Circuits and Systems Exposition Weighted Median Filters : A Tutorial

Weighted Median (WM) filters have attracted a growing number of interest in the past few years. They inherent the robustness and edge preserving capability of the classical median filter and resemble linear FIR filters in certain properties. Furthermore, WM filters belong to the broad class of nonlinear filters called stack filters. This enables the use of the tools developed for the latter class in characterizing and analyzing the behavior and properties of WM filters, e.g. noise attenuation capability. The fact that WM filters are threshold functions allows the use of neural network training methods to obtain adaptive WM filters. In this tutorial paper we trace the development of the theory of WM filtering from its beginnings in the median filter to the recently developed theory of optimal weighted median filtering. The following one and multidimensional applications are presented in this paper: idempotent weighted median filters for speech processing, adaptive weighted median and optimal weighted median filters for image and image sequence restoration, weighted medians as robust predictors in DPCM coding and Quincunx coding, and weighted median filters in scan rate conversion in normal TV and HDTV systems.

[1]  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..

[2]  Moncef Gabbouj,et al.  Deterministic Properties of Center Weighted Median Filters , 1992 .

[3]  Francesco Palmieri,et al.  Hybrid order statistic filters for adaptive image restoration , 1990, Other Conferences.

[4]  Francesco Palmieri.  A backpropagation algorithm for multilayer hybrid order statistic filters , 1989, International Conference on Acoustics, Speech, and Signal Processing,.

[5]  Alan C. Bovik,et al.  Streaking in median filtered images , 1987, IEEE Trans. Acoust. Speech Signal Process..

[6]  J. Andrew Bangham,et al.  Properties of a Series of Nested Median Filters, Namely the Data Sieve , 1993, IEEE Trans. Signal Process..

[7]  D. R. K. Brownrigg,et al.  The weighted median filter , 1984, CACM.

[8]  Moncef Gabbouj,et al.  Median Based Idempotent filters , 1991, J. Circuits Syst. Comput..

[9]  Iickho Song,et al.  Some root properties of recursive weighted median filters , 1991, Signal Process..

[10]  Moncef Gabbouj,et al.  Real Domain Adaptive WOS Filtering using Neural Network Approximations , 1993, IEEE Winter Workshop on Nonlinear Digital Signal Processing.

[11]  Kenneth E. Barner,et al.  Permutation filters: a class of nonlinear filters based on set permutations , 1994, IEEE Trans. Signal Process..

[12]  Sonia Porta,et al.  Weighted Median Filtering StrikingAnalogies to FIR Filters , 1996 .

[13]  Yong Hoon Lee,et al.  Threshold Boolean filters , 1994, IEEE Trans. Signal Process..

[14]  Moncef Gabbouj,et al.  Optimal stack filtering and the estimation and structural approaches to image processing , 1989, Sixth Multidimensional Signal Processing Workshop,.

[15]  Jaakko Astola,et al.  Adaptive multistage weighted order statistic filters based on the backpropagation algorithm , 1994, IEEE Trans. Signal Process..

[16]  T. S. Huang,et al.  Image Sequence Enhancement , 1981 .

[17]  Jaakko Astola,et al.  Adaptive Stack Filtering with Application to Image Processin , 1993, IEEE Trans. Signal Process..

[18]  Moncef Gabbouj,et al.  The Quality of Edge Preservation by Non-Linear Filters , 1992 .

[19]  Kenneth J. Kerpez.  Minimum mean squared error impulse noise estimation and cancellation , 1995, IEEE Trans. Signal Process..

[20]  Pao-Ta Yu,et al.  The classification and associative memory capability of stack filters , 1992, IEEE Trans. Signal Process..

[21]  Eric Goles Ch.,et al.  Decreasing energy functions as a tool for studying threshold networks , 1985, Discret. Appl. Math..

[22]  Sanford Eisenhandler,et al.  Adaptive Median Filtering for Impulse Noise Elimination in Real-Time TV Signals , 1987, IEEE Trans. Commun..

[23]  Yrjö Neuvo,et al.  Improving TV picture quality with linear-median type operations , 1988 .

[24]  Yrjö Neuvo,et al.  Adaptive DPCM With Median Type Predictors , 1991 .

[25]  Moncef Gabbouj,et al.  Minimum Mean Absolute Error Stack Filtering with Structural Constraints and Goals , 1990 .

