Localising discontinuities using weak continuity constraints

Abstract

Many processes in computer vision can be formulated concisely as optimisation problems. In particular the localisation of discontinuities can be regarded as optimisation under weak continuity constraints. A weak constraint is a constraint which can be broken — but at a cost.

In this paper we illustrate the use of weak constraints by considering a simple example — edge detection in 1D. Finite elements are used to discretise the problem. The cost function is minimised using a ‘graduated non-convexity’ algorithm. This gives a local relaxation scheme which could be implemented in parallel.

Results are given for a serial computer implementation of the method. They show that the algorithm does perform as theoretically predicted, and that it is robust in the presence of noise. Results are also given for a 2D version of the method, applied to real images.