Abstract-h this paper, we attack the figure-ground discrimination problem from a combinatorial optimization perspective. In general, the solutions proposed in the past solved this problem only partially: Either the mathematical model encoding the figure-ground problem was too simple, the optimization methods that were used were not efficient enough, or they could not guarantee that the global minimum of the cost function describing the figure-ground model would be found. The method that we devised and is describe-d in this paper is tailored around three main contributions.First, we suggest a mathematical model encoding the figureground discrimination problem that makes explicit a definition of shape (or figure) based on cocircularity, smoothness, proximity, and contrast. This model consists of building a cost function on the basis of image element interactions. Moreover, this cost function fits the constraints of an interacting spin system that, in turn, is a well suited physical model that solves hard combinatorial optimization problems Second, we suggest two combinatorial optimization methods for solving the figure-ground problem, namely i) mean field annealing, which combines mean field approximation theory and annealing, and ii) microcanonical annnealing. Mean field annealing may well be viewed as a deterministic approximation of stochastic methods such as simulated annealing. We describe, in detail, the theoretical bases of these methods, derive computational models, and provide practical algorithms.Third, we provide a comparison of the efficiency of mean field annealing, simulated annealing, and microcanonical annealing algorithms. Within the framework of such a comparison, the figure-ground problem may well be viewed as a benchmark.

In this paper we suggest an optimization approach to visual matching. We assume that the information available in an image may be conveniently represented symbolically in a relational graph. We concentrate on the problem of matching two such graphs. First we derive a cost function associated with graph matching and more precisely associated with relational subgraph isomorphism and with maximum relational subgraph matching. This cost function is well suited for optimization methods such as simulated annealing. We show how the graph matching problem is easily cast into a simulated annealing algorithm. Finally we show some preliminary experimental results and discuss the utility of this graph matching method in computer vision in general.