Generalized Shapes and Point Sets Correspondence and Registration

The theory of shapes, as proposed by David Kendall, is concerned with sets of labeled points in the Euclidean space Rd that define a shape regardless of trans- lations, rotations and dilatations. We present here a method that extends the theory of shapes, where, in this case, we use the term generalized shape for structures of unlabeled points. By using the distribution of distances between the points in a set we are able to define the existence of generalized shapes and to infer the computation of the correspondences and the orthogonal transformation between two elements of the same generalized shape equivalence class. This study is oriented to solve the registration of large set of landmarks or point sets derived from medical images but may be employed in other fields such as computer vision or biological morphometry.