complete classification of roots to one dimensional median and rank order filters the set of roots to the one dimensional median filter is completely determined let 2n 1 be the filter window width it has been shown that if a root contains a monotone segment of length n 1 then it must be locally monotone n 2 for roots with no monotone segment of length n 1 it is proved that the set of such roots is finite and that each such root is periodic the methods used are constructive so given n one can list all possible roots of this type the results developed for the median filter also apply to rank order filters

an efficient method for l filter design for a given or estimated noise type the determination of optimal l filter coefficients is computationally intensive and extremely time consuming thus even though the l filters obtained are very successful in restoring signals corrupted by non gaussian noise their on line design in real time applications is not practical in this paper a new methodology for the design of l filters is presented in which the l filter coefficients are obtained by approximating the covariance matrix of the ordered samples through the use of taylor expansion the paper also presents several examples that compare the proposed approach to the classical approach the simulations show the value of the proposed approach

decision based order statistic filters the authors propose two adaptive order statistic filters which generate outputs based on the results of hypothesis tests to reduce edge shifting of the median filter a modified median filter mmf is proposed first a hypothesis test is applied to detect the edge location then whenever an edge is detected the mmf generates the output by modifying the median output based on a comparison of median value to the current input value the mmf can retain more fine details than the median filter with the same window size as well as reduce the output mean square error which is caused mainly by edge shifting to suppress nonimpulsive noise as well as impulsive noise an l type structure is combined with the mmf the proposed filter generates the output by using asymmetric alpha trimming on the mmf output in addition to noise suppression it can enhance the gradient of blurred edges provided that the size of the window is greater than the transition width of the blurred edges experimental results for one and two dimensional signals are presented

an adaptive order statistic noise filter for gamma corrected image sequences original video signals are often corrupted by a certain amount of noise originating from the camera electronics as a result of the gamma correction in cameras the observed noise is signal dependent we present a spatio temporal order statistic os noise filter that takes into account the gamma correction in the camera the calculation of the filter coefficients requires higher order order statistics hoos of the noise process we make use of a range test rt to determine locally from which neighboring signal values an estimate should be formed the noise filter that we arrive at is adaptive and computationally efficient

error estimates for the combinedh andp versions of the finite element method in theh version of the finite element method convergence is achieved by refining the mesh while keeping the degree of the elements fixed on the other hand thep version keeps the mesh fixed and increases the degree of the elements in this paper we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh in addition it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom even in the presence of corner singularities numerical results comparing theh version p version and combinedh p version for a one dimensional problem are presented

generalization of median root prior reconstruction penalized iterative algorithms for image reconstruction in emission tomography contain conditions on which kind of images are accepted as solutions the penalty term has commonly been a function of pairwise pixel differences in the activity in a local neighborhood such that smooth images are favored attempts to ensure better edge and detail preservation involve difficult tailoring of parameter values or the penalty function itself the previously introduced median root prior mrp favors locally monotonic images mrp preserves sharp edges while reducing locally nonmonotonic noise at the same time quantitative properties of mrp are good because differences in the neighboring pixel values are not penalized as such the median is used as an estimate for a penalty reference against which the pixel value is compared when setting the penalty in order to generalize the class of mrp type of priors the standard median was replaced by other order statistic operations the l and finite impulse response median hybrid fmh filters they allow for smoother appearance as they apply linear weighting together with robust nonlinear operations the images reconstructed using the new mrp l and mrp fmh priors are visually more conventional good quantitative properties of mrp are not significantly altered by the new priors

on morphological operators based on rank filters rank order morphological operators have been designed to better cope with shape variations and noise than the corresponding operators with plain structuring elements in this paper starting from a brief historical overview of rank order filters and erosions dilations in digital image processing rank order based morphological operators are surveyed we also highlight the relations between these operators and show that many are similar if not equivalent an extensive bibliography is provided

locally monotonic regression the concept of local monotonicity appears in the study of the set of root signals of the median filter and provides a measure of the smoothness of the signal the median filter is a suboptimal smoother under this measure of smoothness since a filter pass does necessarily yield a locally monotonic output even if a locally monotonic output does result there is no guarantee that it will possess other desirable properties such as optimal similarity to the original signal locally monotonic regression is a technique for the optimal smoothing of finite length discrete real signals under such a criterion a theoretical framework in which the existence of locally monotonic regression is proved and algorithms for their computation are given regression is considered as an approximation problem in r sub n the criterion of approximation is derived from a semimetric and the approximating set is the collection of signals sharing the property of being locally monotonic

microstatistics in signal decomposition and the optimal filtering problem the author introduces and analyzes a large class of nonlinear filters which are based on signal decomposition and where the estimation in the decomposed space uses linear combinations of either the observation vector the sorted observation vector or in general a nonlinear transformation of the observation vector thus nonlinear filter response characteristics are achieved but with the machinery of linear systems theory available for their optimization and design it is shown that linear filters are a subclass of microstatistic filters and that the optimal linear filter solution is suboptimal in the decomposed signal space the filtering problem reduces to a set of filters operating on the decomposed signals the output is a weighted sum of the decomposed filtered signals the optimal interconnection structure between decomposed signals has complexity o m 1 where m is the cardinality of the decomposition the formulation is given for a radix 1 decomposition and generalized for a radix q decomposition computer simulations illustrate the performance

