1. AdvisorStudent Dr. Jia Li Shaojun Liu Dept. of Computer Science and Engineering, Oakland University 3D Shape Classification Using Conformal Mapping In this paper we propose a new approach for 3D shape classification based on conformal mapping of polygonal mesh. Our approach does not exclude the use of other geometric features besides normal vectors. We want to explore the rotation invariant feature to compare the classification result with that with pose alignment. Available 3D models on Internet and specified database increase dramatically with the advancement of modeling and digitizing techniques, such as computer aided design (CAD) and laser scanning. And shape-based retrieval of 3D data has been an area of research in disciplines such as computer vision, mechanical engineering, artifact searching, molecular biology and chemistry. The performance of 3D shape search engine, however, is left much to be desired as compared with that of text (such as Google search engine), image and audio. The comparison of shape similarity is the basis for shape recognition, matching, classification, and searching. A large mount of 3D methods exist and belong to various classes. For example, Methods based on shape distribution distinguish models in broad categories well but perform poorly on shapes with similar gross shape properties while vastly different detailed shape properties. The spatial map based methods capture the detailed shape properties but still have problems such as center positioning and fine details preservation. We propose a new shape similarity comparison method based on the conformal mapping over unit sphere, which avoids the problem of center positioning. It keeps the fine surface details in the geometry image. And it is more compact as compared to those which encode the shape intercepts at different radial distances. To classify the 3D shapes we use the tool of self-organizing map (SOM), an excellent tool in exploratory phase of data mining. It is a two-level approach, the first level of which is a large set of prototypes that are then combined to form the actual clusters. Usually the prototypes are arranged in 2D grid and each of them is represented by a prototype vector. To visualize SOM result, we use the unified distance matrix, U- matrix, which shows distance between prototype vectors of neighboring map units. Fig. 3 shows the SOM result of 3D shapes downloaded from Internet. And several classes are shown in Fig. 4, Fig. 5 and Fig 6 respectively. The overall approach follows the sequence of pose alignment, conformal mapping, feature extraction, and similarity search as shown in Fig. 1  1. Abstract A new method for 3D shape classification based on conformal mapping of 3D meshes is introduced. We propose to conformally map 3D meshes to the domain of unit sphere, which generates geometry images and normal maps over the sphere. Then the spherical harmonic representation of the normal map is used as the feature vector input of the self-organizing map for shape classification. This method can evade the common object center estimation and preserve the geometric details. The results demonstrate that the proposed method can discriminate the collected 3D shapes very well, and is robust to noise, tesselations and pose difference.  2. Introduction  4. Experimental Results  3. Shape Classification In our pose alignment process we adopt the Continuous Principal Component Analysis (CPCA) method, as it works well with most of the shapes collected from the Internet.  5. Conclusion and Discussion Fig. 1 The procedure of 3D shape classification. Fig. 2 SNI of Ball, cube, cylinder, bunny, goblet and venus. Note that the domain is sphere instead of the square. To extract the geometric features of 3D objects, it is convenient to map the surface onto the region of the plane or sphere first. As the conformal mapping is one-to-one and angle preserving, we take the normal vectors as the geometric feature, and then conformally map them onto sphere. Fig. 2 shows some mapped results, named spherical normal image (SNI). The geometric features in our approach, SNI, is then decomposed by spherical harmonics, whose coefficients constitute a feature vector to be used for shape classification. Fig. 3 The result of SOM. The left side is the U-matrix marked by different colors. The right side is the prototypes with different labels. Blank label means no feature vector is in the prototype. Fig. 4 The 3D models in the prototype with label 'glass1' Fig. 5 The 3D models in the prototype with label '23' Fig. 6 The 3D models in the prototype with label 'Hex0'