Transcript addenda

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
Clouds
Polygon
Soups
Closed
Meshes
Shape Spectrum
Genus-0
Meshes
Retrieval Results
Precision
100%
Sectors and Shells (3D)
Sectors (2D)
Shells (1D)
EGI (2D)
D2 (1D)
50%
0%
0%
50%
Recall
100%
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
1/ 2
1/ 2
 M (v )  M (w )
If v=(v1,…,vn), we have:
M
1/ 2

n
(v ) i   a ijv j
j 1
where
aij  f (d (i , j ))
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
Clouds
Polygon
Soups
Closed
Meshes
Shape Spectrum
Genus-0
Meshes
Retrieval Results
Precision
100%
GEDT (3D)
Sectors and Shells (3D)
Sectors (2D)
Shells (1D)
EGI (2D)
D2 (1D)
50%
0%
0%
50%
Recall
100%