A training framework for stack and Boolean filtering-fast optimal design procedures and robustness case study. 
A training framework is developed in this paper to design optimal nonlinear filters for various signal and image processing tasks.  The targeted families of nonlinear filters are the Boolean filters and stack filters.  The main merit of this framework at the implementation level is perhaps the absence of constraining models, making it nearly universal in terms of application areas.  We develop fast procedures to design optimal or close to optimal filters, based on some representative training set.  Furthermore, the training framework shows explicitly the essential part of the initial specification and how it affects the resulting optimal solution.  Symmetry constraints are imposed on the data and, consequently, on the resulting optimal solutions for improved performance and ease of implementation.  The case study is dedicated to natural images.  The properties of optimal Boolean and stack filters, when the desired signal in the training set is the image of a natural scene, are analyzed.  Specifically, the effect of changing the desired signal (using various natural images) and the characteristics of the noise (the probability distribution function, the mean, and the variance) is analyzed.  Elaborate experimental conditions were selected to investigate the robustness of the optimal solutions using a sensitivity measure computed on data sets.  A remarkably low sensitivity and, consequently, a good generalization power of Boolean and stack filters are revealed.  Boolean-based filters are thus shown to be not only suitable for image restoration but also robust, making it possible to build libraries of "optimal" filters, which are suitable for a set of applications.