mathematical programming algorithms for regression based nonlinear filtering in r sup n this paper is concerned with regression under a sum of partial order constraints examples include locally monotonic piecewise monotonic runlength constrained and unimodal and oligomodal regression these are of interest not only in nonlinear filtering but also in density estimation and chromatographic analysis it is shown that under a least absolute error criterion these problems can be transformed into appropriate finite problems which can then be efficiently solved via dynamic programming techniques although the result does not carry over to least squares regression hybrid programming algorithms can be developed to solve least squares counterparts of certain problems in the class

improving gil werman algorithm for running min and max filters the current best bound on the number of comparison operations needed to compute the running maximum or minimum over a p element sliding data window is approximately three comparisons per output sample this bound is probabilistic for some algorithms and is derived for their complexities on the average for independent identically distributed i i d input signals the worst case complexities of these algorithms are o p the worst case complexity c sub r 3 4 p comparisons per output sample for 1d signals is achieved in the gil werman algorithm 1993 in this correspondence we propose a modification of the gil werman algorithm with the same worst case complexity but with a lower average complexity a theoretical analysis shows that using the proposed modification the complexities of sliding max or min 1d and 2d filters over i i d signals are reduced to c sub 1 2 5 3 5 p 1 p sup 2 and c sub 2 5 7 p 2 p sup 2 comparisons per output sample on the average respectively simulations confirm the theoretical results moreover experiments show that even for highly correlated data namely for real images the behavior of the algorithm remains the same as for i i d signals

weak continuity with structural constraints nonlinear regression and nonlinear regularization are two powerful approaches to segmentation and nonlinear filtering in this correspondence we propose a hybrid approach that effectively combines the best of both worlds and can be efficiently implemented via the viterbi algorithm

a novel pricing method for european options based on fourier cosine series expansions here we develop an option pricing method for european options based on the fourier cosine series and call it the cos method the key insight is in the close relation of the characteristic function with the series coefficients of the fourier cosine expansion of the density function in most cases the convergence rate of the cos method is exponential and the computational complexity is linear its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts we will present the method and its applications in two separate parts the first one is this paper where we deal with european options in particular in a follow up paper we will present its application to options with early exercise features

multistage order statistic filters for image sequence processing the application of multistage order statistic filters mos to the task of noise suppression in time varying imagery is studied it is shown that mos filters efficiently preserve image structures under motion without motion compensation preprocessing in particular the families of multistage median and weighted median filters are considered motion preservation and statistical smoothing measures are derived it is shown that spatiotemporal filtering allows for a significant improvement over both spatial and temporal filtering in terms of output image resolution and noise suppression

on techniques for detecting circumscribed masses in mammograms a method for detecting one type of breast tumor circumscribed masses in mammograms is presented it relies on a combination of criteria used by experts including the shape brightness contrast and uniform density of tumor areas the method uses modified median filtering to enhance mammogram images and template matching to detect the tumors in the template matching step suspicious areas are identified by thresholding the cross correlation values and a percentile method is used to determine a threshold for each film in addition two tests are used to remove false alarms from the resulting candidates the results obtained by applying these techniques to a set of test images are described they are judged encouraging

fast algorithms for training stack filters stack filters constitute a class of nonlinear digital filters possessing a weak superposition property in the paper fast algorithms for training stack filters are introduced the first algorithm is based on a technique previously developed similar to the previous algorithm this improved method requires only simple arithmetic operations increment decrement and local comparison the spl epsiv convergence property of the fast algorithm is established in addition two fast training algorithms are developed for a subclass of stack filters called weighted order statistic filters these fast algorithms are variants of the lms and rls algorithms in linear filters all three fast algorithms developed exploit the repetition property of the training input the factor of speedup is approximately the ratio between the number of quantization levels of the input and the window size

optimal estimation of deterioration from diagnostic image sequence estimation of mechanical structure damage can greatly benefit from the knowledge that the damage accumulates irreversibly over time this paper formulates a problem of estimation of a pixel wise monotonic increasing or decreasing time series of images from noisy blurred image data our formulation includes temporal monotonicity constraints and a spatial regularization penalty we cast the estimation problem as a large scale quadratic programming qp optimization and describe an efficient interior point method for solving this problem the method exploits the special structure of the qp and scales well to problems with more than a million of decision variables and constraints the proposed estimation approach performs well for simulated data we demonstrate an application of the approach to diagnostic images obtained in structural health monitoring experiments and show that it provides a good estimate of the damage accumulation trend while suppressing spatial and temporal noises

