a variational model for polsar data speckle reduction based on the wishart distribution in this paper we propose a variational model for polarimetric synthetic aperture radar polsar data speckle reduction which is based on the complex wishart distribution of the covariance or coherency matrix and multichannel total variation tv regularization defined for complex valued matrices by assuming the tv regularization to be a prior and taking the statistical distribution of the covariance matrix in each resolution element into account the variational model for polsar covariance data speckle suppression named wistv c is derived from the maximum a posteriori estimate a similar variational model for polsar coherency data speckle reduction named wistv t is also obtained as far as we know this is the first variational model for the whole polsar covariance or coherency matrix data despeckling since the model is nonconvex a convex relaxation iterative algorithm is designed to solve the variational problem based on the variable splitting and alternating minimization techniques experimental results on both simulated and real polsar data demonstrate that the proposed approach notably removes speckles in the extended uniform areas and meanwhile better preserves the spatial resolution the details such as edges and point scatterers and the polarimetric scattering characteristics compared with other methods

a critical point theorem and existence results for a nonlinear boundary value problem in this paper the existence of non zero solutions to a neumann boundary value problem is established multiplicity results are also pointed out the approach is based on a novel critical point theorem

critical point theory for nondifferentiable functionals and applications abstract we consider minimization results for locally lipschitzian functionals on a banach space by introducing a compactness condition of the palais smale type which is suggested by the variational principle of ekeland applications are given to some semilinear elliptic problems with discontinuous nonlinearities

a three critical points theorem revisited abstract in this paper the following result is proved let x be a reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional bounded on each bounded subset of x whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that lim x x j x for all i and that there exists r such that sup i inf x x x j x inf x x sup i x j x then there exist a non empty open set a i and a positive real number r with the following property for every a and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x 0 has at least three solutions in x whose norms are less than r

a nonlocal tv based variational method for polsar data speckle reduction in this paper we propose a nonlocal total variation nltv based variational model for polarimetric synthetic aperture radar polsar data speckle reduction this model named wisnltv is obtained based on the wishart fidelity term and the nltv regularization defined for the complex valued fourth order tensor data since the proposed model is non convex an equivalent bi convex model is obtained using the property of conjugate functions then an efficient iteration algorithm is developed to solve the equivalent bi convex model based on the alternating minimization and the forward backward operator splitting technique the proposed iteration algorithm is proved to be convergent under certain conditions theoretically and numerically experimental results on both synthetic and real polsar data demonstrate that the proposed method can effectively reduce speckle noise and meanwhile better preserve the details and the repetitive structures such as textures and edges and the polarimetric scattering characteristics compared with the other methods

multiplicative noise removal using variable splitting and constrained optimization multiplicative noise also known as speckle noise models are central to the study of coherent imaging systems such as synthetic aperture radar and sonar and ultrasound and laser imaging these models introduce two additional layers of difficulties with respect to the standard gaussian additive noise scenario 1 the noise is multiplied by rather than added to the original image 2 the noise is not gaussian with rayleigh and gamma being commonly used densities these two features of multiplicative noise models preclude the direct application of most state of the art algorithms which are designed for solving unconstrained optimization problems where the objective has two terms a quadratic data term log likelihood reflecting the additive and gaussian nature of the noise plus a convex possibly nonsmooth regularizer e g a total variation or wavelet based regularizer prior in this paper we address these difficulties by 1 converting the multiplicative model into an additive one by taking logarithms as proposed by some other authors 2 using variable splitting to obtain an equivalent constrained problem and 3 dealing with this optimization problem using the augmented lagrangian framework a set of experiments shows that the proposed method which we name midal multiplicative image denoising by augmented lagrangian yields state of the art results both in terms of speed and denoising performance

a variational approach to removing multiplicative noise this paper focuses on the problem of multiplicative noise removal we draw our inspiration from the modeling of speckle noise by using a map estimator we can derive a functional whose minimizer corresponds to the denoised image we want to recover although the functional is not convex we prove the existence of a minimizer and we show the capability of our model on some numerical examples we study the associated evolution problem for which we derive existence and uniqueness results for the solution we prove the convergence of an implicit scheme to compute the solution

a further three critical points theorem abstract if x is a real banach space we denote by w x the class of all functionals x r possessing the following property if u n is a sequence in x converging weakly to u x and lim inf n u n u then u n has a subsequence converging strongly to u in this paper we prove the following result let x be a separable and reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional belonging to w x bounded on each bounded subset of x and whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that for each i the functional j is coercive and has a strict local not global minimum say x then for each compact interval a b i for which sup a b x j x there exists r 0 with the following property for every a b and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x has at least three solutions whose norms are less than r

matrix elements for powers of x dependent operators for the hyperbolic poschl teller potentials analytical matrix elements of the position operator xn n 0 and of the first and second order differential operators are derived using the eigenfunctions of two hyperbolic poschl teller potentials these general relations are written in closed form and calculated from the properties of the integral powers of hypergeometric series additionally an explicit expression is derived for the successive derivatives of the beta function convergence and reliability of these results will be discussed and an application to the calculation of the rovibrational energy spectrum of co will be considered

some remarks on a three critical points theorem abstract some remarks on a strict minimax inequality which plays a fundamental role in ricceri s three critical points theorem are presented as a consequence some recent applications of ricceri s theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions

