A nonlinear regularization approach to early vision

We propose a new class of approaches to smooth visual data while preserving significant transitions of these data as clues for segmentation. Formally, the given visual data are represented as a noisy (image) function g, and we present a class of continuously formulated global minimization problems to smooth g. The resulting function u can be characterized as the minimizer of a specific nonquadratic functional or, equivalently, as the result of an associated nonlinear diffusion process. Our approach generalizes the well-known quadratic regularization principle while retaining its attractive properties: For any given g, the solution u to the proposed minimization problem is unique and depends continuously on the data g. Furthermore, convergence of approximate solutions obtained by finite element discretization holds true. We show that the nodal variables of any chosen finite element subspace can be interpreted as computational units whose activation dynamics due to the nonlinear smoothing process evolve like a globally asymptotically stable network. A corresponding analogue implementation is thus feasible and would provide a real time processing stage for the transition preserving smoothing of visual data. Using artificial as well as real data we illustrate our approach by numerical examples. We demonstrate that solutions to our approach improve those obtained by quadratic minimization and show the influence of global parameters which allow for a continuous, scale-dependent, and selective control of the smoothing process.