Determine Similarity between Shapes

• Do two 3D models represent the same or similar shapes?

  

• Represent each model by a shape descriptor, that turns the 3D representation into a high dimensional vectors of some notion of shapes

• Then compare the distances between these vectors

• How are these descriptors created?

  • Can be manually designed
  • Can be learned

• Should be invariant to rigid transforms of the shapes:

  • We can try to align the shapes first and then comput the descriptors

Global shape descriptor: Shape Histogram #

• Store histogram of how much surface area resides within different concentric shells in space

  

• Can also be used as a local descriptor by narrowing the radius

Non-Rigid Shape Matching #

• Consider near-isometric cases (cases where geodesic distances stay constant)

• Find correspondences that preserve geodesic distances

• shape similarity can be measured with the Hausdorff Distance or Gromov-Hausdorff distance

Observation: Near isometries preserve local structure #

• Correspondences can still be established though local descriptors
• Hyperparameter : How large should the local region be?

Heat kernel signature #

• “Simulate” how heat would spread on the surface over time and use this to compare different shapes.
• It is topological invariant to isometric deformations (the thumb could move around but the heat would spread just the same to it)