Shape Analysis and Retrieval Shape Histograms Ankerst et

Shape Histograms • Shape descriptor stores a histogram of how much surface resides at

Boundary Voxel Representation • Represent a model as the (anti-aliased) rasterization of its surface

Boundary Voxel Representation • Properties: – Invertible – 3 D array of information –

Histogram Representations • Challenge: – Histogram comparisons measure overlap, not proximity.

Histogram Representations • Solution: – Quadratic distance form:

Histogram Representations • Solution: – Quadratic distance form: M is a symmetric matrix and

Histogram Representations • Solution: – Quadratic distance form factors: If v=(v 1, …, vn),

Convolving with a Gaussian • The value at a point is obtained by summing

Gaussian EDT • The value at a point is obtained by summing the Gaussian

Gaussian EDT • Properties: – – Invertible 3 D array of information Can be

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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 – 3 D array of information – Can be defined for any model Point Clouds Polygon Soups Closed Meshes Shape Spectrum Genus-0 Meshes

Retrieval Results

Histogram Representations • Challenge: – Histogram comparisons measure overlap, not proximity.

Histogram Representations • Solution: – Quadratic distance form:

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:

Histogram Representations • Solution: – Quadratic distance form factors: If v=(v 1, …, vn), we have: That is, M 1/2(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. ü Distributes the surface into adjacent bins û Blurs the model, loses high frequency information Surface Gaussian convolved surface

Gaussian EDT • The value at a point is obtained by summing the Gaussian of the closest point on the model surface. ü Distributes the surface into adjacent bins ü Maintains high-frequency information max Surface Gaussian EDT [Kazhdan et al. , 2003]

Gaussian EDT • Properties: – – Invertible 3 D 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