adaptive recovery of a chirped signal using the rls algorithm this paper studies the performance of the recursive least squares rls algorithm in the presence of a general chirped signal and additive white noise the chirped signal which is a moving average ma signal deterministically shifted in frequency at rate spl psi can be used to model a frequency shift in a received signal general expressions for the optimum wiener hopf coefficients one step recovery and estimation errors noise and lag misadjustments and the optimum adaptation constant spl beta sub opt are found in terms of the parameters of the stationary ma signal the output misadjustment is shown to be composed of a noise spl xi sub 0 m spl beta 2 and lag term spl kappa spl beta sup 2 spl psi sup 2 and the optimum adaptation constant is proportional to the chirp rate as spl psi sup 2 3 the special case of a chirped first order autoregressive ar1 process with correlation spl alpha is used to illustrate the effect the bandwidth 1 spl alpha of the chirped signal on the adaptation parameters it is shown that unlike for the chirped tone where the spl beta sub opt increases with the filter length m the adaptation constant reaches a maximum for m near the inverse of the signal correlation 1 spl alpha furthermore there is an optimum filter length for tracking the chirped signal and this length is less than 1 spl alpha

multidimensional logical transforms this correspondence reviews the logical transform and pre sents new theoretical and empirical developments novel extensions of the logical transform into spaces of three and four dimensions are de scribed using the hexahedral and octahedral tessellations these new tessellations are important in that they yield sufficiently compact neighborhoods to permit multidimensional cellular logic transforms to be carried out in real time by table lookup applications of logical transforms in these tessellations to data and image analysis are provided

directional morphological filtering we show that a translation invariant implementation of min max filters along a line segment of slope in the form of an irreducible fraction dy dx can be achieved at the cost of 2 k min max comparisons per image pixel where k max dx dy therefore for a given slope the computation time is constant and independent of the length of the line segment we then present the notion of periodic moving histogram algorithm this allows for a similar performance to be achieved in the more general case of rank filters and rank based morphological filters applications to the filtering of thin nets and computation of both granulometries and orientation fields are detailed finally two extensions are developed the first deals with the decomposition of discrete disks and arbitrarily oriented discrete rectangles while the second concerns min max filters along gray tone periodic line segments

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

minimal representations for translation invariant set mappings by mathematical morphology in his 1975 book matheron introduced a pair of dual representations written in terms of erosions and dilations for increasing translation invariant set mappings using the concept of a kernel based on hit miss topology maragos in his 1985 ph d thesis has given sufficient conditions under which the increasing mappings have minimal representations in this paper a pair of dual representations for translation invariant set mappings not necessarily increasing is presented it is shown that under the same sufficient conditions such mappings have minimal representations actually the representations of matheron and maragos are special cases of the proposed ones finally some examples are given to illustrate the theory

enhanced detectability of small objects in correlated clutter using an improved 2 d adaptive lattice algorithm two dimensional 2 d adaptive filtering is a technique that can be applied to many image processing applications this paper will focus on the development of an improved 2 d adaptive lattice algorithm 2 d al and its application to the removal of correlated clutter to enhance the detectability of small objects in images the two improvements proposed here are increased flexibility in the calculation of the reflection coefficients and a 2 d method to update the correlations used in the 2 d al algorithm the 2 d al algorithm is shown to predict correlated clutter in image data and the resulting filter is compared with an ideal wiener hopf filter the results of the clutter removal will be compared to previously published ones for a 2 d least mean square lms algorithm 2 d al is better able to predict spatially varying clutter than the 2 d lms algorithm since it converges faster to new image properties examples of these improvements are shown for a spatially varying 2 d sinusoid in white noise and simulated clouds the 2 d lms and 2 d al algorithms are also shown to enhance a mammogram image for the detection of small microcalcifications and stellate lesions