[26]  Yrjö Neuvo,et al.  Optimal parallel stack filtering under the mean absolute error criterion , 1994, IEEE Trans. Image Process..

[27]  G. Arce,et al.  State description for the root-signal set of median filters , 1982 .

[28]  Parametric analysis of weighted order statistics filters , 1994, IEEE Signal Processing Letters.

[29]  Edward J. Coyle,et al.  Some convergence properties of median filters , 1986 .

[30]  Ioannis Pitas,et al.  Multichannel L filters based on marginal data ordering , 1994, IEEE Trans. Signal Process..

[31]  Jaakko Astola,et al.  An Introduction to Nonlinear Image Processing , 1994 .

[32]  Moncef Gabbouj,et al.  2-D Optimal Weighted Median Filters , 1992 .

[33]  Moncef Gabbouj,et al.  Center weighted median filters: Some properties and their applications in image processing , 1994, Signal Process..

[34]  Petros Maragos,et al.  Morphological filters-Part I: Their set-theoretic analysis and relations to linear shift-invariant filters , 1987, IEEE Trans. Acoust. Speech Signal Process..

[35]  Wei-Hsiang Liao,et al.  Weighted order statistics filters-their classification, some properties, and conversion algorithm , 1994, IEEE Trans. Signal Process..

[36]  Loan Tabu.  Training and model based approaches for optimal stack and boolean filtering with applications in image processing , 1995, Signal Process..

[37]  P. Pirsch.  Design of a DPCM codec for VLSI realization in CMOS technology , 1985, Proceedings of the IEEE.

[38]  B I Justusson,et al.  Median Filtering: Statistical Properties , 1981 .

[39]  Bernard Widrow,et al.  Layered neural nets for pattern recognition , 1988, IEEE Trans. Acoust. Speech Signal Process..

[40]  Yong Hoon Lee,et al.  Design of weighted order statistic filters using the perceptron algorithm , 1993, IEEE Winter Workshop on Nonlinear Digital Signal Processing.

[41]  B. M. Fulk.  MATH , 1992 .

[42]  Lin Yin,et al.  Stack filter design: a structural approach , 1995, IEEE Trans. Signal Process..

[43]  D. Brownrigg.  Generation of representative members of an RrSst weighted median filer class , 1986 .

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

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

[46]  Pao-Ta Yu,et al.  Convergence behavior and root signal sets of stack filters , 1992 .

[47]  Gonzalo R. Arce,et al.  BTC Image Coding Using Median Filter Roots , 1983, IEEE Trans. Commun..

[48]  J. A. Bangham,et al.  Multiscale median and morphological filters used for 2D pattern recognition , 1993 .

[49]  G. Eichmann,et al.  Vector median filters , 1987 .

[50]  Peter Pirsch.  A new predictor design for DPCM coding of TV signals , 1980 .

[51]  Ji-Sang Yoo,et al.  Order Statistic-Based Nonlinear Filters: Stack Filters and Weighted Median Filters , 1993, IEEE Winter Workshop on Nonlinear Digital Signal Processing.

[52]  Gonzalo R. Arce,et al.  Detail-preserving ranked-order based filters for image processing , 1989, IEEE Trans. Acoust. Speech Signal Process..

[53]  Yrjö Neuvo,et al.  FIR-median hybrid filters , 1987, IEEE Trans. Acoust. Speech Signal Process..

[54]  Jaakko Astola,et al.  Optimal generalized weighted-order-statistic filters , 1991, Other Conferences.

[55]  Moncef Gabbouj,et al.  On root structures and convergence properties of weighted median filters , 1995 .

[56]  Pao-Ta Yu,et al.  Convergence behavior and convergence rates of stack filters , 1991, 1991., IEEE International Sympoisum on Circuits and Systems.

[57]  Saburo Muroga,et al.  Threshold logic and its applications , 1971 .

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

[59]  Moncef Gabbouj,et al.  Root-signal sets of morphological filters and their use in variable-length BTC image coding , 2004, Journal of Mathematical Imaging and Vision.

[60]  Edward R. Dougherty,et al.  Morphological methods in image and signal processing , 1988 .

[61]  Bing Zeng,et al.  Optimal median-type filtering under structural constraints , 1995, IEEE Trans. Image Process..

