morphological texture based maximum likelihood pixel classification based on local granulometric moments abstract morphological granulometries are one parameter filter sequences that monotonically decrease image area a size distribution is generated by measuring the residual area after each iteration of the filter sequence normalization yields a probability distribution function whose moments can be employed as image signatures by measuring residual area locally over a window about each point of an image instead of over the entire image local texture features are generated at each pixel and these features can be employed for pixel classification by using several granulometric structuring element generating sequences numerous moment sets result each carrying different textural information a detailed analysis of this pixel classification methodology using a gaussian maximum likelihood classifier is provided included is a statistical study of classification accuracy feature optimization and robustness with respect to various relevant noise models

morphological granulometric estimation of random patterns in the context of parameterized random sets abstract morphological features are used to estimate the state of a random pattern set governed by a multivariate probability distribution the feature vector is composed of granulometric moments and pattern estimation involves feature based estimation of the parameter vector governing the random set under such circumstances the joint density of the features and parameters is a generalized function concentrated on a solution manifold and estimation is determined by the conditional density of the parameters given an observed feature vector the paper explains the manner in which the joint probability mass of the parameters and features is distributed and the way the conditional densities give rise to optimal estimators according to the distribution of probability mass whether constrained or not to the solution manifold the estimation theory is applied using analytic representation of linear granulometric moments the effects of random perturbations in the shape parameter vector is discussed and the theory is applied to random sets composed of disjoint random shapes the generalized density framework provides a proper mathematical context for pattern estimation and gives insight via the distribution of mass on solution manifolds to the manner in which morphological probes discriminate random sets relative to their distributions and the manner in which the use of additional probes can be beneficial for better estimation

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

scalar convergence of convex sets abstract we study the scalar convergence of sequences of convex sets defined by lim sup a n a for all in dual space new properties are given relationship between scalar convergence and other known convergences is examined two natural distinct uniformities on nonvoid closed convex sets define the scalar convergence the associated topology is the weakest such that a d a h is continuous for each hyperplane h

controlled convergence theorems for infinite dimension henstock integrals of fuzzy valued functions based on weak equi integrability abstract in this paper we give the concept of weak henstock equi integrability for a sequence of fuzzy number valued functions under this notion we investigate a new version of the henstock s lemma of fuzzy number valued functions thus it is possible to discuss the controlled convergence theorems of fuzzy henstock integral in sense of vitali covering moreover we prove that a uniform version of sklyarenko s and lusin s integrability condition of fuzzy henstock integrals together with pointwise convergence of a sequence of integrable functions is sufficient for a convergence theorem of fuzzy henstock integrals as the applications of the controlled convergence theorem we discuss the existence theorems of generalized solution for a class of discontinuous fuzzy differential equations

heterogeneous morphological granulometries abstract the most basic class of binary granulometries is composed of unions of openings by structuring elements that are homogeneously scaled by a single parameter these univariate granulometries have previously been extended to multivariate granulometries in which each structuring element is scaled by an individual parameter this paper introduces the more general class of filters in which each structuring element is scaled by a function of its sizing parameter the result being multivariate heterogeneous granulometries owing to computational considerations of particular importance are the univariate heterogeneous granulometries for which scaling is by functions of a single variable the basic morphological properties of heterogeneous granulometries are given analytic and geometric relationships between multivariate and univariate heterogeneous pattern spectra are explored and application to texture classification is discussed the homogeneous granulometric mixing theory both the representation of granulometric moments and the asymptotic theory concerning the distributions of granulometric moments is extended to heterogeneous scaling

least squares approximation with constraints in this paper we study two families of functions fe and fo and show how to approximate the functions in the interval 1 1 the functions are assumed to be real when the argument is real we define

