a general weighted median filter structure admitting negative weights weighted median smoothers which were introduced by edgemore in the context of least absolute regression over 100 years ago have received considerable attention in signal processing during the past two decades although weighted median smoothers offer advantages over traditional linear finite impulse response fir filters it is shown in this paper that they lack the flexibility to adequately address a number of signal processing problems in fact weighted median smoothers are analogous to normalized fir linear filters constrained to have only positive weights it is also shown that much like the mean is generalized to the rich class of linear fir filters the median can be generalized to a richer class of filters admitting positive and negative weights the generalization follows naturally and is surprisingly simple in order to analyze and design this class of filters a new threshold decomposition theory admitting real valued input signals is developed the new threshold decomposition framework is then used to develop fast adaptive algorithms to optimally design the real valued filter coefficients the new weighted median filter formulation leads to significantly more powerful estimators capable of effectively addressing a number of fundamental problems in signal processing that could not adequately be addressed by prior weighted median smoother structures

quadratic weighted median filters for edge enhancement of noisy images quadratic volterra filters are effective in image sharpening applications the linear combination of polynomial terms however yields poor performance in noisy environments weighted median wm filters in contrast are well known for their outlier suppression and detail preservation properties the wm sample selection methodology is naturally extended to the quadratic sample case yielding a filter structure referred to as quadratic weighted median qwm that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers arising in the higher order statistics of environment noise through statistical analysis of higher order samples it is shown that although the parent gaussian distribution is light tailed the higher order terms exhibit heavy tailed distributions the optimal combination of terms contributing to a quadratic system i e cross and square is approached from a maximum likelihood perspective which yields the wm processing of these terms the proposed qwm filter structure is analyzed through determination of the output variance and breakdown probability the studies show that the qwm exhibits lower variance and breakdown probability indicating the robustness of the proposed structure the performance of the qwm filter is tested on constant regions edges and real images and compared to its weighted sum dual the quadratic volterra filter the simulation results show that the proposed method simultaneously suppresses the noise and enhances image details compared with the quadratic volterra sharpener the qwm filter exhibits superior qualitative and quantitative performance in noisy image sharpening

robust frequency selective filtering using weighted myriad filters admitting real valued weights weighted myriad smoothers have been proposed as a class of nonlinear filters for robust non gaussian signal processing in impulsive noise environments however weighted myriad smoothers are severely limited since their weights are restricted to be non negative this constraint makes them unusable in bandpass or highpass filtering applications that require negative filter weights further they are incapable of amplifying selected frequency components of an input signal in this paper we generalize the weighted myriad smoother to a richer structure a weighted myriad filter admitting real valued weights this involves assigning a pair of filter weights one positive and the other negative to each of the input samples equivalently the filter can be described as a weighted myriad smoother applied to a transformed set of samples that includes the original input samples as well as their negatives the weighted myriad filter is analogous to a normalized linear fir filter with real valued weights whose absolute values sum to unity by suitably scaling the output of the weighted myriad filter we extend it to yield the so called scaled weighted myriad filter which includes but is more powerful than the traditional unconstrained linear fir filter finally we derive stochastic gradient based nonlinear adaptive algorithms for the optimization of these novel myriad filters under the mean square error criterion

alpha trimmed means and their relationship to median filters suppose that x is a finite set of n numbers the trimmed mean of x is obtained by sorting x into ascending order removing trimming a fixed fraction alpha 0 leq alpha leq 0 5 from the high and low ends of the sorted set and computing the average of the remaining values when applied to a sliding window of length lw the trimming process is called trim filtering for 0 5 the trimmed mean of a set is the median of the set and the filtering operation is called median filtering repeated application of a median filter to the output of a previous median filter of the same length lw eventually produces a signal which is invariant to median filtering this final signal is called the root signal this paper explains the relationship between trimmed means and median filters derives a simple straightforward and fast algorithm for applying a median filter and provides a new explanation of the convergence of repeated median filtering to the root signal the latter result incorporates an approach which permits generalization of the associated concepts to a larger class of index map filters

