Shape Analysis and Retrieval

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```					Shape Analysis and Retrieval

Shape Histograms
Ankerst et al. 1999

Notes courtesy of
Funk et al., SIGGRAPH 2004
Shape Histograms
• Shape descriptor stores a histogram of how
much surface resides at different bins in
space

Model                 Shape Histogram
(Sectors + Shells)
Boundary Voxel Representation
• Represent a model as the (anti-aliased)
rasterization of its surface into a regular
grid:
– A voxel has value 1 (or area of intersection) if
it intersects the boundary
– A voxel has value 0 if it doesn’t intersect

Model                             Voxel Grid
Boundary Voxel Representation
• Properties:
– Invertible
– 3D array of information
– Can be defined for any model

Point          Polygon        Closed   Genus-0
Clouds           Soups         Meshes   Meshes

Shape Spectrum
Retrieval Results
100%                    Sectors and Shells (3D)
Sectors (2D)
Shells (1D)
EGI (2D)
Precision

D2 (1D)
50%

0%
0%      50%         100%
Recall
Histogram Representations
• Challenge:
– Histogram comparisons measure overlap, not
proximity.
Histogram Representations
• Solution:
– Quadratic distance form:

D (v ,w )  (v w )t M (v w )   with M ij  e  .d (i , j )
Histogram Representations
• Solution:
– Quadratic distance form:

D (v ,w )  (v w )t M (v w )       with M ij  e  .d (i , j )

M is a symmetric matrix and can be expressed as:
M  O t DO
O is a rotation and D is diagonal with positive
entries.
Taking the square root:
M 1/ 2  O t D 1/ 2O
Histogram Representations
• Solution:
– Quadratic distance form factors:
D (v ,w )  (v  w )t M 1/ 2M 1/ 2 (v  w )
 (M 1/ 2 (v  w )) t (M 1/ 2 (v  w ))
2
 M (v )  M (w )
1/ 2         1/ 2

If v=(v1,…,vn), we have:
n
M   1/ 2

(v ) i   a ijv j   where       a ij  f (d (i , j ))
j 1

That is, M1/2(v) is just the convolution of v with some
filter.
Convolving with a Gaussian
• The value at a point is obtained by summing
Gaussians distributed over the surface of the
model.
Distributes the surface into adjacent bins
 Blurs the model, loses high frequency information

Surface             Gaussian                 Gaussian
convolved surface
Gaussian EDT
• The value at a point is obtained by summing
the Gaussian of the closest point on the
model surface.
Distributes the surface into adjacent bins
Maintains high-frequency information

max

Surface              Gaussian              Gaussian EDT
[Kazhdan et al., 2003]
Gaussian EDT
• Properties:
–   Invertible
–   3D array of information
–   Can be defined for any model
–   Difference measures proximity between surfaces

Point           Polygon         Closed        Genus-0
Clouds            Soups          Meshes        Meshes

Shape Spectrum
Retrieval Results
100%                    GEDT (3D)
Sectors and Shells (3D)
Sectors (2D)
Shells (1D)
EGI (2D)
Precision

D2 (1D)
50%

0%
0%       50%        100%
Recall

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