ADAPTIVE RANK ORDER BASED FILTERS

This paper is devoted to the adaptive optimization of a class of nonlinear filters referred to as rank order based filter. This class includes, as special cases, classical rank order filters and morphological filters with flat structuring element. The definition highlights their common origin based on sorting operations. It is shown how to adapt the filter mask and/or the rank parameter in order to minimize a criterion such as the mean square error (MSE) or the mean absolute error (MAE). The proposed algorithms are similar to the LMS algorithm and one of their major advantages is their simplicity. The algorithms derivation is achieved by replacing the input/output relation of the filter involving sorting operations, by an implicit formulation using only sign functions. Several practical examples for one-dimensional signals are described illustrating the convergence properties and the difference between MAE and MSE optimization. Finally, the algorithms are applied to various real images, illustrating the ability of adaptive nonlinear filtering in coping with textures, in removing different types of noise and in dealing with non-stationary signals.