromorphic functions on surfaces: for genus one surfaces, we apply Weierstrass P-functions; for higher genus surfaces, we compute the quotient between two holomorphic 1-forms. Our method of spherical parameterization is theoretically sound and practically efficient. It makes the subsequent applications on high genus surfaces very promising.

36 3 GOAL AND SOLUTION 33 Surface parameterization (for a recent survey, we refer the reader to [Floater and Hormann, 2005]) is a fundamental tool in computer graphics and benefits many digital geometry processing applications such as texture mapping, shape analysis, compression, morphing, remeshing, etc. Some problems become much easier to deal with a uniform parameter domain. Usually in these settings surfaces are represented as triangular meshes, and the maps are required to be at least no-foldovers and low-distortion in terms of area, angle, or both aspects. In graphics, spherical parameterizations for genus zero closed surfaces have been proposed and widely used in the past. Most methods [Gotsman et al., 2003, Gu et al., 2004, Haker et al., 2000b, Sheffer et al., 2004, Praun and Hoppe, 2003] are to directly map the mesh to spherical domain, which is usually formulated as a spherical energy minimization problem, such as conformal, Tutte, Dirichlet, area, spring, stretch energies, or their combinations, as cited in [Floater and Hormann, 2005]. The optimization process is to relax the initial map to reach no-foldovers under specified distortion metric. In medical imaging, spherical parameterizations are broadly applied for brain cortex surface mapping. In this setting, preservation of local shapes are crucial. Therefore, different conformal spherical parameterizations are proposed. Angenent et.al. [Angenent et al., 1999] construct meromorphic functions on the brain surface directly, then lift the mapping onto the sphere using inverse stereographic projections. Gu et.al. [Gu et al., 2004] compute harmonic maps between the brain cortex surface and the unit sphere and use Möbius transformation to adjust the map. Stephenson [Stephenson, 2005] uses circle packing method to construct conformal brain mapping. However, it is well known that the spherical parameterization is limited to genus zero models. To the best of our knowledge, there are few works on high genus surfaces. Recently, Lee et.al. [Lee et al., 2006] present a construction method by boolean operations of positive and negative spheres. This method requires a lot of interactive human recognitions and geometry editing techniques. Furthermore, the results are not conformal. In this work, we aims at automatic generalizing conformal spherical parameterizations for high genus surfaces. Because high genus surfaces and spheres are not topologically equivalent, we allow the existence of branch points. Our method relies on the conformal structure for high genus meshes. There are two ways to compute conformal structures of general surfaces: one method is based on Hodge theory [Gu and Yau, 2002], and the other on discrete surface Ricci flow [Gu et al., 2005b, Jin et al., 2006a,b]. According to Riemann surface theory, a conformal map between a surface and the sphere is equivalent to a meromorphic function defined on the surface. The map wraps the surface onto the sphere by several layers and has several branch points. The number of layers and the branch points are determined by the topology of the surface (by Riemann-Hurwitz theorem). The key is how to construct the meromorphic functions on the input surface. For genus one closed

37 3 GOAL AND SOLUTION 34 Figure 38: Layered Sphere for genus one and genus two cases of our method. From left to right, they are (1) torus mesh with 10,000 vertices and 20,000 faces, (2) layered sphere with four branch points, (3) eight mesh with 12,286 vertices and 24,576 faces, and (4) layered sphere with six branch points. Two layers are connected by branch points where the lines twist together. Figure 39: Spherical conformal parameterization for torus case. From left and to right, each column denotes: (1) conformal parameterization result on layered sphere, (2) initial spherical conformal parameterization result on original surface, (3) spherical conformal parameterization result with Möbius transformation, and (4) curvilinear parameterization of (1), and (5) curvilinear parameterization of (2). surfaces, we construct the well-known Weierstrass P-function. For higher genus surfaces, the quotient between two holomorphic 1-forms is a meromorphic function. Compared with the existing planar parameterization for high genus meshes, the layered sphere (see Figure 38) is more natural domain than the planar domain. Employing the properties of sphere geometry and the existing spherical parameterization related applications on genus zero meshes, the spherical parameterization designed for high genus meshes (see Figure 39) can get more insights on shape analysis, and introduce more possible applications for high genus meshes. The contributions of this work are briefly as follows: 1. To present a novel practical framework to compute conformal spherical parameterizations

