Total variation based image restoration with free local constraints

A new total variation based approach was developed by Rudin, Osher and Fatemi (see Physica D., vol.60, p.259, 1992) to overcome the basic limitations of all smooth regularization algorithms. The TV-based technique use the L¹ norm of the magnitude of a gradient, thus making discontinuous and nonsmooth solutions possible. In TV image restoration, the solution is obtained by solving a time-dependent, nonlinear PDE on a manifold that satisfies the degradation constraints. In practical applications, one assumes a space-varying blurring kernel and signal-dependent (e.g. multiplicative) noise. The evolution part of the TV-based PDE turned out to be related to the curve shortening equation, but scaled by an inverse |grad|. However, even with the degradation constraints enforced, restoration may lose too much valuable, singular information. Most notably, adjacent features are still merged and all geometrical features (such as level sets and edges) are smoothed out. This is not surprising, since the mean curvature evolution of level sets, even scaled as in the TV method, is essentially linear dissipation in the tangential direction. Linearity is at the root of the problem. We seek a solution by nonlinearly minimizing oscillations. But this time, rather than focusing on the oscillations of an image intensity (level sets) we bound the oscillation of other quantities along the “feature curves” and enforce constraints inside the “feature regions”. The feature curves and regions may be formed by level sets and their interior or by a closed edge and a corresponding region