transient properties of a bistable kinetic model with correlations between additive and multiplicative noises mean first passage time the transient properties of a bistable system with correlations between additive and multiplicative noise terms are investigated the explicit expressions of the mean first passage time mfpt are obtained the numerical computations show that the mfpt of the system is affected by ensuremath lambda the strength of correlations between additive intensity ensuremath alpha and multiplicative intensity d noise terms for the case of perfectly correlated noises ensuremath lambda 1 the mfpt corresponding to ensuremath alpha ensuremath gtrsim d and ensuremath alpha d exhibits very different behaviors and the mfpt for ensuremath alpha d diverges to infinity textcopyright 1996 the american physical society

the geometry of basis sets for morphologic closing p maragos 1989 provided a framework for the decomposition of many morphologic operations into orthogonal components or basis sets using this framework a method to find the minimal basis set for the important operation of closing in two dimensions is described the closing basis sets are special because their elements are members of an ordered global set of closing shapes or primitives the selection or design of appropriate individual or multiple structuring elements for image filtering can be better understood and sometimes implemented more easily through consideration of the orthogonal closing decomposition partial closing of images using ordered fractions of a closing basis set may give a finer texture or roughness measure than that obtained from the conventional use of scaled sets of shapes such as the disc the connection between elements of the basis set for closing and the complete minimal representation of arbitrary logic functions is analyzed from a geometric viewpoint

optimal structuring elements for the morphological pattern restoration of binary images in this paper we derive the optimal structuring elements of morphological filters in image restoration the expected pattern transformation of random sets is presented an estimation theory framework for random sets is subsequently proposed this framework is based on the least mean difference lmd estimator the lmd estimator is defined to minimize the cardinality of the expected pattern transformation of the set difference of the parameter and the estimate several important results for the determination of the lmd estimator are derived the lmd structuring elements of morphological filters in image restoration are finally derived

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

theoretical foundations of spatially variant mathematical morphology part i binary images we develop a general theory of spatially variant sv mathematical morphology for binary images in the euclidean space the basic sv morphological operators that is sv erosion sv dilation sv opening and sv closing are defined we demonstrate the ubiquity of sv morphological operators by providing an sv kernel representation of increasing operators the latter representation is a generalization of matheron s representation theorem of increasing and translation invariant operators the sv kernel representation is redundant in the sense that a smaller subset of the sv kernel is sufficient for the representation of increasing operators we provide sufficient conditions for the existence of the basis representation in terms of upper semicontinuity in the hit or miss topology the latter basis representation is a generalization of maragos basis representation for increasing and translation invariant operators moreover we investigate the upper semicontinuity property of the basic sv morphological operators several examples are used to demonstrate that the theory of spatially variant mathematical morphology provides a general framework for the unification of various morphological schemes based on spatially variant geometrical structuring elements for example circular affine and motion morphology simulation results illustrate the theory of the proposed spatially variant morphological framework and show its potential power in various image processing applications

map estimation of finite gray scale digital images corrupted by supremum infimum noise finite gray scale digital images are modeled as realizations of discrete random functions drf and then the estimation of realizations of drf corrupted by a supremum infimum noise model is considered it is proved that morphological operators such as openings closings supremum of openings and infimum of closings are optimal maximum a posteriori map estimators under an appropriate and minimal set of assumptions relating to the structural and statistical constraints on image drf and noise drf these results are obtained for independent identically distributed i i d noise for single and multiframe observation scenarios next the assumption of i i d noise is relaxed and the map optimality and strong consistency of morphological filters for filtering image drf degraded by morphologically smooth noise i e colored noise is proved simulations on actual image data are carried out in support of the validity of theoretical results presented

morphological representation of order statistics filters we propose a comprehensive theory for the morphological bounds on order statistics filters and their repeated iterations conditions are derived for morphological openings and closings to serve as bounds lower and upper respectively on order statistics filters and their repeated iterations under various assumptions morphological open closings and close openings are also shown to serve as tighter bounds lower and upper respectively on iterations of order statistics filters simulations of the application of the results presented to image restoration are finally provided

effect of non gaussian noise sources in a noise induced transition here we study a noise induced transition when the system is driven by a noise source taken as colored and non gaussian we show using both a theoretical approximation and numerical simulations that there is a shift of the transition as the noise departs from the gaussian behavior also we confirm the reentrance effect found for colored gaussian noise and show the behavior of the transition line in the phase like diagram as the noise departs from gaussianity in the large correlation time limit

