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

– Turns arbitrary models to genus-0 surfaces

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 Envelope

Model

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

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

Retrieval Results

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Spherical Extent Function (2D)Gaussian EDT (3D)Shape Histograms (3D)Extended Gaussian Image (2D)D2 (1D)Random

PCA Alignment

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

Sp

dpvpvSVar2

,),(

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

Sp

jiij dpppM

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 [0,2):

cos,sinsin,cossin),(

((, , ))

z

Spherical Functions

Every spherical function can be expressed as the sum of spherical harmonics Yl

m:

Where l is the frequency and m indexes harmonics within a frequency.

l

l

lm

ml

ml Yff ),(),(

imml

ml eP

mlmll

Y )(cos)!(

)!(

4

12),(

Spherical Harmonics

Every spherical function can be expressed as the sum of spherical harmonics Yl

m:

l=1

l=2

l=3

l=0

Spherical Harmonics

Every spherical function can be expressed as the sum of spherical harmonics Yl

m:

Rotation by 0 gives:

imml

ml eP

mlmll

Y )(cos)!(

)!(

4

12),(

),(),( 000 m

limm

l YeY

l

l

lm

ml

imml Yeff ),(),( 0

0

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.

l

l

lm

ml

ml Yff ),(),(

,...,...,,,, 11

01

11

00

mlfffff