Computational Conformal Geometry and Its Applications...1 Introduction Conformal geometry is in the...

Computational Conformal Geometry and ItsApplications

Wei Zeng

Institute of Computing TechnologyChinese Academy of Sciences

[email protected]

Thesis Proposal

Advisor: Harry ShumCo-advisor: Xianfeng Gu

October 11, 2007

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Abstract

Conformal geometry has deep roots in pure mathematics. It is the intersection of com-plex analysis, Riemann surface theory, algebraic geometry, differentialgeometry and al-gebraic topology. Computational conformal geometry plays an important rolein digitalgeometry processing. Recently, theory of discrete conformal geometry and algorithms ofcomputational conformal geometry have been developed. A series of practical algorithmsare presented to compute conformal mapping, which has been broadly applied in a lot ofpractical fields, including computer graphics, computer vision, medical imaging, visualiza-tion, and so on.

The thesis focuses on computational conformal geometry and its applicationson com-puter graphics and visualization, including surface conformal spherical parameterization,3D shape space descriptor, quasiconformal mapping, surface remeshing, and consistentmatching. Practical conformal parameterization methods are generated forspecified pop-ular applications, like human face expressions matching, and colon flattening. The initialexperimental results are very promising.

CONTENTS ii

Contents

1 Introduction 1

2 Background 22.1 Conformal Geometry Theory . . . . . . . . . . . . . . . . . . . . . . . . . .. 3

2.1.1 Conformal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Riemann Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Riemann Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Shape Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.5 General Geometric Structure . . . . . . . . . . . . . . . . . . . . .. . 10

2.2 Computing Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . .122.2.1 Algorithms Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212.3.1 Computer Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Medical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.3 Computer Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.4 Geometric Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Goal and Solution 323.1 Conformal Spherical Parameterization . . . . . . . . . . . . . . .. . . . . . . 323.2 Consistent Surface Matching . . . . . . . . . . . . . . . . . . . . . . . .. . . 353.3 Quasiconformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 403.4 Shape Space Descriptor . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 423.5 Surface Remeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Feasibility Analysis 45

5 Innovations 46

6 Resource and Progress 476.1 Existing Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .476.2 Finished Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.3 Ongoing Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

7 Schedule 48

8 Acknowledgement 48

1 INTRODUCTION 1

1 Introduction

Conformal geometry is in the intersection of many fields in pure mathematics, such as Riemannsurface theory, differential geometry, algebraic curves,algebraic topology, partial differentialgeometry, complex analysis and many other related fields. Ithas long history in pure mathe-matics, and is an active field in both modern geometry and modern physics, for example, theconformal 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 di-mensional digital models are available. The needs for effective methods to represent, process,and utilize the huge amount 3D surfaces become urgent. Digital geometric processing emergesas an inter-disciplinary field, combining computer graphics, computer vision, visualization andgeometry.

Computational conformal geometry plays an important role indigital 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 foun-dation 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 medi-cal imaging, for brain mapping [Gu et al., 2003], virtual colonoscopy [Hong et al., 2006], anddata 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 anatural conformal structure. Therefore, the conformal geometric algorithms are very gen-eral.

• Conformal structure of a general surface is more flexible thanRiemannian metric struc-ture and more rigid than topological structure. It can handles large deformations, whichRiemannian geometry can not efficiently handle; it preservesa lot of geometric informa-tion during the deformation, whereas, topological methodslose too much information.

• Conformal maps are easy to control. For example, the conformal maps between twosimply connected closed surfaces form a six dimensional space, therefore by fixing threepoints, the mapping is uniquely determined. This fact makesconformal geometric methodvery valuable for surface matching and comparison.

• Conformal maps preserve local shapes, therefore it is convenient for visualization pur-poses.

2 BACKGROUND 2

• All surfaces can be classified according to their conformal structures. All the conformalequivalent classes form a finite dimensional manifold. Thismanifold has rich geometricstructures, and can be analyzed and studied. In comparison,the isometric classes ofsurfaces form an infinite dimensional space. It is really difficult to deal with.

• Computational conformal geometric algorithms are based on solving elliptic partial dif-ferential equations, which are easy to solve and the solvingprocess is stable, namely,the solution is insensitive to the noise of the input surfaces. Therefore, computationalconformal geometry method is very practical for real engineering applications.

• In conformal geometry, all surfaces in daily life can be deformed to three canonicalspaces, the sphere, the plane or the disk (the hyperbolic space). In other words, anysurface admits one of the three canonical geometries, spherical geometry, Euclidean ge-ometry or the hyperbolic geometry. Most digital geometric processing tasks in three di-mensional 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 beconformally mapped to each other, they share the same conformal structure. Therefore, com-puting conformal mappings is equivalent to computing conformal structures for surfaces. Ac-cording to Riemann uniformization theorem, all metric surfaces can be conformally deformedto three canonical surfaces, the sphere, the plane and the hyperbolic disk. Different algorithmsare designed to compute the uniformization metrics: (1) forspherical case, harmonic maps arecomputed by using heat flow method; (2) for Euclidean case, holomorphic 1-forms are com-puted; (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 alge-braic geometry, which studies conformal structure of surfaces. Conformal geometry is betweentopology and geometry, softer than geometry and harder thantopology. Conformal geometryplays a fundamental role in nature and the engineering world.

This section introduces major concepts and theorems of conformal geometry and a series ofcomputational algorithms and the related applications in geometric modeling, computer graph-ics, computer vision and medical imaging.

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2.1 Conformal Geometry Theory

The geometric information of a surface has many layers, as shown in Table 1. Higher struc-tures determine lower structures. Riemannian metric structure determines conformal structure.Lower structures confine higher structure. Topological structure determines the total Gaussiancurvature induced by Riemannian metric structure. The inter-relations among structures areprofound and subtle.

Conformal geometry studies theconformal structureof general surfaces. Conformal struc-ture is a natural structure of all surfaces in real life. Riemannian metric is a structure to measurethe lengths of curves on the surface, area of domains on the surface and the intersection an-gles between curves. Conformal structure is a structure to only measure the intersection anglebetween two curves on the surface. Topological structure gives the neighborhood information.Roughly speaking, Conformal structure is more rigid than topological structure and moreflexible than Riemannian metric. Conformal geometry is between topology and Riemanniangeometry.

Conformal geometry originated from the study of natural phenomena in classical physics,such as heat diffusion, electromagnetic field, fluid field andelasticity deformation. Mathemati-cally, conformal geometry is the intersection of many mathematical branches. It has rich struc-tures and abundant theoretic tools, such as differential geometry, algebraic topology, complexanalysis, algebraic geometry, and complex manifold. Conformal field theory is a quantum fieldtheory, and plays important roles in string theory, statistical mechanics and condensed matterphysics. Many engineering applications can not be solved without using conformal geometry.The main work on this field is to convert all surface geometricproblems to special problems onthree canonical domains,S

2, D2, andR

2.Here, we will get further understanding of the above statements through further discussions

around the following aspects.

2.1.1 Conformal Structure

A conformal structure is a structure assigned to a topological manifold, such that angles can bedefined (See Figure 1). It is easy to define angles on the parameter plane. But a manifold cannot be covered by a single coordinate system, instead it is covered by many local coordinatesystems with overlapping (See Figure 9). In relativity, thephysics law is independent of thelocal coordinate system of the observer. If the transition function from one local coordinatesto 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 thatall transition maps are conformal,then angle can be consistently defined on the manifold.

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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 (Uα, φα), whereφα :Uα → R

2. If two charts (Uα, φα) and (Uβ, φβ) overlap, the transition functionφαβ : R2 → R

2

is defined asφαβ = φβ ◦ φα−1. If all transition functions are analytic, then the manifold is a

Riemann surface. The atlas (Uα, φα) is a conformal structure.

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Table 1: Geometric Structure.