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

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

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

effects of time delay on symmetric two species competition subject to noise noise and time delay act simultaneously on real ecological systems the lotka volterra model of symmetric two species competition with noise and time delay was investigated in this paper by means of stochastic simulation we find that i the time delay induces the densities of the two species to periodically oscillate synchronously ii the stationary probability distribution function of the two species densities exhibits a transition from multiple to single stability as the delay time increases iii the characteristic correlation time for the sum of the two species densities squared exhibits a nonmonotonic behavior as a function of delay time our results have the implication that the combination of noise and time delay could provide an efficient tool for understanding real ecological systems

spatially variant morphological restoration and skeleton representation the theory of spatially variant sv mathematical morphology is used to extend and analyze two important image processing applications morphological image restoration and skeleton representation of binary images for morphological image restoration we propose the sv alternating sequential filters and sv median filters we establish the relation of sv median filters to the basic sv morphological operators i e sv erosions and sv dilations for skeleton representation we present a general framework for the sv morphological skeleton representation of binary images we study the properties of the sv morphological skeleton representation and derive conditions for its invertibility we also develop an algorithm for the implementation of the sv morphological skeleton representation of binary images the latter algorithm is based on the optimal construction of the sv structuring element mapping designed to minimize the cardinality of the sv morphological skeleton representation experimental results show the dramatic improvement in the performance of the sv morphological restoration and sv morphological skeleton representation algorithms in comparison to their translation invariant counterparts

stochastic bifurcation for a tumor immune system with symmetric levy noise abstract in this paper we investigate stochastic bifurcation for a tumor immune system in the presence of a symmetric non gaussian levy noise stationary probability density functions will be numerically obtained to define stochastic bifurcation via the criteria of its qualitative change and bifurcation diagram at parameter plane is presented to illustrate the bifurcation analysis versus noise intensity and stability index the effects of both noise intensity and stability index on the average tumor population are also analyzed by simulation calculation we find that stochastic dynamics induced by gaussian and non gaussian levy noises are quite different

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

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

implementing continuous scale morphology via curve evolution abstract a new approach to digital implementation of continuous scale mathematical morphology is presented the approach is based on discretization of evolution equations associated with continuous multiscale morphological operations those equations and their corresponding numerical implementation can be derived either directly from mathematical morphology definitions or from curve evolution theory the advantages of the proposed approach over the classical discrete morphology are demonstrated

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

critical phases of antiferromagnetic potts model on fractals the author studies studies the q state antiferromagnetic potts model on fractals namely the sierpinski carpets 1 df 2 and pastry shells 2 df 3 for both families a critical value qc was found above which the system is always in the paramagnetic phase whereas for q at or less than qc the system exhibits a low temperature critical phase the phase diagrams are analyzed from the geometrical point of view

transitions induced by bounded noise abstract we study noise induced transitions when the system is driven separately by a gaussian colored noise and a non gaussian bounded noise with the same mean and stationary correlation clear differences in the nature of such transitions for these two cases are observed these transitions are shown to be enhanced essentially by the bounded noisy driving the results are obtained through a new approach based on numerical analysis of hyperbolic systems

fast recursive algorithms for morphological operators based on the basis matrix representation a real time implementation method for the most general morphological system the so called grayscale function processing fp system is presented the proposed method is an extension of our previous works 1993 1995 using the matrix representation of the fp system with a basis matrix bm and a block basis matrix bbm composed of grayscale structuring elements gse in order to further improve the computational efficiency of the basis matrix representation we propose recursive algorithms based on the observation of the bm and bbm the efficiency of the proposed algorithms is gained by avoiding redundant steps in computing overlapping local maximum or minimum operations it is shown that with the proposed scheme both opening and closing can be determined in real time by 2n 2 additions and 2n 2 comparisons and oc and co by 4n 4 additions and 4n 4 comparisons when the size of the gse is equal to n it is also shown that the proposed recursive opening and closing require only 3n 3 memory elements

transitions induced by cross correlated bounded noises and time delay in a genotype selection model abstract we present a numerical investigation of occurrence of transitions in a genotype selection model with time delay where two bounded noises are cross correlated stationary probability distribution spd function is obtained it is found that the multiplicative bounded noise can facilitate the gene separation and it plays a constructive role in the genetic selection progress while the additive bounded noise suppresses the gene separation the strong correlation between noises gives a big chance to one type haploid out of the group besides what is more interesting is that the correlation time can induce a new transitions i e the curve of the spd changes from unimodal to bimodal and then to four peaks as the correlation time increases