38 3 GOAL AND SOLUTION 35 for general surfaces; 2. To extend the applications of general spherical parameterization onto that of high genus meshes, including remeshing, morphing, etc; 3. To introduce a systematic method to compute meromorphic functions on general Riemann surfaces. 3.2 Consistent Surface Matching Abstract Surface matching plays an important role in movie production and computer animation, like human face expressions and body motions transferring. Surface parameterization is employed to align feature points for matching. To build the relationship between the source and target surfaces, cross-parameterization with prescribed feature constraints is highly used. In discrete case, the parameter density of points has direct relation to the accuracy of surface matching. Compared with conventional surface parameterization methods, we present a novel conformal parameterization with flexible boundary control, where user can specify which region to be enlarged. This parameterization method is suitable for multi-holed surfaces. The boundaries are parameterized to circular or straight slits, so we call the method, slit parameterization. This method is theoretically guaranteed to be intrinsic without self-overlapping. Based on this, a surface matching algorithm is constructed. The method improves the matching accuracy, which has been illustrated in the application of human face expressions surface matching. For human face expression surfaces sequence, the consistent remeshing is obtained. 3D modeling is a very basic and important technique in the modern movie industry, where face expressions and body motions transferring are the usual method for simulation from the character prototypes especially in animation production. Among these applications, surface matching is required to get consistent meshes between the source and target models for motion transferring, morphing, etc. The problem is formulated as finding the correspondence relation between the source and target. Manual methods directly from pictures are not recommended which costs much labor and time. It is easy to get a sequence of 3D surfaces by 3D scanner, like human face expression surfaces (see Figure 40) from the same person. These surfaces are not topologically consistent when obtained directly from the scanning equipment. The problem to automatically generate 3D face surfaces with consistent connectivity is much desired in modern industry. Because the advantages of angle-preserving and uniqueness, conformal parameterization is widely employed in surface matching, which has many applications including, but not limited to, consistent remeshing, shape deformation analysis, tracking etc.. The problem is especially

39 3 GOAL AND SOLUTION 36 Figure 40: Human face expression surfaces scanned by 3D equipment. hard when the transformation sought is diffeomorphic and non-rigid between the shapes being matched. For the surfaces with minor deformations, like human face expression surfaces, their parameterization results may be much different though under the same parameterization method. Cross-parameterization with feature constraints between the source and target surfaces [Kraevoy and Sheffer, 2004] is presented for surface matching. It computes a low-distortion bijective mapping between models that satisfies user prescribed constraints. Using the mapping, the remeshing algorithm preserves the user-defined feature points correspondence and the shape correlation between the models. We employ this method into our remeshing algorithm. For surface matching, initial alignment of feature points can be specified by user. There is much research around automatic computation of feature points, but it s difficult to find a mature and robust one. Currently, the practical way to get them is by manual markers before scanning. Wang et.al. [Wang et al., 2005b] added the feature correspondence constraints into conformal mappings and presented a fully automatic method for high resolution, non-rigid dense 3D point tracking, which unifies tracking of intensity and geometric features. In addition, human faces have strong symmetrical information, which is importantly useful for human face surface matching, hole-filling, stitching, etc. Here, the conformal map with symmetrical feature correspondence constraints is presented. Given the symmetry, it is easy to find the corresponding part on another half face. The difficulty is to efficiently find the 3D symmetrical plane then construct the feature points correspondence constraints. Compared with conventional surface parameterization methods, we present a novel conformal parameterization, where user can specify which region to be enlarged. This parameterization method is suitable for surfaces with boundaries. The boundaries are parameterized to circular or straight slits, so we call the method, slit parameterization. This method is theoretically guaranteed to be intrinsic without self-overlapping. Based on this, a surface matching algorithm is constructed. The method improves the matching accuracy, which has been illustrated in the application of human face expressions surface matching. For the expression surfaces sequence,