asymptotic granulometric mixing theorem morphological estimation of sizing parameters and mixture proportions abstract if a random set binary image is composed of randomly sized disjoint translates arising as homothetics of a finite number of compact primitives and a granulometry is generated by a convex compact set then the granulometric moments of the random set can be expressed in terms of the granulometric moments of the generating primitives their areas and the radii of the homethetics moreover if there is only a single generating primitive then the granulometric moments of the random set are asymptotically normal relative to an increasing number of grains translates and methods exist to express their asymptotic moments the present paper extends the asymptotic theory to multiple primitive images the extension is significant in two regards first the single primitive asymptotic theory was based on the classical moment theory of cramer the new theory uses an approach based on analytic decomposition on the granulometric moments second not only does the new theory demonstrate asymptotic normality but it also permits method of moment estimation of the parameters of the distributions governing grain sizes and mixture proportions among grain types in the present setting granulometric method of moment estimation is based on asymptotic representation of expectations of granulometric moments in terms of grain sizing model parameters mixture proportions and geometric constants method of moment estimation for sizing parameters and mixture proportions is achieved by replacing granulometric moment expectations with moment estimations from image realizations and then solving the resulting system of equations for the model parameters the theory is applied to both normal and gamma sizing distributions

a decomposition theorem for banach space valued fuzzy henstock integral we establish the following decomposition theorem for fuzzy mappings with values in a banach space a fuzzy mapping is fuzzy henstock integrable if and only if it can be represented as a sum of a fuzzy mcshane integrable fuzzy mapping and of a fuzzy henstock integrable fuzzy mapping generated by a henstock integrable function

random set theory and problems of modeling the three or four dimensional world in which we live is full of objects to be measured and summarized very often a parsimonious finite collection of measurements is enough for scientific investigation into an object s genesis and evolution there is a growing need however to describe and model objects through their form as well as their size the purpose of this article is to show the potentials and limitations of a probabilistic and statistical approach collections of objects the data are assimilated to a random set the model whose parameters provide description and or explanation

joint measurability of mappings induced by a fuzzy random variable abstract the operation of taking the cut of a compact convex fuzzy set is shown to be jointly measurable with respect to both and the fuzzy set as a consequence a number of mappings on product spaces which are induced by a fuzzy random variable are shown to be jointly measurable some applications to the relationships between fuzzy random variables and other imprecise random elements are obtained finally a number of conditions are shown to be equivalent to being a fuzzy random variable at least in the case that the algebra in the sample space is complete and logical implications and equivalences between them are established in the general case

on henstock integrals of interval valued functions and fuzzy valued functions abstract in this paper the h integrals of interval valued functions and fuzzy valued functions are defined and discussed several necessary and sufficient conditions of h integrability for fuzzy number valued functions are given by means of abstract henstock pettis integral theory

optimal linear granulometric estimation for random sets this paper addresses two pattern recognition problems in the context of random sets for the first the random set law is known and the task is to estimate the observed pattern from a feature set calculated from the observation for the second the law is unknown and we wish to estimate the parameters of the law estimation is accomplished by an optimal linear system whose inputs are features based on morphological granulometries in the first case these features are granulometric moments in the second they are moments of the granulometric moments for the latter estimation is placed in a bayesian context by assuming that there exists a prior distribution for the parameters determining the law a disjoint random grain model is assumed and the optimal linear estimator is determined by using asymptotic expressions for the moments of the granulometric moments in both cases the linear approach serves as a practical alternative to previously proposed nonlinear methods granulometric pattern estimation has previously been accomplished by a nonlinear method using full distributional knowledge of the random variables determining the pattern and granulometric features granulometric estimation of the law of a random grain model has previously been accomplished by solving a system of nonlinear equations resulting from the granulometric asymptotic mixing theorem both methods are limited in application owing to the necessity of performing a nonlinear optimization the new linear method avoids this it makes estimation possible for more complex models

fuzzy integral equations abstract we introduce a definition of the integral of a fuzzy valued function that is only slightly different from the usual one yet that is more intuitive and that can be applied to a larger class of functions we show that our definition is equivalent to the extension principle for functions of this class we then apply integration theory of fuzzy valued functions to integral equations we define a measure of the fuzziness of a fuzzy valued function and show that using this measure we can view the integration of fuzzy functions as a fuzzy linear transformation from a certain subspace of fuzzy valued functions to the space of fuzzy numbers we then show how this can be used to study fredholm integral equations of the second kind

