• metric slope
• Gamma-convergence
• direct method
• Mumford-Shah

This thesis deals with a one-dimensional version of the Mumford-Shah functional, that models the problem of the signal segmentation. Let h be a signal; we are looking for signal u which is a "regular approximation" of h. First of all, we identify signals with real-valued functions in [0,1] and we introduce the generalized Mumford-Shah functional in dimension one. Then, we state and prove compactness and lower semicontinuity theorems; so, the functional admits minimum via direct method.
Since we are interested in the expliciting the minimizers, we introduce a family of approximating problems that Gamma-converges toward the generalized Mumford-Shah functional and we prove a compactness theorem. Thanks to a classical results, we obtain the minimum and the minimizers of the generalized Mumford-Shah functional as limits of the sequences of the minima and minimizers of the approximating problems.
The last part of the thesis is devoted to the computation of the descending metric slope of the principal part of the generalized Mumford-Shah functional. We find necessary conditions for the slope to be finite and we give a lower bound for the slope. Surprisingly enough, these conditions turn out to be sufficient for the slope to be finite and we give an upper bound for the slope that coincides with the lower bound, in most cases.