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

potts model on finitely ramified fractals the author presents an exact solution for the ferromagnetic q state potts model on a family of finitely ramified fractal lattices with fractal dimension between 1 and 2 the author obtains the thermal exponent v for the whole fractal family the results stress the connection between geometrical parameters and critical behaviour opening up new possibilities for modelling physical systems showing fractal structures

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

some representation properties of stack filters stack filters a class of nonlinear filters which are based on positive boolean functions as the window operators are considered the representations of these window operations are presented via the structures of on set and off set of the positive boolean functions which can be expressed as a boolean expression containing no complements of the input variables a fast algorithm for finding the representation of stack filters is designed this algorithm can be easily extended to find the representation of morphological filters mentioned by p maragos 1989

a representation theory for morphological image and signal processing a unifying theory for many concepts and operations encountered in or related to morphological image and signal analysis is presented the unification requires a set theoretic methodology where signals are modeled as sets systems signal transformations are viewed as set mappings and translational invariant systems are uniquely characterized by special collections of input signals this approach leads to a general representation theory in which any translation invariant increasing upper semicontinuous system can be presented exactly as a minimal nonlinear superposition of morphological erosions or dilations the theory is used to analyze some special cases of image signal analysis systems such as morphological filters median and order statistic filters linear filters and shape recognition transforms although the developed theory is algebraic its prototype operations are well suited for shape analysis hence the results also apply to systems that extract information about the geometrical structure of signals

further results on map optimality and strong consistency of certain classes of morphological filters morphological openings and closings can be viewed as consistent map estimators of smooth random binary image signals immersed in i i d clutter or suffering from i i d random dropouts we revisit this viewpoint under much more general assumptions and show that quite surprisingly the above interpretation is still valid

renormalisation on symmetric fractals the authors introduce and investigate new classes of sierpinski type fractals they determine their fractal and spectral dimensions using renormalisation procedures and for particular classes they give these dimensions in closed form the spectral dimensions densely fill the interval allowing them to flexibly choose models for applications

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

scale periodicity and its sampling theorem the scalar transform is a new representation for signals offering a perspective that is different from the fourier transform we introduce the notion of a scalar periodic function these functions are then represented through the discrete scale series we also define the notion of a strictly scale limited signal analogous to the shannon interpolation formula we show that such signals can be exactly reconstructed from exponentially spaced samples of the signal in the time domain as an interesting practical application we show how properties unique to the scale transform make it very useful in computing depth maps of a scene

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

bounded noise induced transitions in a tumor immune system interplay by studying a recent biophysical model of tumor growth in the presence of the immune system here we propose that the phenomenon of evasion of tumors from immune control at a temporal mesoscale might in some cases be due to random fluctuations in the levels of the immune system bounded noises are considered but the gaussian approach is also used for analytical reference after showing that in the case of bounded noises there may be multiple attractors in the space of probability densities we numerically show that the velocity of convergence toward asymptotic density is very slow and that a transitory analysis is needed then by simulations using the sine wiener and the tsallis noises we show that if the level of the noise is sufficiently large then there may be the onset of noise induced transitions in the transitory density evaluated at realistic times namely the transitions are from unimodal density centered at low values of tumor burden to bimodal densities that have a second maximum centered at higher values however those transitions depend on the distribution of the noise

implementation of linear digital filters based on morphological representation theory recent advances in mathematical morphology have led to a representation theory that covers increasing linear translation invariant ilti filters as a subclass of morphological filters the representation is based on supremum minimum and addition operations and does not require any multiplications however this representation has not been practical because its essence is to evaluate the supremum of a set with an infinite number of elements based on the previously developed morphological representation we present a finite max min representation for ilti filters the new representation does not require the use of multipliers and is practical in the sense that it consists of a finite number of maximum minimum and addition operations the proposed max min representation is modified to cover fir linear shift invariant filters in general the modified representation requires only one multiplication per output sample and has lower computational complexity than the max min representation other related topics such as duality roundoff noise and implementation are also discussed

median filtering by threshold decomposition median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to further passes of the median filter in this paper it is shown that median filtering an arbitrary level signal to its root is equivalent to decomposing the signal into binary signals filtering each binary signal to a root with a binary median filter and then reversing the decomposition this equivalence allows problems in the analysis and the implementation of median filters for arbitrary level signals to be reduced to the equivalent problems for binary signals since the effects of median filters on binary signals are well understood this technique is a powerful new tool

