Representing and comparing shapes using shape polynomials

The problem of multiresolution 2-D and 3-D shape representation is addressed. 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. To any shape S and every integer d>0 is associated a shape polynomial of degree 2d, whose coefficients are functions of the moments of S. These polynomials are related to the shape S in an affine-invariant way. They yield small values near S and large values far away, and their level sets approximate S. The shape polynomials define two distances between shapes. As asymmetric 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 by a sequence of computationally very fast matrix operations. The distance measures are used for recognition and positioning of objects in occluded environments