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

Chapter 2: Literature Review 20bounding sphere is of radius 32 voxels. They then treat the object as the function in 3Dspace and decompose it into 32 spherical functions by considering spheres of radii 1through 32. They further decompose each of the functions into 16 harmoniccomponents and the 32 x 16 harmonics constitute the harmonic representation of the3D model. They compare two harmonic representations with the Euclidean distance.Zhang and Hebert propose the harmonic shape images as a 2D representation ofsurface patches [Zhang and Hebert, 1999]. The theory of harmonic maps studies themapping between different metric manifolds from the energy minimization point ofview. With the application of harmonic maps, a surface representation called harmonicshape images is generated to represent and match 3D freeform surfaces. The basic ideaof harmonic shape images is to map a 3D surface patch with disc topology to a 2Ddomain and encode the shape information of the surface patch into the 2D image. Thissimplifies the surface-matching problem to a 2D image-matching problem.Shum et al. address the problem of 3D shape similarity between closed surfaces in[Shum et al., 1996]. He defines a shape similarity metric as the L 2 distance betweenthe local curvature distributions over the spherical mesh representations of the twoobjects. He achieves the similarity measure in O (n 2 ) complexity where n is thenumber of tessellations in the object mesh. Their experiments on simple shapes showgood shape similarity measurements.2.3.4 Shape HistogramsThe histogram-based methods reduce the cost of complex matching schemes butsacrifice efficiency and robustness to the methods discussed in Section 2.3.2. Thesemethods compare shapes on the basis of their statistical properties.Ankerst et al. [Ankerst et al., 1999] introduce 3D shape histograms as an intuitivepowerful approach to structural classification of proteins. They decompose a 3Dobject into three models (shell model, sector model and spider web) around an object’scentroid and process model similarity queries based on a filter refinement architecture.A similar search technique for mechanical parts using histograms was proposed in[Kriegel et al., 2003]. The models are normalized into a canonical form and voxelizedinto axis parallel equal partitions. Each of these partitions is assigned to one or severalbins in a histogram depending on the specific similarity model.Besl et al. [Besl et al., 1995] consider histograms of crease angles for all edges in atriangle mesh to describe shape. Their method does not match non-manifold surfacesand is not invariant to changes in mesh tessellation. Osada et al. in [Osada et al., 2002]presents Shape Distributions for a shape similarity search engine by extending Besl’sapproach. According to his technique, random points from the surface of a model are