fusion of 2d grayscale images using multiscale morphology abstract a scheme for fusion of multi sensor 2d images based on multiscale morphology is presented in this paper a point based registration using affine transformation is performed prior to fusion the scale specific features are extracted from both the images using the morphological towers constructed in course of various types of multiscale filtering extracted features are combined to get the fused image a quantitative measure of the degree of fusion is estimated by cross correlation coefficient and the error measure obtained by eigenvector fitting between the fused image and each of the constituting images

transitions induced by time delays and cross correlated sine wiener noises in a tumor immune system interplay abstract a model of tumor immune system interplay with time delays and cross correlated sine wiener noises is investigated by numerical simulations for the stationary probability distribution spd and stationary mean value x s t of tumor cell population results show that i the structure of the spd transfers from bimodal to unimodal as the reaction time of the cell population to environmental constraints increases ii conversely as the time of immune system to develop specific immune competence and the correlation time of noises increase the structure of the spd transfers from unimodal to bimodal iii as the cross correlated intensity between noises increases the transitions induced by are suppressed

a fast thresholded linear convolution representation of morphological operations in this correspondence we present a fast thresholded linear convolution representation of morphological operations the thresholded linear convolution representation of dilation and erosion is first proposed a comparison of the efficiency of the direct implementation of morphological operations and the thresholded linear convolution representation of morphological operations is subsequently conducted mathematical morphology has emerged as a powerful new tool for image processing

some statistical properties of mathematical morphology analyzes the statistical properties of the basic binary and multilevel morphological operations with both 1 d and 2d structuring elements very simple expressions for the output distribution of erosion and opening are derived in the case of any independent identically or nonidentically distributed inputs the probability relations between erosion and dilation also between opening and closing are developed the output expectation bias and variances are analyzed and computed to show the efficiency of morphological operations for noise supression as applications of theoretical results the effects of the morphological operations on noisy signals are illustrated by several examples the study reveals certain interesting phenomena for example the output variances of opening for some input distributions are greater than those of erosion and morphological operations perform better than median filters in edge preservation

map signal estimation in noisy sequences of morphologically smooth images sidiropoulos et al 1994 demonstrated that morphological openings and closings can be viewed as maximum a posteriori map estimators of morphologically smooth signals in signal independent i i d noise the present authors extend these results to the m fold independent observation case and show that the aforementioned estimators are strongly consistent we also demonstrate the validity of a thresholding conjecture sidiropoulos et al 1994 by simulation and use it to evaluate estimator performance taken together these results can help determine the least upper bound m on m which guarantees virtually error free reconstruction of morphologically smooth images

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

shape analysis and reduction of the morphological basis for digital moving average filters a digital moving average filter with nonnegative mask weights that sum to unity is a gray scale morphological filter as such it possesses a representation as a maximum of erosions over its morphological basis the present paper investigates the special properties of the digital moving average morphological basis these properties being for the most part combinatoric in nature shapes of digital structuring elements are related to the filter weights by means of shape classes this is accomplished by defining a characteristic of a signal relative to the filter weights and then characterizing basis elements by means of the characteristic the notion of shape is then extended to a more general class of morphological filters the concept or a morphological quasi average is introduced and the power of such filters to approximate morphological filters is addressed the effect of basis reduction is studied for a rather wide class of morphological filters the aim being to quantify the degree to which a reduce

algorithms for the decomposition of gray scale morphological operations the choice and detailed design of structuring elements plays a pivotal role in the morphologic processing of images a broad class of morphological operations can be expressed as an equivalent supremum of erosions by a minimal set of basis filters diverse morphological operations can then be expressed in a single comparable framework the set of basis filters are data like structures each filter representing one type of local change possible under that operation the data level description of the basis set is a natural starting point for the design of morphological filters this paper promotes the use of the basis decomposition of gray scale morphological operations to design and apply morphological filters a constructive proof is given for the basis decomposition of general gray scale morphological operations as are practical algorithms to find all of the basis set members for these operations

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

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

stochastic resonance driven by two different kinds of colored noise in a bistable system the phenomenon of stochastic resonance in a bistable nonlinear system is investigated when both the multiplicative noise and the coupling between additive and multiplicative noise are colored with different values of noise correlation time ensuremath tau 1 and ensuremath tau 2 combining the functional analysis and unified colored noise approximation the two different kinds of colored noise in the nonlinear system can be simplified the signal to noise ratio is calculated when a weakly periodic signal is added to the system it is found that there appears a transition between one peak and two peaks in the curve of the signal to noise ratio when either the noise correlation time ensuremath tau 1 and ensuremath tau 2 or the coupling strength ensuremath lambda between additive and multiplicative noise is increased the transition between one and two peaks depending on ensuremath tau 1 and ensuremath lambda is more complex than that depending on ensuremath tau 2