40 3 GOAL AND SOLUTION 37 Figure 41: Conformal mappings with free boundaries. Cross-parameterization method [Kraevoy and Sheffer, 2004] is used between columns 1 and 2, and columns 3 and 4. From upper to bottom, they are (1) the original face surfaces, only the fourth face has a boundary around the mouth area, (2) the cross-parameterization of (1) with 74 makers, and (3) the crossparameterization of (1) with symmetrical maker pairs. The parameterization method used here is of free-boundary, so the mouth area looks some strange in parameter domain. the consistent remeshing is obtained. About parameterization methods, we apply the conformal mapping by solving Beltramiequation using free boundary condition, and the conformal mapping of multi-holed annulus to annulus with concentric circular using holomorphic 1-form. The former method is conventional. It is linear and efficient but has less control of the boundaries. The mapping results may have self-overlapping. Extra constraints can be added to enhance the mapping result, such as feature

41 3 GOAL AND SOLUTION 38 Figure 42: Conformal mapping of multi-holed annulus to annulus with concentric circular arcs. There are three holes, two around eyes, and one around mouth. In row 1, the original outer boundary and the boundary of mouth are deformed to outer circle and inner circle respectively, the boundaries of two eyes to two circular slits. In row 2, the rectangle result is obtained by cutting the circle result (row 1) using a straight line through the inner and outer circles. In the current experimental result, cross-parameterization with features correspondence constraints has not been combined into this parameterization method yet. point constraints. When this method is applied onto face surfaces, some area are stretched too much and some overlapped, like the mouth area (see Figure 41, row 2 left). In order to add the freedom of the mapping around mouth area, cutting the mouth is introduced, which can get better results (see Figure 41, row 2 right). Because of the free-boundary condition, the boundary of mouth looks some strange in parameter domain. After the symmetrical information involved, more convincing parameterization results are obtained (see Figure 41, row 3). Compared with the former method, the latter method has much freedom in terms of boundaries. User can specify which boundary to be mapped to the outer boundary for enlargement. In detail, for the multi-holed annulus (e.g., face surface with eye and mouth area cut), special holomorphic 1-forms can be constructed, such that the whole surface is mapped to an annulus with concentric circular arcs. Two boundaries are mapped to the inner and outer boundaries of

42 3 GOAL AND SOLUTION 39 Figure 43: Conformal mapping of multi-holed annulus to annulus with concentric circular arcs. There are three boundaries, two around eyes, and one around mouth. In row 1, the boundaries of two eyes are deformed to outer circle and inner circle respectively, the boundary of mouth and the original outer boundary to two circular slits. Note that, the difference between these two circular results depends on the correspondence between two eye boundaries and two inner and outer circles. In row 2, the rectangle result is obtained by cutting the circle result (row 1) using a straight line through the inner and outer circles. In the current experimental result, cross-parameterization with features correspondence constraints has not been combined into this parameterization method yet. the annulus, other boundaries mapped to circular slits. Computing such holomorphic 1-forms is a linear problem, and the most difficult part is to find harmonic 1-forms. Figure 42 shows the conformal mapping of a three-holed face surface (the mouth is open), the target domain is a unit disk with two concentric circular arcs. The exterior boundary of the face is mapped to the outer circle of the annulus, the mouth boundary mapped to the inner circle. The boundaries of eyes are mapped to the two circular slits. The we conformally mapped the annulus to the band, the outer and inner circles are mapped to parallel lines and the boundaries of eyes are mapped to horizontal slits. If fixing one eye boundary to the outer circle and another eye boundary to the inner circle, we can get another mapping result, see Figure 43. The advantage of this method