Geometric Concept Transformations Data TheoreticStructure Structure Tool

Topological neighborhood homeomorphism connectivity homology,Structure cohomology,

fundamentalgroup

Differential differentiability, diffeomorphism / differentialStructure smoothness, topology,

tangent Morsefunction

Conformal angle, conformal discrete RiemannStructure holo/meromorphic maps holomorphic surface,

forms 1-forms complexgeometry

Riemannian distance, isometry edge RiemannianStructure geodesic, lengths geometry

areaEuclidean position rigid vertex EuclideanStructure motion position geometry,

differentialgeometry

2.1.2 Riemann Mapping

A conformal mapbetween two surfaces preserves angles.Riemann mappingtheorem states thatany simply connected surface with a single boundary (a topological disk) can be conformallymapped to the unit disk. As shown in Figure 3, the frontal partof a human faceS is a topologicaldisk and mapped to the unit diskD by a conformal mappingφ : S → D. Supposeγ1, γ2 are twoarbitrary curves on the face surfaceS, φ maps them toφ(γ1), φ(γ2). If the intersection anglebetweenγ1, γ2 is θ, then the intersection angle betweenφ(γ1) andφ(γ2) is alsoθ. γ1 andγ2 canbe chosen arbitrarily. Therefore, we sayφ is conformal, meaning angle-preserving.

The conformality can be visualized usingtexture mappingtechniques in computer graphics.Figure 4 illustrates the idea. A texture refers to an image onthe plane. First a conformalmapping between the face surface (a) to the unit disk (b) is established. Then cover the disk

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θθ

φ

S D

γ1γ2 φ(γ1)φ(γ2)

Figure 3: A Riemann mapping from a human face to the unit disk, the mapping is angle-preserving.

(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 bedirectly visualized. If the texture is a checker board, all the right angles of the corners of thecheckers are preserved on the human face as shown in (c). If replacing the texture by circlepacking pattern, then planar circles are mapped to circles on the surface, the tangency relationamong circles are preserved as shown in (d).

Figure 5 shows the conformal mappings of a multi-holed annulus. The planar domains arecircular disk with circular holes, which is computed by Ricciflow. The planar domains aredetermined by the conformal structure of the original surface.

A conformal map, also called an angle-preserving map, is a transformation that preserveslocal angles. Two surfaces areconformal equivalent, if there exists a bijective conformal mapbetween them. Surfaces can be classified by conformal equivalence relation, where each con-formal equivalent class is also called aRiemann surface. For example, all genus zero closed

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γ0

γ1

γ1γ2

γ0

γ1

γ2 γ3

Figure 5: Conformal mappings of a multi-holed annulus.

surfaces can be conformally mapped to the unit sphere. Therefore, they are the same Rie-mann surface. Conformal maps are stronger than harmonic maps. Harmonic maps between twohomeomorphic surfaces exist, but conformal maps may not exist.

Algorithm (Riemann Mapping): Given a topological disk surface, it can be conformallymapped to the unit disk as follows:

1. Double covering.

2. Conformally map the doubled surface to the unit sphere.

3. Use the sphere Mobius transformation to make the mapping symmetric.

4. Use stereographic projection to map each hemisphere to the unit disk.

2.1.3 Riemann Uniformization

The conformal map between two planar domains is the conventional analytic function, or holo-morphic function. From this point of view, conformal mappings are the generalization of holo-morphic functions, and Riemann surfaces are the generalization of complex plane. All surfacesin real life are real surfaces. The derivative of a analytic function is called aholomorphic dif-ferential. Holomorphic differentials can be defined on surfaces directly. They can be visualizedusing the same technique as the visualization of conformal mapping. By integrating the holo-morphic differentials, the surface can be locally mapped tothe plane, the mapping is conformaland visualized by checker board texture mapping (see Figures 6 and 7).

Holomorphic, meromorphic functions and holomorphic, meromorphic differentials on thesurface form special groups, the group structure is governed by Riemann-Roch theorem, whichis a profound theory, connecting geometry, topology and partial differential geometry.

Riemann uniformization theoremstates that all surfaces in real life can be conformallymapped to three canonical shapes, the unit sphere, the Euclidean plane and the hyperbolic space.

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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 hyper-bolic 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 with-out 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 tothe plane. In details, the wholekitten surface in the middle column of the figure can be conformally mapped to a parallelogramon the plane, the repetition of the parallelogram forms a tiling of the whole plane. The thirdcolumn shows a exotic bottle, it has two handles. The surfacehas no self-intersection and isembedded in the Euclidean space. It can be conformally periodically mapped to the unit disk,

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(a)χ > 0 (b) χ = 0 (c) χ < 0

Figure 8: Uniformization Theorem: all surfaces with Riemannian metric can be conformallyembedded 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 ismapped to a hyperbolic octagon,the repetition of the octagon forms a tiling of the whole hyperbolic space.

2.1.4 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 surfacesare conformal equivalent. Roughly speaking, for closed surfaces with one handle, if the shapesof the parallelograms, then they are conformal equivalent.Same result holds for surfaces withmore handles. For closed surfaces with two handles, if theirhyperbolic octagon are congruentin the hyperbolic space, then they are conformal equivalent.

The conformal equivalence classes form a finite dimensionalspace, which is called the Te-ichmuller 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 under-stand the topological and geometric structures of the shapespace is the most active research in

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Figure 9: Riemann Surface: The manifold is covered by a set of charts (Uα, φα), whereφα :Uα → R

2. If two charts (Uα, φα) and (Uβ, φβ) overlap, the transition functionφαβ : R2 → R

2

is defined asφαβ = φβ ◦ φα−1. If all transition functions are analytic, then the manifold is a

Riemann surface. The atlas (Uα, φα) 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 describedin [Jin et al., 2007a].

2.1.5 General Geometric Structure

A conformal structure is a structure assigned to a topological manifold, such that angles can bedefined (See Figure 1). It is easy to define angles on the parameter plane. But a manifold cannot be covered by a single coordinate system, instead it is covered by many local coordinatesystems with overlapping (See Figure 9). In relativity, thephysics law is independent of thelocal coordinate system of the observer. If the transition function from one local coordinatesto 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 thatall 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 collectionof open sets to cover the surface and map each open set to the plane, we then use the planarcoordinates as the local coordinate system for the corresponding open set. Such kind of localcoordinate system is call anatlasof the surface. One point on the surface may be covered bymultiple local coordinates, the transformation from one local coordinates system to another iscalled thetransition functionor coordinates change(See Figure 9).

SupposeX is a topological space,G is the transformation group ofX, a (X,G) structureis an atlas, such that the local coordinates are inX, the transition functions are inG. For

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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, allthe transition functions are rotations. Figure 8 can be interpreted as the visualization of sphericalstructure, Euclidean structure and the hyperbolic structure respectively.

According to Flix Klein’s Erlangen program, different geometries study the invariants underdifferent transformation groups. For example, letX be the Euclidean plane, ifG is rigid motion,then the geometry is Euclidean geometry, the invariants arelengths, angles, area etc. IfG isaffine transformation group, then the geometry is the affine geometry, the major invariant is theratio for three points on a line, parallelism. IfG is the real projective transformation group, thenthe corresponding geometry is real projective geometry. The major invariant is cross ratio forfour points on a line.

Figure 11: Visualization of the hyperbolic structure and the real projective structure of a genustwo vase model. [Jin et al., 2006b]

If a surface admits a(X,G) structure, then the corresponding geometry can be defined onthe surface directly. For example, in automobile industry and mechanics engineering fields,surfaces are represented as splines, which are piecewise rational polynomials. Conventional

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splines are defined on the Euclidean plane and constructed based on affine invariants. Thefundamental problem in computer aided geometric design (CAD) field is to construct splinesdefined on arbitrary surfaces. If the surface admits anaffine structure, then splines can bedefined on the surface without any difficulty. Unfortunately, very few surfaces admits affinestructure. This fact causes intrinsic difficulty for applications in CAD field. Fortunately, allsurfaces admitsreal projective structure. How to construct splines based on real projectivegeometry is an active research direction in geometric modeling today.

Like conformal structure, the shape space of all conformal structures has rich topologicaland geometric properties. The understanding of the shape space of all(X,G) structures iswidely open also.

2.2 Computing Conformal Mapping

Figure 12: Face geometries with different expressions of the same person, scanned using realtime high speed high resolution scanner.

