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

fuzzy probability measures abstract in 4 hohle has defined fuzzy measures on g fuzzy sets 2 where g stands for a regular boolean algebra consequently since the unit interval is not complemented fuzzy sets in the sense of zadeh 8 do not fit in this framework in a straightforward manner it is the purpose of this paper to continue the work started in 5 which deals with 0 1 fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space 5 we give necessary and sufficient conditions for such a measure to be a classical integral as in 9 in the case the space is generated a counterexample in the general case is also presented finally it is shown that a fuzzy probability measure is always an integral if the space is generated if we replace the operations and by the t norm to and its dual s0 see 6

interval random sets and entropy abstract the problem of proper choice from the class of interval random sets generated by some fuzzy set is considered this problem is determined for the maximum entropy criterion the class of interval random sets is studied

morphologically constrained grfs applications to texture synthesis and analysis a new class of gibbs random fields grfs is proposed capable of modeling geometrical constraints in images by means of mathematical morphology the proposed approach known as morphologically constrained grfs models images by means of their size density since the size density is a multiresolution statistical summary morphologically constrained grfs explicitly incorporate multiresolution information into image modeling important properties are studied and their implication to texture synthesis and analysis is discussed statistical inference can be easily implemented by means of mathematical morphology this allows the design of a computationally simple morphological bayes classifier which produces excellent results in classifying natural textures

applications of product space algebra of conditional events and one point random set representations of fuzzy sets to the development of conditional fuzzy sets abstract the field of conditional event algebra is a newly emerging approach to establishing algebraic bases for probabilistic conditioning this has been further enhanced more recently by the development of a conditional event algebra based upon a countable product space construction this paper considers a new rigorous approach to the development of conditional fuzzy sets based upon the further extension of conditional event algebra to a fuzzy setting using the previously established one point random set coverage probability representation of fuzzy sets

a note on random sets and the thurstonian scaling methods abstract we argue that if a vague concept is represented by a random set then it is justifiable to use the law of comparative judgment to estimate the means of the membership function distributions these means may be taken as the type 1 membership values

homogeneous almost quaternion hermitian manifolds an almost quaternion hermitian structure on a riemannian manifold m 4n g is a reduction of the structure group of m to mathrm sp n mathrm sp 1 subset text so 4n in this paper we show that a compact simply connected homogeneous almost quaternion hermitian manifold of non vanishing euler characteristic is either a wolf space or mathbb s 2 times mathbb s 2 or the complex quadric text so 7 mathrm u 3

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

computational aspects of the coarsening at random model and the shapley value this paper is concerned with computational aspects of coarse data in information technology a method for calculating the car coarsening at random model of a finite random set is obtained and a formula for computing the shapley value using the distribution of the random set is given the relationship between the car model solution and the shapley value is also discussed several examples are given to illustrate our results

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

on the convergence of monte carlo maximum likelihood calculations summary monte carlo maximum likelihood for normalized families of distributions can be used for an extremely broad class of models given any family he 0 e 0 of non negative integrable functions maximum likelihood estimates in the family obtained by normalizing the functions to integrate to 1 can be approximated by monte carlo simulation the only regularity conditions being a compactification of the parameter space such that the evaluation maps 0 h0 x remain continuous then with probability 1 the monte carlo approximant to the log likelihood hypoconverges to the exact log likelihood its maximizer converges to the exact maximum likelihood estimate approximations to profile likelihoods hypoconverge to the exact profile and level sets of the approximate likelihood support regions converge to the exact sets in painleve kuratowski set convergence the same results hold when there are missing data if a wald type integrability condition is satisfied asymptotic normality of the monte carlo error and convergence of the monte carlo approximation to the observed fisher information are also shown

classification of textures using gaussian markov random fields the problem of texture classification arises in several disciplines such as remote sensing computer vision and image analysis in this paper we present two feature extraction methods for the classification of textures using two dimensional 2 d markov random field mrf models it is assumed that the given m m texture is generated by a gaussian mrf model in the first method the least square ls estimates of model parameters are used as features in the second method using the notion of sufficient statistics it is shown that the sample correlations over a symmetric window including the origin are optimal features for classification simple minimum distance classifiers using these two feature sets yield good classification accuracies for a seven class problem

clifford structures on riemannian manifolds we introduce the notion of even clifford structures on riemannian manifolds which for rank r 2 and r 3 reduce to almost hermitian and quaternion hermitian structures respectively we give the complete classification of manifolds carrying parallel rank r even clifford structures kahler quaternion kahler and riemannian products of quaternion kahler manifolds for r 2 3 and 4 respectively several classes of 8 dimensional manifolds for 5 r 8 families of real complex and quaternionic grassmannians for r 8 6 and 5 respectively and rosenfeld s elliptic projective planes op 2 c o p 2 h o p 2 and o o p 2 which are symmetric spaces associated to the exceptional simple lie groups f4 e6 e7 and e8 for r 9 10 12 and 16 respectively as an application we classify all riemannian manifolds whose metric is bundle like along the curvature constancy distribution generalizing well known results in sasakian and 3 sasakian geometry 2000 mathematics subject classification primary 53c26 53c35 53c10 53c15

single value simulation of fuzzy variable some further results abstract the problem of a proper choice from the class of random sets equivalent to a fuzzy set under one point coverages is considered the selected random variable is used for construction of a random variable probability distribution serving as a random generator of a value of the fuzzy variable with values restricted by the fuzzy set the problem is determined for two selection criteria i maximum probability possibility consistency and ii maximum entropy in the case of the entropy criterion the problem has been solved first by goodman and here only another proof of the result is presented

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

random intervals as a model for imprecise information random intervals constitute one of the classes of random sets with a greater number of applications in this paper we regard them as the imprecise observation of a random variable and study how to model the information about the probability distribution of this random variable two possible models are the probability distributions of the measurable selections and those bounded by the upper probability we prove that under some hypotheses the closures of these two sets in the topology of the weak convergence coincide improving results from the literature moreover we provide examples showing that the two models are not equivalent in general and give sufficient conditions for the equality between them finally we comment on the relationship between random intervals and fuzzy numbers

some new results concerning random sets and fuzzy sets abstract this paper is a continuation of past work showing direct connections between fuzzy set theory and classical probability theory through the use of random sets this includes characterizations of random intervals one point coverage equivalent to fuzzy sets determination of all nested random sets equivalent to fuzzy sets the solution of the one and multiple point coverage problems for random sets in finite spaces and the use of entropy for ordering random sets within the class of one point coverage equivalent ones for any given fuzzy set finally a connection between random variables random sets and fuzzy sets is pointed out

a mass assignment theory of the probability of fuzzy events abstract mass assignment theory techniques for processing uncertainty in fril are reviewed the notion of the probability of a fuzzy event is introduced together with the t norm definition of conditional probabilities the latter is then shown to be probability possibility inconsistent an alternative theory of conditional probabilities based on mass assignments is presented together with a number of results illustrating some intuitive properties in particular the mass assignment theory of conditional probabilities is shown to be probability possibility consistent

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

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

representation theorems for l fuzzy quantities abstract in this paper representation theorems are given for l fuzzy quantities which permit a better understanding of fuzziness in particular representation theorems due to negoita and ralescu 17 and to sherwood and taylor 23 are extended to the scope of complete and brouwerian lattices