Spherical Extent Functions
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Transcript of Spherical Extent Functions
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Spherical Extent Functions
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Spherical Extent Function
![Page 3: Spherical Extent Functions](https://reader035.fdocuments.us/reader035/viewer/2022062321/56812d17550346895d9204b3/html5/thumbnails/3.jpg)
Spherical Extent Function
![Page 4: Spherical Extent Functions](https://reader035.fdocuments.us/reader035/viewer/2022062321/56812d17550346895d9204b3/html5/thumbnails/4.jpg)
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
![Page 5: Spherical Extent Functions](https://reader035.fdocuments.us/reader035/viewer/2022062321/56812d17550346895d9204b3/html5/thumbnails/5.jpg)
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
![Page 6: Spherical Extent Functions](https://reader035.fdocuments.us/reader035/viewer/2022062321/56812d17550346895d9204b3/html5/thumbnails/6.jpg)
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
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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
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Retrieval Results
0%
50%
100%
0% 50% 100%
Spherical Extent Function (2D)Gaussian EDT (3D)Shape Histograms (3D)Extended Gaussian Image (2D)D2 (1D)Random
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PCA Alignment
Treat a surface as a collection of points and define the variance function:
Sp
dpvpvSVar2
,),(
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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
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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.
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Spherical Functions
Parameterize points on the sphere in terms of angles [0,] and [0,2):
cos,sinsin,cossin),(
((, , ))
z
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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),(
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Spherical Harmonics
Every spherical function can be expressed as the sum of spherical harmonics Yl
m:
l=1
l=2
l=3
l=0
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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
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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