finding of optimal stack filter by graphic searching methods an efficient process to filter noise via an optimal stack filter is proposed the graphic search based techniques are employed to speed up the finding of the optimal stack filter experimental results and performance evaluation are demonstrated to show the efficiency of our proposed method

optimal morphological pattern restoration from noisy binary images a theoretical analysis of morphological filters for the optimal restoration of noisy binary images is presented the problem is formulated in a general form and an optimal solution is obtained by using fundamental tools from mathematical morphology and decision theory consideration is given to the set difference distance function as a measure of comparison between images this function is used to introduce the mean difference function as a quantitative measure of the degree of geometrical and topological distortion introduced by morphological filtering it is proved that the class of alternating sequential filters is a set of parametric smoothing morphological filters that best preserve the crucial structure of input images in the least mean difference sense

swarm based translation invariant morphological prediction method for financial time series forecasting in this paper we present a method to overcome the random walk rw dilemma for financial time series forecasting called swarm based translation invariant morphological prediction stmp method it consists of a hybrid model composed of a modular morphological neural network mmnn combined with a particle swarm optimizer pso which searches for the best time lags to optimally describe the time series phenomenon as well as estimates the initial sub optimal parameters of the mmnn weights architecture and number of modules an additional optimization is performed with each particle of the pso population a distinct mmnn using the back propagation bp algorithm after the mmnn parameters adjustment we use a behavioral statistical test and a phase fix procedure to adjust time phase distortions observed in financial time series finally we conduct an experimental analysis with the proposed method using four real world stock market time series where five well known performance metrics and a fitness function are used to assess the prediction performance the obtained results are compared with those generated by classical models presented in the literature

aperiodic stochastic resonance stochastic resonance sr is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise recently we presented a method and theory for characterizing sr type behavior in excitable systems with aperiodic i e broadband input signals phys rev e 52 r3321 1995 we coined the term aperiodic stochastic resonance asr to describe this general type of behavior in that earlier study we demonstrated asr in the fitzhugh nagumo neuronal model here we demonstrate asr in three additional systems a bistable well system an integrate and fire neuronal model and the hodgkin huxley hh neuronal model we present computational and theoretical results for each system in the context of the hh model we develop a general theory for asr in excitable membranes this work clearly shows that sr type behavior is not limited to systems with periodic inputs thus in general noise can serve to enhance the response of a nonlinear system to a weak input signal regardless of whether the signal is periodic or aperiodic textcopyright 1996 the american physical society

theoretical foundations of spatially variant mathematical morphology part ii gray level images in this paper we develop a spatially variant sv mathematical morphology theory for gray level signals and images in the euclidean space the proposed theory preserves the geometrical concept of the structuring function which provides the foundation of classical morphology and is essential in signal and image processing applications we define the basic sv gray level morphological operators that is sv gray level erosion dilation opening and closing and investigate their properties we demonstrate the ubiquity of sv gray level morphological systems by deriving a kernel representation for a large class of systems called v systems in terms of the basic sv gray level morphological operators a v system is defined to be a gray level operator which is invariant under gray level vertical translations particular attention is focused on the class of sv flat gray level operators the kernel representation for increasing v systems is a generalization of maragos kernel representation for increasing and translation invariant function processing systems a representation of v systems in terms of their kernel elements is established for increasing and upper semicontinuous v systems this representation unifies a large class of spatially variant linear and nonlinear systems under the same mathematical framework the theory is used for analyzing special cases of signal and image processing systems such as sv order rank filters and linear time varying systems finally simulation results show the potential power of the general theory of gray level sv mathematical morphology in several image analysis and computer vision applications

anisotropic morphological filters with spatially variant structuring elements based on image dependent gradient fields this paper deals with the theory and applications of spatially variant discrete mathematical morphology we review and formalize the definition of spatially variant dilation erosion and opening closing for binary and gray level images using exclusively the structuring function without resorting to complement this theoretical framework allows to build morphological operators whose structuring elements can locally adapt their shape and orientation across the dominant direction of the structures in the image the shape and orientation of the structuring element at each pixel are extracted from the image under study the orientation is given by means of a diffusion process of the average square gradient field which regularizes and extends the orientation information from the edges of the objects to the homogeneous areas of the image and the shape of the orientated structuring elements can be linear or it can be given by the distance to relevant edges of the objects the proposed filters are used on binary and gray level images for enhancement of anisotropic features such as coherent flow like structures results of spatially variant erosions dilations and openings closings based filters prove the validity of this theoretical sound and novel approach