a family of nonlinear filters with data dependent coefficients a new family of data dependent nonlinear filters ddnl is presented coefficients are computed locally and the absolute distance between signal samples is used to determine a rank order dependent weighting function the new filters have very good performance regarding noise suppression and impulse rejection and at the same time provide better edge preservation than the averager and other linear smoothers

application of partition based median type filters for suppressing noise in images an adaptive median based filter is proposed for removing noise from images specifically the observed sample vector at each pixel location is classified into one of m mutually exclusive partitions each of which has a particular filtering operation the observation signal space is partitioned based an the differences defined between the current pixel value and the outputs of cwm center weighted median filters with variable center weights the estimate at each location is formed as a linear combination of the outputs of those cwm filters and the current pixel value to control the dynamic range of filter outputs a location invariance constraint is imposed upon each weighting vector the weights are optimized using the constrained lms least mean square algorithm recursive implementation of the new filter is then addressed the new technique consistently outperforms other median based filters in suppressing both random valued and fixed valued impulses and it also works satisfactorily in reducing gaussian noise as well as mixed gaussian and impulse noise

the wmmr filters a class of robust edge enhancers the authors develop a class of filters called weighted majority of m values with minimum range wmmr sup m that have the same impulse rejection properties as the median they demonstrate a subclass of these filters wmmr that may be optimized for edge enhancement in one dimension in that their output converges to the closest perfect edge one of these filters is shown to restore a class of noisy edges to the closest perfect edge on one pass applications in one and two dimensions are discussed and a two dimensional simulation is provided comparing the wmmr to other filters for smoothing and edge enhancement

a class of rank order smoothers smoothers or nonlinear filters that make estimates purely on the basis of rank and temporal order determinations among members of the input data are inherently robust and compatible with vlsi implementations the classic example is the moving median smoother but it is possible to define others which have significantly different attributes both in relation to the statistical estimation problem and in relation to hardware implications attributes offered in the latter category which are especially desirable in systolic vlsi include essentially recursive operation and elimination of the interactions between rank and temporal ordering requirements which are a feature of median smoothers experimental results indicate significantly improved response to noisy steps for these smoothers in comparison to median smoothers of comparable rectangular pulse detection capabilities

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

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

dual control monte carlo method for tight bounds of value function in regime switching utility maximization abstract in this paper we study the dual control approach for the optimal asset allocation problem in a continuous time regime switching market we find the lower and upper bounds of the value function that is a solution to a system of fully coupled nonlinear partial differential equations these bounds can be tightened with additional controls to the dual process we suggest a monte carlo algorithm for computing the tight lower and upper bounds and show the method is effective with a variety of utility functions including power non hara and yaari utilities the latter two utilities are beyond the scope of any current methods available in finding the value function

rank order operators and the mean absolute error criterion the author shows that among all randomizing stack filters whose output decisions are based on the sum of the bits in the window on each level of the threshold decomposition architecture rank order operators are optimal under an error criterion which is the sum of the absolute error incurred on each level of the threshold decomposition architecture he also shows that if the filtering class being optimized obeys a deterministic stacking property then the traditional multilevel mean absolute error criterion reduces to the sum of the absolute error incurred to each level this leads to an efficient algorithm for choosing the rank order filter which minimizes the multilevel mean absolute error criterion

morphological filters part i their set theoretic analysis and relations to linear shift invariant filters this paper examines the set theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations which involve supremum infimum operations and additions binary signals are classified as sets and multilevel signals as functions two set theoretic representations of signals are reviewed filters are classified as set processing sp or function processing fp conditions are provided for certain fp filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as sp filters the basic morphological operations of set erosion dilation opening and closing are related to minkowski set operations and are used to construct fp morphological filters emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions the latter case allows the definition of morphological convolutions and correlations toward this goal various properties of fp morphological filters are also examined linear shift invariant filters due to their translation invariance are uniquely characterized by their kernel which is a special collection of input signals increasing linear filters are represented as the supremum of erosions by their kernel functions if the filters are also discrete and have a finite extent impulse response they can be represented as the supremum of erosions only by their minimal with respect to a signal ordering kernel functions stable linear filters can be represented as the sum of at most two weighted suprema of erosions these results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal shaping strategies

robust iterative fitting of multilinear models parallel factor parafac analysis is an extension of low rank matrix decomposition to higher way arrays also referred to as tensors it decomposes a given array in a sum of multilinear terms analogous to the familiar bilinear vector outer products that appear in matrix decomposition parafac analysis generalizes and unifies common array processing models like joint diagonalization and esprit it has found numerous applications from blind multiuser detection and multidimensional harmonic retrieval to clustering and nuclear magnetic resonance the prevailing fitting algorithm in all these applications is based on alternating least squares which is optimal for gaussian noise in many cases however measurement errors are far from being gaussian in this paper we develop two iterative algorithms for the least absolute error fitting of general multilinear models the first is based on efficient interior point methods for linear programming employed in an alternating fashion the second is based on a weighted median filtering iteration which is particularly appealing from a simplicity viewpoint both are guaranteed to converge in terms of absolute error performance is illustrated by means of simulations and compared to the pertinent crame spl acute r rao bounds crbs

