Noise-corrupted signals and images can be reconstructed by minimization of a Hamiltonian. Often the Hamiltonian is a non-convex functional. The solution of minimum energy can then be approximated by the Graduated Non-Convexity (GNC) algorithm developed for the weak membrane by Blake and Zisserman. The GNC approximates the non-convex solution space by a convex solution space, and varies the solution space slowly towards the non-convex solution space. Earlier work used Mean Field Annealing (MFA) to relax the Hamiltonian in the general case. It is often claimed that MFA leads to a GNC algorithm. It is shown that this is not necessarily the case, and especially not the case for earlier MF approximations of the weak membrane. In the case of the weak membrane, MFA might lead to predictable and inexpedient results. Two automatic and proved GNC-generating methods are presented. One is using a Gaussian filtering of the smoothness term and is called Smoothness Focusing (SF). The other is using a Gaussian filtering of the a priori distribution of the derivative in a Maximum A Posteriori estimation scheme, and is called Probability Focusing (PF). The algorithms are experimentally compared to the Blake-Zisserman GNC and shown competitive.