adaptive weighted myriad filter algorithms for robust signal processing in spl alpha stable noise environments stochastic gradient based adaptive algorithms are developed for the optimization of weighted myriad filters wmyfs wmyfs form a class of nonlinear filters motivated by the properties of spl alpha stable distributions that have been proposed for robust non gaussian signal processing in impulsive noise environments the weighted myriad for an n long data window is described by a set of nonnegative weights w sub i sub i l sup n and the so called linearity parameter k 0 in the limit as k spl rarr spl infin the filter reduces to the familiar weighted mean filter which is a constrained linear fir filter necessary conditions are obtained for optimality of the filter weights under the mean absolute error criterion an implicit formulation of the filter output is used to find an expression for the gradient of the cost function using instantaneous gradient estimates an adaptive steepest descent algorithm is then derived to optimize the weights this algorithm involves a very simple update term that is computationally comparable to the update in the classical lms algorithm the robust performance of this adaptive algorithm is demonstrated through a computer simulation example involving lowpass filtering of a one dimensional chirp type signal in impulsive noise

affine order statistic filters medianization of linear fir filters this paper introduces a novel data adaptive filtering framework affine order statistic filters affine order statistic filters operate effectively on a wide range of signal statistics are sensitive to the dispersion of the observed data and are therefore particularly useful in the processing of nonstationary signals these properties result from the introduction of a tunable affinity function that measures the affinity or closeness of observation samples in their natural order to their corresponding order statistics the obtained affinity measures are utilized to control the influence of individual samples in the filtering process depending on the spread of the affinity function which is controlled by a single parameter spl gamma affine order statistic filters operate effectively in various environments ranging from gaussian to impulsive the class of affine order statistic filters subsumes the family of weighted order statistic wos affine filters and the class of fir affine filters we focus on a representative of the wos affine filter class the median affine filter whose behavior can be tuned from that of a linear fir filter to that of a robust median filter by narrowing the affinity function to a process referred to as medianization the superior performance of affine order statistic filters is demonstrated in two applications

recursive weighted median filters admitting negative weights and their optimization a recursive weighted median rwm filter structure admitting negative weights is introduced much like the sample median is analogous to the sample mean the proposed class of rwm filters is analogous to the class of infinite impulse response iir linear filters rwm filters provide advantages over linear iir filters offering near perfect stopband characteristics and robustness against noise unlike linear iir filters rwm filters are always stable under the bounded input bounded output criterion regardless of the values taken by the feedback filter weights rwm filters also offer a number of advantages over their nonrecursive counterparts including a significant reduction in computational complexity increased robustness to noise and the ability to model resonant or vibratory behavior a novel recursive decoupling adaptive optimization algorithm for the design of this class of recursive wm filters is also introduced several properties of rwm filters are presented and a number of simulations are included to illustrate the advantages of rwm filters over their nonrecursive counterparts and iir linear filters

weighted median filters with sigma delta modulation encoding digital decimation filters play a fundamental role in oversampled sigma delta a d decoders in this paper we first show that weighted median wm filtering of a demodulated sequence at the nyquist rate can be implemented concurrently in the a d decoder through a simple modification of the binary time series outputted by the a d modulator the sequence obtained after the sigma delta modulation sdm decoder is shown to be equivalent to wm filtering the multilevel sequence at the nyquist rate second we show that wm filters can be used for sdm decimation filters and that these filters are readily implemented in the sdm binary domain a very promising characteristic of sdm converters equipped with wm decimating filters is that sharp discontinuities edges can be preserved and acquired thus the bandlimited constraint imposed on the input signals can be relaxed making sdm more attractive to a d conversion of signals containing sharp transitions the proposed signal processing algorithms in essence combine a d sigma delta converters and wm filters into a single programmable system