Recently,3D scanning technology is developing extremely fast. Figure 12 shows severalfacial surfaces with different expressions of the same person, scanned by a scanner based onphase-shifting method. The scanning speed is as fast as180 frames per second, each framehas640 × 480 samples. The scanner can capture dynamic facial expressions in real time. Itis challenging to process the huge amount geometric data efficiently and robustly. Conformalgeometry offers powerful tools to tackle the problem. The main strategy is to use conformal

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mappings to transform 3D surfaces to canonical 2D domains, and convert 3D geometric prob-lems to 2D ones.

In computer graphics and discrete mathematics, much sound research has focused on dis-crete conformal parameterizations. Here, we briefly overview related works, and refer readersto [Floater and Hormann, 2005, Kraevoy and Sheffer, 2004] for thorough surveys.

All parameterizations can be classified according to the type of output produced, whichcan be a vector valued function, i.e. a mapping, a holomorphic differential form or a flat Rie-mannian metric. In general, the derivative of a conformal map is a holomorphic 1-form; eachholomorphic 1-form induces a flat metric. Therefore, methods which compute metrics are themost general, although they are more expensive to compute.

1. Mappings. First order finite element approximations of the Cauchy-Riemann equationswere introduced by Levy et al. [Levy et al., 2002]. Discrete intrinsic parameterization byminimizing Dirichlet energy was introduced by [Desbrun et al., 2002], which is equiv-alent to least-squares conformal mapping [Levy et al., 2002]. Discrete harmonic mapswere computed using the cotan-formula in [Pinkall and Polthier, 1993]. Mean value coor-dinates were introduced in [Floater, 2003]; these generalize the cotan-formula. All theselinear methods can easily incorporate free boundary conditions to improve the quality ofthe parameterization produced, such as the methods in [Desbrun et al., 2002] and [Zayeret 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 computeglobal conformal surface parameterizations for high genussurfaces. Discrete holomor-phy was introduced in [Mercat, 2001] using discrete exterior calculus [Hirani, 2003]. Theproblem of computing optimal holomorphic 1-forms to reducearea distortion was con-sidered in [Jin et al., 2004]. Gortler et al. [Gortler et al.,2006] generalized 1-forms to thediscrete case, using them to parameterize genus one meshes.Recently, Tong et al. [Tonget al., 2006] generalized the 1-form method to incorporate cone singularities.

3. Metrics. There are three major methods for computing edgelengths (or equivalently theangles): angle based flattening, circle packing, and circlepatterns. Sheffer and Sturler[Sheffer and de Sturler, 2001] introduced the angle based mesh flattening method. Thisworks by posing a constrained quadratic minimization problem seeking to find corner an-gles which are close to desired angles in a weighted L2 norm. The efficiency and stabilityof ABF are improved in [Sheffer et al., 2005] by using advancednumerical algorithmand hierarchical method. The circle packing method was introduced in [Thurston, 1976].Continuous conformal mappings can be characterized as mapping infinitesimal circles toinfinitesimal circles. Circle packings replace infinitesimal circles with finite circles. In

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the limit of refinement the continuous conformal maps are recovered [Rodin and Sullivan,1987]. Collins and Stephenson [Collins and Stephenson, 2003]have implemented theseideas in their software CirclePack. The first variational principle for circle packings, waspresented in a seminal paper by Colin de Verdiere [de Verdiere, 1991]. Circle patternsbased on those in Bobenko and Springborn [Bobenko and Springborn, 2004] have beenapplied for parameterization in [Kharevych et al., 2006]. Springborn [Springborn, 2003]shows that in theory, circle packing and circle patterns areequivalent.

4. Ricci Flow. Recently, a novel curvature flow method in geometric analysis is introducedto prove the Poincare conjecture, the Ricci Flow. Ricci flow refers to conformally deformthe Riemannian metric of a surface by its Gaussian curvature,such that the curvatureevolves according to a heat diffusion process. Ricci flow is a powerful tool to computethe Riemannian metric by the curvature. It can be applied for discrete conformal pa-rameterizations. The connection between circle packing and smooth surface Ricci flow[Hamilton, 1988] was discovered in [Chow and Luo, 2003]. Conventional circle pack-ing only considers combinatorics. The discrete Ricci flow method was introduced in [Jinet al., 2006b, Gu et al., 2007b], which incorporate geometric information and was appliedfor computing hyperbolic and projective structure and manifold splines.

2.2.1 Algorithms Overview

According to Riemann uniformization theorem, all metric surfaces can be conformally mappedto three canonical shapes, the sphere, the plane and the hyperbolic disk. The mappings areperiodic and reflect the intrinsic symmetries of the surfaces.

Figure 8 shows three kinds of algorithms for computing conformal structure.

• For genus zero surfacein the first column, the mapping can be computed using sphericalharmonic maps, in paper [Gu et al., 2003]. Spherical geometry can be defined on thesurface.

• For genus one surfacein the second column, the mapping can be computed using holo-morphic 1-forms, in paper [Gu and Yau, 2003b]. Another algorithm is to use EuclideanRicci flow, in paper [Jin et al., 2006a]. Euclidean geometry can be defined on the surface.

• Forhigher genus surfacein the third column, the mapping can be computed using hyper-bolic Ricci flow, in paper [Jin et al., 2006b]. Hyperbolic geometry can be defined on thesurface.

The followings are the detailed descriptions for the major algorithms of computing confor-mal mappings of surfaces with various topologies.

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Harmonic Maps for Topological Disks The harmonic map between simply connected sur-faces with a single boundary and a convex planar domain can becomputed by solving Dirichleltproblem. First fix the boundary on a convex planar curve, and compute the interior by minimiz-ing harmonic energy. The Euler-Lagrange equation of the critical point of the harmonic energyis the Laplace equation. By using finite element method, the Laplace equation is formulatedas a symmetric positive definite linear system. The problem is a linear problem. The energyoptimization can be performed using conjugate gradient method efficiently. Figure 13 shows aharmonic 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 sphereand the canonical unit sphere is automatically conformal. The computational algorithm is basedon 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 alongthe tangential component of the Laplacian. Because of the projection to the tangential space,the heat diffusion process becomes non-linear. Different solutions differ by Mobius transfor-mations. Therefore, normalization conditions are necessary. Figure 14 shows one example ofconformal mapping of a topological sphere.

Riemann Mappings of Topological Disks Harmonic maps between a topological disk to theunit disk may not necessarily be conformal. First compute the double covering of the topolog-ical disk, which is a topological sphere, then compute a conformal map between the doubledsurface and the unit sphere, such that, each copy of the topological disk is mapped to a hemi-sphere. Then use stereo-graphic to project the unit sphere onto the whole plane, the lowerhemisphere is mapped to the unit disk. This induces the mapping from the surface to the unit

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Figure 14: Harmonic maps for topological spheres.

disk, and the map is conformal. Figure 15 shows a Riemann mapping from a human face to adisk on the plane.

Figure 15: Riemann mapping.

Conformal Mappings with Free boundaries Conformal mappings with free boundaries canbe 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 disadvan-tage is the less control of the boundaries. It mainly handle genus zero surfaces. The mappingresults may have self-overlapping. Extra constraints can be added to enhance the mapping re-sult, such as feature point constraints. Figure 16 shows oneexample of conformal mappingwith free boundary from a human face to the plane.

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Figure 16: Solving Beltrami equation using free boundary condition.

Holomorphic 1-forms All metric surfaces are Riemann surfaces, which admit special com-plex differential forms,holomorphic 1-forms. The group of the holomorphic 1-forms has specialstructure, the generators can be explicitly calculated. A holomorphic 1-form has zero points, thenumber of zero points equals to the absolution value of the Euler number. In the neighborhoodof normal points, holomorphic 1-form induces conformal maps between the neighborhood tothe complex domains. Iso-parametric curves through zero points can be used to segment thesurface. Figure 17 illustrates a conformal texture mappinginduced 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 re-fer to the atlases, such that all the local coordinates changes belong to the special transformationgroups. Most popular spline schemes are constructed based on affine invariants, therefore canbe 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

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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. Ingeneral cases, extraordinary points have to be introduced.Conventional subdivision surfacesare splines defined on the surfaces with extraordinary points. Holomorphic 1-forms naturallyinduce affine structures on the surfaces. The extraordinarypoints are the zero points of theholomorphic 1-form, as show in the Figure 18, the centered octagons are zero points. Thenumber of zero points equals to the Euler number of the surface. This pave the way to definingvarious planar splines on general surface domains.