43 3 GOAL AND SOLUTION 40 is that we can specify the boundary to be mapped to the outer circle for enlargement, thus the matching accuracy of the area around this boundary can be enhanced. About matching algorithm, the source and target surfaces are flattened onto 2D canonical domains through the conformal parameterization methods above. Through the cross-parameterization, the feature points is well aligned, then the correspondence of other parts is constructed. The consistent connectivity can be generated by sampling and lifting from 2D domain to 3D domain. The key difficulty is to find an alignment with features consistent everywhere. In order to advance the accuracy of alignment and the subsequent surface matching, we present a novel parameterization method with flexible boundary control which allows the enlargement of userspecified region (see Figures 42 and 43). Combining with other conventional parameterization methods (see Figure 41), we implement the consistent matching algorithm for a sequence surface with minor deformations. For human face expressions application, the matchings of different local regions, like mouth, eyes, or brown area, are performed by different parameterized methods. Applying consistent remeshing into the video of 3D objects can save much space and accelerate computation speed, which has much potential on the fields of movie production and computer animation. The contributions of this work are proposed to be: 1. To present a novel conformal planar parameterization method for multi-holed surfaces, which can make much control on the boundaries for local enlargement; 2. To carry out the consistent remeshing for human face expression surfaces; 3. To reproduce the video of human face expressions. 3.3 Quasiconformal Mapping Abstract Quasiconformal mapping is a generalized conformal mapping. In mathematics, it can be achieved by solving Beltrami equation. Many mathematical solution schemes have been presented so far. This work is to implement its discrete solution on triangular meshes, aiming to get the quasiconformal mapping for 3D surfaces. Except the angular metric, area factor is considered, which balances the original information between shape and size, and is much desired in medical imaging, like polyp detection through colon surface flattening. Quasiconformal mapping is a generalized conformal map. We refer the readers to [Ahlfors, 1966]. The notion of a quasiconformal mapping, but not the name, was introduced by H. Grötzsch in If Q is a square and R is a rectangle, not a square, there is no conformal mapping of Q on R which maps vertices on vertices. Instead, Grötzsch asks for the most nearly conformal mapping of this kind. This calls for a measure of approximate conformality, and

44 3 GOAL AND SOLUTION 41 in supplying such a measure Grötzsch took the first step toward the creation of a theory of quasiconformal mappings. It has wide applications in many fields, like in medical imaging, including brain flattening, colon flattening, etc. In mathematics, the concept of quasiconformal mapping, introduced as a technical tool in complex analysis, has blossomed into subject all its own. A conformal mapping in the plane sends small discs to other discs (to first order). A quasiconformal mapping on an open set is a continuous homeomorphism that sends small discs to small ellipses, in which the ratio of major axis to minor axis is bounded. Such mappings are in general not holomorphic functions, but play an auxiliary role in questions about such functions. Quasiconformal mappings are mappings of the complex plane to itself that are almost conformal. That is, they do not distort angles arbitrarily and this distortion is uniformly bounded throughout their domain of definition. Alternatively one can think of quasiconformal mappings as mappings which take infinitesimal circles to infinitesimal ellipses. For example invertible linear maps are quasiconformal. More rigorously, suppose f is a mapping of the complex plane to itself, and here we will only consider sense preserving mappings, which is mappings with a positive Jacobian, while negative for reversing mappings. Definition 1 Dilatation. Define the dilatation of the mapping f at the point z as D f (z) := f z + f z f z f z 1, and define the maximal dilatation of the mapping as K f := supd f (z). z Now we are ready to define what it means for f to be quasiconformal. Definition 2 Quasiconformal. For f as above, we will call f quasiconformal if the maximal dilatation of f is finite. We will say that f is K-quasiconformal mapping if the maximal dilatation of this mapping is K. Note that sometimes the term K-quasiconformal is used to mean that the dilatation is K or lower. It is easy to see that a conformal sense preserving mapping has a dilatation of 1 since f z = 0. We can further define several other related quantities. Definition 3 Small Dilatation. For f as above, define the small dilatation as d f (z) := f z f z. Again for sense preserving maps this quantity is less than 1 and it is equal to 0 if the mapping is conformal. Some authors call a map k-quasiconformal if the small dilatation is bounded by k. It is however not ambiguous as the large dilatation is always greater than or equal to 1. Furthermore this is related to the large dilatation by d f := D f 1 D f + 1.

