On the robustness of the class of stack filters

A widely held view in the nonlinear signal processing community is that the class of stack filters is robust. Although this is supported by extensive experimental evidence, no systematic theoretical justification exists, despite the availability of analytical tools for studying robustness of individual stack filters. We focus on rank selection probabilities (RSPs) as measures of robustness as it is well known that other statistical characterizations of stack filters, such as output distributions, breakdown probabilities and output distributional influence functions can be represented in terms of RSPs. We show, in a very general sense, that the class of stack filters is highly robust. It is also shown that almost all stack filters have very similar output distributions for independent and identically distributed (i.i.d.) input signals and, thus, very similar statistical behavior.

[1]  Moncef Gabbouj,et al.  On the LP which finds a MMAE stack filter , 1991, IEEE Trans. Signal Process..

[2]  Moncef Gabbouj,et al.  A training framework for stack and Boolean filtering-fast optimal design procedures and robustness case study , 1996, IEEE Trans. Image Process..

[3]  John W. Tukey,et al.  Exploratory Data Analysis. , 1979 .

[4]  Gonzalo R. Arce,et al.  Stack filters, stack smoothers, and mirrored threshold decomposition , 1999, IEEE Trans. Signal Process..

[5]  Karen O. Egiazarian,et al.  The use of sample selection probabilities for stack filter design , 2000, IEEE Signal Processing Letters.

[6]  Jaakko Astola,et al.  Block-median pyramidal transform: analysis and denoising applications , 2001, IEEE Trans. Signal Process..

[7]  Jaakko Astola,et al.  Boolean derivatives, weighted Chow parameters, and selection probabilities of stack filters , 1996, IEEE Trans. Signal Process..

[8]  Edward J. Coyle,et al.  A fast algorithm for designing stack filters , 1999, IEEE Trans. Image Process..

[9]  M. K. Prasad,et al.  Stack filters and selection probabilities , 1990, IEEE International Symposium on Circuits and Systems.

[10]  Karen O. Egiazarian,et al.  Output Distributions of Stack Filters Using Ordered Binary Decision Diagrams , 2000 .

[11]  N. Gallagher,et al.  An overview of median and stack filtering , 1992 .

[12]  Edward J. Coyle,et al.  Stack filters and the mean absolute error criterion , 1988, IEEE Trans. Acoust. Speech Signal Process..

[13]  Jaakko Astola,et al.  Optimal stack filters under rank selection and structural constraints , 1995, Signal Process..

[14]  Chin-Chuan Han,et al.  Finding of optimal stack filter by graphic searching methods , 1997, IEEE Trans. Signal Process..

[15]  Jaakko Astola,et al.  Analysis of the properties of median and weighted median filters using threshold logic and stack filter representation , 1991, IEEE Trans. Signal Process..

[16]  Koichi Yamamoto.  Note on the Order of Free Distributive Lattices , 1953 .

[17]  J. Astola,et al.  Fundamentals of Nonlinear Digital Filtering , 1997 .

[18]  Gonzalo R. Arce,et al.  A general weighted median filter structure admitting negative weights , 1998, IEEE Trans. Signal Process..

[19]  Pao-Ta Yu,et al.  Fuzzy stack filters-their definitions, fundamental properties, and application in image processing , 1996, IEEE Trans. Image Process..

[20]  Thomas Sellke,et al.  Stack Filters and Free Distributive Lattices , 1995 .

[21]  Ioannis Pitas,et al.  Nonlinear Digital Filters - Principles and Applications , 1990, The Springer International Series in Engineering and Computer Science.

[22]  J. Fitch,et al.  Median filtering by threshold decomposition , 1984 .

[23]  Алексей Дмитриевич Коршунов,et al.  Монотонные булевы функции@@@Monotone Boolean functions , 2003 .

[24]  Sari Peltonen,et al.  Output distributional influence function , 2001, IEEE Trans. Signal Process..

[25]  Ilya Shmulevich.  Properties and applications of monotone Boolean functions and stack filters , 1997 .

[26]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[27]  Gonzalo R. Arce,et al.  Output distributions of stack filters based on mirrored threshold decomposition , 2001, IEEE Trans. Signal Process..

[28]  T. Sellke,et al.  Adaptive stack filtering under the mean absolute error criterion , 1990 .

[29]  C. L. Mallows,et al.  Some Theory of Nonlinear Smoothers , 1980 .

[30]  Jaakko Astola,et al.  Output distributions of recursive stack filters , 1999, IEEE Signal Processing Letters.

[31]  Ilya Shmulevich,et al.  On the distribution of the number of monotone Boolean functions relative to the number of lower units , 2002, Discret. Math..

[32]  Edward J. Coyle,et al.  Stack filters , 1986, IEEE Trans. Acoust. Speech Signal Process..

[33]  Harold G. Longbotham,et al.  Fuzzy logic and mathematical morphology: implementation by stack filters , 1996, IEEE Trans. Signal Process..

[34]  Leonid P. Yaroslavsky,et al.  Nonlinear Filters in Image Processing , 2004 .

[35]  Moncef Gabbouj,et al.  Weighted median filters: a tutorial , 1996 .

[36]  Pertti Koivisto,et al.  Breakdown probabilities of recursive stack filters , 2001, Signal Process..