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Spherical Extent Functions

Spherical Extent Functions. Spherical Extent Function. Spherical Extent Function. 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

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Spherical Extent Functions

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  1. Spherical Extent Functions

  2. Spherical Extent Function

  3. Spherical Extent Function

  4. 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

  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 Model Star-Shaped Envelope

  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 Matching Nearest 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 Ylm: Where l is the frequency and m indexes harmonics within a frequency.

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

  15. Spherical Harmonics Every spherical function can be expressed as the sum of spherical harmonics Ylm: 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: • Invariant to rotation by 0. • Invariant to axial flips about the x-, y-, and z-axes.

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