Annals of Mathematics and Artificial Intelligence

, Volume 13, Issue 3–4, pp 281–300

# Smooth curve extraction by mean field annealing

• Laurent Hérault
Article

## Abstract

In 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 or the optimization methods that were used were not efficient enough or they could not guarantee to find the global minimum of the cost function describing the figure — ground model. The method that we devised and which is described in this paper is tailored around the following contributions. First, we suggest a mathematical model encoding the figure — ground 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 aninteracting spin system, which in turn is a well suited physical model to solve hard combinatorial optimization problems. Second, we suggest a combinatorial optimization method for solving the figure — ground problem, namely mean field annealing which combines the mean field approximation and annealing. 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 this method, derive a computational model, and provide a practical algorithm. Finally, some experimental results are shown for both synthetic and real images.

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