Calibrating Parameters of Cost Functionals

We propose a new framework for calibrating parameters of energy functionals, as used in image analysis. The method learns parameters from a family of correct examples, and given a probabilistic construct for generating wrong examples from correct ones. We introduce a measure of frustration to penalize cases in which wrong responses are preferred to correct ones, and we design a stochastic gradient algorithm which converges to parameters which minimize this measure of frustration. We also present a first set of experiments in this context, and introduce extensions to deal with data-dependent energies. keywords: Learning, variational method, parameter estimation, image reconstruction, Bayesian image models 1 Description of the Method Many problems in computer vision are addressed through the minimization of a cost functional U . This function is typically defined on a large, finite, set Ω (for example the set of pictures with fixed dimensions), and the minimizer of x 7→ U(x) is supposed to conciliates several properties which are generally antithetic. Indeed, the energy is usually designed as a combination of several terms, each of them corresponding to a precise property which must be satisfied by the optimal solution. As an example among many others, let us quote probably the most studied cost functional in computer vision, namely the Mumford/Shah energy (cf. [5]), which is used to segment and smooth an observed picture. Expressed in a continuous setting, it is the combination of three terms, one which ensures that the smoothed picture x, defined on a set D ⊂ IR is not too different from the observed one ξ, another which states that the derivative of the smoothed picture is small, except, possibly, on a discontinuity set ∆, and a last one which ensures that the discontinuity set has small length. These terms are weighted by parameters, yielding an energy function of the kind