45 3 GOAL AND SOLUTION 42 Definition 4 Complex Dilatation. For f as above, define the complex dilatation as µ f (z) := f z f z. The complex dilatation now appears in the Beltrami differential equation f z (z) = µ f (z)f z (z). This means that a quasiconformal mapping is a solution to the Beltrami equation where a nonnegative measurable µ f is uniformly bounded by some k < 1. The above results are stated for f : C C, but the statements are exactly the same if you take f : G C C for an open set G. Regular solutions of the Laplace-Beltrami equation are generalizations of harmonic functions and are usually called harmonic functions on the surface R (cf. also Harmonic function). These solutions are interpreted physically like the usual harmonic functions, e.g. as the velocity potential of the flow of an incompressible liquid flowing over the surface R, or as the potential of an electrostatic field on R, etc. Harmonic functions on a surface retain the properties of ordinary harmonic functions. A generalization of the Dirichlet principle is valid for them. In mathematics, quasiconformal mapping is formulated as the problem of solving Beltrami equation. There have been many mathematical and theoretical methods around it, but no corresponding implementation attempts on discrete triangular surface so far. The difficulty lies in how to solve a high nonlinear partial differential equation in discrete case. In practice, the concept of circle packing is used to carry out the quasiconformal mapping [Hurdal et al., 1999] for brain surface flattening, under a bounded amount of angular distortion. In this work, we aim to the following goals: 1. To present the discrete algorithm of solving Beltrami equation on triangular meshes; 2. To carry out a quasiconformal mapping algorithm on triangular meshes; 3. To apply the quasiconformal mapping for colon flattening. 3.4 Shape Space Descriptor Abstract A novel shape descriptor, shape space coordinates, is introduced, which is intrinsic, and invariant of similarity transformation. It can be computed for 2-manifolds with arbitrary structure. Initial experiments show that it is efficient and effective for shape retrieval and classification. In order to get more accurate results, a large amount of manifolds need to be created for classification experiments. And performance comparison with other shape descriptors will be explored on both idea and experiment. In general, surfaces are classified by different transformation groups, such as homeomorphism, which preserve topologies, and isometries, which preserve Riemannian metrics. Topological classification is too coarse for real applications; metric classification is too refined and

46 3 GOAL AND SOLUTION 43 Figure 44: Kitten and Torus. From topological classification, they all belong to the same class; from geometrical classification, none of them are in the same class; from conformal classification, the original kitten (or torus) and the deformed kitten (or torus) belong to the same conformal class, which can be visualized by the texture transferred from original model to deformed model without distortion of angles. [Jin et al.2007] expensive to compute, and the curvatures are too sensitive to local noises. In this work, we propose a novel classification, conformal classification, which overcomes the shortcomings of topological or metric ones. Surfaces can be classified by conformal equivalence relation, where each conformal equivalent class is also called a Riemann surface. For example, all genus zero closed surfaces can be conformally mapped to the unit sphere. Therefore, they are the same Riemann surface. As shown in Figure 44, all surfaces are topologically equivalent, but geometrically inequivalent. But under conformal classification, the kitten surfaces are equivalent to each other, same as the torii. This demonstrates that Teichmüller coordinates are much more discriminating than topological invariants. Furthermore, conformal classification is much more efficient than metric classification and the conformal invariants are much stabler than curvatures. This work proposes to classify surfaces using conformal mappings. Given the topology of the surfaces, all conformal equivalent classes form a finite dimensional manifold, the so-called Teichmüller shape space [Buser, 1992]. In this shape space, each point represents a class of surfaces, and a curve represents a deformation process from one class to the other. Two surfaces corresponds to different points if they can not be mapped to each other by a conformal map. Namely, they have different conformal structures. The geodesics between them can be explicitly computed, which indicates the most natural deformation with minimal distortion energy. The geodesic distance measures the similarity between the surfaces. Therefore, Teichmüller shape space is a good framework for the study of shape classification, shape comparison and deformation. The coordinates in the Teichmüller shape space have explicit geometric meanings. Basi-

