Graduated Non-Convexity by Smoothness Focusing

Noise-corrupted signals and images can be reconstructed by regularization. If discontinuities must be preserved in the reconstruction, a non-convex solution space is implied. The solution of minimum energy can be approximated by the Graduated Non-Convexity (GNC) algorithm. 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. This work provides a method of finding the convex approximation to the solution space, and the convergent series of solution spaces. The same methodology can be used on a wide range of regularization schemes. The approximation of the solution space is carried out by a scale space extension of the smoothness measure, in which a coarse-to-fine analysis can be performed. It is proven, that this scale space extension yields a convex solution space. GNC by smoothness focusing is tested against the Blake-Zisserman formulation and is shown to yield better results in most cases. Furthermore, is it pointed out that Mean Field Annealing (MFA) of the weak string does not necessarily imply GNC, but behaves in a predictable and inexpedient manner.