1. Shape Analysis and Retrieval
Shape Histograms
Ankerst et al. 1999
Notes courtesy of Funk et al., SIGGRAPH 2004

2. Shape Histogram (Sectors + Shells)
Shape descriptor stores a histogram of how much surface resides at different bins in space
Model
Shape Histogram (Sectors + Shells)

3. 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

4. Boundary Voxel Representation
Properties:
Invertible
3D array of information
Can be defined for any model
Point Clouds
Polygon Soups
Closed Meshes
Genus-0 Meshes
Shape Spectrum

5. Retrieval Results

6. Histogram Representations
Challenge:
Histogram comparisons measure overlap, not proximity.

7. Histogram Representations
Solution:
Quadratic distance form:

8. Histogram Representations
Solution:
Quadratic distance form:
M is a symmetric matrix and can be expressed as: O is a rotation and D is diagonal with positive entries.
Taking the square root:

9. Histogram Representations
Solution:
Quadratic distance form factors:
If v=(v1,…,vn), we have:
That is, M1/2(v) is just the convolution of v with some filter.

10. 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

11. 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]

12. 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
Genus-0 Meshes
Shape Spectrum

13. Retrieval Results