enhanced gradient for training restricted boltzmann machines restricted boltzmann machines rbms are often used as building blocks in greedy learning of deep networks however training this simple model can be laborious traditional learning algorithms often converge only with the right choice of metaparameters that specify for example learning rate scheduling and the scale of the initial weights they are also sensitive to specific data representation an equivalent rbm can be obtained by flipping some bits and changing the weights and biases accordingly but traditional learning rules are not invariant to such transformations without careful tuning of these training settings traditional algorithms can easily get stuck or even diverge in this letter we present an enhanced gradient that is derived to be invariant to bit flipping transformations we experimentally show that the enhanced gradient yields more stable training of rbms both when used with a fixed learning rate and an adaptive one

new approach to study splines by blossoming method and application to the construction of a bivariate c 1 quartic quasi interpolant this work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions splines which uses the blossoming approach some existing results in the literature are reformulated such as the smoothness conditions between polynomials of a spline by using the affinity property of the blossom some definitions of sub splines are proposed which can be very useful in the study and the construction of splines such as macro elements or quasi interpolants as an application of the proposed results a c 1 quartic spline quasi interpolant with optimal approximation order is defined without using any mask for smoothness or b spline basis numerical results are presented and compared with other methods given in the literature

estimation of generalized multisensor hidden markov chains and unsupervised image segmentation this paper attacks the problem of generalized multisensor mixture estimation a distribution mixture is said to be generalized when the exact nature of components is not known but each of them belongs to a finite known set of families of distributions estimating such a mixture entails a supplementary difficulty one must label for each class and each sensor the exact nature of the corresponding distribution such generalized mixtures have been studied assuming that the components lie in the pearson system we propose a more general procedure with applications to estimating generalized multisensor hidden markov chains our proposed method is applied to the problem of unsupervised image segmentation the method proposed allows one to 1 identify the conditional distribution for each class and each sensor 2 estimate the unknown parameters in this distribution 3 estimate priors and 4 estimate the true class image

the grothendieck teichmuller group action on differential forms and formality morphisms of chains it is known that one can associate a kontsevich type formality morphism to every drinfeld associator we show that this morphism may be extended to a kontsevich shoikhet formality morphism of cochains and chains by describing the action of the grothendieck teichmuller group on such objects up to homotopy

contextual estimators of mixing probabilities for markov chain random fields abstract this paper discusses the estimation of proportions of classes from an image classification methods are compared with likelihood methods and the importance of contextual information is discussed the joint distribution of classes at neighbouring sites is modelled by a markov chain random field the class attributes are estimated from training sets and unclassified observations the effect of biased class means is reduced with a stochastic model of the bias contextual likelihood methods yield better results than non contextual methods

parametric inference for imperfectly observed gibbsian fields this paper presents a maximum likelihood estimation method for imperfectly observed gibbsian fields on a finite lattice this method is an adaptation of the algorithm given in younes 28 presentation of the new algorithm is followed by a theorem about the limit of the second derivative of the likelihood when the lattice increases which is related to convergence of the method some practical remarks about the implementation of the procedure are eventually given

hyperparameter estimation for satellite image restoration using a mcmc maximum likelihood method abstract the satellite image deconvolution problem is ill posed and must be regularized herein we use an edge preserving regularization model using a function involving two hyperparameters our goal is to estimate the optimal parameters in order to automatically reconstruct images we propose to use the maximum likelihood estimator mle applied to the observed image we need sampling from prior and posterior distributions since the convolution prevents use of standard samplers we have developed a modified geman yang algorithm using an auxiliary variable and a cosine transform we present a markov chain monte carlo maximum likelihood mcmcml technique which is able to simultaneously achieve the estimation and the reconstruction

formality theorem for gerbes abstract the main result of the present paper is an analogue of kontsevich formality theorem in the context of the deformation theory of gerbes we construct an l deformation of the schouten algebra of multi vectors which controls the deformation theory of a gerbe