parallel processing architectures for rank order and stack filters many architectures have been proposed for rank order and stack filtering to achieve additional speedup in these structures requires the use of parallel processing techniques such as pipelining and block processing pipelining is well understood but few block architectures have been developed for rank order and stack filtering block processing is essential for additional speedup when the original architecture has reached the throughput limits caused by the underlying technology a trivial block structure simply repeats a single input single output structure to generate a multiple input multiple output structure therefore the architecture can achieve speedups equal to the number of multiple outputs or the block size however unlike linear filters the rank order and stack filter outputs are calculated using comparisons it is possible to share these comparisons within the block structure and thus substantially reduce the size of the block structure the authors introduce a systematic method for applying block processing to rank order filters and stack filters this method takes advantage of shared comparisons within the block structure to generate a block filter with shared substructures whose complexity is reduced by up to one third compared to the original filter structure times the block size furthermore block processing is important for the generation of low power designs a block structure can trade its increased speedup for a throughput equal to the original single output architecture but with a significantly lower power requirement the power reduction in the trivial block structures is limited by the power supply voltage they demonstrate how block structures with shared substructures allow them to continue decreasing the power consumption beyond the limit imposed by the supply voltage

ξ filters a new family of nonlinear digital convolution filters is described having unusual and valuable properties the response is constant phase over the entire input frequency range and in some forms no signal whatsoever is passed beyond the cutoff frequency and side lobes are identically zero by simply iterating the transform upon which a filter is based without changing the size of the convolution kernel the cutoff frequency may be varied use of filters is analyzed and demonstrated as applied to ultrahigh speed image processing using table lookup computation

adaptive lattice filter for multiple sinusoids in white noise in this paper we present an analytical study of the lattice filter for an input time series consisting of multiple sinusoids in white noise for this model the method of undetermined coefficients is applied to derive the convergent solution further expressions for the bias and misadjustment of the stochastic gradient sg lattice filter coefficients show that for inputs of moderate tonal to noise power ratios the bias is negligible and the misadjustment is comparable to that of the lms transversal filter a heuristic approach to the derivation of the filter convergence time is presented

a fuzzy morphological hit or miss transform for grey level images the translation of the hit or miss transform hmt for grey level images from the hmt for binary images is not simple initially established as a powerful tool for morphological binary image processing some generalizations for grey level images have been proposed in the literature the difficulty lies in the definition of the complement of a given grey level image in this paper after performing a detailed review of the different approaches proposed in the literature including those based on fuzzy logic we propose the definition of the hit or miss transform for grey level images in the framework of the fuzzy mathematical morphology based on fuzzy conjunctions and implications under the duality with respect to a fuzzy negation approach some theoretical properties of this operator are studied when a general fuzzy conjunction is considered in particular we prove that this fuzzy morphological hmt reduces to the binary one when it is applied to binary images after that we focus on the fuzzy morphological hmt derived from t norms and we introduce the concept of part of an image which will guide the study of the desirable properties of our operator and the interpretability of the obtained results some preliminary experimental results provide evidence of the potential of this tool to be feasible in design algorithms for detecting patterns moreover some comparisons with other hit or miss transforms proposed in the literature are performed

an efficient technique for multi frequency acoustic analysis by boundary element method a new technique for multi frequency acoustic analysis by boundary element method is proposed this new technique is based on the elimination of the frequency dependent character of the boundary element coefficient matrices by making use of the algebraic polynomials and taking out the frequency term from integrands numerical integration involved in setting up the coefficient matrices is only confined to a frequency independent part and the boundary element global coefficient matrices obtained by this technique are frequency independent the final global coefficient matrices at any frequency can be simply formed by a summation of the frequency independent global matrices and the need to compute the numerical integration at each frequency is eliminated the technique therefore seems to be especially interesting when the bulk of the computational effort is spent on the integration phase of the solution process for a multi frequency analysis the test case of sound radiation from a pulsating sphere is given to demonstrate the technique the numerical results show that this new technique can significantly save the cpu time for a multi frequency problem

stack filters the median and other rank order operators possess two properties called the threshold decomposition and the stacking properties the first is a limited superposition property which leads to a new architecture for these filters the second is an ordering property which allows an efficient vlsi implementation of the threshold decomposition architecture motivated by the success of rank order filters in a wide variety of applications and by the ease with which they can now be implemented we consider in this paper a new class of filters called stack filters they share the threshold decomposition and stacking properties of rank order filters but are otherwise unconstrained they are shown to form a very large class of easily implemented nonlinear filters which includes the rank order operators as well as all compositions of morphological operators the convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters the results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals the output distribution for these filters is also found all the stack filters of window width 3 are determined along with their convergence properties among these filters are found two which we have named asymmetric median filters they share all the properties of median filters except that they remove impulses of one sign only that is one removes only positive going edges the other removes only negative going edges while the median filter removes impulses of both signs this investigation of the properties of stack filters thus produces several new useful and easily implemented filters