47 3 GOAL AND SOLUTION 44 cally, the coordinates of a point in the shape space are the geodesic lengths of special curves on the surface under the uniformization metric. For the simplest case, let S be a three punctured sphere with boundaries b 0,b 1,b 2, then under the uniformization metric, it has Gaussian curvature -1 everywhere, and the boundaries become geodesics. The Teichmüller coordinates of S are the lengths of b 0,b 1,b 2. Hence, the Teichmüller shape space of 3-punctured sphere is 3 dimensional. As shape descriptors, Teichmüller coordinates have many advantages, invaluable for practical applications. Teichmüller coordinates are general for all manifold surfaces with arbitrary topologies. They are intrinsic, invariant under rotation, translation and scaling. Furthermore, the descriptors are invariant under isometric deformation. They are stable, for deformations with small area stretching, like the posture change of a human skin surface, which changes slightly. They are efficient, easy to compute and compare. The major goal of this work is to develop rigorous and practical algorithms to compute Teichmüller coordinates for arbitrary surfaces with negative Euler numbers. First we compute the uniformization metric of the surface. Then, we measure the geodesic lengths of a set of curves with algebraic methods. The geodesic lengths are the shape space coordinates of the surface. The major contributions of this work are 1. To propose a theoretical framework to model all surfaces in a shape space, Teichmüller shape space. The framework has deep roots in modern geometry and is practical for computation. It offers novel views and tools for tackling engineering problems; 2. To introduce a series of practical algorithms for computing Teichmüller shape space coordinates for surfaces with complicated topologies, which are conformal shape descriptors; 3. To apply Teichmüller coordinates for real applications, such as shape comparison, identification and retrieval. 3.5 Surface Remeshing Abstract A practical algorithm is introduced to approximate surface Delaunay triangulations using planar Delaunay triangulations of sampling points on conformal parametric domains. The method produces good approximation results and is practical for real applications. In geometric modeling and processing, computer graphics and computer vision, smooth surfaces are approximated by discrete triangular meshes reconstructed from sample points on the surfaces. A fundamental problem is to design rigorous algorithms to guarantee the geometric approximation accuracy by controlling the sampling density and triangulation method.

48 4 FEASIBILITY ANALYSIS 45 Figure 45: Approximated surface Delaunay triangulation by global conformal parameterization for a genus-zero closed surface. The sample points are 10k. From the left to right, they are (1) original surface, (2) spherical conformal mapping, (3) Delaunay triangulation, front view, and (4) Delaunay triangulation, back view. [Dai et al. 2007] Theoretical surface Delaunay triangulation is impractical, because the geodesic circles are difficult to compute. In this work, we propose to use global conformal parameterization [Gu and Yau, 2002, Gu et al., 2005b, Jin et al., 2006a,b] to map the surface to the canonical domains, generate random samples on the conformal parametric domains, and compute the planar Delaunay triangulation to approximate the surface Delaunay triangulation. Delaunay triangulations maximize the minimal angle, conformal parameterization preserves angles, therefore, Delaunay triangulations on conformal parametric domains approximate the surface Delaunay triangulations faithfully. The result can be seen in Figure 45. This method was proposed in [Alliez et al., 2002] for topological disk case. In current work, we generalize the algorithm for arbitrary surfaces. The work is included as one part of the work [Dai et al., 2007], which gives explicit formulae of approximation error bounds for both Hausdorff distance and normal distance in terms of principle curvature and the radii of geodesic circum-circle of the triangles. These formulae can be directly applied to design sampling density for data acquisitions and surface reconstructions. 4 Feasibility Analysis 1. Conformal Spherical Parameterization. Conformal spherical parameterization for high genus surfaces can be obtained by meromorphic function theory. The branch points can not be located accurately on discrete case. There is much overlapping and flipping around branch points. Therefore, there is much technical difficulty. 2. Quasiconformal Mapping. The quasiconformal mapping is obtained by solving Beltrami