hybrid polynomial filters for gaussian and non gaussian noise environments traditional polynomial filtering theory based on linear combinations of polynomial terms is able to approximate important classes of nonlinear systems the linear combination of polynomial terms however yields poor performance in environments characterized by gaussian and heavy tailed distributions weighted median and weighted myriad filters in contrast are well known for their outlier suppression and detail preservation properties it is shown here that the weighted median and weighted myriad methodologies are naturally extended to the polynomial sample case yielding hybrid filter structures that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers for both gaussian and heavy tailed distributions environments moreover the introduced hybrid polynomial filter classes are well motivated by analysis of cross and square term statistics of gaussian and heavy tailed distributions a presented asymptotic tail mass analysis shows that polynomial terms both under gaussian and heavy tailed noise statistics have heavier tails than the observed samples indicating that robust combination methods should be utilized to avoid undue influence of outliers further analysis shows weighted median processing of polynomial terms for the gaussian noise case and weighted median and weighted myriad processing of cross and square terms respectively for the heavy tailed noise case are justified from a maximum likelihood perspective filters parameter optimization procedures are also presented finally the effectiveness of hybrid filters is demonstrated through simulations that include temporal spectrum and bispectrum analysis

generalized mean median filtering for robust frequency selective applications huber proposed the pepsi family of epsi contaminated normal distributions to model environments characterized by heavy tailed distributions based on this two component mixture distribution mean median mem filters were proposed the mem filter output is a combination of the sample mean and the sample median where observation samples are weighted uniformly this property of mem filters constrains them to the class of smoothers lacking frequency selective filtering capabilities this paper extends mem filtering to the weighted sum median wsm filtering structure admitting real valued weights thereby enabling more general filtering characteristics i e bandpass and high pass filtering the proposed filter structure is also well motivated from a presented maximum likelihood ml estimate analysis under epsi contaminated statistics the ml analysis demonstrates the need for a combination of weighted sum ws and weighted median wm type filters for processing of signals corrupted by epsi contaminated noise the wsm filter is statistically analyzed through the determination of filter output variance and breakdown probability the combination parameter alpha is optimized to minimize the filter output variance which is a measure of noise attenuation capability moreover filter design procedures that yield a desired spectral response are detailed finally the proposed wsm filter structure is tested utilizing signal processing applications including low pass bandpass and high pass filtering and image processing applications including image sharpening and denoising evaluating and comparing the wsm filter performance to that of the ws wm and mem filters

twist in an exactly solvable directed lattice ribbon we investigate the transition between a twisted regime and a disordered regime in a directed ribbon model on a cubic lattice a fugacity corresponding to an interaction which models half twists in the ribbon is introduced and the interacting model is solved exactly our results suggest that conformational entropy and a local interaction which induces twist are key ingredients to model qualitatively the crossover behavior between a twisted helical regime and a denatured regime in duplex biopolymers such as dna

adaptive alpha trimmed mean filters under deviations from assumed noise model alpha trimmed mean filters are widely used for the restoration of signals and images corrupted by additive non gaussian noise they are especially preferred if the underlying noise deviates from gaussian with the impulsive noise components the key design issue of these filters is to select its only parameter spl alpha optimally for a given noise type in image restoration adaptive filters utilize the flexibility of selecting spl alpha according to some local noise statistics in the present paper we first review the existing adaptive alpha trimmed mean filter schemes we then analyze the performance of these filters when the underlying noise distribution deviates from the gaussian and does not satisfy the assumptions such as symmetry specifically the clipping effect and the mixed noise cases are analyzed we also present a new adaptive alpha trimmed filter implementation that detects the nonsymmetry points locally and applies alpha trimmed mean filter that trims out the outlier pixels such as edges or impulsive noise according to this local decision comparisons of the speed and filtering performances under deviations from symmetry and gaussian assumptions show that the proposed filter is a very good alternative to the existing schemes

