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

Chapter 5: Analysis and Results 66that makes use of an equation involving quartiles to output a single number as theoptimal bandwidth. We have observed that it sometimes gives us under-smoothed datacompared to the cross validation methods. We have decided to use the plug-in methodfor bandwidth optimization because we want our algorithm to be fully automaticwithout us having to interfere. Another important parameter with the densitydistribution that decides accuracy of the estimate is the number of points at which wecalculate the density.Another small but significant implementation issue that we would like to throw lightupon is the difference between continuous random variables and discrete randomvariables. The discrete density function is not a sampled form of the continuous densityfunction. We note that the density at each point of a discrete random variable is less thanor equal to one and the sum of the densities is unity.Since some values of the density function estimated are possibly zero and since we areusing a logarithmic information measure, we should get around the zero points of thedensity function. We do not compute entropy at the zero points.5.2 State-of-the-Art Shape DescriptorsThe analysis in this section is the backbone of our CVM algorithm. We haveimplemented a few state-of-the-art algorithms to better understand the process of shapeextraction from triangle meshes and also to know about the existing curvature-basedmetrics. Now that we have accurate measures of principle curvature and Gaussiancurvature, we are able to identify curvedness and shape index used by Dorai in her“COSMOS” framework for shape recognition on range images.In Figure 5.5 we show curvedness, shape index and the Gaussian curvature color codedmodels of the fan disk. (Model source of the fan disk: Hughes Hoppe, MicrosoftResearch.) By color coding we mean we have attributed color in the RGB spectrum toeach vertex of that model. The cosine color coding is proportional to the value of theparameter that we have computed at that vertex. For example, in Figure 5.5(d) eachvertex is color coded to Gaussian curvature. We have chosen the fan disk model becauseit is the one that has a combination of flat and curved surfaces and is not too simple ortoo complex.