Spherical Extent Functions Spherical Extent Function Spherical Extent

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Spherical Extent Functions

Spherical Extent Functions

Spherical Extent Function

Spherical Extent Function

Spherical Extent Function

Spherical Extent Function

Spherical Extent Function A model is represented by its star-shaped envelope: – The minimal

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

Spherical Extent Function A model is represented by its star-shaped envelope: – The minimal

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

Spherical Extent Function Properties: – Invertible for star-shaped models – 2 D array of

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

Spherical Extent Function Properties: – Can be defined for most models – Invertible for

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

Retrieval Results

Retrieval Results

PCA Alignment Treat a surface as a collection of points and define the variance

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

PCA Alignment Define the covariance matrix M: Find the eigen-values and align so that

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

PCA Alignment Limitation: – Eigen-values are only defined up to sign! PCA alignment is

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.

Spherical Functions Parameterize points on the sphere in terms of angles [0, ] and

Spherical Functions Parameterize points on the sphere in terms of angles [0, ] and [0, 2 ): z ( , )

Spherical Functions Every spherical function can be expressed as the sum of spherical harmonics

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.

Spherical Harmonics Every spherical function can be expressed as the sum of spherical harmonics

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

Spherical Harmonics Every spherical function can be expressed as the sum of spherical harmonics

Spherical Harmonics Every spherical function can be expressed as the sum of spherical harmonics Ylm: Rotation by 0 gives:

Spherical Harmonics If f is a spherical function: Then storing just the absolute values:

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.