an efficient boundary element algorithm for multi frequency acoustical analysis in this article a new boundary element algorithm in acoustics is proposed this new algorithm uses a so called green s function interpolation technique so that it is ideally suited to a multi frequency analysis in the numerical implementation of this new algorithm the green s function used in the boundary integral equation is interpolated by a set of shape functions numerical integration is then confined to a frequency independent part only and hence is carried out only once at the first frequency of a multi frequency run at subsequent frequencies only the nodal values of the green s function have to be re calculated for the formation of the boundary element coefficient matrices compared to the recently developed frequency interpolation technique this new algorithm requires much less disk space and can be easily extended to half space problems and problems with planes of symmetry

edge detection in machine vision using a simple l 1 norm template matching algorithm abstract template matching based on the sum of absolute errors the l 1 norm is an effective means of edge detection in certain controlled imaging environments where the form of the edges to be detected is known the algorithm employs a 1 d edge profile as the template edges are detected by computing the l 1 norm of an error vector obtained by subtracting an edge template from the image data this paper evaluates the performance of the l 1 norm template matching algorithm and draws comparisons to classical correlation matching the l 1 norm template matching algorithm has potential for integrated circuit and printed circuit board inspection and other inspection applications where the lighting and view aspect can be controlled

order configuration functions mathematical characterizations and applications to digital signal and image processing we call a function f in n variables an order configuration function if for any x1 xn such that xi1 xin we have f x1 xn xt where t is determined by the n tuple i1 in corresponding to that ordering equivalently it is a function built as a minimum of maxima or a maximum of minima well known examples are the minimum the maximum the median and more generally rank functions or the composition of rank functions such types of functions are often used in nonlinear processing of digital signals or images for example in the median or separable median filter min max filters rank filters etc in this paper we study the mathematical properties of order configuration functions and of a wider class of functions that we call order subconfiguration functions we give several characterization theorems for them we show through various examples how our concepts can be used in the design of digital signal filters or image transformations based on order configuration functions

least squares arma modeling of linear time varying systems lattice filter structures and fast rls algorithms the problem of modeling linear time varying systems is one of considerable importance in many varied and diverse applications this paper looks at ar ma and arma modeling of linear time varying systems with lattice filters there are two parts to this paper the first part develops the lattice filter structures in particular a new arma n m lattice filter structure is presented n ar order m ma order which is fully consistent with the geometric characteristics of the ar and ma lattice filter structures in that it evaluates all optimal arma i j filters of order lower than n m and each such filter is realized in terms of a fully orthogonal set of realization vectors for uncorrelated input this arma lattice filter simplifies greatly resulting in the white input arma lattice filter the second part of the paper develops new fast rls algorithms for the evaluation of the lattice filter coefficients there are two classes of algorithms presented one class is based on projection error sample averaging while the other uses input output sample averaging and some orthogonality relations this latter approach is equivalent to the exact rls formulation the algorithms of the former class are the most efficient modeling algorithms known to the authors and the algorithms of the latter class are of comparable or better efficiency than other algorithms in the same class

a unifying view for stack filter design based on graph search methods stack filters are operators that commute with the thresholding operation i e thresholding a signal applying the binary filter on each thresholded binary signals and then summing up stacking the results yields the same result as applying the multi level gray scale filter on the original signal several approaches for designing optimal stack filters from training data where optimality is characterized in terms of costs based on input output joint observations have been proposed this work considers stack filter design from training data under a general statistical framework developed in the context of morphological image operator design this framework 1 provides a common point of view for distinct design approaches being useful for comparative analysis or for emphasizing differences 2 clearly answers the issue of why binary signals from different threshold levels although following distinct distributions can be pooled together in the cost estimation process and 3 helps to show that several stack filter design approaches based on lattice diagrams search methods share a common underlying formulation

