Optimal median-type filtering under structural constraints

This paper presents a method for the design of median-type filters that achieve the maximum noise attenuation under structural constraints imposed by the requirement of preserving certain signal or image features. As compared with the design of optimal weighted median (WM) filters that calls for solving a set of linear inequalities, this method is extremely simple yet general enough. The filter is obtained by modifying directly the Boolean function of a median filter, and an analytic, closed-form representation of its Boolean function can be obtained. Furthermore, it is proven theoretically that under the same set of structural constraints, the filter designed in this way will never do worse in removing i.i.d. noise with any distribution than the optimal WM filter-improvements as high as 41, 46, and 52% have been achieved in simulations in a 2-D case for the uniform, Gaussian, and double-exponential distributions, respectively. Also, the improvement in the impulsive noise environment is very significant, as is demonstrated by an image enhancement application.<
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