Surface Classification Using Conformal StructuresSurface Classification Using Conformal StructuresXianfeng GuXianfeng Gu11, Shing-Tung Yau, Shing-Tung Yau22

1.1. Computer and Information Science and Engineering, University of Florida Computer and Information Science and Engineering, University of Florida 2.2. Mathematics Department, Harvard University Mathematics Department, Harvard Universitygu@cise.ufl.edu, yau@math.harvard.edu.gu@cise.ufl.edu, yau@math.harvard.edu.

AbstractAbstract

3D surface classification is a fundamental problem in computer vision and computational geometry. Surfaces can can be classified by different transformation groups. Traditional classification methods mainly use topological transformation group and Euclidean transformation group. This paper introduces a novel method to classify surfaces by conformal transformation group. Conformal equivalent class is refiner than topological equivalent class and coarser than isometric equivalent class, making it suitable for practical classification purposes. For general surfaces, the gradient fields of conformal maps form a vector space, which has a natural structure invariant under conformal transformations. We present an algorithm to compute this conformal structure, which can be represented as matrices, and use it to classify surfaces. The result is intrinsic to the geometry, invariant to triangulation and insensitive to resolution. To the best of our knowledge, this is the first paper to classify surfaces with arbitrary topologies by global conformal invariants. The method introduced here can also be used for surface matching problems.

AbstractAbstract 3D surface classification is a fundamental problem in computer vision and

computational geometry. Surfaces can can be classified by different transformation groups. Traditional classification methods mainly use topological transformation group and Euclidean transformation group. This paper introduces a novel method to classify surfaces by conformal transformation group. Conformal equivalent class is refiner than topological equivalent class and coarser than isometric equivalent class, making it suitable for practical classification purposes. For general surfaces, the gradient fields of conformal maps form a vector space, which has a natural structure invariant under conformal transformations. We present an algorithm to compute this conformal structure, which can be represented as matrices, and use it to classify surfaces. The result is intrinsic to the geometry, invariant to triangulation and insensitive to resolution. To the best of our knowledge, this is the first paper to classify surfaces with arbitrary topologies by global conformal invariants. The method introduced here can also be used for surface matching problems. Conformal Map A conformal map is a map which only scales the first fundamental forms, hence

preserving angles. Locally, shape is preserved and distances and areas are only changed by a scaling

factor

Conformal Map A conformal map is a map which only scales the first fundamental forms, hence

preserving angles. Locally, shape is preserved and distances and areas are only changed by a scaling

factor

Riemann Surface and Conformal Structure Any surface is a Riemann surface, namely they can be covered by holomorphic

coordinate charts. Let M be a closed surface of genus g, and B={e1,e2,…,e2g} be an arbitrary basis of its

homology group. We define the entries of the intersection matrix C of B as where the dot denotes the algebraic number of intersections. A holomorphic basis is defined to be dual of

B if

Define matrix S as having entries

The matrix R defined as CR = S satisfies where I is the identity matrix, and R is called the period matrix of M with respect to the

homology basis B. We call (R,C) the conformal structure of M. The conformal structure are the complete invariants under conformal transformation

group and can be represented as matrices. For two surfaces M1 and M2 with conformal structure (R1,C1) and (R2,C2)respectively,

M1 and M2 are conformal equivalent if and only if there exists an integer matrix N such that

Riemann Surface and Conformal Structure Any surface is a Riemann surface, namely they can be covered by holomorphic

coordinate charts. Let M be a closed surface of genus g, and B={e1,e2,…,e2g} be an arbitrary basis of its

homology group. We define the entries of the intersection matrix C of B as where the dot denotes the algebraic number of intersections. A holomorphic basis is defined to be dual of

B if

Define matrix S as having entries

The matrix R defined as CR = S satisfies where I is the identity matrix, and R is called the period matrix of M with respect to the

homology basis B. We call (R,C) the conformal structure of M. The conformal structure are the complete invariants under conformal transformation

group and can be represented as matrices. For two surfaces M1 and M2 with conformal structure (R1,C1) and (R2,C2)respectively,

M1 and M2 are conformal equivalent if and only if there exists an integer matrix N such that

jiij eec

},...,,{ 221 gB

ij

e

j ci

Re

ij

e

j si

Im

IR 2

21211 ; CNCNRNRN T

Original SurfaceOriginal Surface Checkerboard textureCheckerboard texture Texture mapped surfaceTexture mapped surfaceConformal map to the planeConformal map to the plane

Torus oneTorus one Conformal map to a parallelogram

Conformal map to a parallelogram

Torus twoTorus two Conformal map to a parallelogram

Conformal map to a parallelogram

Topological equivalent but not conformal equivalentTopological equivalent but not conformal equivalent

Surface with 4K facesSurface with 4K faces Holomorphic 1-form Holomorphic 1-form Surface with 34K facesSurface with 34K faces Holomorphic 1-formHolomorphic 1-form

Conformal structure is only dependent on geometry, independent of triangulation and insensitive to resolution Conformal structure is only dependent on geometry, independent of triangulation and insensitive to resolution

Algorithm at a GlanceComputing homology Computing harmonic one-formsComputing holomorphic one-forms Computing period matrixDouble covering for surface with boundaries Surface classification and matching method