robust image modeling techniques with an image restoration application a robust parameter estimation algorithm for a nonsymmetric half plane nshp autoregressive model where the driving noise is a mixture of a gaussian and an outlier process is presented the convergence of the estimation algorithm is proved an algorithm to estimate parameters and original image intensity simultaneously from the impulse noise corrupted image where the model governing the image is not available is also presented the robustness of the parameter estimates is demonstrated by simulation finally an algorithm to restore realistic images is presented the entire image generally does not obey a simple image model but a small portion e g 8 8 of the image is assumed to obey an nshp model the original image is divided into windows and the robust estimation algorithm is applied for each window the restoration algorithm is tested by comparing it to traditional methods on several different images

on the statistical optimality of locally monotonic regression locally monotonic regression is a recently proposed technique for the deterministic smoothing of finite length discrete signals under the smoothing criterion of local monotonicity locally monotonic regression falls within a general framework for the processing of signals that may be characterized in three ways regressions are given by projections that are determined by semimetrics the processed signals meet shape constraints that are defined at the local level and the projections are optimal statistical estimates in the maximum likelihood sense the authors explore the relationship between the geometric and deterministic concept of projection onto generally nonconvex sets and the statistical concept of likelihood with the object of characterizing projections under the family of the p semi metrics as maximum likelihood estimates of signals contaminated with noise from a well known family of exponential densities

threshold superposition in morphological image analysis systems it is shown that four composite morphological systems namely morphological edge detection peak valley extraction skeletonization and shape size distributions obey a weak linear superposition called threshold linear superposition the output image signal or measurement from each system is shown to be the sum of outputs due to input binary images that result from thresholding the input gray level image at all levels these results are generalized to a vector space formulation e g to any finite linear combination of simple morphological systems thus many such systems processing gray level images are reduced to corresponding binary image processing systems which are easier to analyze and implement

on solenoidal high degree polynomial approximations to solutions of the stationary navier stokes equations the cell discretization algorithm a nonconforming extension of the finite element method is used to obtain approximations to the navier stokes equations for viscous steady state incompressible flow an implementation using polynomial bases is described that permits the use of approximations that exactly satisfy the solenoidal requirement a galerkin basis is constructed and starting with the solution of the associated stokes problem newton s method can be used to generate an approximate solution a unique solution exists if the viscosity is sufficiently large here an a posteriori computation is described that tests if uniqueness is likely to hold and gives an error estimate showing closeness of the approximation tests of the algorithm applied to problems in polygonal domains are described where approximations are 10th degree polynomials

the approximation of solutions of elliptic boundary value problems via the p version of the finite element method the approximation theory developed in 11 is used to determine the piecewise polynomial approximability of solutions of elliptic problems on polygonal domains in mathbb r 2 and polyhedra in mathbb r 3 from these estimates convergence orders for the p version of the finite element method applied to such problems are readily obtained the critical issue is the approximation of the singularities which occur at the nonsmooth parts of the domain boundaries it is seen that the estimates of 11 involving the weighted sobolev spaces z l s are well suited for treating such singular functions yielding directly the optimal approximation degree up to an arbitrarily small varepsilon numerical results for two problems from two dimensional linear elasticity are also presented the computations show that the predicted order of convergence is achieved even for low values of p moreover in contrast to the usual h version of the finite element method the point at which the p version enters the asym

utility maximization with discretionary stopping utility maximization problems of mixed optimal stopping control type are considered which can be solved by reduction to a family of related pure optimal stopping problems sufficient conditions for the existence of optimal strategies are provided in the context of continuous time ito process models for complete markets the mathematical tools used are those of optimal stopping theory continuous time martingales convex analysis and duality theory several examples are solved explicitly including one which demonstrates that optimal strategies need not always exist

dual control monte carlo method for tight bounds of value function under heston stochastic volatility model abstract the aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the heston stochastic volatility model with general utility functions it is well known there is a closed form solution to the hjb equation for power utility due to its homothetic property it is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the hjb equation for the heston model in this paper we propose an efficient dual control monte carlo method for computing tight lower and upper bounds of the value function we identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions including power non hara and yaari utilities finally we perform some numerical tests to see the efficiency accuracy and robustness of the method the numerical results support strongly our proposed scheme

multichannel distance filter a nonlinear multichannel digital filter is presented in this correspondence the output is a weighted sum of all samples in the filter window with a single parameter controlling the filter nonlinearity although input data ordering is not required performance can surpass the performance of other ordering based multichannel filters

deterministic properties of analog median filters two key deterministic properties of analog median filters are established the first is that root signals are characterized by a special class of locally monotone functions and the second is that the repeated application of a median filter produces a root signal which retains significant features in the original data both results are extensions of well known facts for discrete median filters in addition it is shown that these properties do not extend to a natural multidimensional version of the median filter

