On the Minimization of a Circular Function on the Isomorphic Shrunk Subset

In this paper a circular shape representation is-used to define a circular set for a pair of fixed shapes. This paper is an extension of an approach to shape comparison based on skeleton isomorphism proposed in. The main advantage over existing approaches is mathematically correctly defined shape metrics via Hausdorff distance with the concurrent use of topology features of a shape. The circular function is proposed on the defined circular set. It has two parameters as two given shapes and two variables as two circulars. The circular set could be reduced to a special shrunk isomorphic subset. The minimization of a circular function problem on a shrunk subset for a pair of shapes is proposed. The existence and reachability of minimum on this shrunk subset is proved. Effective monotone subsets are constructed to reduce the searching of an optimal problem´s solution. The latter reduces the number of all possible subgraphs where to search the best pair to logarithmic (by skeleton edges) computational complexity comparing to the exponential total number of possible subgraphs. All the results are mathematically correct and may be useful in shape comparison and shape analysis applications where pairs of shapes are considered.