Conformal Mapping of Multi-Holed Annuli to Annulus with Conce ntric Circular ArcsSpecial holomorphic 1-forms can be constructed on a multi-holed annulus, such that the wholesurface is mapped to an annulus with concentric circular arcs. Two boundaries are mappedto the inner and the outer boundaries of the annulus, the other boundaries are mapped to theslits. Computing such holomorphic 1-forms is a linear problem, and the most difficult part isto find harmonic 1-forms. ChapterHolomorphic Forms algorithmsexplains the details. Figure19 shows the conformal mapping of a three-holed face surface, (the mouth is cut open), thetarget domain is a unit disk with two concentric circular arcs. The exterior boundary of theface is mapped to the outer circle of the annulus, the mouth boundary is mapped to the innercircle. The boundaries of eyes are mapped to the two circularslits. Then conformally mappingthe annulus to the rectangle, the outer and inner circles aremapped to parallel lines and theboundaries of eyes are mapped to horizontal slits.

Euclidean Ricci Flow for Genus One Surface Euclidean Ricci flow method computes spe-cial metrics of the surface conformal to the original metricwith prescribed target curvature. Forgenus one closed surface, set the target Gaussian curvatureto be zero everywhere, and computethe flat metric, which is conformal to the original induced Euclidean metric. The universal cov-

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Figure 19: Conformal Mapping between a multi-holed annulus to an annulus with concentriccircular arcs and a rectangle with slits.

ering space of the surface can be isometrically embedded on the plane. Figure 20 shows oneexample. The kitten surface is of genus one, the universal covering space is embedded on theplane. 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 aunique Riemannian metric, which is conformal to the originalRiemannian metric, and in-duces constant Gaussian curvature everywhere, the constant is −1. Such kind of metric canbe computed using hyperbolic Ricci flow. The universal covering space of the surface can beisometrically embedded on the hyperbolic space. Figure 21 demonstrates the embedding of theuniversal covering space of a genus two surface on the Poincare model of hyperbolic space.

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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, suchthat all chart transitions are real projective transformations. Such kind of the projective atlas canbe 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 coveringspace on the Poincare model of the hyperbolic space, finally transform the Poincare model to theKlein model, where all the rigid motions are real projectivetransformations. This embeddinginduces 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 thecurvatures of a surface are essentially of one-to-one correspondence. The conformal metric canbe 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 planardomains specified by curvature on the boundaries. The curvature of interior points are zero

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Figure 23: Conformal flat metrics are designed by the target curvature.

everywhere. The conformal mapping induced by the metric is fully controlled by the prescribedcurvatures. It is also possible to concentrate all the curvature of a surface with arbitrary topologyto a single point.

2.3 Applications

Computational conformal geometric methods are valuable fora broad range application in ge-ometric modeling, computer graphics, computer vision, visualization medical imaging and sci-entific computing and many other engineering fields. In the following, we briefly browse somemost direct applications of conformal geometric methods.

2.3.1 Computer Graphics

Conformal geometry has numerous applications in computer graphics, including surface param-eterization, mesh repairing, texture mapping and synthesis, surface re-meshing, mesh matching,mesh-spline conversion, geometric morphing, efficient rendering, animation and many other ap-plications.

Global Conformal Parameterization In computer graphics, surface Parameterizations referto the process to map the surface onto 2D planar domains. To avoid the problems associated

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with discontinuous boundaries, global conformal parametrization, which preserves conformal-ity 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 computingglobal conformal param-eterizations for general surfaces, with nontrivial topologies, with or without boundaries, usingthe structure of the cohomology group of holomorphic 1-forms. Jin et. al. [Jin et al., 2004] pro-vided an explicit method for finding optimal global conformal parameterizations of arbitrarysurfaces. Figure 24 illustrates the whole process.

As a powerful geometric tool to compute the uniformization metric, Ricci flow has been in-troduced 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 structuresin an unified way using Euclidean Ricci flow. Jin et. al. [Jin et al., 2006b] introduced discretevariational Ricci flow to compute the hyperbolic structure and real projective structure for gen-eral surfaces with negative Euler characteristic numbers.The method is efficient and robust inpractice.

Texture Mapping In computer graphics, surfaces are approximated by triangular meshes(polygonal surfaces, each face is a triangle), which can be supported by graphics hardwaredirectly. The rendering efficiency of the hardware depends on the resolution of the mesh. Forreal time applications, time is critical, therefore low resolution meshes are highly preferred.Small geometric details, and material properties are modeled as texture images. The parame-terization process map each vertex to the planar domain, andobtain its 2D coordinates, whichis called the texture coordinates. Then graphics hardware will glue the texture to the meshesusing the texture coordinates. Figure 25 illustrates the process.

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(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 retessellatethe triangular meshes to quad meshes. Because most popular spline schemes are based ontensor-product. First parameterize the triangular mesh onto the planar region, and use regulargrids to tessellate the planar image of the surface, this induces the tessellation of the originalsurface and convert it to a quad mesh. Figure 26 demonstratesone example for remeshing atriangular 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 in-formation represented as the coordinates of the vertices. After remeshing, the quadmesh con-nectivity is regular, it is unnecessary to encode the connectivity any more and just record thecoordinates 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

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meshes, the other one handles texture images. Geometry image unifies both geometry andtexture, which has the potential to simplify the graphics hardware. Geometry images can beapplied for efficient rendering. The details of geometry image is presented in [Gu et al., 2002].

2.3.2 Medical Imaging

With the rapid development of medical imaging technologies, vast medical imaging data areavailable today. In order to fuse medical images acquired from different modalities, extract sur-faces or volumes, register, fuse and compare different geometric data sets, conformal geometricalgorithms have been developed and proven to be valuable forreal 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 otherextraneous 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 anddatabasing of brain maps.Computational problems arise when integrating and comparing brain data. The cortex surfaceof a brain is highly convoluted and anatomical structures varies from person to person. Oneway to analyze and compare brain data is to map them into a canonical space while retaininggeometric information on the original cortex surface as faras possible.

Cortical surfaces are of genus zero, therefore, they can be conformally mapped onto the unitsphere. All such conformal mappings differ by Mobius transformations of the sphere, whichform a6 dimensional group. For genus zero closed surfaces, harmonic maps are also conformal.A conformal mapping can be obtained by optimizing the harmonic energy. Further constraintsare added to ensure that the conformal map is unique. Empirical tests on magnetic resonanceimaging (MRI) data show that the mappings preserve angular relationships, are stable in MRIs

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acquired at different times, and are robust to differences in data triangulation, and resolution.Figure 2.3.2 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 packingto compute an approximation to the conformal map of a cortical surface. For genus zero sur-faces, a unique mapping between any two genus zero manifoldscan be found by minimizingthe harmonic energy of the map. Conformal brain mapping usingnonlinear heat diffusion isintroduced in [Gu et al., 2003] and [Gu et al., 2004]. Spherical conformal mapping is presentedin [Gotsman et al., 2003]. Conformal brain mapping based on Riemann surface structure isexplained in [Wang et al., 2007c].

Computational and visualization tools are needed to interact with these conformal flat mapsto gain information about spatial and functional relationships that might not be apparent. Suchinformation can contribute to earlier diagnostic tools fordiseases and improved treatment.

Conformal Colon Flattening Colon cancer is one of the leading causes of cancer deathsin the United States. Computed tomographic (CT) colonographyis a technique used in thedetection of colonic polyps, the precursors of colorectal carcinoma, with CT of the cleansedand air-distended colon. Virtual colonoscopy has been successfully demonstrated to be moreconvenient and efficient than the real optical colonoscopy.However, because of the length ofthe 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) Conformalreconstructed from CT flattening images parameterization

Figure 28: Conformal colon flattening. [Hong et al., 2006].

