Weighted Myriad Filters - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 151

5

Weighted M yriad Filters

Gonzalo R. Arce, Juan G. Gonzalez, and Yinbo Li

CONTENTS
5.1

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Introduction 151

5.2 α -Stable Distributions ............................................... 153

5.3 Running Myriad Smoothers ......................................... 154

5.4 Optimality of the Sample Myriad in the α -Stable Model ........... 163

5.5 Weighted Myriad Smoothers ........................................ 165

5.6 Fast Weighted Myriad Computation ................................ 171

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5.7 Weighted Myriad Smoother Design 174

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5.8 Weighted Myriad Filters with Real-Valued Weights 178

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5.9 Fast Real-Valued Weighted Myriad Computation 180

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5.10 Weighted Myriad Filter Design 182

5.11 Conclusions .......................................................... 191

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References 192

5.1 Introduction

In recent years, there has been considerable interest in signal processing based
on

-stable distributions. The motivations are simple yet profound. First,
good empirical fits are often found through the use of stable distributions on
data exhibiting skewness and heavy tails. Second, there is solid theoretical
justification that non-Gaussian stable processes emerge in practice, e.g., mul-
tiple access interference in a Poisson distributed communication network, 1
Internet traffic, 2 and numerous other examples as described in Uchaikin and
Zolotarev 3 and in Feller. 4 The third argument for modeling with stable distri-
butions is perhaps the most significant and compelling. Stable distributions
satisfy an important generalization of the central limit theorem, which states
that the only possible limit of normalized sums of independent and identi-
cally distributed terms is stable. 5 A wide variety of impulsive processes found
in these applications arise as the superposition of many small independent
effects. While Gaussian models are clearly inappropriate, stable distributions

α