a nonequilibrium entropy for dynamical systems it is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure these are chosen by considering the totality of future possibilities for observation on the system it is shown that this entropy is necessarily a nondecreasing function of the timet there is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal under suitable conditions which are stated explicitly this entropy approaches the equilibrium entropy ast and the fine grained entropy ast in particular the conditions can always be satisfied if the system is ak system as in the sinai billiard models some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition and at the same time elucidate the connection between our concept of entropy and the entropy invariant of kolmogorov and sinai

some statistical properties of alpha trimmed mean and standard type m filters the statistical properties of two classes of filters generalizing the median filter are considered the two classes are the alpha trimmed mean alpha tm filter and the standard type m stm filter both of which are special cases of the l filter results are developed to quantify the white noise suppression and edge preservation characteristics of the filters by considering their output sequence error statistics it is shown that the alpha tm and stm filters perform better than the running mean and median filters in white noise suppression while they can be designed to be comparable to the median filter in edge preservation in the presence of noise

steerable weighted median filters a filter is steerable if transformed i e rotated scaled etc versions of its impulse response can be expressed as linear combinations of a fixed set of basis functions steerability is important for numerous image processing applications however it is a property presently shared only by a specific class of linear filters on the other hand several classes of nonlinear filters such as weighted median filters wmfs may offer certain advantages over linear filters such as robustness and edge preserving capabilities in this paper the concept of steerability is extended to encompass wmfs it will be shown that in general a steerable wmf design technique needs to be capable of handling negative weights although methods that allow the design of wmfs admitting negative weights have already been proposed such methods do not necessarily produce filters that are steerable as opposed to the approach presented in this work experimental results illustrate the applicability of steerable wmfs in two applications namely edge detection and orientation analysis

weighted iterative truncated mean filter the iterative truncated arithmetic mean itm filter was proposed recently it offers a way to estimate the sample median by simple arithmetic computing instead of the time consuming data sorting in this paper a rich class of filters named weighted itm witm filters are proposed by iteratively truncating the extreme samples the output of the witm filter converges to the weighted median proper stopping criterion makes the witm filters own merits of both the weighted mean and median filters and hence outperforms the both in some applications three structures are designed to enable the witm filters being low band and high pass filters properties of these filters are presented and analyzed experimental evaluations are carried out on both synthesis and real data to verify some properties of the witm filters

spectral design of weighted median filters a general iterative approach a new design strategy for weighted median wm filters admitting real and complex valued weights is presented the algorithms are derived from mallows theory for nonlinear selection type smoothers which states that the closest linear filter to a selection type smoother in the mean square error sense is the one having as coefficients the sample selection probabilities ssps of the smoother the new design method overcomes the severe limitations of previous approaches that require the construction of high order polynomial functions and high dimensional matrices as such previous approaches could only provide solutions for filters of very small sizes the proposed method is based on a new closed form function used to derive the ssps of any wm smoother this function allows for an iterative approach to wm filter design from the spectral profile of a linear filter this method is initially applied to solve the median filter design problem in the real domain and then it is extended to the complex domain the final optimization algorithm allows the design of robust weighted median filters of arbitrary size based on linear filters having arbitrary spectral characteristics

tuning the smoothness of the recursive median filter the median filter is a special case of nonlinear filters used for smoothing signals since the output of the median filter is always one of the input samples it is conceivable that certain signals could pass through the median filter unaltered these signals define the signature of a filter and are referred to as root signals median filters are known to possess the convergence property meaning that by repeating median filtering a root signal will be found starting from any input signal by associating the nonlinear operation of median filtering with a two terms cost function an optimization process that minimizes that function is obtained cost functions of the same type are associated with different recursive median filtering schemes by replacing the actual values inside the filter s window with the original signal the convergence behavior of these filters and their smoothness are studied by changing the positions of the replacements during filtering a tuning effect of the smoothness is obtained simulation results show that the proposed filtering schemes provide improved performance over the standard recursive median filter succeeding in preserving small details and fine textures

