Shape Spaces

• The shape of a collection of N points is defined as "whatever is left after the effects of rotation, translation and scaling have been ruled out" [Kendall, 1984].
  If X are the coordinates of the N points (a 3xN matrix) relative to any reference frame, and X are the corresponding normalized coordinates (a point on a 3N-1 dimensional sphere), then a shape s is obtained by centering and re-orienting the reference system so as to align it with the principal axes of the cloud of points.
  Therefore, a shape is just a re-positioning of the coordinates on a special coordinate frame intrinsic to the object
  s = p X
  p is a rigid change of coordinates (a point on the Lie group SE(3)) that we call pose .

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• A Shape Space can be represented geometrically as the base of a "fiber bundle", where each fiber describes all (scaled) sets of points which can be transformed rigidly one to the other.
  Kendall, Le, Carne and others have unraveled the Riemannian geometric structure of shape spaces.

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• Problems:
  • Object with a different number of points belong to different shape-spaces
    Problems with occlusions.
  • Local coordinatization of the shape space is ill-conditioned in the presence of symmetric objects [Carne 1992].