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

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

3. Spherical Parameterizations
Start with the EGI

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

5. 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.

6. Spherical Parameterizations
Convolving with a Gaussian
Initial Function
Filter
Convolution

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

8. 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.