maximization and minimization of the rank and inertia of the hermitian matrix expression a bx bx with applications abstract we give in this paper a group of closed form formulas for the maximal and minimal ranks and inertias of the linear hermitian matrix function a bx bx with respect to a variable matrix x as applications we derive the extremal ranks and inertias of the matrices x x where x is a solution to the matrix equation axb c and then give necessary and sufficient conditions for the matrix equation axb c to have hermitian definite and re definite solutions in addition we give closed form formulas for the extremal ranks and inertias of the difference x 1 x 2 where x 1 and x 2 are hermitian solutions of two matrix equations a 1 x 1 a 1 c 1 and a 2 x 2 a 2 c 2 and then use the formulas to characterize relations between hermitian solutions of the two equations

new texture features based on the complexity curve abstract a new set of texture features based on the complexity curve is proposed for texture analysis the complexity curve is obtained by computing the number of black to white transitions observed in a binary image for all possible threshold values in a gray level texture image the new features characterize the first order statistics of the complexity curve and therefore give a second order description of texture the set of features is applied to texture classification and to capture the directionality and the periodicity of texture for texture classification two experiments are performed first some brodatz textures are used to evaluate the classification performance of the features some of the traditional texture features extracted from the co occurrence matrix have also been used for comparison secondly the features are applied to the classification of transmission electron micrographs of premalignant and malignant cell nuclei in a controlled mouse liver carcinogenesis experiment directionality and periodicity are two high level texture features that guide the process of perceptual grouping of texture images therefore two additional experiments have been performed to evaluate the performance of the complexity curve features in estimating the texture directionality and periodicity the experimental results demonstrate that the new texture features based on the complexity curve are very useful tools for texture analysis

flots et series de taylor stochastiques we study the expansion of the solution of a stochastic differential equation as an infinite sum of iterated stochastic stratonovitch integrals this enables us to give a universal and explicit formula for any invariant diffusion on a lie group in terms of lie brackets as well as a universal and explicit formula for the brownian motion on a riemannian manifold in terms of derivatives of the curvature tensor the first of these formulae contains and extends to the non nilpotent case the results of doss 6 sussmann 17 yamato 18 fliess and normand cyrot 7 krener and lobry 19 and kunita 11 on the representation of solutions of stochastic differential equations

the hermitian solution of a x a b subject to cxc d in this paper we first establish some necessary and sufficient conditions for the existence of hermitian solution of a x a b subject to cxc d where b and d are hermitian matrices furthermore a general expression for this hermitian constrained solution is derived several special cases are also considered

nonnegative definite and positive definite solutions to the matrix equationaxa b revisited abstract the general hermitian nonnegative definite solution to the matrix equation axa b is established in a form which can be viewed as a corrected version of that derived by c g khatri and s k mitra siam j appl math 31 1976 579 585 and an alternative version to that derived by j k baksalary linear and multilinear algebra 16 1984 133 139 the new representation admits an easy way to obtain solutions of minimal and maximal rank respectively

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

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 hermitian least squares solution of the matrix equation axb c subject to inequality restrictions this paper gives some closed form formulas for computing the maximal and minimal ranks and inertias of p x with respect to x where email protected c h n is given and x is a hermitian least squares solution to the matrix equation axb c we derive as applications necessary and sufficient conditions for x p in the lowner partial ordering in addition we give necessary and sufficient conditions for the existence of a hermitian positive negative nonpositive nonnegative definite least squares solution to axb c

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

a representation of the general common solution to the matrix equations a1xb1 c1 and a2xb2 c2 with applications we give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations a1xb1 c1 and a2xb2 c2 and derive a new representation of the general common solution to these two equations we apply this result to determine new necessary and sufficient conditions for the existence of an hermitian solution and a representation of the general hermitian solution to the matrix equation axb c

positive definite solutions of some matrix equations in this paper we investigate some existence questions of positive semi definite solutions for certain classes of matrix equations known as the generalized lyapunov equations we present sufficient and necessary conditions for certain equations and only sufficient for others

cubature on wiener space it is well known that there is a mathematical equivalence between solving parabolic partial differential equations pdes and the integration of certain functionals on wiener space monte carlo simulation of stochastic differential equations sdes is a naive approach based on this underlying principle in finite dimensions it is well known that cubature can be a very effective approach to integration we discuss the appropriate extension of this idea to wiener space in the process we develop high order numerical schemes valid for high dimensional sdes and semi elliptic pdes

the rank constrained hermitian nonnegative definite and positive definite solutions to the matrix equation axa b abstract a simple representation of the general rank constrained hermitian nonnegative definite positive definite solution to the matrix equation axa b is derived as medium steps the general hermitian solution and the general hermitian nonnegative definite positive definite solution to the matrix equation are also obtained the proposed approach is different from those which we have known and possesses good numerical reliability since it mainly involves only two singular value decompositions and inverses of two positive definite diagonal matrices the presented example illustrates the proposed approach

