To the Graduate Council: I am submitting herewith a thesis written by ...

Chapter 2: Literature Review 22Gotsman et al. describe the fundamentals of spherical parameterization for 3D meshes[Gotsman et al., 2003]. They argue that closed manifold genus-zero meshes aretopologically equivalent to a sphere and assign a 3D position on the unit sphere toeach of the mesh vertices. They use barycentric coordinates for the planarparameterization. Leibowitz et al. [Leibowitz et al., 1999] share their memoryintensive experience in implementing geometric hashing for the comparison of proteinmolecules represented as 3D atomic structures.In [McWherter et al., 2001] model signature graphs have been proposed fortopological comparison of solid models. They extend attribute adjacency graphs,mentioned in [Joshi and Chang, 1998], to consider curved surfaces. Model signaturegraphs are constructed from boundary representation of the solid. This graph forms theshape signature of the solid model. Once a model signature graph is constructed, thesolid models are compared using spectral graph theory [Chung, 1997]. The eigenvalues of the Laplacian matrix are used in the comparison. The eigen values of theLaplacian are strongly related to other graph properties such as the graph diameter.The graph diameter is the largest number of vertices, which must be traversed, totravel from one vertex to another in the graph. Another technique proposed forcomparing the graphs is the use of graph invariance vectors [McWherter et al., 2001].The vectors are then compared using L 2 norm to determine similarity between thegraphs and hence the solid models. The graph invariants that form the graphinvariance vectors include node and edge count, minimum and maximum degree ofthe nodes, median and mode degree of the nodes, and diameter of the graph. The useof graph invariance vectors improves the efficiency of the method. However it resultsin decrease in the accuracy of comparison. This technique has been applied tomechanical parts and is applicable to product design and manufacturing domain. Thepaper [Cardone et al., 2003] is a comprehensive survey on shape-similarity basedassessment for product design applications.Multi-Resolution Reeb Graphs presented in [Hilaga et al., 2001] have been used formodeling 3D shapes. The Reeb graph is derived from the triangle mesh models bydefining a suitable function such as the geodesic curvature. The choice of the functiondepends on the topological properties selected. The range of the function over theobject is split into smaller bins. The number of bins is the resolution of the Reebgraph. Each connected region in the bin will map into a node of the Reeb graph, andthe adjacent nodes will be connected by edges. The Reeb graph construction has atime complexity of O (N log N), N being the number of vertices in the mesh. The Reebgraphs of two objects can be used for maximizing a similarity function atcorresponding nodes. This technique is not invariant to Euclidean transformations.We have very briefly described some of the key methods for shape analysis on 2Dintensity images and 3D mesh models. In the next section we present two tables that