Surface Classification Using Conformal Structures Xianfeng Gu 1, Shing-Tung Yau 2 1. Computer and Information Science and Engineering, University of Florida 2. Mathematics Department, Harvard University Surface Classification Using Conformal Structures Xianfeng Gu 1, Shing-Tung Yau 2 1. Computer and Information Science and Engineering, University of Florida 2. Mathematics Department, Harvard University Abstract can 3D surface classification is a fundamental problem in computer vision and computational geometry. Surfaces 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.Abstract Conformal Map l A conformal map is a map which only scales the first fundamental forms, hence preserving angles. l Locally, shape is preserved and distances and areas are only changed by a scaling factor Conformal Map l A conformal map is a map which only scales the first fundamental forms, hence preserving angles. l Locally, shape is preserved and distances and areas are only changed by a scaling factor Riemann Surface and Conformal Structure l Any surface is a Riemann surface, namely they can be covered by holomorphic coordinate charts. l 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. l The conformal structure are the complete invariants under conformal transformation group and can be represented as matrices. l 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 l Any surface is a Riemann surface, namely they can be covered by holomorphic coordinate charts. l 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. l The conformal structure are the complete invariants under conformal transformation group and can be represented as matrices. l 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 Original Surface Checkerboard texture Texture mapped surface Conformal map to the plane Torus one Conformal map to a parallelogram Torus two Conformal map to a parallelogram Topological equivalent but not conformal equivalent Surface with 4K faces Holomorphic 1-form Surface with 34K faces Holomorphic 1-form Conformal structure is only dependent on geometry, independent of triangulation and insensitive to resolution Algorithm at a Glance Computing homology Computing harmonic one-forms Computing holomorphic one-forms Computing period matrix Double covering for surface with boundaries Surface classification and matching method Algorithm at a Glance Computing homology Computing harmonic one-forms Computing holomorphic one-forms Computing period matrix Double covering for surface with boundaries Surface classification and matching method Algorithm Details l 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 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: l Holomorphic one-forms Holomorphic one-forms are the gradient fields of conformal maps, which can be formulated as Algorithm Details l 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 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: l Holomorphic one-forms Holomorphic one-forms are the gradient fields of conformal maps, which can be formulated as Genus 3 surface Holomorphic 1-form dual to the first handle Holomorphic 1-form dual to the second 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 handle l 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. l 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. Brief Reference I. 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 II. X. Gu and S.-T. Yau. Computing conformal structures of surafces. Communication of Informtion and Systems, December Brief Reference I. 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 II. X. Gu and S.-T. Yau. Computing conformal structures of surafces. Communication of Informtion and Systems, December Discussion l We are the first group to systematically use conformal structure for surface classification problems. l The method is intrinsic to the geometry, independent of triangulation and insensitive to resolution. l The conformal invariants are global features of surfaces, hence they are robust to noises. l The conformal equivalent classification is refiner than topological classification and coarser than isometric classification, making it suitable for surface classifications and matching. Discussion l We are the first group to systematically use conformal structure for surface classification problems. l The method is intrinsic to the geometry, independent of triangulation and insensitive to resolution. l The conformal invariants are global features of surfaces, hence they are robust to noises. l 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) MeshAngleLength ratioverticesfaces Torus Knot Knot Rocker Teapot Genus one and Genus two surfaces with different conformal structures l 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: l 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: l 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. l 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 Surface Double Covering Double Covering Holomorphic one-form Reversed Orientation