granulometric moments and corneal endothelium status abstract specular microscopy is a common practice in ophthalmology the corneal endothelium status is usually evaluated by means of the density the hexagonality the mean the standard deviation and the coefficient of variation of cell areas we propose to replace the cell area moments by the corresponding moments of a different probability distribution the granulometric size distribution associated to a disc all cells touching the frame are ignored by the area moments but used by the granulometric moments twenty images have been analyzed when the size of the focused region is reduced the area moments show a greater variation than the corresponding granulometric moments

cubature on wiener space in infinite dimension we prove a stochastic taylor expansion for stochastic partial differential equations spdes and apply this result to obtain cubature methods i e high order weak approximation schemes for spdes

sensitivity analysis using ito malliavin calculus and martingales and application to stochastic optimal control we consider a multidimensional diffusion process x alpha t 0 leq t leq t whose dynamics depends on a parameter alpha our first purpose is to write as an expectation the sensitivity nabla alpha j alpha for the expected cost j alpha mathbb e f x alpha t in order to evaluate it using monte carlo simulations this issue arises for example from stochastic control problems where the controller is parameterized which reduces the control problem to a parametric optimization one or from model misspecifications in finance previous evaluations of nabla alpha j alpha using simulations were limited to smooth cost functions f or to diffusion coefficients not depending on alpha see yang and kushner siam j control optim 29 1991 pp 1216 1249 in this paper we cover the general case deriving three new approaches to evaluate nabla alpha j alpha which we call the malliavin calculus approach the adjoint approach and the martingale approach to accomplish this we leverage ito calculus malliavin calculus and martingale arguments in the second part of this work we provide discretization procedures to simulate the relevant random variables then we analyze their respective errors this analysis proves that the discretization error is essentially linear with respect to the time step this result which was already known in some specific situations appears to be true in this much wider context finally we provide numerical experiments in random mechanics and finance and compare the different methods in terms of variance complexity computational time and time discretization error

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

calculating the greeks by cubature formulae we provide cubature formulae for the calculation of derivatives of expected values in the spirit of terry lyons and nicolas victoir in financial mathematics derivatives of option prices with respect to initial values so called greeks are of particular importance as hedging parameters the proof of existence of cubature formulae for greeks is based on universal formulae which lead to the calculation of greeks in an asymptotic sense even

computing symmetric solutions of general sylvester matrix equations via lanczos version of biconjugate residual algorithm abstract many problems in control theory can be studied by obtaining the symmetric solution of linear matrix equations in this investigation we deal with the symmetric solutions x y and z of the general sylvester matrix equations a 1 x b 1 c 1 y d 1 e 1 z f 1 g 1 a 2 x b 2 c 2 y d 2 e 2 z f 2 g 2 a t x b t c t y d t e t z f t g t the lanczos version of biconjugate residual bcr algorithm is generalized to compute the symmetric solutions of the general sylvester matrix equations the convergence properties of this algorithm are discussed and it is proven that it smoothly converges to the symmetric solutions of the general sylvester matrix equations in a finite number of iterations in the absence of round off errors finally the comparative numerical results are carried out to support that the current algorithm may be more efficient than other iterative algorithms

extremal ranks of matrix expression of a bxc with respect to hermitian matrix abstract the extremal ranks of matrix expression of a bxc with respect to x h x have been discussed by applying the quotient singular value decomposition q svd and some rank equalities of matrices in this paper

equalities and inequalities for inertias of hermitian matrices with applications abstract the inertia of a hermitian matrix is defined to be a triplet composed of the numbers of the positive negative and zero eigenvalues of the matrix counted with multiplicities respectively in this paper we show some basic formulas for inertias of 2 2 block hermitian matrices from these formulas we derive various equalities and inequalities for inertias of sums parallel sums products of hermitian matrices submatrices in block hermitian matrices differences of outer inverses of hermitian matrices as applications we derive the extremal inertias of the linear matrix expression a bxb with respect to a variable hermitian matrix x in addition we give some results on the extremal inertias of hermitian solutions to the matrix equation ax b as well as the extremal inertias of a partial block hermitian matrix

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

rank constrained matrix best approximation problem with respect to skew hermitian matrices in this literature we study a rank constrained matrix approximation problem in the frobenius norm minr x kbxbaf2 where k is a nonnegative integer a and x are skew hermitian matrices by using the singular value decomposition and the spectrum decomposition we derive some conditions for the existence of skew hermitian solutions and establish general forms for the skew hermitian solutions to this matrix approximation problem