a property of the minimum vectors of a regularizing functional defined by means of the absolute norm we consider a regularizing functional defined by means of the l sub 1 norm where the regularization is obtained using first differences as is well known such a functional can be put in relation with recursive median filters of appropriate window length we show that at least one of the minima is reached at a vector whose components have values over the same discrete set of the given signal this suggests a simple method to refine the approximate solution to the regularization problem which can be obtained with recursive median filters of increasing order we also report an example of application where the refinement method is employed for a signal detection problem

a generalization of median filtering using linear combinations of order statistics we consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence assuming a constant signal in white noise the coefficients in the linear combination are chosen to minimize the output mse for several noise distributions it is shown that the optimal order statistic filter osf tends toward the median filter as the noise becomes more impulsive the optimal osf is applied to an actual noisy image and is shown to perform well combining properties of both the averaging and median filters a more general design scheme for applications involving nonconstant signals is also given

the treatment of nonhomogeneous dirichlet boundary conditions by thep version of the finite element method the paper addresses the problem of the implementation of nonhomogeneous essential dirichlet type boundary conditions in thep version of the finite element method

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

adaptive algorithms for non gaussian noise environments the order statistic least mean square algorithms convergence properties are studied for a class of gradient based adaptive filters known as order statistic least mean square oslms algorithms these algorithms apply an order statistic filtering operation to the gradient estimate of the standard least mean square lms algorithm the order statistic operation in oslms algorithms can reduce the variance of the gradient estimate relative to lms when operating in non gaussian noise environments a consequence is that in steady state the excess mean square error can be reduced it is shown that when the input signals are iid and symmetrically distributed the coefficient estimates for the oslms algorithms converge on average to a small area around their optimal values simulations provide supporting evidence for algorithm convergence as a measurement of performance the mean squared coefficient error of oslms algorithms has been evaluated under a range of noise distributions and os operators guidelines for selection of the os operator are presented based on the expected noise environment

the nearest neighbor median filter some deterministic properties and implementations abstract further analysis of the nearest neighbor median filter nnmf is carried out it is proved that the nnmf has the threshold decomposition property and the stacking property using these properties it is formally shown that the nnmf is a subclass of the weighted median filter wmf and the weight coefficients of the equivalent wmf are derived it is shown that the nnmf is more efficient than the equivalent wmf from the viewpoint of computational complexity two theorems about the effect of repeated operations of nnmf are proved these theorems are described in the style of dealing with local configurations of image moreover hardware implementations of the nnmf are presented

mrl filters a general class of nonlinear systems and their optimal design for image processing a class of morphological rank linear mrl filters is presented as a general nonlinear tool for image processing they consist of a linear combination between a morphological rank filter and a linear filter a gradient steepest descent method is proposed to optimally design these filters using the averaged least mean squares lms algorithm the filter design is viewed as a learning process and convergence issues are theoretically and experimentally investigated a systematic approach is proposed to overcome the problem of nondifferentiability of the nonlinear filter component and to improve the numerical robustness of the training algorithm which results in simple training equations image processing applications in system identification and image restoration are also presented illustrating the simplicity of training mrl filters and their effectiveness for image signal processing

edge sensitive image restoration using order constrained least squares methods in this paper we consider a novel technique for the restoration of noise corrupted images using order constrained least squares methods the restoration process uses a moving cross shaped filter window within which two operations are combined the first operation consists of simple hypothesis tests for the presence of an edge of some minimal height crossing the center of the window the second operation computes the window output as the order constrained least squares fit of the windowed values if an edge is deemed to be present or simply the average if no edge is present the new technique is applied to some actual noise corrupted images and the results are compared to the results of applying similarly configured median and averaging filters some computational considerations and comparisons are discussed at the end of the paper

an edge gradient enhancing adaptive order statistic filter we consider an adaptive order statistic os filter that can enhance gradients of edges while suppressing impulsive and some nonimpulsive noise components in particular comparison and selection cs filters are introduced in which the output is determined by comparing the sample mean and median values inside each window it is shown theoretically that cs filters can enhance gradients of a variety of edges including noisy blurred ones some root properties of cs filters are also shown finally experimental results are presented to illustrate the performance characteristics of these filters

lum filters a class of rank order based filters for smoothing and sharpening a new class of rank order based filters called lower upper middle lum filters is introduced the output of these filters is determined by comparing a lower and an upper order statistic to the middle sample in the filter window these filters can be designed for smoothing and sharpening or outlier rejection the level of smoothing done by the filter can range from no smoothing to that of the median filter this flexibility allows the lum filter to be designed to best balance the tradeoffs between noise smoothing and signal detail preservation lum filters for enhancing edge gradients can be designed to be insensitive to low levels of additive noise and to remove impulsive noise furthermore lum filters do not cause overshoot or undershoot some statistical and deterministic properties of the lum filters are developed and a number of experimental results are presented to illustrate the performance these experiments include applications to 1d signals and to images

