Meridian Filtering for Robust Signal Processing

Abstract

A broad range of statistical processes is characterized by the generalized Gaussian statistics. For instance, the Gaussian and Laplacian probability density functions are special cases of generalized Gaussian statistics. Moreover, the linear and median filtering structures are statistically related to the maximum likelihood estimates of location under Gaussian and Laplacian statistics, respectively. In this paper, we investigate the well-established statistical relationship between Gaussian and Cauchy distributions, showing that the random variable formed as the ratio of two independent Gaussian distributed random variables is Cauchy distributed. We also note that the Cauchy distribution is a member of the generalized Cauchy distribution family. Recently proposed myriad filtering is based on the maximum likelihood estimate of location under Cauchy statistics. An analogous relationship is formed here for the Laplacian statistics, as the ratio of Laplacian statistics yields the distribution referred here to as the Meridian. Interestingly, the Meridian distribution is also a member of the generalized Cauchy family. The maximum likelihood estimate under the obtained statistics is analyzed. Motivated by the maximum likelihood estimate under meridian statistics, meridian filtering is proposed. The analysis presented here indicates that the proposed filtering structure exhibits characteristics more robust than that of median and myriad filtering structures. The statistical and deterministic properties essential to signal processing applications of the meridian filter are given. The meridian filtering structure is extended to admit real-valued weights utilizing the sign coupling approach. Finally, simulations are performed to evaluate and compare the proposed meridian filtering structure performance to those of linear, median, and myriad filtering.