Algorithm at a GlanceComputing homology Computing harmonic one-formsComputing holomorphic one-forms Computing period matrixDouble covering for surface with boundaries Surface classification and matching method Algorithm Details

Homology group Boundary operators The homology group is defined as the quotient space The homology bases are the eigenvectors of the kernel space of the linear operator L

Harmonic one-forms Below is called the harmonic energy of Harmonic one-forms have zero Laplacian

Compute a dual basis of the harmonic one-forms by the following linear systems: Holomorphic one-forms Holomorphic one-forms are the gradient fields of conformal maps, which can be

formulated as

Algorithm Details Homology group Boundary operators The homology group is defined as the quotient space The homology bases are the eigenvectors of the kernel space of the linear operator L

Harmonic one-forms Below is called the harmonic energy of Harmonic one-forms have zero Laplacian

Compute a dual basis of the harmonic one-forms by the following linear systems: Holomorphic one-forms Holomorphic one-forms are the gradient fields of conformal maps, which can be

formulated as

i

ippi

i

ipip p 2,1,

2

11 ),(

img

KerZMH

TTL 2211

2

,, ),(

2

1)(

1

vukEKvu

vu

kvu

vu vuku,

, ),()(

je

iji

i

i

0

0

,*1 n*

Genus 3 surfaceGenus 3 surface Holomorphic 1-form dual to the first handle

Holomorphic 1-form dual to the first handle

Holomorphic 1-form dual to the second handle

Holomorphic 1-form dual to the second handle

Holomorphic 1-form dual to the third handle

Holomorphic 1-form dual to the third handle

Holomorphic one-form basis of a genus 3 surfaces. Each holomorphic base is dual to one handleHolomorphic one-form basis of a genus 3 surfaces. Each holomorphic base is dual to one handle

Period matrix For a higher genus surface, suppose we have computed a canonical homology basis is the Kronecker symbol, and constructed a dual holomorphic

differential basis then the matrices C and S have

entries: Then R is computed as (R,C) are the conformal

invariants.

Period matrix For a higher genus surface, suppose we have computed a canonical homology basis is the Kronecker symbol, and constructed a dual holomorphic

differential basis then the matrices C and S have

entries: Then R is computed as (R,C) are the conformal

invariants.

},,...,,{~

221 geeeB gi

jgi

jji gjgiee ,21,

}*1,...,*1,*1{ 222211*

ggB

Brief ReferenceI. X. Gu, Y.Wang, T. Chan, P. Tompson, and S.-T. Yau. Genus zero surface conformal mapping and its application to brain surface mapping.

Information Processing Medical Imaging, July 2003.

II. X. Gu and S.-T. Yau. Computing conformal structures of surafces. Communication of Informtion and Systems, December 2002.

Brief ReferenceI. X. Gu, Y.Wang, T. Chan, P. Tompson, and S.-T. Yau. Genus zero surface conformal mapping and its application to brain surface mapping.

Information Processing Medical Imaging, July 2003.

II. X. Gu and S.-T. Yau. Computing conformal structures of surafces. Communication of Informtion and Systems, December 2002.

Discussion We are the first group to systematically use conformal structure for surface classification problems. The method is intrinsic to the geometry, independent of triangulation and insensitive to resolution. The conformal invariants are global features of surfaces, hence they are robust to noises. The conformal equivalent classification is refiner than topological classification and coarser than

isometric classification, making it suitable for surface classifications and matching.

Discussion We are the first group to systematically use conformal structure for surface classification problems. The method is intrinsic to the geometry, independent of triangulation and insensitive to resolution. The conformal invariants are global features of surfaces, hence they are robust to noises. The conformal equivalent classification is refiner than topological classification and coarser than

isometric classification, making it suitable for surface classifications and matching.

Experimental Results (Conformal invariants of genus one surfaces)Experimental Results (Conformal invariants of genus one surfaces)

Mesh Angle Length ratio vertices faces

Torus 89.987 2.2916 1089 2048

Knot 85.1 31.150 5808 11616

Knot2 89.9889 25.2575 2050 3672

Rocker 85.432 4.9928 3750 7500

Teapot 89.95 3.0264 17024 34048

Genus one and Genus two surfaces with different conformal structuresGenus one and Genus two surfaces with different conformal structures

Surface classification and matching method If two surfaces M1 and M2 with conformal structures (R1, C1) and (R2, C2)

respectively are conformal equivalent, the sufficient and necessary conditions are:

Surface classification and matching method If two surfaces M1 and M2 with conformal structures (R1, C1) and (R2, C2)

respectively are conformal equivalent, the sufficient and necessary conditions are:

2121

121

12222

11111 ,,,, CNCNPPNPPRPPR T

ii e

jij

e

jij sc *, SCR 1 Double covering Given a surface M with boundaries, we make a copy of it, then reverse the orientation of its

copy. We simply glue M and its copy together along their corresponding boundaries, the obtained is a closed surface and called the double covering of M.

Double covering Given a surface M with boundaries, we make a copy of it, then reverse the orientation of its

copy. We simply glue M and its copy together along their corresponding boundaries, the obtained is a closed surface and called the double covering of M.

Original

Surface

Original

SurfaceDouble

Covering

Double

Covering

Holomorphic one-form

Holomorphic one-form

Reversed Orientation

Reversed Orientation