Transcript addenda

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
Model
Star-Shaped Envelope
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
Shape Spectrum
Genus-0
Meshes
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
100%
Spherical Extent Function (2D)
Gaussian EDT (3D)
Shape Histograms (3D)
Extended Gaussian Image (2D)
D2 (1D)
Random
50%
0%
0%
50%
100%
PCA Alignment
Treat a surface as a collection of points and define the
variance function:
Var (S ,v ) 
p ,v

p S

2
dp
PCA Alignment
Define the covariance matrix M:
M ij 
p i p j dp

p S

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 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):
( , )  sin  cos , sin  sin , cos 
z


(, )
Spherical Functions
Every spherical function can be expressed as the sum
of spherical harmonics Ylm:
f ( ,  )  
l
l
m m
f
 l Yl ( , )
m  l
Where l is the frequency and m indexes harmonics
within a frequency.
2l  1 (l  m )! m
Y l ( ,  ) 
Pl (cos  )e im
4 (l  m )!
m
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 Ylm:
2l  1 (l  m )! m
m
Y l ( ,  ) 
Pl (cos  )e im
4 (l  m )!
Rotation by 0 gives:
Y l m ( ,   0 )  e im Y l m ( , 0 )
0
f ( ,    0 )  
l
 f
l
m  l
l
m
e im Y l m ( ,  )
0
Spherical Harmonics
If f is a spherical function:
f ( ,  )  
l
l
m m
f
 l Yl ( , )
m  l
Then storing just the absolute values:
f
0
0

, f 11 , f 10 , f 11 ,..., f l m ,...
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.