Virtual dissection is an efficient visualization techniquefor polyp detection, in which theentire 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 centralpath and unfolding the cross sections. Paik et al. [Paik et al., 2000] use cartographic projections

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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 cylinderwhich is mapped finally to theimage. Haker et al. [Haker et al., 2000a] propose a method based on the discretization ofthe Laplace-Beltrami operator to flatten the colon surface onto the plane in a manner whichpreserves angles. Hong et al. [Hong et al., 2006] conformally map the 3D colon surface to a 2Drectangle using conformal structure, which is general and can handle high genus surfaces (seeFigure 28).

2.3.3 Computer Vision

Conformal geometry has been applied in computer vision for surface matching, shape compar-ison, shape classification, geometric analysis and tracking.

Shape Classification Recognition, retrieval, and classification are the common applicationsin computer vision field. With graphics hardware getting faster and 3D scanning hardware get-ting cheaper, the number of 3D geometric models (see Figures29 and 30) in online repositoriesis growing dramatically, and the demand for effective retrieval of models is continuously in-creasing. The primary challenge in building a shape-based classification and retrieval system isto find a computational representation of shape descriptorsfor which an index can be built, andsimilarity queries can be answered efficiently.

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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 asto volume models. The corresponding shape searching methods are evaluated with respect toseveral requirements of content-based 3D shape retrieval,such as: (1) shape representation re-quirements, (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 clas-sification methods mainly use topological transformation groups and Euclidean transformationgroups. In recent years, conformal geometry has been studied and applied for shape classifica-tion analysis. For many shape classification problems basedon geometric features, conformalinvariants can offer sufficient information to differentiate the different shapes.

[Gu and Yau, 2003a] introduces a novel method to classify surfaces by conformal trans-formation groups. Conformal equivalent class is refiner thantopological equivalent class andcoarser than isometric equivalent class. Also, conformal invariants are concise and efficientto compute, and can be used as search keys conveniently. Hence conformal classification ismore suitable for practical surface classification problems, such as human face surface match-ing [Wang et al., 2006], and human brain surface matching [Guand Vemuri, 2004], which isthe first paper to classifies surfaces with arbitrary topologies by global conformal invariants.

Surface Matching Surface matching is a fundamental task for computer vision,graphicsand medical imaging. Figure 31 shows the basic idea of using conformal parameterizationsto convert 3D matching problems to 2D ones. SupposeS1 and S2 are two surfaces inR3.π1 : S1 → D andφ2 : S2 → D are conformal mappings to map surfaces to the canonical planardomain.φ : D → D is a a map fromD to itself, this is a 2D matching process. Then

φ = π−1

2◦ φ ◦ π1, S1 → S2,

is the desired 3D matching.If S1 andS2 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 ofcorresponding feature points will be close to each other on the planar images. In Figure 31, wecan 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. Itis deformed to get another surface shown in (c). Their shapesare acquired using 3D scannerto acquire their shapes, denoted asS1 andS2. Then use conformal mapφ1 : S1 → R

2 andφ2 : S2 → R

2 with the only constraints that the images of all the boundaries are circles. The

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f

f

φ1 φ2

Figure 31: Surface matching using conformal mapping. [Wanget 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) preservesconformal structures, their planarimages 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 calculatedautomatically by the conformal geometric algorithms. Their planar images are shown in (b) and(d), which are identical. Then the 2D mapφ is the identity of the two hole annulus, the 3D mapφ = π−1

2◦ π1, which is exactly the isometric deformation. This example shows that conformal

geometric methods can recover isometric maps automatically. Therefore, for the purpose of sur-face matching, conformal geometric methods reduce the dimensionality and recover isometricmaps, furthermore, they can handle surfaces with arbitrarytopologies.

In Figure 33, two genus two surfaces and two genus six surfaces are matched using hy-perbolic Ricci flow method. The matching is visualized by transferring the textures from thedomain surfaces to the range surfaces. Figure 34 shows the surface matching between twogenus two surface by a geometric morphing.

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Figure 33: Surface matching between two high genus surfacesusing conformal geometric meth-ods.

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 3Dsurfaces undergoing non-rigiddeformations. 3D faces are captured using 3D scanner. Each face has approximately80K 3Dpoints with both shape and texture information available. The subjects were not asked to keeptheir head and facial expression still during the 3D face scanning.

An important property of conformal mappings is that they canmap a 3D shape to a 2D do-main in a continuous manner with minimized local angle distortion. This implies that conformalmappings are not sensitive to surface deformations, e.g., if there is not too much stretching be-tween two faces with different expressions, they will induce similar planar images. Therefore,matching on the planar images of conformal mappings are morereliable and accurate than di-rect matching in 3D. For partial surface matching, extra constraints are needed, such as featurepoints, feature lines, or area distortion factors.

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(a) (c) (e) (g)

(b) (d) (f) (h)

Figure 35:An example of surface alignment and stitching: (a,b) Two original 3D faceswith texture indifferent poses and deformations. (c,d) Original 3D faces without texture. (e,f) The conformal mappingimages of the faces. (g) The aligned planar images of the two faces. (h) The resulting3D face bystitching a part of (c) into (d). [Wang et al., 2007b]

2.3.4 Geometric Modeling

The surfaces obtained by 3D scanners are represented as point clouds. After geometric pro-cessing means, triangular meshes are constructed. In geometric modeling fields, surfaces areusually represented as piecewise polynomials or rational polynomials with higher order con-tinuity, called splines. In order to convert meshes to splines, conformal geometric method isa most useful tool. For the purpose of generalizing splines from planar domain to manifolddomain, special atlas needs to be constructed using conformal geometric methods.

Manifold Splines Conventional splines are defined on planar domains. Manifoldsplines de-fine polar forms directly on manifolds. Constructing splines, whose parametric domain is anarbitrary manifold, and effectively computing such splines in real-world applications are of fun-damental importance in solid and shape modeling, geometricdesign, graphics, etc. [Gu et al.,2005a] shows the existence of manifold splines is equivalent to the existence of affine structureof the manifold, which is obstructed by topology. Figure 36 demonstrates the key componentsof manifold spline. Practical methods to compute affine atlas for general surfaces and generalize

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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 boundariesadmits an affine structure and only genus one closed surfacesadmit affine structure. In the fig-ure, two holes are punched on the genus two surface, and then an affine atlas using holomorphicdifferential forms and manifold splines are constructed. Holomorphic differential forms induceaffine atlas covering the whole surface except several zero points, the affine atlas can be usedto construct the manifold splines. Figure 37 shows a manifold triangular B-Spline defined on a

3 GOAL AND SOLUTION 32

genus three surface.The theoretic framework of manifold splines is establishedin [Gu et al., 2005a] and [Gu

et al., 2006]. Then the theory is applied to generalized manyspline schemes on manifolds,such as manifold T-Splines [He et al., 2006b], triangular B-Spline [He et al., 2006a], polycubesplines [Wang et al., 2007a]. Especially, manifold splineswith 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 appli-cations 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 experimen-tal results.

3.1 Conformal Spherical Parameterization

Abstract Surface parameterization establishes bijective maps froma surface onto a topologi-cally equivalent standard domain. It is well known that the spherical parameterization is limitedto genus-zero surfaces. In this work, we design a new parameter domain, two-layered sphere,and present a framework for mapping high genus surfaces ontosphere. This setup allows us totransfer the existing applications based on general spherical parameterization to the field of highgenus surfaces, such as remeshing, consistent parameterization, shape analysis, and so on. Ourmethod 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 com-pute the quotient between two holomorphic 1-forms. Our method of spherical parameterizationis theoretically sound and practically efficient. It makes the subsequent applications on highgenus surfaces very promising.


Surface parameterization (for a recent survey, we refer thereader to [Floater and Hormann,2005]) is a fundamental tool in computer graphics and benefits many digital geometry process-ing applications such as texture mapping, shape analysis, compression, morphing, remeshing,etc. Some problems become much easier to deal with a uniform parameter domain. Usually inthese settings surfaces are represented as triangular meshes, and the maps are required to be atleast no-foldovers and low-distortion in terms of area, angle, or both aspects.

