Set-valued maps for image segmentation

Abstract.

The goal of this paper is an analytical basis of a region growing method using set-valued maps. They are to provide an approach to image segmentation that does not use a priori conditions on the final segment.

Seizing a suggestion of Demongeot and Leitner ([8]), we start with a compact subset of the grey-valued image, and the region growing method is based on the continuous deformation of sets for decreasing some error functional. Avoiding any further restrictions on these sets leads to describing the process as a set-valued map.

The ansatz of the deforming sets utilizes reachable sets of differential inclusions that admit more than one velocity of propagation at each point. So set-valued maps underlie a mathematical segmentation problem posed and solved in the first part. Then we present a suggestion how to apply the analytical results to a very simple computer implementation.