tracking properties and steady state performance of rls adaptive filter algorithms adaptive signal processing algorithms derived from ls least squares cost functions are known to converge extremely fast and have excellent capabilities to track an unknown parameter vector this paper treats analytically and experimentally the steady state operation of rls recursive least squares adaptive filters with exponential windows for stationary and nonstationary inputs a new formula for the estimation noise has been derived involving second and fourth order statistics of the filter input as well as the exponential windowing factor and filter length furthermore it is shown that the adaptation process associated with lag effects depends solely on the exponential weighting parameter in addition the calculation of the excess mean square error due to the lag for an assumed markov channel provides the necessary information about tradeoffs between speed of adaptation and steady state error it is also the basis for comparison to the simple lms algorithm in a simple case of channel identification it is shown that the lms and rls adaptive filters have the same tracking behavior finally in the last part we present new rls restart procedures applied to transversal structures for mitigating the disastrous results of the third source of noise namely finite precision arithmetic

boundary element acoustics fundamentals and computer codes fundamentals of linear acoustics the helmholtz integral equation two dimensional problems three dimensional problems the normal derivative integral equation indirect variational boundary element method in acoustics acoustic eigenvalue analysis by boundary element methods time domain three dimensional analysis extended kirchhoff integral formulations

size density spectra and their application to image classification we develop a density opening operator that is shown to satisfy the properties of an algebraic opening this density opening enables the development of a number of variants of pattern spectra which quantify the size or density information of a blob arrangement within the image in contrast to regular morphological pattern size spectra the proposed pattern spectra are spatially sensitive and robust to noise distortions a pattern size density spectrum or 2d signature was introduced and used for solving image classification tasks its application to the classification of real world medical images is illustrated

theory of matrix morphology the current concept of mathematical morphology called scalar morphology here is extended to a matrix morphology formalism a formal definition of matrix morphology on binary images is followed by a complete pictorial example of a nontrivial application binary matrix morphology is extended to gray scale matrix morphology using a traditional development general window transforms are expressed in terms of matrices an application to character recognition is given

an advanced computational method for radiation and scattering of acoustic waves in three dimensions the method proposed in this paper provides a computational method for implementing the helmholtz integral formula for acoustic radiation and scattering problems associated with arbitrary shaped three dimensional bodies in particular an isoparametric element formulation is used in which both the surface geometry and the acoustic variables on the surface of the body are represented by second order shape functions within the local coordinate system a general formula for the surface velocity potential and the exterior field is derived this result is applicable to nonsmooth bodies i e it includes the case where the surface may have a nonunique normal e g at the edge of a cube test cases are shown involving spherical cylindrical and cubical geometry

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

mathematical morphology a modern approach in image processing based on algebra and geometry mathematical morphology is a theory of image transformations and image functionals which is based on set theoretical geometrical and topological concepts the methodology is particularly useful for the analysis of the geometrical structure in an image the main goal of this paper is to give an impression of the underlying philosophy and the mathematical theories which are relevant to this field the following topics are discussed introduction to mathematical morphology generalization to complete lattices morphological filters and their construction by iteration geometrical aspects of morphology e g convexity distance geodesic operators granulometries metric dilations distance transform cost functions and extension of binary operators to grey scale images

numerically robust least squares lattice ladder algorithms with direct updating of the reflection coefficients new time recursive equations are derived for the reflection coefficients and the ladder gains in the a priori and a posteriori forms of the exact least squares ls lattice ladder filtering algorithms the numerical accuracy of the ls lattice ladder algorithms obtained by use of these new direct time update equations is analyzed and compared to the accuracy resulting from the conventional ls lattice ladder algorithms the analysis and a number of simulation results which are presented lead us to conclude that the new a priori and a posteriori forms of the ls lattice ladder algorithms yield superior performance

behavior of the partial correlation coefficients of a least squares lattice filter in the presence of a nonstationary chirp input this paper studies the performance of the a posteriori recursive least squares lattice filter in the presence of a nonstationary chirp signal the forward and backward partial correlation parcor coefficients for a wiener hopf optimal filter are shown to be complex conjugates for the general case of a nonstationary input with constant power such an optimal filter is compared to a minimum mean square error based least squares lattice adaptive filter expressions are found for the behavior of the first stage of the adaptive filter based on the least squares algorithm for the general nth stage the parcor coefficients of the previous stages are assumed to have attained wiener hopf optimal steady state the parcor coefficients of such a least squares adaptive filter are compared with the optimal coefficients for such a nonstationary input the optimal lattice fitter is seen to track a chirp input without any error and the tracking lag in such an adaptive filter is due to the least squares update procedure the expression for the least squares based parcor coefficients are found to contain two terms a decaying convergence term due to the weighted estimation procedure and a tracking component that asymptotically approaches the optimal coefficient value the rate of convergence is seen to depend inversely on the forgetting factor the tracking lag of the filter is derived as a function of the rate of nonstationarity and the forgetting factor it is shown that for a given chirp rate there is a threshold adaptation constant below which the total tracking error is negligible for forgetting factors above this threshold the error increases nonlinearly further this threshold forgetting factor decreases with increasing chirp rate simulations are presented to validate the analysis