on the use of lagrange multipliers in domain decomposition for solving elliptic problems the primal hybrid method for solving second order elliptic equa tions is extended from finite element approximations to general bases varia tional techniques are used to show convergence of approximations to the so lution of the homogeneous dirichlet problem for selfadjoint equations error estimates are obtained and examples are given

moving window based double haar wavelet transform for image processing image denoising is a lively research field the classical nonlinear filters used for image denoising such as median filter are based on a local analysis of the pixels within a moving window recently the research of image denoising has been focused on the wavelet domain compared to the classical nonlinear filters it is based on a global multiscale analysis of images apparently the wavelet transform can be embedded in a moving window thus a moving window based local multiscale analysis is obtained in this paper based on the haar wavelet a class of nonorthogonal multichannel filter bank with its corresponding wavelet shrinkage called lee shrinkage is derived as a special case of this filter bank the double haar wavelet transform is introduced examples show that it is suitable for a moving window based local multiscale analysis used for image denoising edge detection and edge enhancement

nonparametric combinatorial regression for shape constrained modeling we introduce a unified approach for calculating nonparametric shape constrained regression enforcement of the shape constraint often accounts for the impact of a physical phenomenon or a specific property it also improves the model s predicability and facilitates subsequent optimizations the regression models are built by transforming the problem into the combinatorial domain where the shape constraints are imposed by bounding the combinatorial search space we start by addressing isotonicity shape constraint using a dynamic programming algorithm and demonstrate how the problem can be mapped to the graph combinatorics domain next we show how a number of other important shape constraints including unimodality convexity limited level set and limited slope can be addressed using the same framework the flexibility of proposed framework enables solving the shape constrained regression problem with an arbitrary user defined error metric this flexibility is exploited to add robustness against outliers to the model the algorithms are described in detail and their computational complexity is established the performance and effectiveness of the shape constrained regression is evaluated on traces of temperature and humidity measurements from a deployed sensor network where a high degree of accuracy and robustness is demonstrated

a robust running window detector and estimator for step signals in contaminated gaussian noise an n point window is applied to noisy data to recover stepped signals in non gaussian noise robust measures of signal step level and noise distribution spread are used to detect sequential clusters of data points which are statistically significantly different thereby detecting the step using conventional analysis of variance methods but with robust parameter estimates false alarm probabilities are set reasonably accurately and miss probabilities and signal level estimates are shown by simulation to yield good results applications to kalman filtering seismic and well log data and image processing are indicated

on using the cell discretization algorithm for mixed boundary value problems and domain decomposition the cell discretization algorithm is used to approximate solutions to self adjoint elliptic equations with general nonhomogeneous dirichlet or neumann or mixed boundary values error estimates are obtained showing general convergence this provides the framework for a nonoverlapping iterative algorithm for domain decomposition the domain of a dirichlet problem is partitioned into two subdomains solutions on the two subdomains can be patched together to form a solution to the original problem provided the solutions agree across the common interface of the subdomains and the normal derivatives there have equal absolute values and are opposite in sign we generate such a solution by alternating between imposing neumann and dirichlet conditions on the interface with boundary data adapted from the results of the previous approximation a posteriori error estimates show that we have global convergence provided computed interface errors are sufficiently small

error estimates using the cell discretization method for some parabolic problems the cell discretization algorithm is used to approximate solutions to parabolic problems with coefficients independent of time error estimates are obtained showing general convergence for homogeneous problems a polynomial implementation of the algorithm is described that can be viewed as a nonconforming extension of the finite element method that can also produce the continuous approximations of an h p finite element method numerical tests that confirm the theoretical estimates are discussed

adaptive stack filtering under the mean absolute error criterion an adaptive filter algorithm is developed for the class of stack filters which is a class of nonlinear filters obeying a weak superposition property the adaptation algorithm can be interpreted as a learning algorithm for a group of decision making units the decisions of which are subject to a set of constraints called the stacking constraints under a rather weak statistical assumption on the training inputs the decision strategy adopted by the group which evolves according to the proposed learning algorithm is shown to converge to an optimal strategy in the sense that it corresponds to an optimal stack filter under the mean absolute error criterion this adaptive algorithm requires only increment decrement and comparison operations and only local interconnections between the learning units implementation of the algorithm in hardware is therefore very feasible an example is provided to show how the adaptive stack filtering algorithm can be used in an application in image processing

the effect of median filtering on edge estimation and detection in this paper we consider the effect of median prefiltering on the subsequent estimation and detection of edges in digital images where possible a quantitative statistical comparison is made for a number of filters defined with two dimensional geometries in some cases one dimensional analyses are required to illustrate certain points noise images prefiltered by median filters defined with a variety of windowing geometries are used to support the analysis and it is found that median prefiltering improves the performance of both thresholding and zero crossing based edge detectors

