3D OBJECT REPRESENTATION WITH TOPO-GEOMETRIC SHAPE MODELS Sajjad Baloch † , Hamid Krim † , Irina Kogan ‡ , and Dmitry Zenkov ‡ North Carolina State University, Raleigh, NC 27695 ABSTRACT We propose a new method for 3D object representation using weighted skeletal graphs. The geometry of an object is captured by assigning weights to the skeletal graph of the object, which in turn represents its topology. The weights provide necessary information for object reconstruction. The method is rotation, translation, and scaling invariant. Applications include shape representa- tion, compression, and object recognition. 1. INTRODUCTION In this paper, a 3D object to be analyzed is replaced with its 2D boundary, which is assumed to be a compact two-dimensional manifold embedded in R 3 . It is further assumed that all manifolds of interest are smooth. A short overview of various techniques for 3D shape modeling is given below. Shinagawa et al. [15] proposed an algorithm for mod- eling the topology of 3D objects using Reeb graphs based on the height function. This approach completely ignores the geometric information of an object. In Os- ada et al. [13], and Hamza and Krim [3], 3D objects are represented through shape distributions, and then a dissimilarity measure for distributions is employed to classify them. Lazarus et al. [8] proposed skeletonization based on the geodesic distance from a manually chosen source point and called their graphs level set diagrams. Hilaga et al. [5] extended this approach by eliminating the need for manual selection of a source point and proposing a matching algorithm based on multiresolution Reeb graphs. Although this achieved rotational invariance, the algorithm was computationally intensive. In Kazhdan et al. [6] [7], 3D shapes were modeled through a reflective symmetry descriptor defined over a canonical parame- terization. Kazhdan and Funkhouser [7] used rotation invariant spherical harmonics as a shape descriptor. Our approach to capturing the geometry of an object begins with the calculation of a skeletal graph of the object’s boundary. We follow [15], [5], [3] in this calcula- tion; however, our choice of a Morse function is different. †Department of Electrical and Computer Engineering. ‡Department of Mathematics. This work was supported by AFOSR F49620-98-1-0190 and NSF CCR-9984067 grants. Fig. 1. Weighted skeletal graph: Graph itself captures the topology while graph coupled with the weights W i encodes the geometry of the object. We use the distance function that produces tranlation, rotation, and scale invariant skeletal graphs. Next, we assign weights to the graph, which is illustrated in Fig. 1. These weights record the information about the shapes of intersections of the object’s boundary and the level surfaces of the distance function. Details are given in Section 3. This model is complete in the sense that the original object can be reconstructed, with given precision, from its weighted skeletal graph. 2. TOPOLOGICAL MODEL In this paper, we concentrate on the study of 2D sur- faces embedded in R 3 . The points of R 3 are represented by their position vectors, which are typed in bold. 2.1. Some Definitions Consider the distance function d : p →‖p‖ in R 3 . Given a generic surface M⊂ R 3 , the restriction of the distance function on M, d : M→ R + , (1) is a Morse function, i.e., all critical points of d on M are non-degenerate (see e.g. [12] and [10]). One can thus use the distance function for constructing a skeletal graph of the surface M. To analyze and encode a compact surface using the Morse function (1), we start at the origin and gradually increase the value of the distance function in K steps to a sufficiently large number which we denote b. The integer K is called the resolution of the skeletal graph. Making K larger increases the precision of captured structural changes in the level sets of the distance function. Recall that such changes occur only at critical level sets.