[62]  M. K. Prasad,et al.  Weighted median filters: generation and properties , 1989, IEEE International Symposium on Circuits and Systems,.

[63]  Risto Wichman,et al.  Matched median detectors for multiple-access channels , 1994, Signal Process..

[64]  T. Loupas,et al.  An adaptive weighted median filter for speckle suppression in medical ultrasonic images , 1989 .

[65]  Yrjö Neuvo,et al.  Vector median operation for color image processing , 1990, Other Conferences.

[66]  Jaakko Astola,et al.  Matched median filtering , 1992, IEEE Trans. Commun..

[67]  Pao-Ta Yu,et al.  The convergence behavior of symmetric weighted median filters , 1993, IEEE Winter Workshop on Nonlinear Digital Signal Processing.

[68]  Jaakko Astola,et al.  Performance of Detail-Preserving Weighted Median Filters for Image Processing , 1993, IEEE Winter Workshop on Nonlinear Digital Signal Processing.

[69]  Yrjö Neuvo,et al.  Motion adaptive scan rate up-conversion , 1992, Multidimens. Syst. Signal Process..

[70]  Moncef Gabbouj,et al.  Fast algorithms for analyzing and designing weighted median filters , 1995, Signal Process..

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

[72]  S. Thomas Alexander,et al.  Adaptive Signal Processing , 1986, Texts and Monographs in Computer Science.

[73]  Francesco Palmieri,et al.  Ll-filters-a new class of order statistic filters , 1989, IEEE Trans. Acoust. Speech Signal Process..

[74]  G. Wise,et al.  A theoretical analysis of the properties of median filters , 1981 .

[75]  John W. Tukey,et al.  Nonlinear (nonsuperposable) methods for smoothing data , 1974 .

[76]  Moncef Gabbouj,et al.  Root properties of morphological filters , 1993, Signal Process..

[77]  Saleem A. Kassam,et al.  A structure for adaptive order statistics filtering , 1994, IEEE Trans. Image Process..

[78]  M. Gabbouj,et al.  A unified design method for rank order, stack, and generalized stack filters based on classical Bayes decision , 1991 .

[79]  Jaakko Astola,et al.  Optimal weighted median filtering under structural constraints , 1995, IEEE Trans. Signal Process..

[80]  Yrjö Neuvo,et al.  Video signal processing using vector median , 1990, Other Conferences.

[81]  Edward J. Coyle,et al.  Adaptive stack filtering under the mean absolute error criterion , 1990, Other Conferences.

[82]  Edward J. Coyle,et al.  From median filters to optimal stack filtering , 1991, 1991., IEEE International Sympoisum on Circuits and Systems.

[83]  Yrjö Neuvo,et al.  Fast adaptation and performance characteristics of FIR-WOS hybrid filters , 1994, IEEE Trans. Signal Process..

[84]  Thomas S. Huang,et al.  The Effect of Median Filtering on Edge Estimation and Detection , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[85]  N. Jayant,et al.  Digital Coding of Waveforms: Principles and Applications to Speech and Video , 1990 .

[86]  Edward J. Coyle,et al.  Minimum mean absolute error estimation over the class of generalized stack filters , 1990, IEEE Trans. Acoust. Speech Signal Process..

[87]  Josef Kittler,et al.  Pattern recognition : a statistical approach , 1982 .

[88]  Kenneth E. Barner,et al.  Rank conditioned rank selection filters for signal restoration , 1994, IEEE Trans. Image Process..

[89]  T. Viero,et al.  3-D Median Structures for Image Sequence Filtering and Coding , 1993 .

[90]  T. Viero,et al.  Non-moving regions preserving median filters for image sequence filtering , 1991, IEEE 1991 International Conference on Systems Engineering.

[91]  Peter D. Wendt.  Nonrecursive and recursive stack filters and their filtering behavior , 1990, IEEE Trans. Acoust. Speech Signal Process..

[92]  Yong Hoon Lee,et al.  Stack filters and selection probabilities , 1994, IEEE Trans. Signal Process..

[93]  T. Nodes,et al.  Median filters: Some modifications and their properties , 1982 .

[94]  E. Gilbert.  Lattice Theoretic Properties of Frontal Switching Functions , 1954 .

[95]  Jaakko Astola,et al.  Optimal weighted order statistic filters under the mean absolute error criterion , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.