Abstract
Linear filtering theory has been largely motivated by the characteristics of Gaussian signals. In the same manner, the proposed Myriad Filtering methods are motivated by the need for a flexible filter class with high statistical efficiency in non-Gaussian impulsive environments that can appear in practice. Myriad filters have a solid theoretical basis, are inherently more powerful than median filters, and are very general, subsuming traditional linear FIR filters. The foundation of the proposed filtering algorithms lies in the definition of the myriad as a tunable estimator of location derived from the theory of robust statistics. We prove several fundamental properties of this estimator and show its optimality in practical impulsive models such as the α-stable and generalized-t. We then extend the myriad estimation framework to allow the use of weights. In the same way as linear FIR filters become a powerful generalization of the mean filter, filters based on running myriads reach all of their potential when a weighting scheme is utilized. We derive the normal equations for the optimal myriad filter, and introduce a suboptimal methodology for filter tuning and design. The strong potential of myriad filtering and estimation in impulsive environments is illustrated with several examples.
Highlights
A large number of filtering algorithms used in practical applications are limited to the cases of Gaussian noise and/or linear operation, presenting serious performance degradation in the presence of impulsive contamination
An important shortcoming that has hampered their use in other fields is that their output is always constrained, by definition, to one of the samples in the input window. This “selection” characteristic is very desirable in image processing applications [1], it gives efficiency losses that are unacceptable for many other practical applications
In the same way as linear and median filters are related to Gaussian and Laplacian distributions, respectively, myriad filter theory is based on the definition of the sample myriad as the maximum likelihood location estimator of the Cauchy distribution—the only non-Gaussian symmetric α-stable distribution for which a closed-form density is available
Summary
A large number of filtering algorithms used in practical applications are limited to the cases of Gaussian noise and/or linear operation, presenting serious performance degradation in the presence of impulsive contamination. We introduce a novel class of M-filters motivated by the need for a flexible filtering framework with high statistical efficiency in distribution families which, like the α-stable, can appear in engineering practice The foundation of these Myriad Filters, as we propose to call them, lies in the definition of the sample myriad as an M-estimator derived from tunable cost functions of the form ρ(x) = log k2 + x2 ,. This rich variety of operation modes is the key concept explaining important optimality properties of the myriad in the class of symmetric α-stable distributions. EURASIP Journal on Applied Signal Processing at the models that motivate the choice of the cost functions in (1)
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