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, which depends on a parameter T, thus obtaining a sequence of functions FT(f) converging to a function F(f) that implicitly refers to the line process. In the case of a non-interacting line process, function F(f) coincides with the one derived for the weak membrane problem. The minimum of F(f) is computed through a GNC-type algorithm which minimizes in sequence the various FT(f)'s using gradient descent techniques. When generalized to the case of self-interacting line elements, the method is flexible in introducing any kind of constraint on the configurations of the discontinuity field. The results of simulations highlight that the method improves the quality of the reconstruction when constraints on the line process are introduced, without any increase in the computational costs with respect to the case where there are no self-interactions between lines.