MRL-filters and their adaptive optimal design for image processing

In this paper a general class of nonlinear systems, named MRL-filters, is formulated. They consist of a convex combination between a morphological/rank filter and a linear filter. A steepest descent method is then proposed to optimally design these filters, using the averaged LMS algorithm. The filter design is viewed as a learning process, and convergence issues are investigated. To overcome the problem of non-differentiability of the nonlinear filter component, and to improve the numerical robustness of the training algorithm, an alternative approach is also proposed, resulting in very simple training equations. Image processing applications in system identification and noise cancellation are finally presented. The results not only support the proposed algorithm, but also illustrate the potential of MRL-filters and their training algorithm as important tools for nonlinear signal and image processing.