estimation of generalized mixtures and its application in image segmentation we introduce the notion of a generalized mixture and propose some methods for estimating it along with applications to unsupervised statistical image segmentation a distribution mixture is said to be generalized when the exact nature of the components is not known but each belongs to a finite known set of families of distributions for instance we can consider a mixture of three distributions each being exponential or gaussian the problem of estimating such a mixture contains thus a new difficulty we have to label each of three components there are eight possibilities we show that the classical mixture estimation algorithms expectation maximization em stochastic em sem and iterative conditional estimation ice can be adapted to such situations once as we dispose of a method of recognition of each component separately that is when we know that a sample proceeds from one family of the set considered we have a decision rule for what family it belongs to considering the pearson system which is a set of eight families the decision rule above is defined by the use of skewness and kurtosis the different algorithms so obtained are then applied to the problem of unsupervised bayesian image segmentation we propose the adaptive versions of sem em and ice in the case of blind i e pixel by pixel segmentation global segmentation methods require modeling by hidden random markov fields and we propose adaptations of two traditional parameter estimation algorithms gibbsian em gem and ice allowing the estimation of generalized mixtures corresponding to pearson s system the efficiency of different methods is compared via numerical studies and the results of unsupervised segmentation of three real radar images by different methods are presented

kontsevich deformation quantization and flat connections in torossian j lie theory 12 2 597 616 2002 the second author used the kontsevich deformation quantization technique to define a natural connection n on the compactified configuration spaces overline c n 0 of n points on the upper half plane connections n take values in the lie algebra of derivations of the free lie algebra with n generators in this paper we show that n is flat

estimation of generalized mixture in the case of correlated sensors this paper deals with unsupervised bayesian classification of multidimensional data we propose an extension of a previous method of generalized mixture estimation to the correlated sensors case the method proposed is valid in the independent data case as well as in the hidden markov chain or field model case with known applications in signal processing particularly speech or image processing the efficiency of the method proposed is shown via some simulations concerning hidden markov fields with application to unsupervised image segmentation

thermodynamics and statistical analysis of gaussian random fields gaussian fields are considered as gibbsian fields thermodynamic functions are calculated for them and the variational principle is proved as an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for gaussian markov fields

noise contrastive estimation of unnormalized statistical models with applications to natural image statistics we consider the task of estimating from observed data a probabilistic model that is parameterized by a finite number of parameters in particular we are considering the situation where the model probability density function is unnormalized that is the model is only specified up to the partition function the partition function normalizes a model so that it integrates to one for any choice of the parameters however it is often impossible to obtain it in closed form gibbs distributions markov and multi layer networks are examples of models where analytical normalization is often impossible maximum likelihood estimation can then not be used without resorting to numerical approximations which are often computationally expensive we propose here a new objective function for the estimation of both normalized and unnormalized models the basic idea is to perform nonlinear logistic regression to discriminate between the observed data and some artificially generated noise with this approach the normalizing partition function can be estimated like any other parameter we prove that the new estimation method leads to a consistent convergent estimator of the parameters for large noise sample sizes the new estimator is furthermore shown to behave like the maximum likelihood estimator in the estimation of unnormalized models there is a trade off between statistical and computational performance we show that the new method strikes a competitive trade off in comparison to other estimation methods for unnormalized models as an application to real data we estimate novel two layer models of natural image statistics with spline nonlinearities

a stochastic method for bayesian estimation of hidden markov random field models with application to a color model we propose a new stochastic algorithm for computing useful bayesian estimators of hidden markov random field hmrf models that we call exploration selection estimation ese procedure the algorithm is based on an optimization algorithm of o francois called the exploration selection e s algorithm the novelty consists of using the a posteriori distribution of the hmrf as exploration distribution in the e s algorithm the ese procedure computes the estimation of the likelihood parameters and the optimal number of region classes according to global constraints as well as the segmentation of the image in our formulation the total number of region classes is fixed but classes are allowed or disallowed dynamically this framework replaces the mechanism of the split and merge of regions that can be used in the context of image segmentation the procedure is applied to the estimation of a hmrf color model for images whose likelihood is based on multivariate distributions with each component following a beta distribution meanwhile a method for computing the maximum likelihood estimators of beta distributions is presented experimental results performed on 100 natural images are reported we also include a proof of convergence of the e s algorithm in the case of nonsymmetric exploration graphs

a general method for constructing quasi interpolants from b splines a general method for constructing quasi interpolation operators based on b splines is developed given a b spline f in r s s 1 normalized by i z email protected i 1 the classical structure q f email protected i z email protected i f i for a quasi interpolation operator q is considered a minimization problem is derived from an estimate of the quasi interpolation error associated with q when lf is a linear combination of values of f at points in some neighbourhood of the support of f or a linear combination of values of f and some of its derivatives at some points in this set or a linear combination of weighted mean values of the function to be approximated that linear functional is defined to produce a quasi interpolant exact on the space of polynomials of maximal total degree included in the space spanned by the integer translates of f the solution of that minimization problem is characterized in terms of specific splines which do not depend on l but only on f

