Statistically-Efficient Filtering in Impulsive Environments: Weighted Myriad Filters
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tatistically-Efficient Filtering in Impulsive Environments: Weighted Myriad Filters Juan G. Gonzalez Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA Email: [email protected]
Gonzalo R. Arce Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA Email: [email protected] Received 1 August 2001 and in revised form 15 November 2001 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. Keywords and phrases: weighted myriad filters, weighted median filters, impulsive noise, heavy tails, alpha-stable distributions, Cauchy distribution, phase-locked loop.
1. INTRODUCTION 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. The need for a flexible theory of robust nonlinear filtering that can be efficiently applied in real impulsive environments has been repeatedly acknowledged in the signal processing community. Significant research efforts, especially in the field of image processing, have concentrated on finding suitable alternatives to the linear filter that are robust or resistant to the presence of impulsive noise. Among these, the approach that has received considerable attention is that of median filters. Today, due to their sound underlying theory, they are being increasingly used in image processing commercial products. 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. Although this “selection” characteristic is very desirable in image processing
applications [1], it gives efficiency losses that are unacceptable for many oth
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