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Surface Classification Using Conformal Structures Surface Classification Using Conformal Structures Xianfeng Gu Shing-Tung Yau Department of CISE Department of Mathematics University of Florida Harvard University Gainesville, FL 32611 Cambridge, MA 02138 [email protected]fl.edu [email protected] Abstract novel viewpoint: treating them as Riemann surfaces with conformal structures, or namely one dimensional complex 3D surface classification is a fundamental problem in manifolds. A Riemann surface is a surface covered by holo- computer vision and computational geometry. Surfaces can morphic coordinate charts. The conformal structure is the be classified by different transformation groups. Traditional complete invariants under conformal transformations and classification methods mainly use topological transforma- can be represented as matrices. tion groups and Euclidean transformation groups. This pa- Compared to other surface classification methods, con- per introduces a novel method to classify surfaces by con- formal classification has some advantages. Theoretically, formal transformation groups. Conformal equivalent class conformal geometry has a sound foundation. Conformal is refiner than topological equivalent class and coarser than equivalent classes are much refiner than the topological isometric equivalent class, making it suitable for practical equivalent classes and much coarser than the isometric classification purposes. For general surfaces, the gradient classes. Each topological equivalent class has infinite con- fields of conformal maps form a vector space, which has formal equivalent classes, each conformal equivalent class a natural structure invariant under conformal transforma- has infinite isometric classes. The conformal structures tions. We present an algorithm to compute this conformal are intrinsic to the geometry, independent of triangulation, structure, which can be represented as matrices, and use it insensitive to resolution and local features, and robust to to classify surfaces. The result is intrinsic to the geometry, noises. Also, conformal invariants are concise and efficient invariant to triangulation and insensitive to resolution. To to compute, and can be used as search keys conveniently. the best of our knowledge, this is the first paper to classify Hence conformal classification is more suitable for practi- surfaces with arbitrary topologies by global conformal in- cal surface classification problems. variants. The method introduced here can also be used for Conformal invariants can also be used for general sur- surface matching problems. face matching problems. In nature, it is highly unlikely for different shapes to share the same conformal structure. For many surface matching problems based on geometric fea- 1. Introduction tures in the Euclidean space, conformal invariants can offer sufficient information to differentiate shapes. The compu- 3D surface classification and matching are fundamental tation of conformal invariants is much cheaper and more problems in computer vision, computational geometry and stable than computing geometric features. computer aided geometric design. Recent developments in To the best of our knowledge, although conformal struc- modelling and digitizing techniques have led to an increas- ture is well known, we are the first group to systematically ing accumulation of 3D models. This has highlighted the use it for surface classification problems. need for efficient 3D objects searching techniques in large We introduce previous work and the theoretic back- scale databases. ground in the following part of this section, followed by Many methods have been developed based on the topo- detailed explanation of the algorithms in Section two. Our logical and geometric features of the surfaces in order to surface classification method is introduced in Section three. describe shapes. In general, all the methods treat the sur- Experimental results are reported in Section four. Finally, a face as a two dimensional real manifold embedded in R3, brief summary and conclusion appears in Section five, fol- with the induced Euclidean metric structure. lowed by a discussion of topics for future work in Section In this paper, we view the surfaces from a completely six. Proceedings of the Ninth IEEE International Conference on Computer Vision (ICCV 2003) 2-Volume Set 0-7695-1950-4/03 $17.00 © 2003 IEEE 1.1 Previous work 1.2 Theoretic background The algorithms introduced in this paper are based on the 3D shape classification and recognition is a core prob- theories of Riemann surfaces, especially the Abel-Jacobi lem in computer vision. Due to the apparent difference in theory as introduced in [4, 13]. We treat surfaces as Rie- nature, 2D shape classification methods can not be easily mann surfaces, and compute their conformal structures rep- extended to 3D shape classification problems. To develop a resented as holomorphic one-forms and period matrices. 