on approximating the solution of the non stationary stokes equations using the cell discretization algorithm the cell discretization algorithm a nonconforming extension of the finite element method is used to obtain approximations to the velocity and pressure satisfying the nonstationary stokes equations error estimates show convergence of the approximations an implementation using polynomial bases is described that permits the use of the continuous approximations of the h p finite element method and exactly satisfies the solenoidal requirement we express the error estimates in terms of the diameter h of a cell and the degree p of the approximation on each cell results of an experiment with p 10 are presented that confirm the theoretical estimates

on the convergence rate of the cell discretization algorithm for solving elliptic problems error estimates for the cell discretization algorithm are obtained for polynomial bases used to approximate both h k and analytic solutions to selfadjoint elliptic problems the polynomial implementation of this algorithm can be viewed as a nonconforming version of the h p finite element method that also can produce the continuous approximations of the h p method the examples provided by our experiments provide discontinuous approximations that have errors similar to the finite element results

an optimal nonlinear extension of linear filters based on distributed arithmetic distributed arithmetic da based implementation of linear filters relies on the linear nature of this operation and has been suggested as a multiplication free solution in this work we introduce a nonlinear extension of linear filters optimizing under mean square error criterion the memory function mf multivariate boolean function with not only binary output which is in the core of da based implementation such an extension will improve the filtering of noise which may contain non gaussian components without increasing the complexity of implementation experiments on real images have shown the superiority of the proposed filters over the optimal linear filters different versions of these filters are also considered for an impulsive noise removal faster processing and filtering using large input data windows

the viterbi optimal runlength constrained approximation nonlinear filter simple nonlinear filters are often used to enforce hard syntactic constraints while remaining close to the observation data e g in the binary case it is common practice to employ iterations of a suitable median or a one pass recursive median openclose or closeopen filter to impose a minimum symbol runlength constraint while remaining faithful to the observation unfortunately these filters are in general suboptimal motivated by this observation we pose the following optimization given a finite alphabet sequence of finite extent y y n sub n 0 sup n 1 find a sequence x x n sub n 0 sup n 1 that minimizes d x y spl sigma sub n 0 sup n 1 d sub n y n x n subject to the following x is piecewise constant of plateau run length spl ges m we show how a suitable reformulation of the problem naturally leads to a simple and efficient viterbi type optimal algorithmic solution we call the resulting nonlinear input output operator the viterbi optimal runlength constrained approximation vorca filter the method can be easily generalized to handle a variety of local syntactic constraints the vorca is optimal computationally efficient and possesses several desirable properties e g idempotence we therefore propose it as an attractive alternative to standard median stack and morphological filtering we also discuss some applications

streaking in median filtered images this paper presents a probabilistic analysis of the streaking or blotching effect commonly observed in median filtered signals in both one and two dimensions the effcts are identified as runs of equal or nearly equal values which create visual impressions that have no visual correlate for one dimensional discrete iid random signals with continuous input probability densities the probability of a streak of length l occurring is computed and shown to be independent of the input probability distribution expressions for the first and second moments of the streak length are also derived and certain asymptotic results are given as the analysis and definition of the analogous effect in two dimensions is less tractable the probability that medians taken from distinct overlapping windows will take the same value is derived for various filter geometries the analytic results are supported by examples using both one and two dimensional signals

a class of order statistic lms algorithms conventional gradient based adaptive filters as typified by the well known lms algorithm use an instantaneous estimate of the error surface gradient to update the filter coefficients such a strategy leaves the algorithm extremely vulnerable to impulsive interference a class of adaptive algorithms employing order statistic filtering of the sampled gradient estimates is presented these algorithms dubbed order statistic least mean squares oslms are designed to facilitate adaptive filter performance close to the least squares optimum across a wide range of input environments from gaussian to highly impulsive three specific oslms filters are defined the median lms the average lms and the trimmed mean lms the properties of these algorithms are investigated and the potential for improvement demonstrated finally a general adaptive oslms scheme in which the nature of the order statistic operator is also adapted in response to the statistics of the input signal is presented it is shown that this can facilitate performance gains over a wide range of input data types

an optimal portfolio model with stochastic volatility and stochastic interest rate we consider a portfolio optimization problem under stochastic volatility as well as stochastic interest rate on an infinite time horizon it is assumed that risky asset prices follow geometric brownian motion and both volatility and interest rate vary according to ergodic markov diffusion processes and are correlated with risky asset price we use an asymptotic method to obtain an optimal consumption and investment policy and find some characteristics of the policy depending upon the correlation between the underlying risky asset price and the stochastic interest rate