weighted median image sharpeners for the world wide web a class of robust weighted median wm sharpening algorithms is developed in this paper unlike traditional linear sharpening methods weighted median sharpeners are shown to be less sensitive to background random noise or to image artifacts introduced by jpeg and other compression algorithms these concepts are extended to include data dependent weights under the framework of permutation weighted medians leading to tunable sharpeners that in essence are insensitive to noise and compression artifacts permutation wm sharpeners are subsequently generalized to smoother sharpener structures that can sharpen edges and image details while simultaneously filter out background random noise a statistical analysis of the various algorithms is presented theoretically validating the characteristics of the proposed sharpening structures a number of experiments are shown for the sharpening of jpeg compressed images and sharpening of images with background film grain noise these algorithms can prove useful in the enhancement of compressed or noisy images posted on the world wide web www as well as in other applications where the underlying images are unavoidably acquired with noise

optimality of the myriad filter in practical impulsive noise environments linear filtering theory has been largely motivated by the characteristics of gaussian signals in the same manner the proposed myriad filtering methods are motivated by the need for a flexible filter class with high statistical efficiency in non gaussian impulsive environments that can appear in practice we introduce several important properties of the myriad filter and prove its optimality in the family of spl alpha stable distributions

iterative truncated arithmetic mean filter and its properties the arithmetic mean and the order statistical median are two fundamental operations in signal and image processing they have their own merits and limitations in noise attenuation and image structure preservation this paper proposes an iterative algorithm that truncates the extreme values of samples in the filter window to a dynamic threshold the resulting nonlinear filter shows some merits of both the fundamental operations some dynamic truncation thresholds are proposed that guarantee the filter output starting from the mean to approach the median of the input samples as a by product this paper unveils some statistics of a finite data set as the upper bounds of the deviation of the median from the mean some stopping criteria are suggested to facilitate edge preservation and noise attenuation for both the long and short tailed distributions although the proposed iterative truncated mean itm algorithm is not aimed at the median it offers a way to estimate the median by simple arithmetic computing some properties of the itm filters are analyzed and experimentally verified on synthetic data and real images

a new class of detail preserving filters for image processing a new class of median type filters for image processing is proposed in the filters linear fir substructures are used in conjunction with the median operation the root signals and noise attenuation properties of the fir median hybrid filters are analyzed and compared to representative edge preserving filtering operations the concept of multilevel median operation is introduced to improve the detail preserving property of conventional median and the fir median hybrid filters in the multilevel filters there exists a tradeoff between noise attenuation and detail preservation the analysis and examples indicate that fir median hybrid filters preserve details better and are computationally much more efficient than the conventional median and the k nearest neighbor averaging filters

a general class of nonlinear normalized adaptive filtering algorithms the normalized least mean square nlms algorithm is an important variant of the classical lms algorithm for adaptive linear filtering it possesses many advantages over the lms algorithm including having a faster convergence and providing for an automatic time varying choice of the lms stepsize parameter that affects the stability steady state mean square error mse and convergence speed of the algorithm an auxiliary fixed step size that is often introduced in the nlms algorithm has the advantage that its stability region step size range for algorithm stability is independent of the signal statistics in this paper we generalize the nlms algorithm by deriving a class of nonlinear normalized lms type nlms type algorithms that are applicable to a wide variety of nonlinear filter structures we obtain a general nonlinear nlms type algorithm by choosing an optimal time varying step size that minimizes the next step mse at each iteration of the general nonlinear lms type algorithm as in the linear case we introduce a dimensionless auxiliary step size whose stability range is independent of the signal statistics the stability region could therefore be determined empirically for any given nonlinear filter type we present computer simulations of these algorithms for two specific nonlinear filter structures volterra filters and the previously proposed class of myriad filters these simulations indicate that the nlms type algorithms in general converge faster than their lms type counterparts