In graphics, spherical parameterizations for genus zero closed surfaces have been proposedand widely used in the past. Most methods [Gotsman et al., 2003, Gu et al., 2004, Hakeret al., 2000b, Sheffer et al., 2004, Praun and Hoppe, 2003] are to directly map the mesh tospherical domain, which is usually formulated as a spherical energy minimization problem,such as conformal, Tutte, Dirichlet, area, spring, stretchenergies, or their combinations, ascited in [Floater and Hormann, 2005]. The optimization process is to relax the initial map toreach no-foldovers under specified distortion metric.

In medical imaging, spherical parameterizations are broadly applied for brain cortex surfacemapping. In this setting, preservation of local shapes are crucial. Therefore, different conformalspherical parameterizations are proposed. Angenent et.al. [Angenent et al., 1999] constructmeromorphic functions on the brain surface directly, then lift the mapping onto the sphereusing inverse stereographic projections. Gu et.al. [Gu et al., 2004] compute harmonic mapsbetween the brain cortex surface and the unit sphere and use Mobius transformation to adjustthe map. Stephenson [Stephenson, 2005] uses circle packingmethod to construct conformalbrain mapping.

However, it is well known that the spherical parameterization is limited to genus zero mod-els. To the best of our knowledge, there are few works on high genus surfaces. Recently, Leeet.al. [Lee et al., 2006] present a construction method by boolean operations of positive andnegative spheres. This method requires a lot of interactivehuman recognitions and geometryediting techniques. Furthermore, the results are not conformal.

In this work, we aims at automatic generalizing conformal spherical parameterizations forhigh genus surfaces. Because high genus surfaces and spheresare not topologically equivalent,we allow the existence of branch points.

Our method relies on the conformal structure for high genus meshes. There are two waysto 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 betweena surface and the sphere isequivalent to a meromorphic function defined on the surface.The map wraps the surface ontothe sphere by several layers and has several branch points. The number of layers and the branchpoints are determined by the topology of the surface (by Riemann-Hurwitz theorem). The keyis how to construct the meromorphic functions on the input surface. For genus one closed


Figure 38: Layered Sphere for genus one and genus two cases ofour method. From left toright, they are (1) torus mesh with 10,000 vertices and 20,000 faces, (2) layered sphere with fourbranch points, (3) eight mesh with 12,286 vertices and 24,576 faces, and (4) layered sphere withsix branch points. Two layers are connected by branch pointswhere the lines twist together.

Figure 39: Spherical conformal parameterization for toruscase. From left and to right, eachcolumn denotes: (1) conformal parameterization result on layered sphere, (2) initial sphericalconformal parameterization result on original surface, (3) spherical conformal parameterizationresult with Mobius transformation, and (4) curvilinear parameterization of (1), and (5) curvilin-ear parameterization of (2).

surfaces, we construct the well-known Weierstrass P-function. For higher genus surfaces, thequotient between two holomorphic 1-forms is a meromorphic function.

Compared with the existing planar parameterization for highgenus meshes, the layeredsphere (see Figure 38) is more natural domain than the planardomain. Employing the propertiesof sphere geometry and the existing spherical parameterization related applications on genuszero 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 highgenus meshes.

The contributions of this work are briefly as follows:

1. To present a novel practical framework to compute conformal spherical parameterizations


for general surfaces;

2. To extend the applications of general spherical parameterization onto that of high genusmeshes, including remeshing, morphing, etc;

3. To introduce a systematic method to compute meromorphic functions on general Riemannsurfaces.

3.2 Consistent Surface Matching

Abstract Surface matching plays an important role in movie production and computer ani-mation, like human face expressions and body motions transferring. Surface parameterizationis employed to align feature points for matching. To build the relationship between the sourceand target surfaces, cross-parameterization with prescribed feature constraints is highly used.In discrete case, the parameter density of points has directrelation to the accuracy of surfacematching. Compared with conventional surface parameterization methods, we present a novelconformal parameterization with flexible boundary control, where user can specify which re-gion to be enlarged. This parameterization method is suitable for multi-holed surfaces. Theboundaries are parameterized to circular or straight slits, so we call the method,slit parameter-ization. This method is theoretically guaranteed to be intrinsic without self-overlapping. Basedon this, a surface matching algorithm is constructed. The method improves the matching accu-racy, which has been illustrated in the application of humanface 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, whereface expressions and body motions transferring are the usual method for simulation from thecharacter prototypes especially in animation production.Among these applications, surfacematching is required to get consistent meshes between the source and target models for motiontransferring, morphing, etc. The problem is formulated as finding the correspondence relationbetween the source and target. Manual methods directly frompictures are not recommendedwhich costs much labor and time. It is easy to get a sequence of3D surfaces by 3D scanner, likehuman face expression surfaces (see Figure 40) from the sameperson. These surfaces are nottopologically consistent when obtained directly from the scanning equipment. The problem toautomatically generate 3D face surfaces with consistent connectivity is much desired in modernindustry.

Because the advantages of angle-preserving and uniqueness,conformal parameterization iswidely employed in surface matching, which has many applications including, but not limitedto, consistent remeshing, shape deformation analysis, tracking etc.. The problem is especially


Figure 40: Human face expression surfaces scanned by 3D equipment.

hard when the transformation sought is diffeomorphic and non-rigid between the shapes be-ing matched. For the surfaces with minor deformations, likehuman face expression surfaces,their parameterization results may be much different though under the same parameterizationmethod. 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 bi-jective mapping between models that satisfies user prescribed constraints. Using the mapping,the remeshing algorithm preserves the user-defined featurepoints correspondence and the shapecorrelation 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 ismuch research around automatic computation of feature points, but it’s difficult to find a matureand 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 conformalmappings and presented a fully automatic method for high resolution, non-rigid dense 3D pointtracking, which unifies tracking of intensity and geometricfeatures.

In addition, human faces have strong symmetrical information, which is importantly usefulfor human face surface matching, hole-filling, stitching, etc. Here, the conformal map withsymmetrical feature correspondence constraints is presented. Given the symmetry, it is easyto find the corresponding part on another half face. The difficulty is to efficiently find the 3Dsymmetrical plane then construct the feature points correspondence constraints.

Compared with conventional surface parameterization methods, we present a novel confor-mal parameterization, where user can specify which region to be enlarged. This parameteriza-tion method is suitable for surfaces with boundaries. The boundaries are parameterized to cir-cular or straight slits, so we call the method,slit parameterization. This method is theoreticallyguaranteed to be intrinsic without self-overlapping. Basedon this, a surface matching algorithmis constructed. The method improves the matching accuracy,which has been illustrated in theapplication of human face expressions surface matching. For the expression surfaces sequence,


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. Fromupper to bottom, they are (1) the original face surfaces, only the fourth face has a boundaryaround the mouth area, (2) the cross-parameterization of (1) with 74 makers, and (3) the cross-parameterization of (1) with symmetrical maker pairs. The parameterization method used hereis 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 Beltrami-

equation using free boundary condition, and the conformal mapping of multi-holed annulus toannulus 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 haveself-overlapping. Extra constraints can be added to enhance the mapping result, such as feature


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 outerboundary 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, therectangle result is obtained bycutting the circle result (row 1) using a straight line through the inner and outer circles. In thecurrent experimental result, cross-parameterization with features correspondence constraintshas not been combined into this parameterization method yet.

point constraints. When this method is applied onto face surfaces, some area are stretched toomuch and some overlapped, like the mouth area (see Figure 41,row 2 left). In order to addthe freedom of the mapping around mouth area, cutting the mouth is introduced, which can getbetter results (see Figure 41, row 2 right). Because of the free-boundary condition, the boundaryof 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 bound-aries. 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), specialholomorphic 1-forms can be constructed, such that the wholesurface is mapped to an annuluswith concentric circular arcs. Two boundaries are mapped tothe inner and outer boundaries of


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 boundariesof two eyes are deformed to outer circle and inner circle respectively, the boundary of mouthand the original outer boundary to two circular slits. Note that, the difference between thesetwo circular results depends on the correspondence betweentwo eye boundaries and two innerand outer circles. In row 2, the rectangle result is obtainedby cutting the circle result (row1) 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 intothis parameterization method yet.

the annulus, other boundaries mapped to circular slits. Computing such holomorphic 1-formsis a linear problem, and the most difficult part is to find harmonic 1-forms. Figure 42 shows theconformal mapping of a three-holed face surface (the mouth is open), the target domain is a unitdisk with two concentric circular arcs. The exterior boundary of the face is mapped to the outercircle of the annulus, the mouth boundary mapped to the innercircle. The boundaries of eyesare mapped to the two circular slits. The we conformally mapped the annulus to the band, theouter and inner circles are mapped to parallel lines and the boundaries of eyes are mapped tohorizontal slits. If fixing one eye boundary to the outer circle and another eye boundary to theinner circle, we can get another mapping result, see Figure 43. The advantage of this method


is that we can specify the boundary to be mapped to the outer circle for enlargement, thus thematching accuracy of the area around this boundary can be enhanced.