3D shape classification method, which makes use of 3D ob- A conformal map is a map which only scales the first ject topological and geometric features that is independent fundamental forms, hence preserving angles. If a map- of tessellation and resolution, becomes desirable. Roughly, ping f : M1 → M2 is conformal, where M1 and M2 are the current 3D shape classification methods fall into the fol- two surfaces, suppose (u1,u2) are local parameters and the lowing categories. 2 Riemann metric (first fundamental form) ofM1 is ds = i j 2 = ˜ i j ij gijdu du , the metric of M2 is ds ijgij du du , ∗˜ 1. Statistical properties based methods. The simplest ap- then the induced metric f gij satisfies proach represents objects with feature vectors in a mul- ( 1 2)= ( 1 2) ∗˜ ( 1 2) tidimensional space where the axes encode global ge- gij u ,u λ u ,u f gij u ,u . (1) ometric properties. Ankerst et al. [1] proposed shape Figure 1 illustrates a conformal map from a real female histogram decomposing shells and sectors around a face surface to a square. All the rights angles on the texture model’s centroid. Osada et al. [12] represented shapes are preserved on the surface, which is shown in (c) and (d). with probability distributions of geometric properties Two surfaces are called conformal equivalent if there ex- computed for points randomly sampled on an object’s ists a conformal diffeomorphism between them. Conformal surface. However, these statistical methods are not equivalent surfaces share the same conformal invariants, discriminating enough to make subtle distinctions be- which can be represented as a matrix. tween shapes. Figure 2 shows two genus one surfaces. Although they are topologically equivalent, they are not conformal equiv- alent. Each torus can be cut open and conformally mapped 2. Topology based methods. Hilaga et al. [8] computed to a planar parallelogram. The two tori can be conformally 3D shape similarity by comparing Multiresolutional mapped to each other, if and only if one such parallelogam Reeb Graphs(MRGs) which encodes the skeletal and can be exactly matched to the other by translation, rotation topological structure at various levels of resolution. and scaling. In other words, the shape of such parallelo- The MRG is constructed using a continuous function gram indicates the conformal equivalent class for the torus. on the 3D shape, preferably a function of geodesic dis- We use the non-obtuse angle (right angle in this case) of the tance. These methods can not describe the geometric parallelogram and the length ratio between the two adjacent distinctions. edges to represent the conformal invariants of the genus one surfaces. We call them shape factors. From (b) and (d), it is 3. Geometry based methods. Novotni et al. [11] describe clear that the two tori have different shape factors, they are a method based on calculating a volumetric error be- not conformal equivalent. Hence conformal classification is tween one object and a sequence of offset hulls of refiner than topological classification. the other object. Tangelder et al. [15] represent the For higher genus surfaces, the conformal invariants are 3D shape by a signature representing a weighted point more complicated. Basically, handles of the surface can be set. A shape similarity measurement based on weight cut open and conformally mapped to parallelograms with transportation is used to compute the similarity be- different shapes. The shape factors of all the handles in- tween two shapes. Funkhouser et al. [3] developed a dicate the conformal class. Figure 3 demonstrates a genus 3D matching algorithm that uses spherical harmonics three surface, where each handle is conformally mapped to to compute discriminating similarity measures. Kazh- a parallelogram. The shapes of the three parallelograms dan et al.[9] introduced a reflective symmetry descrip- are the conformal invariants. The rigorous representation tor that represents a measure of reflective symmetry of conformal invariants of a high genus surface, through pe- for an arbitrary 3D model for all planes through the riod matrices, is explained below. model’s center of mass. These methods take into ac- If a surface is mapped to the complex plane, and the map- count of the embedding of the geometric shapes. The ping is conformal everywhere on the surface, then we call shape descriptors are represented as functions, incon- the complex gradient vector field of the mapping a holomor- venient for searching. The classification is also too re- phic one-form. All the holomorphic one-forms on the sur- strictive. face form a real vector space, which we call holomorphic Proceedings of the Ninth IEEE International
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