on comparing the writhe of a smooth curve to the writhe of an inscribed polygon we find bounds on the difference between the writhing numbers of a smooth curve and a polygonal curve inscribed within the proof is based on an extension of fuller s difference of writhe formula to the case of polygonal curves the results establish error bounds useful in the numerical computation of writhe in terms of bounds on the edge lengths of the polygon and the derivatives of the curve the bounds are adaptive in the sense that they improve when regions of the smooth curve with larger derivatives are approximated by shorter edges of the polygon

entanglement complexity of lattice ribbons we consider a discrete ribbon model for double stranded polymers where the ribbon is constrained to lie in a three dimensional lattice the ribbon can be open or closed and closed ribbons can be orientable or nonorientable we prove some results about the asymptotic behavior of the numbers of ribbons withn plaquettes and a theorem about the frequency of occurrence of certain patterns in these ribbons we use this to derive results about the frequency of knots in closed ribbons the linking of the boundary curves of orientable closed ribbons and the twist and writhe of ribbons we show that the centerline and boundary of a closed ribbon are both almost surely knotted in the infinite n limit for an orientable ribbon the expectation of the absolute value of the linking number of the two boundary curves increases at least as fast as n and similar results hold for the twist and writhe

generalizing ribbons and the twist of the lattice ribbon for a standard or canonicalribbon from differential geometry the topological white s theorem connects the linking number writhe and total twist of the ribbon here we provide an integral expression analog to the total twist of a canonicalribbon that connects linking number and writhe of two curves that do not necessarily form a canonical ribbon first we apply this integral expression to derive an expression for the writhe of a polygonal curve second but importantly we revisit the lattice ribbon lattice ribbons were introduced some time ago to enable simulation of physical systems modeled by double stranded polymers application of the integral expression yields an algorithm for determining the twist of the lattice ribbon an interesting relation between writhe of the center line of a lattice ribbon and its linking number follows

detail preserving ranked order based filters for image processing the theoretical analysis of multistage median filters is developed it is shown that multistage median filters are a combination of max median and min median filters since multistage median filters belong to the class of two dimensional stack filters they have threshold decomposition attributes making their theoretical analysis simple statistical threshold decomposition is applied to derive the statistical characteristics of these filters and the results are used to evaluate the performance of these two types of multistage filters finally a quantitative and qualitative comparison of the multistage filters and of other efficient detail preserving filters is presented the comparisons are made using the mean squared error and the mean absolute error criteria

switching bilateral filter with a texture noise detector for universal noise removal in this paper we propose a switching bilateral filter sbf with a texture and noise detector for universal noise removal operation was carried out in two stages detection followed by filtering for detection we propose the sorted quadrant median vector sqmv scheme which includes important features such as edge or texture information this information is utilized to allocate a reference median from sqmv which is in turn compared with a current pixel to classify it as impulse noise gaussian noise or noise free the sbf removes both gaussian and impulse noise without adding another weighting function the range filter inside the bilateral filter switches between the gaussian and impulse modes depending upon the noise classification result simulation results show that our noise detector has a high noise detection rate as well as a high classification rate for salt and pepper uniform impulse noise and mixed impulse noise unlike most other impulse noise filters the proposed sbf achieves high peak signal to noise ratio and great image quality by efficiently removing both types of mixed noise salt and pepper with uniform noise and salt and pepper with gaussian noise in addition the computational complexity of sbf is significantly less than that of other mixed noise filters

spl lscr m s filters for image restoration applications a new filtering architecture is proposed generalizing some previously introduced multilevel median filters an efficient design procedure for the new filtering architecture is demonstrated for image restoration application simulation results show a good noise rejection performance combined with a fine detail preservation capability