About matching algorithm, the source and target surfaces are flattened onto 2D canonicaldomains through the conformal parameterization methods above. Through the cross-parameterization,the feature points is well aligned, then the correspondenceof other parts is constructed. Theconsistent connectivity can be generated by sampling and lifting from 2D domain to 3D do-main. The key difficulty is to find an alignment with features consistent everywhere. In orderto advance the accuracy of alignment and the subsequent surface matching, we present a novelparameterization method with flexible boundary control which allows the enlargement of user-specified region (see Figures 42 and 43). Combining with otherconventional parameterizationmethods (see Figure 41), we implement the consistent matching algorithm for a sequence sur-face with minor deformations. For human face expressions application, the matchings of dif-ferent local regions, like mouth, eyes, or brown area, are performed by different parameterizedmethods. Applying consistent remeshing into the video of 3Dobjects can save much space andaccelerate computation speed, which has much potential on the fields of movie production andcomputer 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, itcan be achieved by solving Beltrami equation. Many mathematical solution schemes have beenpresented so far. This work is to implement its discrete solution on triangular meshes, aimingto get the quasiconformal mapping for 3D surfaces. Except the angular metric, area factoris considered, which balances the original information between shape and size, and is muchdesired 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.Grotzsch in 1928. IfQ is a square andR is a rectangle, not a square, there is no conformalmapping ofQ onRwhich maps vertices on vertices. Instead, Grotzsch asks for the most nearlyconformal mapping of this kind. This calls for a measure of approximate conformality, and


in supplying such a measure Grotzsch took the first step toward the creation of a theory ofquasiconformal 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 incomplex analysis, has blossomed into subject all its own. A conformal mapping in the planesends small discs to other discs (to first order). A quasiconformal mapping on an open set is acontinuous homeomorphism that sends small discs to small ellipses, in which the ratio of majoraxis to minor axis is bounded. Such mappings are in general not holomorphic functions, butplay an auxiliary role in questions about such functions.

Quasiconformal mappings are mappings of the complex plane to itself that are almost con-formal. That is, they do not distort angles arbitrarily and this distortion is uniformly boundedthroughout their domain of definition. Alternatively one can think of quasiconformal mappingsas mappings which take infinitesimal circles to infinitesimal ellipses. For example invertiblelinear maps are quasiconformal.

More rigorously, supposef is a mapping of the complex plane to itself, and here we willonly consider sense preserving mappings, which is mappingswith a positive Jacobian, whilenegative for reversing mappings.

Definition 1 Dilatation. Define the dilatation of the mappingf at the pointz asDf (z) :=|fz| + |fz|

|fz| − |fz|≥ 1, and define the maximal dilatation of the mapping asKf := sup

z

Df (z).

Now we are ready to define what it means forf to be quasiconformal.

Definition 2 Quasiconformal.Forf as above, we will callf quasiconformal if the maximal di-latation off is finite. We will say thatf is K-quasiconformal mapping if the maximal dilatationof this mapping isK.

Note that sometimes the termK-quasiconformal is used to mean that the dilatation isK orlower. It is easy to see that a conformal sense preserving mapping has a dilatation of1 since|fz| = 0. We can further define several other related quantities.

Definition 3 Small Dilatation.Forf as above, define the small dilatation asdf (z) :=|fz|

|fz|.

Again for sense preserving maps this quantity is less than 1 and it is equal to 0 if the map-ping is conformal. Some authors call a mapk-quasiconformal if the small dilatation is boundedby 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 bydf :=Df − 1

Df + 1.


Definition 4 Complex Dilatation.Forf as above, define the complex dilatation asµf (z) :=fz

fz

.

The complex dilatation now appears in the Beltrami differential equationfz(z) = µf (z)fz(z).This means that a quasiconformal mapping is a solution to theBeltrami equation where a non-negative measurableµf is uniformly bounded by somek < 1.

The above results are stated forf : C → C, but the statements are exactly the same if youtakef : G ⊂ C → C for an open setG.

Regular solutions of the Laplace-Beltrami equation are generalizations of harmonic func-tions and are usually called harmonic functions on the surfaceR (cf. also Harmonic function).These solutions are interpreted physically like the usual harmonic functions, e.g. as the velocitypotential of the flow of an incompressible liquid flowing overthe surfaceR, or as the potential ofan electrostatic field onR, etc. Harmonic functions on a surface retain the propertiesof ordinaryharmonic functions. A generalization of the Dirichlet principle is valid for them.

In mathematics, quasiconformal mapping is formulated as the problem of solving Beltramiequation. There have been many mathematical and theoretical methods around it, but no cor-responding implementation attempts on discrete triangular surface so far. The difficulty liesin how to solve a high nonlinear partial differential equation in discrete case. In practice, theconcept 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 arbitrarystructure. Initial experiments show that it is efficient andeffective for shape retrieval and clas-sification. In order to get more accurate results, a large amount of manifolds need to be createdfor classification experiments. And performance comparison with other shape descriptors willbe explored on both idea and experiment.

In general, surfaces are classified by different transformation groups, such as homeomor-phism, which preserve topologies, and isometries, which preserve Riemannian metrics. Topo-logical classification is too coarse for real applications;metric classification is too refined and


Figure 44: Kitten and Torus. From topological classification, they all belong to the same class;from geometrical classification, none of them are in the sameclass; from conformal classifica-tion, the original kitten (or torus) and the deformed kitten(or torus) belong to the same confor-mal class, which can be visualized by the texture transferred from original model to deformedmodel without distortion of angles. [Jin et al.2007]

expensive to compute, and the curvatures are too sensitive to local noises. In this work, wepropose a novel classification,conformal classification, which overcomes the shortcomings oftopological or metric ones. Surfaces can be classified by conformal equivalence relation, whereeach conformal equivalent class is also called aRiemann surface. For example, all genus zeroclosed surfaces can be conformally mapped to the unit sphere. Therefore, they are the sameRiemann surface. As shown in Figure 44, all surfaces are topologically equivalent, but geo-metrically inequivalent. But under conformal classification, the kitten surfaces are equivalentto each other, same as the torii. This demonstrates that Teichmuller coordinates are much morediscriminating than topological invariants. Furthermore, conformal classification is much moreefficient than metric classification and the conformal invariants are much stabler than curva-tures.

This work proposes to classify surfaces using conformal mappings. Given the topology ofthe surfaces, all conformal equivalent classes form a finitedimensional manifold, the so-calledTeichmuller shape space [Buser, 1992]. In this shape space, each point represents a class of sur-faces, and a curve represents a deformation process from oneclass to the other. Two surfacescorresponds 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 explic-itly computed, which indicates the most natural deformation with minimal distortion energy.The geodesic distance measures the similarity between the surfaces. Therefore, Teichmullershape space is a good framework for the study of shape classification, shape comparison anddeformation.