convergence properties of an adaptive digital lattice filter convergence properties of a continuously adaptive digital lattice filter used as a linear predictor are investigated for both an unnormalized and a normalized gradient adaptation algorithm the parcor coefficient mean values and the output mean square error mse are approximated and a simple model is described which approximates these quantities as functions of time calculated curves using this model are compared with simulation results results obtained for a two stage lattice are then compared with the two stage least mean square lms transversal filter algorithm demonstrating that it is possible but unlikely for the transversal filter to converge faster than the analogous lattice filter

recursive least squares ladder estimation algorithms recursive least squares ladder estimation algorithms have attracted much attention recently because of their excellent convergence behavior and fast parameter tracking capability compared to gradient based algorithms we present some recently developed square root normalized exact least squares ladder form algorithms that have fewer storage requirements and lower computational requirements than the unnormalized ones a hilbert space approach to the derivations of magnitude normalized signal and gain recursions is presented the normalized forms are expected to have even better numerical properties than the unnormalized versions other normalized forms such as joint process estimators e g adaptive line enhancer and arma pole zero models will also be presented applications of these algorithms to fast or zero startup equalizers adaptive noise and echo cancellers non gaussian event detectors and inverse models for control problems are also mentioned

grey level hit or miss transforms part i unified theory the hit or miss transform hmt is a fundamental operation on binary images widely used since 40 years as it is not increasing its extension to grey level images is not straightforward and very few authors have considered it moreover despite its potential usefulness very few applications of the grey level hmt have been proposed until now part i of this paper developed hereafter is devoted to the description of a theory leading to a unification of the main definitions of the grey level hmt mainly proposed by ronse and soille respectively part ii will deal with the applicative potential of the grey level hmt which will be illustrated by its use for vessel segmentation from 3d angiographic data in this first part we review the previous approaches to the grey level hmt especially the supremal one of ronse and the integral one of soille the latter was defined only for flat structuring elements ses but it can be generalized to non flat ones we present a unified theory of the grey level hmt which is decomposed into two steps first a fitting associates to each point the set of grey levels for which the ses can be fitted to the image as in soille s approach this fitting step can be constrained next a valuation associates a final grey level value to each point we propose three valuations supremal as in ronse integral as in soille and binary

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

the bitonic filter linear filtering in an edge preserving morphological framework a new filter is presented which has better edge and detail preserving properties than a median noise reduction capability similar to a gaussian and is applicable to many signal and noise types it is built on a definition of signal as bitonic i e containing only one local maxima or minima within the filter range this definition is based on data ranking rather than value hence the bitonic filter comprises a combination of non linear morphological and linear operators it has no data level sensitive parameters and can locally adapt to the signal and noise levels in an image precisely preserving both smooth and discontinuous signals of any level when there is no noise but also reducing noise in other areas without creating additional artifactual noise both the basis and the performance of the filter are examined in detail and it is shown to be a significant improvement on the gaussian and median it is also compared over various noisy images to the image guided filter anisotropic diffusion non local means the grain filter and self dual forms of leveling and rank filters in terms of signal to noise the bitonic filter outperforms all these except non local means and sometimes anisotropic diffusion however it gives good visual results in all circumstances with characteristics which make it appropriate particularly for signals or images with varying noise or features at varying levels the bitonic has very few parameters does not require optimization nor prior knowledge of noise levels does not have any problems with stability and is reasonably fast to implement despite its non linearity it hence represents a very practical operation with general applicability