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

polynomial weighted median filtering this paper extends weighted median wm filters to the class of polynomial weighted median pwm filters traditional polynomial filtering theory based on linear combinations of polynomial terms is able to approximate important classes of nonlinear systems the linear combination of polynomial terms however yields poor performance in environments characterized by heavy tailed distributions weighted median filters in contrast are well known for their outlier suppression and detail preservation properties the weighted median sample selection methodology is naturally extended to the polynomial sample case yielding a filter structure that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers moreover the pwm filter class is well motivated by an analysis of cross and square term statistics a presented probability density function analysis shows that these terms have heavier tails than the observed samples indicating that robust combination methods should be utilized to avoid undue influence of outliers further analysis shows weighted median processing of polynomial terms is justified from a maximum likelihood perspective the established pwm filter class is statistically analyzed through the determination of the filter output distribution and breakdown probability filter parameter optimization procedures are also presented finally the effectiveness of pwm filters is demonstrated through simulations that include temporal spectrum and bispectrum analysis

design of linear combination of weighted medians this paper introduces a novel nonlinear filtering structure the linear combination of weighted medians lcwm the proposed filtering scheme is modeled on the structure and design procedure of the linear phase fir highpass hp filter in that the linear phase fir hp filter can be obtained by changing the sign of the filter coefficients of the fir lowpass lp filter in the odd positions the hp filter can be represented as the difference between two lp subfilters that have all positive coefficients this representation of the fir hp filter is analogous to the difference of estimates doe such as the difference of medians dom the dom is essentially a nonlinear hp filter that is commonly used in edge detection based on this observation we introduce a class of lcwm filters whose output is given by a linear combination of weighted medians of the input sequence we propose a method of designing the 1 d and 2 d lcwm filters satisfying required frequency specifications the proposed method adopts a transformation from the fir filter to the lcwm filter we show that the proposed lcwm filter can offer various frequency filtering characteristics including lp bandpass bp and hp responses

weighted median filters for multichannel signals weighted medians over multichannel signals are not uniquely defined due to its simplicity astola s vector median vm has received considerable attention particularly in image processing applications in this paper we show that the vm and its direct extension the weighted vm are limited as they do not fully utilize the cross channel correlation in fact vm treats all sub channel components independent of each other by revisiting the principles of maximum likelihood estimation of location in a multivariate signal space we propose two new and conceptually simple multichannel weighted median filters which can capture cross channel information effectively their optimal filter derivations are also presented followed by a series of simulations from color image denoising to array signal processing where the advantages of the new filtering structures are illustrated

analysis of the properties of median and weighted median filters using threshold logic and stack filter representation the deterministic properties of weighted median wm filters are analyzed threshold decomposition and the stacking property together establish a unique relationship between integer and binary domain filtering the authors present a method to find the weighted median filter which is equivalent to a stack filter defined by a positive boolean function because the cascade of wm filters can always be expressed as a single stack filter this allows expression of the cascade of wm filters as a single wm filter a direct application is the computation of the output distribution of a cascade of wm filters the same method is used to find a nonrecursive expansion of a recursive wm filter as applications of theoretical results several interesting deterministic and statistical properties of wm filters are derived

second order heavy tailed distributions and tail analysis this correspondence studies the second order distributions of heavy tail distributed random variables rvs two models for the heavy tailed distributions are considered power law and epsi contaminated distributions special cases of the models considered include 1 rvs formed by the product of two independent but not necessarily identically distributed heavy tailed rvs x and y such that z xy and 2 rvs formed through squaring a heavy tail distributed rv x such that w x2 tail analysis of the rvs their cross terms and square values shows the ordering of their tail heaviness the following results hold strictly for power law distributions and under mild conditions for epsi contaminated distributions the tail of f z x is heavier than that of fx x and the tail of fw x is heavier than that of fz x where f x denotes the probability density distribution of the corresponding random variable the heaviness of the tails indicates that robust methods of sample combination and output determination should be utilized to avoid undue influence of outliers and degradation in performance as examples the denoising and frequency selective filtering problems under the derived cross and square statistics for hyperbolic tailed and epsi contaminated models are considered simulation results indicate that the weighted myriad wmy filter outperforms the weighted median wm filter and the wm filter outperforms the weighted sum finite impulse response fir filter the results may be exploited in higher order applications of heavy tailed distributions in networking such as data traffic modeling and in nonlinear signal and image processing such as polynomial and volterra filtering