The coordinates in the Teichmuller shape space have explicit geometric meanings. Basi-


cally, the coordinates of a point in the shape space are the geodesic lengths of special curveson the surface under the uniformization metric. For the simplest case, letS be a three punc-tured sphere with boundariesb0, b1, b2, then under the uniformization metric, it has Gaussiancurvature -1 everywhere, and the boundaries become geodesics. The Teichmuller coordinatesof S are the lengths ofb0, b1, b2. Hence, the Teichmuller shape space of3-punctured sphere is3dimensional.

As shape descriptors, Teichmuller coordinates have many advantages, invaluable for prac-tical applications. Teichmuller coordinates are general for all manifold surfaces with arbitrarytopologies. They are intrinsic, invariant under rotation,translation and scaling. Furthermore,the descriptors are invariant under isometric deformation. They are stable, for deformationswith small area stretching, like the posture change of a human skin surface, which changesslightly. They are efficient, easy to compute and compare.

The major goal of this work is to develop rigorous and practical algorithms to computeTeichmuller coordinates for arbitrary surfaces with negative Euler numbers. First we computethe uniformization metric of the surface. Then, we measure the geodesic lengths of a set ofcurves with algebraic methods. The geodesic lengths are theshape space coordinates of thesurface.

The major contributions of this work are

1. To propose a theoretical framework to model all surfaces in a shape space, Teichmullershape space. The framework has deep roots in modern geometryand is practical forcomputation. It offers novel views and tools for tackling engineering problems;

2. To introduce a series of practical algorithms for computing Teichmuller shape space coor-dinates for surfaces with complicated topologies, which are conformal shape descriptors;

3. To apply Teichmuller coordinates for real applications, such as shape comparison, iden-tification and retrieval.

3.5 Surface Remeshing

Abstract A practical algorithm is introduced to approximate surfaceDelaunay triangulationsusing planar Delaunay triangulations of sampling points onconformal parametric domains. Themethod produces good approximation results and is practical for real applications.

In geometric modeling and processing, computer graphics and computer vision, smoothsurfaces are approximated by discrete triangular meshes reconstructed from sample points onthe surfaces. A fundamental problem is to design rigorous algorithms to guarantee the geometricapproximation accuracy by controlling the sampling density and triangulation method.

4 FEASIBILITY ANALYSIS 45

Figure 45: Approximated surface Delaunay triangulation byglobal conformal parameterizationfor 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 aredifficult to compute. In this work, we propose to use global conformal parameterization [Guand Yau, 2002, Gu et al., 2005b, Jin et al., 2006a,b] to map thesurface to the canonical do-mains, generate random samples on the conformal parametricdomains, and compute the planarDelaunay triangulation to approximate the surface Delaunay triangulation. Delaunay triangu-lations maximize the minimal angle, conformal parameterization preserves angles, therefore,Delaunay triangulations on conformal parametric domains approximate the surface Delaunaytriangulations faithfully. The result can be seen in Figure45.

This method was proposed in [Alliez et al., 2002] for topological disk case. In currentwork, we generalize the algorithm for arbitrary surfaces. The work is included as one partof the work [Dai et al., 2007], which gives explicit formulaeof approximation error boundsfor both Hausdorff distance and normal distance in terms of principle curvature and the radiiof geodesic circum-circle of the triangles. These formulaecan be directly applied to designsampling density for data acquisitions and surface reconstructions.

4 Feasibility Analysis

1. Conformal Spherical Parameterization. Conformal spherical parameterization for highgenus surfaces can be obtained by meromorphic function theory. The branch points cannot be located accurately on discrete case. There is much overlapping and flipping aroundbranch points. Therefore, there is much technical difficulty.

2. Quasiconformal Mapping. The quasiconformal mapping is obtained by solving Beltrami

5 INNOVATIONS 46

equation for triangular meshes, towards which the discretealgorithm is reuired. Thealgorithm efficiency is much desired for applications.

3. Consistent Matching. Through conformal parameterization, the human face expressionsurfaces can be flattened to 2D domain. After feature points alignment, the consistentconnectivity can be obtained by sampling and lifting from 2Dto 3D. The key problemis to find an alignment with feature consistent everywhere. The proposed approach is tocombine several 2D parameterization methods which may be suitable for different localregions, like mouth, eyes, or brown.

5 Innovations

The followings are the results and innovations proposed:

1. The algorithm of conformal spherical parameterization for high genus surfaces is pre-sented. The innovation is to break up the limit of the spherical parameterization, whichwas only for genus-zero surfaces. And the related research can be performed, such asconsistent spherical parameterization [Asirvatham et al., 2005] and its corresponding ap-plications for high genus surfaces.

2. A consistent matching method for a sequence of human face expression surfaces withcontinuous deformation is designed. In order to get the bestalignment of feature points,several conformal parameterization methods are analyzed and combined. A novel confor-mal parameterization method with for flexible boundary control is presented. The videoof human face expressions is reproduced.

3. A quasiconformal mapping algorithm for 3D surfaces is designed based on the discretesolutions of the Beltrami equation on triangular meshes. Except the angular metric, areafactor is considered, which balances the original information between shape and size,and is much desired in medical imaging. As an application, the colon surface flatteningalgorithm is designed for efficient polyps detection.

4. A 3D shape internet search engine based on shape space theory. A new shape descriptoris introduced. A shape repository of manifolds is built and the corresponding evaluationand measurement are made in detail.

5. A practical algorithm is introduced to approximate surface Delaunay triangulations usingplanar Delaunay triangulations on conformal parametric domains. The method producesgood approximation results and is practical for real applications.

6 RESOURCE AND PROGRESS 47

6 Resource and Progress

6.1 Existing Resources

There are some research fundamental algorithms, such as:

1. Global conformal surface parameterization based on Ricciflow Jin et al. [2006a,b], holo-morphic 1-forms Gu and Yau [2002, 2003b], Jin et al. [2004].

2. Shape space coordinates based on Teichmuller shape space theory Jin et al. [2007d].About 50 manifold models have been created.

6.2 Finished Work

The following work items have been done:

1. Conformal spherical parameterization for high genus surfaces.

2. Shape classification based on shape space descriptor.

3. Surface remeshing using planar Delaunay triangulation on conform parametric domains.

6.3 Ongoing Work

The current work items are as follows:

1. Solutions to Beltrami equation and quasiconformal mapping algorithm.

2. Consistent matching and remeshing for human face expression surfaces.

6.4 Publications

According to the thesis proposed, related publications are:

1. Zeng, W., Li, X., Yau, S.-T., AND Gu, X. 2007. Conformal Spherical Parameterizationfor High Genus Surfaces.In Submission.

2. Jin, M., Zeng, W. AND Gu, X. 2007. Computing Shape Space.In Submission.

3. Dai, J., Luo, W., Jin, M., Zeng, W., He, Y., Yau, S.-T. AND Gu, X. 2007. Geometric Ac-curacy Analysis for Discrete Surface Approximation.Computer Aided Geometric Design24, 6, 323-338.

7 SCHEDULE 48

4. Zeng, W., Meng, X.,Yang C., AND Huang L. 2006. Feature Extraction for Online Hand-written Characters Using Delaunay Triangulation.International Journal of Computers &Graphics 30, 5, 779-786.

7 Schedule

The proposed graduation date is summer in 2008. The brief schedule is planned as follows:

1. 2006/12 - 2007/02 Shape classification based on shape space theory. In charge of back-ground survey and build up the 3D shape database.

2. 2007/03 - 2007/06 Conformal spherical parameterization for high genus surfaces usingthe concept of meromorphic function.

3. 2007/07 - 2007/10 Consistent matching and remeshing for human face expression sur-faces. Paper writing and submission.

4. 2007/11 - 2007/03 Solutions to Beltrami equation and quasiconformal mapping algo-rithm. Paper writing and submission.

5. 2008/03 - 2008/04 Thesis paper writing.

8 Acknowledgement

I would like to appreciate my advisor Harry Shum and co-advisor Xianfeng Gu for their sug-gestions and kind help on my research. I also would like to thank Baining Guo, Kun Zhouand Xin Tong for being my thesis committee member. And I wouldlike to thank help fromall colleague in Visualization lab of Stony Brook Universityand Internet Graphics Group ofMicrosoft Research Asia.

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