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

Chapter 5: Analysis and Results 74We observe that the results with the super quadrics are interesting. We chose superquadrics to perform our experiments because they provide us a scheme of slowlyvarying shapes (that can be controlled by a parameter) and smoothly varyingcurvature. Though we are unable to cluster or classify shapes based on a singlenumber, we would like to point out the success with super quadrics. We however hadto deal with another major problem. The asymptotic behavior of the divergencemeasure as the resolution tended to infinity. We end up with an impulse function onthe sphere in the continuous case as a reference to a curvature density of another 3Dmodel. Though the divergence measures are defined for the continuous randomvariables, our shape complexity measure becomes unstable with resolutionapproaching the continuous case. We also are not able to justify what it means for twocompletely different shapes having the same measure. The magnitude of the measurethough can be understood as the number of bits that are required to describe the shapecomplexity of the object; it is not very convincing for the application of shapeclassification or clustering.Our focus is to make the CVM independent of resolution and support it withtheoretical consistency. We hence decided to change the reference from the sphere thatrepresented the object with least information to the abstract, most complex object atthat resolution. We have extended the Shannon’s definition to a resolution normalizedentropy form as shown in Equation 5.9.H∆( X ) p( x )logR p( x )(5.9)= −where R is the resolution of the datasets under consideration and p(x) is the probabilitydensity of the curvature. We achieve two things with this measure of information. Themeasure is normalized. It has a minimum value of zero and a maximum value of one.The measure is in a logarithmic scale and is resolution independent. In Figure 5.11 weshow how curvature on surfaces acts a descriptor with the spherical cup, saddle andthe monkey saddle surfaces. We would like to point out that the broader theprobability density function the higher the complexity. In Figure 5.12 we perform amulti-resolution experiment with our CVM shape signature. We resample the monkeysaddle without obvious change in shape to show that our measure is now independentof resolution. N refers to the number of vertices in that surface and F is the number offaces.We recollect the experience with the curvature-based descriptors. Curvature alone is nota sufficient feature for shape description because we have lost more than twodimensions of description in trying to represent 3D into a 1D function of curvature.However, now that we have verified the surface description capabilities of our measurewe propose to describe objects that can be broken down into surface patches. We makeuse of curvedness for this task. We identify the sharp edges and creases and use it forsegmentation of the triangle meshes.