robust techniques for edge detection in multiplicative weibull image noise abstract recent studies of the approximately multiplicative noise present in images from coherent radar systems suggest that the first order statistics can be broadly modeled by the two parameter weibull probability density function w y y 1 exp y this paper defines and compares some novel techniques for the robust detection of sustained image irradiance changes or edges in images immersed in multiplicative weibull noise it is shown that edge detection in multiplicative noise is accomplished effectively by comparing ratios of locally adjacent image irradiance estimates to an appropriate threshold several such estimates are considered local irradiance averages optimal single order statistics and best linear unbiased estimates using order statistics the results of a statistical monte carlo performance analysis suggest that edge detection using ratios of single order statistics ros detector offers the best compromise among computational convenience edge localization and robust performance while the ratio of blues detector rob detector yields the best overall statistical and empirical noise suppression the ratio of averages detector roa detector while statistically optimal in some cases often yields inferior edge localization relative to the other schemes images corrupted by synthetically generated weibull noise are used to support the analysis

edge preserving filtering by combining nonlinear mean and median filters a very general class of nonlinear filters called mapping order statistics filters mosfs is introduced a subclass of the mosfs called l sub p mean median filters l sub p mmfs is discussed the l sub p mmf shows better properties in preserving edges and attenuating positive and negative impulse noise simultaneously than do the nonlinear mean filters in the analysis of the deterministic properties of the l sub p mmf the concept of a monotonic trend region is introduced a necessary condition for a signal to be a root signal is given this necessary condition is also valid for the known finite impulse response median hybrid filter

nonlinear smoothing filters based on rank estimates of location a class of nonlinear filters that are based on the rank estimates r estimates of location parameters in statistical theory is introduced it is shown how moving window rank filters r filters can be defined starting from rank estimates of location these filters utilize the relative ranks of the observations in each window to produce an output value the idea of rank winsorization is extended to that of averaging only observations which lie within small temporal neighborhoods this leads to a definition of the class of generalized wilcoxon gw filters which are parameterized by three parameters namely the degrees of temporal and rank winsorization and the degree of averaging the gw filters can be defined to have desirable characteristics of edge preservation detail retention and impulse rejection in addition to the property of gaussian noise smoothing performance characteristics of these filters are considered through analysis and simulation the filters show that all three well known classes of robust location estimates the l m and r estimates can be applied to nonlinear smoothing of signals

a robust approach to image enhancement based on fuzzy logic in this paper we propose a robust approach to image enhancement based on fuzzy logic that addresses the seemingly conflicting goals of image enhancement i removing impulse noise ii smoothing out nonimpulse noise and iii enhancing or preserving edges and certain other salient structures we derive three different filters for each of the above three tasks using the weighted or fuzzy least squares ls method and define the criteria for selecting each of the three filters the criteria are based on the local context and they constitute the antecedent clauses of the fuzzy rules the overall result of the fuzzy rule based system is the combination of the results of the individual filters where each result contributes to the degree that the corresponding antecedent clause is satisfied this approach gives us a powerful and flexible image enhancement paradigm results of the proposed method on several types of images are compared with those of other standard techniques

noise reduction of image sequences using motion compensation and signal decomposition in this paper a new spatio temporal filtering method for removing noise from image sequences is proposed this method combines the use of motion compensation and signal decomposition to account for the effects of object motion because of object motion image sequences are temporally nonstationary which requires the use of adaptive filters by motion compensating the sequence prior to filtering nonstationarities i e parts of the signal that are momentarily not stationary can be reduced significantly however since not all nonstationarities can be accounted for by motion a motion compensated signal still contains nonstationarities an adaptive algorithm based on order statistics is described that decomposes the motion compensated signal into a noise free nonstationary part and a noisy stationary part an rls filter is then used to filter the noise from the stationary signal our new method is experimentally compared with various noise filtering approaches from literature

fast digital locally monotonic regression locally monotonic regression is the optimal counterpart of iterated median filtering in a previous paper restrepo and bovik see ibid vol 41 no 9 p 2796 2810 1993 developed an elegant mathematical framework in which they studied locally monotonic regressions in r sup n the drawback is that the complexity of their algorithms is exponential in n we consider digital locally monotonic regressions in which the output symbols are drawn from a finite alphabet and by making a connection to viterbi decoding provide a fast o a sup 2 spl alpha n algorithm that computes any such regression where a is the size of the digital output alphabet a stands for lomo degree and n is the sample size this is linear in n and it renders the technique applicable in practice

fir median hybrid filters with excellent transient response in noisy conditions two classes of fir median hybrid fmh filters are introduced which are called in place growing fir median hybrid ipg fmh and recursive in place growing fir median hybrid ripg fmh filters this new extended family of filters exhibits a very small bias error in the vicinity of an edge an yet provides good noise attenuation it is shown that the ipg fmh filters can be expressed as weighted fmh wfmh filters

error estimates using the cell discretization method for steady state convection diffusion equations the cell discretization algorithm is applied to generate approximate solutions for some second order non self adjoint elliptic equations general convergence for homogeneous problems is shown by obtaining suitable error estimates the method is applied using polynomial bases this provides a nonconforming extension of the finite element method that can also produce the continuous approximations of an h p finite element method numerical tests on convection diffusion problems are made that confirm the theoretical estimates and methods for dealing with boundary layer problems are illustrated