tracking model of an adaptive lattice filter for a linear chirp fm signal in noise the paper studies the behavior of the partial correlation parcor coefficients and the output misadjustment of the stochastic gradient adaptive lattice filter in response to a complex linear chirp fm signal in white gaussian noise analytic expressions for the optimal parcor coefficients of the filter are derived analytic as well as iterative models for a three stage filter are also derived the analytic expressions show that the tracking and convergence properties of the filter are separate phenomena simulation results also show that the spectral contents of the parcor coefficients for the stochastic gradient update algorithm consist of a stationary and a linearly swept component a single stage model is developed to explain this behavior finally output misadjustment plots for the filter show that an optimum value for the forgetting factor can be obtained to minimize the misadjustment but the value required to achieve local minimum misadjustment varies with each stage of the filter it is shown that in applications where the input has a high signal to noise ratio snr the misadjustment decreases rapidly at each successive stage thus implying that relatively short filter lengths are sufficient to provide effective tracking

recursive least squares lattice algorithms a geometrical approach several time recursive least squares algorithms have been developed in recent years in this paper a geometrical formalism is defined which utilizes a nested family of metric spaces indexed by the data time interval this approach leads to a simplified derivation of the so called recursive least squares lattice algorithms recursive in time and order in particular it is found that the resulting structure provides a single framework which encompasses an entire family of fairly complex algorithms as well as providing geometrical insight into their behavior

morphological filters part ii their relations to median order statistic and stack filters this paper extends the theory of median order statistic os and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions dilations openings closings and open closings that commute with thresholding the max min representation of os filters is introduced by showing that any median or other os filter is equal to a maximum of erosions moving local minima and also to a minimum of dilations moving local maxima thus os filters can be computed by a closed formula that involves a max min on prespecified sets of numbers and no sorting stack filters are established as the class of filters that are composed exactly of a finite number of max min operations the kernels of median os and stack filters are collections of input signals that uniquely represent these filters due to their translation invariance the max min functional definitions of these nonlinear iliters is shown to be equivalent to a maximum of erosions by minimal with respect to a signal ordering kernel elements and also to a minimum of dilations by minimal kernel elements of dual filters the representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum of products expressions of boolean functions it is also shown that median filtering and its iterations of any signal by convex 1 d windows is bounded below by openings and above by closings a signal is a root fixed point of the median iff it is a root of both an opening and a closing the open closing and clos opening yield median roots in one pass suppress impulse noise similarly to the median can discriminate between positive and negative noise impulses and are computationally less complex than the median some similar results are obtained for 2 d median filtering

adaptive recovery of a chirped sinusoid in noise ii performance of the lms algorithm the authors study the ability of the exponentially weighted recursive least square rls algorithm to track a complex chirped exponential signal buried in additive white gaussian noise power p sub n the signal is a sinusoid whose frequency is drifting at a constant rate psi lt is recovered using an m tap adaptive predictor five principal aspects of the study are presented the methodology of the analysis proof of the quasi deterministic nature of the data covariance estimate r k a new analysis of rls for an inverse system modeling problem a new analysis of rls for a deterministic time varying model for the optimum filter and an evaluation of the residual output mean square error mse resulting from the nonoptimality of the adaptive predictor the misadjustment in terms of the forgetting rate beta of the rls algorithm it is shown that the misadjustment is dominated by a lag term of order beta sup 2 and a noise term of order beta thus a value beta sub opt exists which yields a minimum misadjustment it is proved that beta sub opt m 1 rho psi sup 2 sup 1 3 and the minimum misadjustment is equal to 3 4 p sub n m 1 beta sub opt where rho is the input signal to noise ratio snr

template matching based on a grayscale hit or miss transform this paper introduces a class of rank order based template matching criteria that are multiplier free and independent of the dc variations of the image the core component of these criteria is a grayscale morphological hit or miss transform ghmt experimental results show that the ghmt features sharp and robust indications in the presence of gaussian noise the idea of the ghmt is used to develop more general forms of matching criteria that are robust to both gaussian and impulsive noises

an efficient indirect boundary element technique for multi frequency acoustic analysis an efficient indirect boundary element solution procedure for the analysis of multi frequency acoustic problems is developed by incorporating techniques that improve the efficiency of the integration and matrix solution phases of the computing process the integration phase is made efficient by computing the system matrices at few predetermined key frequencies only and then evaluating the matrices at other intermediate frequencies by quadratic interpolation the matrix solution process is made efficient by iterating the solutions using the factored form of the key frequency matrices the effectiveness of the present development is confirmed by solving a number of example problems copyright 1999 john wiley sons ltd