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

the shape boltzmann machine a strong model of object shape a good model of object shape is essential in applications such as segmentation detection inpainting and graphics for example when performing segmentation local constraints on the shapes can help where object boundaries are noisy or unclear and global constraints can resolve ambiguities where background clutter looks similar to parts of the objects in general the stronger the model of shape the more performance is improved in this paper we use a type of deep boltzmann machine salakhutdinov and hinton international conference on artificial intelligence and statistics 2009 that we call a shape boltzmann machine sbm for the task of modeling foreground background binary and parts based categorical shape images we show that the sbm characterizes a strong model of shape in that samples from the model look realistic and it can generalize to generate samples that differ from training examples we find that the sbm learns distributions that are qualitatively and quantitatively better than existing models for this task

on the coefficients of the alekseev torossian associator abstract this paper explains a method to calculate the coefficients of the alekseev torossian associator as linear combinations of iterated integrals of kontsevich weight forms of lie graphs

comparison theorems for gibbs measures the dobrushin comparison theorem is a powerful tool to bound the difference between the marginals of high dimensional probability distributions in terms of their local specifications originally introduced to prove uniqueness and decay of correlations of gibbs measures it has been widely used in statistical mechanics as well as in the analysis of algorithms on random fields and interacting markov chains however the classical comparison theorem requires validity of the dobrushin uniqueness criterion essentially restricting its applicability in most models to a small subset of the natural parameter space in this paper we develop generalized dobrushin comparison theorems in terms of influences between blocks of sites in the spirit of dobrushin shlosman and weitz that substantially extend the range of applicability of the classical comparison theorem our proofs are based on the analysis of an associated family of markov chains we develop in detail an application of our main results to the analysis of sequential monte carlo algorithms for filtering in high dimension

efficient recursions for general factorisable models let n s valued categorical variables be jointly distributed according to a distribution known only up to an unknown normalising constant for an unnormalised joint likelihood expressible as a product of factors we give an algebraic recursion which can be used for computing the normalising constant and other summations a saving in computation is achieved when each factor contains a lagged subset of the components combining in the joint distribution with maximum computational efficiency as the subsets attain their minimum size if each subset contains at most r 1 of the n components in the joint distribution we term this a lag r model whose normalising constant can be computed using a forward recursion in o sr 1 computations as opposed to o sn for the direct computation we show how a lag r model represents a markov random field and allows a neighbourhood structure to be related to the unnormalised joint likelihood we illustrate the method by showing how the normalising constant of the ising or autologistic model can be computed

near minimally normed spline quasi interpolants on uniform partitions spline quasi interpolants qis are local approximating operators for functions or discrete data we consider the construction of discrete and integral spline qis on uniform partitions of the real line having small infinity norms we call them near minimally normed qis they are exact on polynomial spaces and minimize a simple upper bound of their infinity norms we give precise results for cubic and quintic qis also the qi error is considered as well as the advantage that these qis present when approximating functions with isolated discontinuities

truncation artifact reduction in magnetic resonance imaging by markov random field methods a new statistical method is proposed for reduction of truncation artifacts when reconstructing a function by a finite number of its fourier series coefficients following the bayesian approach it is possible to take into account both the errors induced by the truncation of the fourier series and some specific characteristics of the function a suitable markov random field is used for modeling these characteristics furthermore in applications like magnetic resonance imaging where these coefficients are the measured data the experimental random noise in the data can also be taken into account monte carlo markov chain methods are used to make statistical inference parameter selection in the bayesian model is also addressed and a solution for selecting the parameters automatically is proposed the method is applied successfully to both simulated and real magnetic resonance images

minimizing the quasi interpolation error for bivariate discrete quasi interpolants we define a class of discrete quasi interpolants based on bivariate box splines by imposing the exactness on a space of polynomials of total degree depending on the box spline and minimizing a constant appearing in the leading term of an appropriate quasi interpolation error estimate we give some c 1 quadratic and c 2 quartic examples and compare them with other well known quasi interpolants

