1. Spherical Extent Functions

2. Spherical Extent Function

4. A model is represented by its star-shaped envelope: –The minimal surface containing the model such that the center sees every point on the surface –Turns arbitrary models to genus-0 surfaces

5. Spherical Extent Function A model is represented by its star-shaped envelope: –The minimal surface containing the model such that the center sees every point on the surface –Turns arbitrary models to genus-0 surfaces Star-Shaped EnvelopeModel

6. Spherical Extent Function Properties: –Invertible for star-shaped models –2D array of information –Can be defined for most models Point Clouds Polygon Soups Closed Meshes Genus-0 Meshes Shape Spectrum

7. Spherical Extent Function Properties: –Can be defined for most models –Invertible for star-shaped models –2D array of information Limitations: –Distance only measures angular proximity Spherical Extent MatchingNearest Point Matching

8. Retrieval Results

9. PCA Alignment Treat a surface as a collection of points and define the variance function:

10. PCA Alignment Define the covariance matrix M: Find the eigen-values and align so that the eigen-values map to the x-, y-, and z-axes

11. PCA Alignment Limitation: –Eigen-values are only defined up to sign! PCA alignment is only well-defined up to axial flips about the x-, y-, and z-axes.

12. Spherical Functions Parameterize points on the sphere in terms of angles  [0,  ] and  [0,2  ):    ( ,  ) z

13. Spherical Functions Every spherical function can be expressed as the sum of spherical harmonics Y l m : Where l is the frequency and m indexes harmonics within a frequency.

14. Spherical Harmonics Every spherical function can be expressed as the sum of spherical harmonics Y l m : l=1l=1 l=2l=2 l=3l=3 l=0l=0

15. Spherical Harmonics Every spherical function can be expressed as the sum of spherical harmonics Y l m : Rotation by  0 gives:

16. Spherical Harmonics If f is a spherical function: Then storing just the absolute values: gives a representation of f that is: 1.Invariant to rotation by  0. 2.Invariant to axial flips about the x-, y-, and z-axes.