Shape Analysis and Retrieval Spherical Parameterizations Notes courtesy of Funk et al., SIGGRAPH 2004

Spherical Parameterizations Goal Get a one-to-one mapping between points on the surface and points on the sphere.

Spherical Parameterizations Start with the EGI

Spherical Parameterizations Limitation Different points on the surface can map to the same point on the sphere. Circular Angle Model Point

Spherical Parameterizations Approach Smooth the function, by convolving with a Gaussian: The value of a point is obtained by taking a weighted average of the surrounding points.

Spherical Parameterizations Convolving with a Gaussian Initial Function Filter Convolution

Laplacian Smoothing Given a function f define the family of functions that you would get by smoothing f by different amounts Taking derivatives gives:

Laplacian Smoothing Taking derivatives gives: So that smoothing a little amounts to adding the second derivative (Laplacian). This is important when you want to smooth in a space where there is no direct way to convolve.