computing the range of values of real functions using b spline form abstract in this paper we present an easy and efficient method for computing the range of a function by using spline quasi interpolation we exploit the close relationship between the spline function and its control polygon and use tight subdivision technique in order to obtain monotonic splines which make the range of the spline easy to compute the proposed method is useful in case of given scattered data generated by some unknown function f or scientific measurements several numerical examples are given for cubic and quintic quasi interpolant approximant to illustrate the efficiency and the performance of our method

effortless quasi interpolation in hierarchical spaces we present a general and simple procedure to construct quasi interpolants in hierarchical spaces such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement the proposed hierarchical quasi interpolants are described in terms of the so called truncated hierarchical basis assuming a quasi interpolant is selected for each space associated with a particular level in the hierarchy the hierarchical quasi interpolants are obtained without any additional manipulation the main properties like polynomial reproduction of the quasi interpolants selected at each level are locally preserved in the hierarchical construction we show how to construct hierarchical local projectors and the local approximation order of the underling hierarchical space is also investigated the presentation is detailed for the truncated hierarchical b spline basis and we discuss its extension to a more general framework

the dimension and basis of spaces of multivariate splines abstract the purpose of this paper is to study the recent development of certain aspects of multivariate spline spaces more specifically we will survey the existence of bivariate spline the dimension and basis of the bivariate spline spaces s k with various partitions furthermore we will also introduce some results on the higher dimensional splines

equivalence of formalities of the little discs operad we show that kontsevich s formality of the little disk operad obtained using graphs is homotopic to tamarkin s formality for a special choice of a drinfeld associator the associator is given by parallel transport of the alekseev torossian connection

point and differential c1 quasi interpolation on three direction meshes abstract in this paper we construct and analyse c 1 cubic and quartic quasi interpolating splines on type 1 triangulations approximating regularly distributed data without using minimal determining sets and without defining the approximating splines as linear combinations of compactly supported bivariate spanning functions in particular the c 1 cubic splines are directly determined by setting their bernstein bezier coefficients to appropriate combinations of the given data values without using prescribed derivatives at any point of the domain in such a way that the c 1 smoothness conditions are satisfied and approximation order three is guaranteed for smooth functions we also propose some numerical tests that confirm the theoretical results then from the above c 1 cubic splines we obtain c 1 quartic splines exact on p 3 achieving approximation order four the associated differential quasi interpolation operator involves the values of the first partial derivatives in its definition

quasi interpolants associated with h splines on a three direction mesh h splines are b splines with regular hexagonal supports on a three direction mesh of the plane this family contains in particular classical box splines which are convolution powers of the piecewise affine pyramid in this paper we construct discrete or integral qua si interpolants having the best possible approximation order for the smooth functions the construction is based on properties of hexagonal sequences and associated difference operators and of discrete laplacian schemes

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

stochastic approximation algorithms for partition function estimation of gibbs random fields we present an analysis of previously proposed monte carlo algorithms for estimating the partition function of a gibbs random field we show that this problem reduces to estimating one or more expectations of suitable functionals of the gibbs states with respect to properly chosen gibbs distributions as expected the resulting estimators are consistent certain generalizations are also provided we study computational complexity with respect to grid size and show that monte carlo partition function estimation algorithms can be classified into two categories e type algorithms that are of exponential complexity and p type algorithms that are of polynomial complexity turing reducible to the problem of sampling from the gibbs distribution e type algorithms require estimating a single expectation whereas p type algorithms require estimating a number of expectations with respect to gibbs distributions which are chosen to be sufficiently close to each other in the latter case the required number of expectations is of polynomial order with respect to grid size we compare computational complexity by using both theoretical results and simulation experiments we determine the most efficient e type and p type algorithms and conclude that p type algorithms are more appropriate for partition function estimation we finally suggest a practical and efficient p type algorithm for this task

a simulation based estimator for hidden markov random fields an estimator for estimating the parameters of a markov random field x from inaccurate observations is introduced the author considers first a markov gibbs random field x x sub i j on a lattice l i j i 1 2 n j 1 2 m the marginal distributions of x sub i j x sub i u j v u v 1 0 1 are first estimated from an image then random fields x are simulated with the probability of x sub i u j v b nearly equal to the estimate of p x sub i j x sub i u b a simulation method similar to the gibbs sampler is used the parameters of the markov random field model are estimated from the x s with the pseudolikelihood method

