• discrete approximations
• gamma-convergence
• metric slope
• Mumford-Shah functional

In this thesis we consider the one dimensional version of the functional introduced by D. Mumford and J. Shah (CPAM 1989) in the context of vision theory.
We present a discrete variational approximation of the Mumford-Shah functional introduced by A. Chambolle (SIAM 1995) that is related to the weak membrane model (A. Blake and A. Zissermann, 1987), also known as softening effect.
The approximating functionals F_n are the discrete version of the norm of the derivative truncated at level n.

In the first part of the thesis we prove
• the Gamma-convergence of the approximating functionals F_n to MS (the proof we
provide is different from the original one),
• a property of equicoerciveness which yields the convergence of minima and minimizers.

In the second part we recall the definition of descending metric slope and then
• we relate the Gamma-limit of the slopes to the slope of the Gamma-limit in a general metric setting,
• we compute the slope of MS and of a suitable regularization of F_n,
• we show that the Gamma-limit (over sequences with bounded energy) of the slopes of F_n is the slope of MS.