Nonlinear Signal Processing: A Statistical Approach -- Preface -- Acknowledgments -- Contents -- Acronyms -- 1 Introduction -- 1.1 NonGaussian Random Processes -- 1.1.1 Generalized Gaussian Distributions and Weighted Medians -- 1.1.2 Stable Distributions and Weighted Myriads -- 1.2 Statistical Foundations -- 1.3 The Filtering Problem -- 1.3.1 Moment Theory -- Part I Statistical Foundations -- 2 NonGaussian Models -- 2.1 Generalized Gaussian Distributions -- 2.2 Stable Distributions -- 2.2.1 Definitions -- 2.2.2 Symmetric Stable Distributions -- 2.2.3 Generalized Central Limit Theorem -- 2.2.4 Simulation of Stable Sequences -- 2.3 Lower-Order Moments -- 2.3.1 Fractional Lower-Order Moments -- 2.3.2 Zero-Order Statistics -- 2.3.3 Parameter Estimation of Stable Distributions -- Problems -- 3 Order Statistics -- 3.1 Distributions Of Order Statistics -- 3.2 Moments Of Order Statistics -- 3.2.1 Order Statistics From Uniform Distributions -- 3.2.2 Recurrence Relations -- 3.3 Order Statistics Containing Outliers -- 3.4 Joint Statistics Of Ordered And NonOrdered Samples -- Problems -- 4 Statistical Foundations of Filtering -- 4.1 Properties of Estimators -- 4.2 Maximum Likelihood Estimation -- 4.3 Robust Estimation -- Problems -- Part II Signal Processing with Order Statistics -- 5 Median and Weighted Median Smoothers -- 5.1 Running Median Smoothers -- 5.1.1 Statistical Properties -- 5.1.2 Root Signals (Fixed Points) -- 5.2 Weighted Median Smoothers -- 5.2.1 The Center-Weighted Median Smoother -- 5.2.2 Permutation-Weighted Median Smoothers -- 5.3 Threshold Decomposition Representation -- 5.3.1 Stack Smoothers -- 5.4 Weighted Medians in Least Absolute Deviation (LAD) Regression -- 5.4.1 Foundation and Cost Functions -- 5.4.2 LAD Regression with Weighted Medians -- 5.4.3 Simulation -- Problems -- 6 Weighted Median Filters.

6.1 Weighted Median Filters With Real-Valued Weights -- 6.1.1 Permutation-Weighted Median Filters -- 6.2 Spectral Design of Weighted Median Filters -- 6.2.1 Median Smoothers and Sample Selection Probabilities -- 6.2.2 SSPs for Weighted Median Smoothers -- 6.2.3 Synthesis of WM Smoothers -- 6.2.4 General Iterative Solution -- 6.2.5 Spectral Design of Weighted Median Filters Admitting Real-Valued Weights -- 6.3 The Optimal Weighted Median Filtering Problem -- 6.3.1 Threshold Decomposition For Real-Valued Signals -- 6.3.2 The Least Mean Absolute (LMA) Algorithm -- 6.4 Recursive Weighted Median Filters -- 6.4.1 Threshold Decomposition Representation of Recursive WM Filters -- 6.4.2 Optimal Recursive Weighted Median Filtering -- 6.5 Mirrored Threshold Decomposition and Stack Filters -- 6.5.1 Stack Filters -- 6.5.2 Stack Filter Representation of Recursive WM Filters -- 6.6 Complex-Valued Weighted Median Filters -- 6.6.1 Phase-Coupled Complex WM Filter -- 6.6.2 Marginal Phase-Coupled Complex WM Filter -- 6.6.3 Complex threshold decomposition -- 6.6.4 Optimal Marginal Phase-Coupled Complex WM -- 6.6.5 Spectral Design of Complex-Valued Weighted Medians -- 6.7 Weighted Median Filters for Multichannel Signals -- 6.7.1 Marginal WM filter -- 6.7.2 Vector WM filter -- 6.7.3 Weighted Multichannel Median Filtering Structures -- 6.7.4 Filter Optimization -- Problems -- 7 Linear Combination of Order Statistics -- 7.1 L-Estimates of Location -- 7.2 L-Smoothers -- 7.3 Ll-Filters -- 7.3.1 Design and Optimization of Ll filters -- 7.4 Ljl Permutation Filters -- 7.5 Hybrid Media/Linear FIR Filters -- 7.5.1 Median and FIR Affinity Trimming -- 7.6 Linear Combination of Weighted Medians -- 7.6.1 LCWM Filters -- 7.6.2 Design of LCWM filters -- 7.6.3 Symmetric LCWM Filters -- Problems -- Part III Signal Processing with the Stable Model -- 8 Myriad Smoothers -- 8.1 FLOM Smoothers.

8.2 Running Myriad Smoothers -- 8.3 Optimality of the Sample Myriad -- 8.4 Weighted Myriad Smoothers -- 8.5 Fast Weighted Myriad Computation -- 8.6 Weighted Myriad Smoother Design -- 8.6.1 Center-Weighted Myriads for Image Denoising -- 8.6.2 Myriadization -- Problems -- 9 Weighted Myriad Filters -- 9.1 Weighted Myriad Filters With Real-Valued Weights -- 9.2 Fast Real-valued Weighted Myriad Computation -- 9.3 Weighted Myriad Filter Design -- 9.3.1 Myriadization -- 9.3.2 Optimization -- Problems -- References -- Appendix A Software Guide -- Index.