an adaptive gaussian model for satellite image deblurring the deconvolution of blurred and noisy satellite images is an ill posed inverse problem which can be regularized within a bayesian context by using an a priori model of the reconstructed solution since real satellite data show spatially variant characteristics we propose here to use an inhomogeneous model we use the maximum likelihood estimator mle to estimate its parameters and we show that the mle computed on the corrupted image is not suitable for image deconvolution because it is not robust to noise we then show that the estimation is correct only if it is made from the original image since this image is unknown we need to compute an approximation of sufficiently good quality to provide useful estimation results such an approximation is provided by a wavelet based deconvolution algorithm thus a hybrid method is first used to estimate the space variant parameters from this image and then to compute the regularized solution the obtained results on high resolution satellite images simultaneously exhibit sharp edges correctly restored textures and a high snr in homogeneous areas since the proposed technique adapts to the local characteristics of the data

a general spline differential quadrature method based on quasi interpolation the differential quadrature method is a numerical discretization technique for the approximation of derivatives the classical method is polynomial based and there is a natural restriction in the number of grid points involved a general spline based method is proposed to avoid this problem for any degree a lagrangian spline interpolant is defined having a fundamental function with small support a quasi interpolant is used to achieve the optimal approximation order that two stage scheme is detailed for the cubic quartic quintic and sextic cases and compared with another methods that appear in the literature

increasing the approximation order of spline quasi interpolants abstract in this paper we show how by a very simple modification of bivariate spline discrete quasi interpolants we can construct a new class of quasi interpolants which have remarkable properties such as high order of regularity and polynomial reproduction more precisely given a spline discrete quasi interpolation operator q d which is exact on the space p m of polynomials of total degree at most m we first propose a general method to determine a new differential quasi interpolation operator q r d which is exact on p m r q r d uses the values of the function to be approximated at the points involved in the linear functional defining q d as well as the partial derivatives up to the order r at the same points from this result we then construct and study a first order differential quasi interpolant based on the c 1 cubic b spline on the equilateral triangulation with a hexagonal support when the derivatives are not available or extremely expensive to compute we approximate them by appropriate finite differences to derive new discrete quasi interpolants q d we estimate with small constants the quasi interpolation errors f q r d f and f q d f in the infinity norm finally numerical examples are used to analyze the performance of the method

cardinal interpolation by multivariate splines the purpose of this paper is to investigate cardinal interpolation using locally supported piecewise polynomials in particular the notion of a commutator is introduced and its connection with the marsden identity is observed the order of a commutator is shown to be equivalent to the strang and fix conditions that arise in the study of the local approxima tion orders using quasi interpolants we also prove that scaled cardinal interpolants give these local approximation orders introduction cardinal interpolation by bivariate box splines was first studied by de boor hollig and riemenschneider 3 the purpose of our paper is to investigate the cardinal interpolation problem from a different point of view in particular the notion of a commutator is introduced it will be shown that this notion generalizes the marsden identity for univariate splines to the multivariate setting the order of a commutator will be shown to be equivalent to the strang and fix conditions used in the study of the order of controlled approximation by dahmen and micchelli 8 or local approximation by de boor and jia 4 an application to obtain approximation orders through the constructive method of scaled cardinal interpolation will also be studied in this paper 1 preliminaries this section consists of preliminary material for multivariate cardinal interpolation our approach is motivated by the work in 5 and 13 where cardinal interpolation in l2 was connected with certain convolution operators lr let z denote the set of integers and z the nonnegative ones for any given complex sequence by

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

theoremes de convergence presque sure pour une classe d algorithmes stochastiques a pas decroissant the paper studies the pathwise asymptotic behaviour of stochastic algorithms of the following general form theta n 1 theta n gamma n 1 f theta n y n 1 the hypotheses allowing discontinuities on the adaptation termf the process y n n 0 is a markov chain controlled by n for each fixed the markov chain y n n 0 is essentially of positive recurrent type

near best quasi interpolants associated with h splines on a three direction mesh spline quasi interpolants with optimal approximation orders and small norms are useful in several applications in this paper we construct the so called near best discrete and integral quasi interpolants based on h splines i e b splines with regular hexagonal supports on the uniform three directional mesh of the plane these quasi interpolants are obtained so as to be exact on some space of polynomials and to minimize an upper bound of their infinite norms which depend on a finite number of free parameters we show that this problem has always a solution which is not unique in general concrete examples of these types of quasi interpolants are given in the two last sections

