We address the problem of multiresolution 2D and 3D shape representation. Shape is defined as a probability measure with compact support. Both object representations, typically sets of curves and/or surface patches, and observations, sets of scattered data, can be represented in this way. Global properties of shapes are defined as expectations (statistical averages) of certain functions. In particular, the moments of the shapes are global properties. For any shape Ë � and every integer � � � we associate a shape polynomial of degree � � whose coefficients are functions of the moments of Ë. These polynomials are related to the shape Ë in an affine invariant way. They yield small values near Ë and large values far away and their level sets approximate Ë. With the shape polynomials we define two distances between shapes. An asymmetric distance measures how well one shape fits as a subset of another one; a symmetric version indicates how equal two shapes are. The evaluation of these distance measures is determined via a sequence of computationally very fast matrix operations. The distance measures are used for recognition and positioning of objects in occluded environments
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