exponential operators and the algebraic description of quantum confined systems we study the relations and transformations produced by exponential operators the argument of which are dependent on the basic algebraic elements of supersymmetric and shape invariant potential systems and obtain explicit expressions we apply our results to self similar potential systems and to a set of translational shape invariant systems including the morse poschl teller scarf and rosen morse potentials and obtain closed form expressions we show that our results reproduce those obtained for the harmonic oscillator in the appropriate limits

an integrated approach to ladder and shift operators for the morse oscillator radial coulomb and radial oscillator potentials the morse oscillator radial coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods these problems correspond to different realizations of the so 2 1 algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems the resultant transition operators act as ladder or energy changing operators in the cases of the coulomb and harmonic oscillator potentials whereas they act as shift operators acting at constant energy in the case of the morse potential this is a consequence of the so 2 1 dynamical symmetry whereby the morse hamiltonian is expressible solely in terms of the casimir operator of the algebra an alternative algebraic approach the use of the method of supersymmetric quantum mechanics or factorization produces in each case a set of shift operators relations between the various ladder and shift operators may be identified by means of the appropriate mappings and these results can be generalized so as to relate the one dimensional morse oscillator to the radial coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions

an su 1 1 dynamical algebra for the poschl teller potential an su 1 1 dynamical algebra to describe the continuum part of the spectrum for the modified poschl teller potential is proposed the space associated with this algebra is given in terms of a family of orthonormal functions n characterized by the parameter this set is constructed from polynomials which are orthogonal with respect to a weighting function corresponding to a poschl teller ground state an analysis of the associated algebra is investigated in detail the functions are identified with poschl teller like functions associated with different potential depths we prove that by choosing appropriately the parameter it is possible to decouple the discrete states of a given parity from the continuum part of the spectrum a discussion of the matrix elements of the dipole function operator is also included

a critical point theorem via the ekeland variational principle abstract the aim of this paper is to establish the existence of a local minimum for a continuously g teaux differentiable function possibly unbounded from below without requiring any weak continuity assumption several special cases are also emphasized moreover a novel definition of palais smale condition which is more general than the usual one is presented and a mountain pass theorem is pointed out as a consequence multiple critical points theorems are then established finally as an example of applications an elliptic dirichlet problem with critical exponent is investigated

a general variational principle and some of its applications abstract in this paper given a reflexive real banach space x and two sequentially weakly lower semicontinuous functionals on x with strongly continuous and coercive we are mainly interested in the existence of infinitely many local minima of the functional for each sufficiently large r

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

on high order denoising models and fast algorithms for vector valued images variational techniques for gray scale image denoising have been deeply investigated for many years however little research has been done for the vector valued denoising case and the very few existent works are all based on total variation regularization it is known that total variation models for denoising gray scaled images suffer from staircasing effect and there is no reason to suggest this effect is not transported into the vector valued models high order models on the contrary do not present staircasing in this paper we introduce three high order and curvature based denoising models for vector valued images their properties are analyzed and a fast multigrid algorithm for the numerical solution is provided ams subject classifications 68u10 65f10 65k10

matrix elements of u and p for the modified poschl teller potential closed analytical expressions in terms of a single sum are obtained for the matrix elements of the momentum and the natural variable u tanh x in the basis of the modified poschl teller mpt bound eigenstates these matrix elements are first expressed in terms of franck condon factors which thereafter are substituted for analytic expressions expansions of the variables p and u in terms of creation and annihilation operators associated with the mpt bound eigenfunctions are also presented

non differentiable functionals and applications to elliptic problems with discontinuous nonlinearities abstract multiple critical points theorems for non differentiable functionals are established applications both to elliptic variational hemivariational inequalities and eigenvalue problems with discontinuous nonlinearities are then presented

convergence of the alternating minimization algorithm for blind deconvolution abstract blind deconvolution refers to the image processing task of restoring the original image from a blurred version without the knowledge of the blurring function one approach that has been proposed recently t chan c wong ieee trans image process 7 1998 370 375 y you m kaveh ieee trans image process 5 1996 416 428 is a joint minimization model in which an objective function is set up consisting of three terms the data fitting term and the regularization terms for the image and the blur this model implicitly defines a one parameter family of blurred images and point spread functions psfs from which the user can decide usually using additional information which is the best restored image to find a local minimum of the objective function we use an alternating minimization am procedure y you m kaveh ieee trans image process 5 1996 416 428 in which we fix either the blur or the image and minimize respect to the other variable each step of which is a standard non blind deconvolution problem while the model is not convex and thus allows multiple solutions we have found that the am procedure always converges globally but with the converged solution depending on the initial guess in this paper we give an analysis of the am procedure which explains the convergence behavior and the observed robustness of the method

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data

alternating minimization algorithm for speckle reduction with a shifting technique speckles multiplicative noise in synthetic aperture radar sar make it difficult to interpret the observed image due to the edge preserving feature of total variation tv variational models with tv regularization have attracted much interest in reducing speckles algorithms based on the augmented lagrangian function have been proposed to efficiently solve speckle reduction variational models with tv regularization however these algorithms require inner iterations or inverses involving the laplacian operator at each iteration in this paper we adapt tseng s alternating minimization algorithm with a shifting technique to efficiently remove the speckle without any inner iterations or inverses involving the laplacian operator the proposed method is very simple and highly parallelizable therefore it is very efficient to despeckle huge size sar images numerical results show that our proposed method outperforms the state of the art algorithms for speckle reduction variational models with a tv regularizer in terms of central processing unit time