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

m kontsevich s graph complex and the grothendieck teichmuller lie algebra we show that the zeroth cohomology of m kontsevich s graph complex is isomorphic to the grothendieck teichmuller lie algebra mathfrak grt 1 the map is explicitly described this result has applications to deformation quantization and duflo theory we also compute the homotopy derivations of the gerstenhaber operad they are parameterized by mathfrak grt 1 up to one class or two depending on the definitions more generally the homotopy derivations of the non unital e n operads may be expressed through the cohomology of a suitable graph complex our methods also give a second proof of a result of h furusho stating that the pentagon equation for mathfrak grt 1 elements implies the hexagon equation

optimal bivariate c1 cubic quasi interpolation on a type 2 triangulation in a guessab o nouisser g schmeisser multivariate approximation by a combination of modified taylor polynomials j comput appl math 196 2006 162 179 a general method is proposed to increase the approximation order of approximation operators in this work by using these enhancement techniques we introduce and study new schemes based on a c 1 spline quasi interpolant on a type 2 triangulation they are designed for approximating real valued functions defined on r 2 the proposed method is based on the following idea from a discrete quasi interpolant defined by the quadratic box spline exact on p 2 and by judiciously choosing the first order taylor coefficients we derive a cubic differential quasi interpolant yielding optimal approximation order in addition when the derivatives are not available or are extremely expensive to compute we approximate them by finite difference approximations having the desired accuracy to derive a new class of discrete quasi interpolants as an essential difference to some of the existing methods we only use the given data values and then we do not modify the original triangulation finally we present some numerical tests which confirm the efficiency of the newly quasi interpolant and demonstrate good visual quality in particular we compare it with a differential quasi interpolant done by lai m j lai approximation order from bivariate c 1 cubics on a four directional mesh is full comput aided geom design 11 2 1994 215 223 which is also exact on p 3 but uses third order partial derivatives

estimation of fuzzy gaussian mixture and unsupervised statistical image segmentation this paper addresses the estimation of fuzzy gaussian distribution mixture with applications to unsupervised statistical fuzzy image segmentation in a general way the fuzzy approach enriches the current statistical models by adding a fuzzy class which has several interpretations in signal processing one such interpretation in image segmentation is the simultaneous appearance of several thematic classes on the same site we introduce a new procedure for estimating of fuzzy mixtures which is an adaptation of the iterative conditional estimation ice algorithm to the fuzzy framework we first describe the blind estimation i e without taking into account any spatial information valid in any context of independent noisy observations then we introduce in a manner analogous to classical hard segmentation the spatial information by two different approaches contextual segmentation and adaptive blind segmentation in the first case the spatial information is taken into account at the segmentation step level and in the second case it is taken into account at the parameter estimation step level the results obtained with the iterative conditional estimation algorithm are compared to those obtained with expectation maximization em and the stochastic em algorithms on both parameter estimation and unsupervised segmentation levels via simulations the methods proposed appear as complementary to the fuzzy c means algorithms

hierarchical reconstruction using geometry and sinogram restoration the authors describe and demonstrate a hierarchical reconstruction algorithm for use in noisy and limited angle or sparse angle tomography the algorithm estimates an object s mass center of mass and convex hull from the available projections and uses this information along with fundamental mathematical constraints to estimate a full set of smoothed projections the mass and center of mass estimates are made using a least squares estimator derived from the principles of consistency of the radon transform the convex hull estimate is produced by first estimating the positions of support lines of the object from each available projection and then estimating the overall convex hull using prior shape information estimating the position of two support lines from a single projection is accomplished using a generalized likelihood ratio technique for estimating jumps in linear systems results for simulated objects in a variety of measurement situations are shown and several possible extensions to this work are discussed

uniform lan condition of planar gibbsian point processes and optimality of maximum likelihood estimators of soft core potential functions a parametric model of planar point patterns in a bounded region is constructed using grand canonical gibbsian point processes with soft core potential functions a simple and explicit condition that this model becomes a uniform locally asymptotic normal ulan family will be given from this result we can conclude that the maximum likelihood estimator of the potential function is asymptotically efficient for a wide class of loss functions

