Sigmoidal approximations for self-interacting line processes in edge-preserving image restoration

Image restoration is formulated as the problem of minimizing a non-convex cost function E(f,l) in which a binary self-interacting line process is introduced. Each line element is then approximated by a sigmoidal function of the local intensity gradient, depending on a parameter T, thus obtaining a sequence of functions FT(f) converging to a function F(f) which implicitly refers to the line process. In the case of a non-interacting line process, function F(f) coincides with that derived for the weak membrane problem. The minimum of F(f) is computed through a ONC-type algorithm that minimizes in turn the various FT (f)s by gradient descent techniques. When generalized to the case of self-interacting line elements, the method is flexible with respect to the introduction of any kind of constraints on the configurations of the discontinuity field. The results of simulations evidence that the method produces improvement of the quality of the reconstructions when constraints on the line process are introduced, without any increase of the computational costs with respect to the case of absence of self-interactions of the lines.