Shape Analysis and Retrieval

1
Shape Analysis and Retrieval

• Shape Histograms
• Ankerst et al. 1999

Notes courtesy of Funk et al., SIGGRAPH 2004
2
Shape Histograms

• 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 doesnt 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
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Histogram Representations

• Challenge
• Histogram comparisons measure overlap, not
  proximity.

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Histogram Representations

• Solution
• Quadratic distance form

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Histogram Representations

• Solution
• Quadratic distance form
• M is a symmetric matrix and can be expressed
  asO is a rotation and D is diagonal with
  positive entries.
• Taking the square root

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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
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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
Gaussian EDT
Surface
Gaussian
Kazhdan et al., 2003
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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
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Retrieval Results