quasi interpolation by c1 quartic splines on type 1 triangulations abstract in this paper we construct two new families of c 1 quartic quasi interpolating splines on type 1 triangulations approximating regularly distributed data the splines are directly determined by setting their bernstein bezier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain the quasi interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation reproduce cubic polynomials and yield approximation order four for smooth functions we also propose some numerical tests that confirm the theoretical results

on near best discrete quasi interpolation on a four directional mesh spline quasi interpolants are practical and effective approximation operators in this paper we construct qis with optimal approximation orders and small infinity norms called near best discrete quasi interpolants which are based on w splines i e b splines with octagonal supports on the uniform four directional mesh of the plane these quasi interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters we show that this problem has always a solution in general nonunique concrete examples of such quasi interpolants are given in the last section

a multinomial spline approximation scheme using spline quasi interpolants abstract this paper presents a multinomial spline approximation scheme based on spline quasi interpolants the scheme can be considered as an extension of the usual bernstein approximation for complex exponentials error estimates and numerical examples are given to show that this new scheme could produce highly accurate results

logarithms and deformation quantization we prove the statement conjecture of m kontsevich on the existence of the logarithmic formality morphism mathcal u mathrm log this question was open since 1999 and the main obstacle was the presence of dr r type singularities near the boundary r 0 in the integrals over compactified configuration spaces the novelty of our approach is the use of local torus actions on configuration spaces of points in the upper half plane it gives rise to a version of stokes formula for differential forms with singularities at the boundary which implies the formality property of mathcal u mathrm log we also show that the logarithmic formality morphism admits a globalization from mathbb r d to an arbitrary smooth manifold

on the construction of trivariate near best quasi interpolants based on c 2 quartic splines on type 6 tetrahedral partitions the construction of new quasi interpolants qis having optimal approximation order and small infinity norm and based on a trivariate c 2 quartic box spline is addressed in this paper these quasi interpolants called near best qis are obtained in order to be exact on the space of cubic polynomials and to minimize an upper bound of their infinity norm which depends on a finite number of free parameters in a tetrahedral sequence defining the coefficients of the qis we show that this problem has always a unique solution which is explicitly given we also prove that the sequence of the resulting near best quasi interpolants converges in the infinity norm to the schoenberg operator

superconvergent local quasi interpolants based on special multivariate quadratic spline space over a refined quadrangulation abstract in this paper we first recall some results concerning the construction and the properties of quadratic b splines over a refinement of a quadrangulation of a planar domain introduced recently by lamnii et al then we introduce the b spline representation of hermite interpolant in the special space s 2 1 0 of any polynomial or any piecewise polynomial over refined quadrangulation of after that we use this b representation for constructing several superconvergent discrete quasi interpolants the new results that we present in this paper are an improvement and a generalization of those developed in the above cited paper

on the estimation of noisy binary markov random fields possolo technical report no 77 department of statistics gn 22 university of washington seattle 1986 and derin and elliott ieee trans pattern analysis mach intell pami 9 39 55 1987 proposed for the estimation of binary and more general m ary markov random fields the logit method based on histogramming an image the authors applied the method to noise free markov random fields for estimation of noisy images one might recursively implement the logit method within an iterative restoration algorithm such as the iterated conditional modes icm method of besag j r statist soc b48 259 302 1986 by alternating parameter estimation and restoration it is noted that failure to smooth zero histogram counts for the purposes of estimation can cause icm to cycle indefinitely

construction techniques for multivariate modified quasi interpolants with high approximation order in this paper we propose several approximations of a multivariate function by quasi interpolants on non uniform data and we study their properties in particular we characterize those that preserve constants via the partition of unity approach as one of the main results we show how by a very simple modification of a given quasi interpolant it is possible to construct new quasi interpolants with remarkable properties we also provide some results regarding bivariate c 2 quintic spline quasi interpolation finally numerical tests are presented to show the approximation power of these quasi interpolants

semi cardinal interpolation and difference equations from cubic b splines to a three direction box spline construction this paper considers the problem of interpolation on a semi plane grid from a space of box splines on the three direction mesh building on a new treatment of univariate semi cardinal interpolation for natural cubic splines the solution is obtained as a lagrange series with suitable localization and polynomial reproduction properties it is proved that the extension of the natural boundary conditions to box spline semi cardinal interpolation attains half of the approximation order of the cardinal case

