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

Chapter 5: Analysis and Results 67Fan Disk Model CurvednessShape Index Gaussian Curvature2 2κ1+ κ1 12−1κ1+ κ2 κ = κΚ =η = − tan1κ 222 π κ1− κ2(a) (b) (c) (d)Figure 5.5: Curvature-based descriptors.We would like to make the following conclusions from Figure 5.5. Curvedness provesto be a good descriptor that detects abrupt change in curvature. Curvedness is consistentat low resolutions but with bad triangulation it produces erroneous results. We attributethis result however to the curvature estimation method that assumes good and uniformtriangulation. While on the other hand, we see how shape index is colorful indicatingsurface variation along the flat surface facing us in the diagram. The definition of shapeindex assumes uniform topology of meshes as in range images. That is why the shapeindex of a spherical cap and a spherical cup which look the same visually possessdifferent shape indices. We see in Figure 5.5(d) that the Gaussian curvature clearlyshows variation in curvature in each of the surface patches and no or very little variationin the flat surfaces.We have also implemented a recent method for shape classification and descriptioncalled Shape Distributions [Osada, 2002]. The approach represents shapes ashistograms. Randomly sampled points on the surface of a triangle mesh is used toextract several features such as the centroidal profile, distance between two points, anglebetween three random points. These features are binned into a histogram. Thishistogram is used for object detection and classification. Results show similar shapeshaving similar feature histograms. This algorithm was implemented for shape searchingand retrieval on the web. We have tested this algorithm with our automotive parts andhave come to realize that several 1D features cannot represent completely the 3Dinformation in an object. We show our implementation of Shape Distributions in Figure5.6 and our experience in representing automotive components in Figure 5.7. Wedemonstrate the lack of uniqueness in description with the fan disk, disc brake andmuffler models. These models have the same bounding box but we see that the discbrake and the fan disk though extremely different in shape have a similar histogram,while mufflers though similar in shape have noticeable amount of variation.