non homothetic granulometric mixing theory with application to blood cell counting a granulometry is a family of morphological openings by scaled structuring elements as the scale increases increasing image area is removed normalizing removed area by the total area yields the pattern spectrum of the image the pattern spectrum is a probability distribution function and its moments are known as granulometric moments modeling the image as a random set the pattern spectrum is a random function and its moments are random variables the original granulometric mixing theory provides closed form representation of the granulometric moments shows that the distributions of the moments are asymptotically normal and gives asymptotic expressions for the means and variances of the granulometric moments the theory applies to random set models formed as disjoint unions of randomly scaled image primitives grains the scaled grains are known as homothetics the theory can be used in a method of moments fashion to estimate the parameters governing the random scaling factors and mixture proportions application is limited by the homothetic assumption this paper drops the homothetic requirement and provides a mixing theory for a disjoint union of fully randomized primitives whereas the original asymptotic theory gives expressions for the moments themselves and is applied by taking expectations afterwards the non homothetic theory involves the granulometric size density which is the mean of the original size distribution prior to normalization hence the representation concerns expectations not random variables nonetheless a similar method of moments approach can be used to estimate mixture proportions a large part of the paper is devoted to estimate blood cell proportions that correspond to cell age categories each cell class is represented by a random grain and the problem is to estimate the proportions of cells occurring in the various age categories the random behavior of the cells in each category makes the non homothetic theory appropriate because the estimation strategy leads to a system of nonlinear equations whose solution presents computational difficulties the estimation is accomplished via a divide and conquer strategy in which the full mixture problem is partitioned into smaller problems and the solutions of these problems are joined to solve the full problem

a decomposition theorem for the fuzzy henstock integral we study the fuzzy henstock and the fuzzy mcshane integrals for fuzzy number valued functions the main purpose of this paper is to establish the following decomposition theorem a fuzzy number valued function is fuzzy henstock integrable if and only if it can be represented as a sum of a fuzzy mcshane integrable fuzzy number valued function and of a fuzzy henstock integrable fuzzy number valued function generated by a henstock integrable function

on retarded fuzzy functional differential equations and nonabsolute fuzzy integrals abstract we prove a generalized convergence theorem for fuzzy henstock integrals based on the sequence of fuzzy number valued functions which are weak generalized uniformly absolute continuous u a c g to widen the applications of this convergence theorem we provide some existence theorems for the generalized solution for retarded fuzzy functional differential equations and the dependence of the solutions on a parameter under the assumption of the strong generalized derivative of the solution our results generalize existence results in a kaleva integral setting to a fuzzy henstock integral setting

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures numerical methods for strongly oscillatory and singular functions are given in this paper beside a summary of standard methods and product integration rules we consider a class of complex integration methods several applications of quadrature processes in problems in telecom munications and physics are also presented 1998 elsevier science ltd all rights reserved integration oscillatory kernel singular kernel orthogonal polynomials product rules gaussian quaclratures error function bessel functions legendre functions

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

spatial size distributions applications to shape and texture analysis this paper proposes new descriptors for binary and gray scale images based on newly defined spatial size distributions ssd the main idea consists of combining a granulometric analysis of the image with a comparison between the geometric covariograms for binary images or the auto correlation function for gray scale images of the original image and its granulometric transformation the usual granulometric size distribution then arises as a particular case of this formulation examples are given to show that in those cases in which a finer description of the image is required the more complex descriptors generated from the ssd could be advantageously used it is also shown that the new descriptors are probability distributions so their intuitive interpretation and properties can be appropriately studied from the probabilistic point of view the usefulness of these descriptors in shape analysis is illustrated by some synthetic examples and their use in texture analysis is studied various cases of ssd and several former methods for texture classification are compared

on henstock integral of fuzzy number valued functions i abstract in this paper we define and discuss the fh integral for fuzzy number valued functions using a concrete structure into which we embed the fuzzy number space e 1 several necessary and sufficient conditions of integrability for fuzzy number valued functions are given by means of abstract function theory

strong integral of multifunctions relative to a fuzzy measure abstract in this paper we define and study an integral for measurable multifunctions taking values in the family of all nonvoid subsets of 0 with respect to a fuzzy measure unlike various ways of defining such an integral e g aumann way using different selections dunford way riemann way we use the sugeno method directly applied to multifunctions the originality of our definition is that the value of the integral is a real scalar and not usually a set some classical integral properties a convergence theorem and relationships with other integrals are presented