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

decay of correlations under dobrushin s uniqueness condition and its applications an estimate on the correlation of functionals of gibbs fields satisfying dobrushin s uniqueness condition is given as a consequence a result of gross saying that the truncated pair correlation function decays in the same weighted summability sense as the potential can be extended to the whole dobrushin uniqueness region applications to the central limit theorem and the second derivative of the pressure are also given

construction of spherical spline quasi interpolants based on blossoming a general theory of quasi interpolants based on quadratic spherical powell sabin splines on spherical triangulations of a sphere like surface s is developed by using polar forms as application various families of discrete and differential quasi interpolants reproducing quadratic spherical bezier bernstein polynomials or the whole space of the spherical powell sabin quadratic splines of class c 1 are presented

a cumulant expansion technique for simultaneous markov random field image restoration and hyperparameter estimation we investigate hyperparameter estimation for incomplete data in markov random field image restoration assuming linear dependence of energies with respect to hyperparameters framework we use a cumulant expansion technique widely known in statistical physics and signal processing new insight is given on maximum likelihood estimation of hyperparameters of the prior regularization and contour probability distribution functions pdfs for an explicit joint boundary pixel process aimed to preserve discontinuities in particular the case where the prior regularization potential is an homogeneous function of pixels is fully analyzed a generalized stochastic gradient gsg algorithm with a fast sampling technique is devised aiming to achieve simultaneous hyperparameter estimation and pixel restoration image restoration performances of posterior mean performed during gsg convergence and of simulated annealing performed after gsg convergence are compared experimentally results and perspectives are given

combining monte carlo and mean field like methods for inference in hidden markov random fields issues involving missing data are typical settings where exact inference is not tractable as soon as nontrivial interactions occur between the missing variables approximations are required and most of them are based either on simulation methods or on deterministic variational methods while variational methods provide fast and reasonable approximate estimates in many scenarios simulation methods offer more consideration of important theoretical issues such as accuracy of the approximation and convergence of the algorithms but at a much higher computational cost in this work we propose a new class of algorithms that combine the main features and advantages of both simulation and deterministic methods and consider applications to inference in hidden markov random fields hmrfs these algorithms can be viewed as stochastic perturbations of variational expectation maximization vem algorithms which are not tractable for hmrf we focus more specifically on one of these perturbations and we prove their almost sure convergence to the same limit set as the limit set of vem in addition experiments on synthetic and real world images show that the algorithm performance is very close and sometimes better than that of other existing simulation based and variational em like algorithms

cyclic operad formality for compactified moduli spaces of genus zero surfaces the framed little 2 discs operad is homotopy equivalent to the kimura stasheff voronov cyclic operad of moduli spaces of genus zero stable curves with tangent rays at the marked points and nodes we show that this cyclic operad is formal meaning that its chains and its homology the batalin vilkovisky operad are quasi isomorphic cyclic operads to prove this we introduce a new complex of graphs in which the differential is a combination of edge deletion and contraction and we show that this complex resolves bv as a cyclic operad

bivariate box splines and smooth pp functions on a three direction mesh abstract let s denote the space of bivariate piecewise polynomial functions of degree k and smoothness on the regular mesh generated by the three directions 1 0 0 1 1 1 we construct a basis for s in terms of box splines and truncated powers this allows us to determine the polynomials which are locally contained in s and to give upper and lower bounds for the degree of approximation for 2k 2 3 k 2 3 the case of minimal degree k for given smoothness we identify the elements of minimal support in s and give a basis for s loc f s supp f with a convex subset of r 2

fuzzy markov random fields versus chains for multispectral image segmentation this paper deals with a comparison of recent statistical models based on fuzzy markov random fields and chains for multispectral image segmentation the fuzzy scheme takes into account discrete and continuous classes which model the imprecision of the hidden data in this framework we assume the dependence between bands and we express the general model for the covariance matrix a fuzzy markov chain model is developed in an unsupervised way this method is compared with the fuzzy markovian field model previously proposed by one of the authors the segmentation task is processed with bayesian tools such as the well known mpm mode of posterior marginals criterion our goal is to compare the robustness and rapidity for both methods fuzzy markov fields versus fuzzy markov chains indeed such fuzzy based procedures seem to be a good answer e g for astronomical observations when the patterns present diffuse structures moreover these approaches allow us to process missing data in one or several spectral bands which correspond to specific situations in astronomy to validate both models we perform and compare the segmentation on synthetic images and raw multispectral astronomical data