Order Structure, Correspondence, and Shape Based Categories

We propose a general method for finding pointwise correspondence between 2-D shapes based on the concept of order structure and using geometric hashing. The problem of finding correspondence and the problem of establishing shape equivalence can be considered as one and the same problem. Given shape equivalence, we can in general find pointwise correspondence and the existence of a unambiguous correspondence mapping can be used as a rule for deciding shape equivalence. As a measure of shape equivalence we will use the concept of order structure which in principle can be defined for arbitrary geometric configurations such as points lines and curves. The order structure equivalence of subsets of points and tangent directions of a shape is will be used to establish pointwise correspondence. The finding of correspondence between different views of the same object and different instances of the same object category can be used as a foundation for establishment and recognition of visual categories.