generalized besicovitch spaces and applications to deterministic homogenization the purpose of the present work is to introduce a framework which enables us to study nonlinear homogenization problems the starting point is the theory of algebras with mean value very often in physics from very simple experimental data one gets complicated structure phenomena these phenomena are represented by functions which are permanent in mean but complicated in detail in addition the functions are subject to the verification of a functional equation which in general is nonlinear the problem is therefore to give an interpretation of these phenomena using functions having the following qualitative properties they are functions that represent a phenomenon on a large scale and which vary irregularly undergoing nonperiodic oscillations on a fine scale in this work we study the qualitative properties of spaces of such functions which we call generalized besicovitch spaces and we prove general compactness results related to these spaces we then apply these results in order to study some new homogenization problems one important achievement of this work is the resolution of the generalized weakly almost periodic homogenization problem for a nonlinear pseudo monotone parabolic type operator we also give the answer to the question raised by frid and silva in their paper 35 h frid j silva homogenization of nonlinear pde s in the fourier stieltjes algebras siam j math anal 41 4 2009 1589 1620 as regards whether there exist or do not exist ergodic algebras that are not subalgebras of the fourier stieltjes algebra

variational bayesian image restoration based on a product of t distributions image prior image priors based on products have been recognized to offer many advantages because they allow simultaneous enforcement of multiple constraints however they are inconvenient for bayesian inference because it is hard to find their normalization constant in closed form in this paper a new bayesian algorithm is proposed for the image restoration problem that bypasses this difficulty an image prior is defined by imposing student t densities on the outputs of local convolutional filters a variational methodology with a constrained expectation step is used to infer the restored image numerical experiments are shown that compare this methodology to previous ones and demonstrate its advantages

a new twist two step iterative shrinkage thresholding algorithms for image restoration iterative shrinkage thresholding 1st algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems this class of problems results from combining a linear observation model with a nonquadratic regularizer e g total variation or wavelet based regularization it happens that the convergence rate of these 1st algorithms depends heavily on the linear observation operator becoming very slow when this operator is ill conditioned or ill posed in this paper we introduce two step 1st twist algorithms exhibiting much faster convergence rate than 1st for ill conditioned problems for a vast class of nonquadratic convex regularizers lscrp norms some besov norms and total variation we show that twist converges to a minimizer of the objective function for a given range of values of its parameters for noninvertible observation operators we introduce a monotonic version of twist mtwist although the convergence proof does not apply to this scenario we give experimental evidence that mtwist exhibits similar speed gains over ist the effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples

a connes amenable dual banach algebra need not have a normal virtual diagonal let g be a locally compact group and let wap g denote the space of weakly almost periodic functions on g we show that if g is a sinj group but not compact then the dual banach algebra wap g does not have a normal virtual diagonal consequently whenever g is an amenable non compact sin group wap g is an example of a connes amenable dual banach algebra without a normal virtual diagonal on the other hand there are amenable non compact locally compact groups g such that wap g does have a normal virtual diagonal

complex valued matrix differentiation techniques and key results a systematic theory is introduced for finding the derivatives of complex valued matrix functions with respect to a complex valued matrix variable and the complex conjugate of this variable in the framework introduced the differential of the complex valued matrix function is used to identify the derivatives of this function matrix differentiation results are derived and summarized in tables which can be exploited in a wide range of signal processing related situations

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

the minimum reconstruction error choice of regularization parameters some more efficient methods and their application to deconvolution problems for the simple problem of estimating a vector bf x 0 from a noisy data vector bf y b bf x 0 bf e where b is a known ill conditioned m times n matrix and bf e is an unknown white noise vector a classical regularized solution say bf x tau where tau 0 is the regularization parameter can be satisfactory provided tau is well chosen standard data based methods for choosing tau like generalized cross validation or gcv are known to give a good estimate of the value of tau which minimizes the prediction error b bf x tau b bf x 0 2 in this paper we focus on the minimization of the estimation or reconstruction error bf x tau bf x 0 2 we give sufficient conditions for the existence of two unbiased estimators of the expectation of the inner product langle bf x 0 bf x tau rangle this provides two estimates of the tau which minimizes bf x tau bf x 0 2 the first one was proposed

identification problem of two operators for nonlinear systems in banach spaces abstract we consider the identification problem of two operators having different properties for the systems governed by nonlinear evolution equations for the identification problem we show the existence of optimal solutions and present necessary optimality conditions we illustrate the approach on two examples

bm3d frames and variational image deblurring a family of the block matching 3 d bm3d algorithms for various imaging problems has been recently proposed within the framework of nonlocal patchwise image modeling in this paper we construct analysis and synthesis frames formalizing bm3d image modeling and use these frames to develop novel iterative deblurring algorithms we consider two different formulations of the deblurring problem i e one given by the minimization of the single objective function and another based on the generalized nash equilibrium gne balance of two objective functions the latter results in the algorithm where deblurring and denoising operations are decoupled the convergence of the developed algorithms is proved simulation experiments show that the decoupled algorithm derived from the gne formulation demonstrates the best numerical and visual results and shows superiority with respect to the state of the art in the field confirming a valuable potential of bm3d frames as an advanced image modeling tool

on some bayesian regularization methods for image restoration methods are reviewed for choosing regularized restorations in image processing in particular a method developed by galatsanos and katsaggelos see ibid vol 1 p 322 336 1992 is given a bayesian interpretation and is compared with other bayesian and non bayesian alternatives a small illustrative example is provided and a complement is provided for the discussion of noise variance estimation of galatsanos et al

a regularization parameter in discrete ill posed problems the tikhonov regularization method for discrete ill posed problems is considered for the practical choice of the regularization parameter alpha some authors use a plot of the norm of the regularized solution versus the norm of the residual vector for all alpha considered this paper contains an analysis of the shape of this plot and gives a theoretical justification for choosing the regularization parameter so it is related to the l corner of the plot considered in the logarithmic scale moreover a new criterion for choosing alpha is introduced independent of the shape of the plot which gives a new interpretation of the corner criterion mentioned above the existence of l corner is discussed

blind deconvolution using a variational approach to parameter image and blur estimation following the hierarchical bayesian framework for blind deconvolution problems in this paper we propose the use of simultaneous autoregressions as prior distributions for both the image and blur and gamma distributions for the unknown parameters hyperparameters of the priors and the image formation noise we show how the gamma distributions on the unknown hyperparameters can be used to prevent the proposed blind deconvolution method from converging to undesirable image and blur estimates and also how these distributions can be inferred in realistic situations we apply variational methods to approximate the posterior probability of the unknown image blur and hyperparameters and propose two different approximations of the posterior distribution one of these approximations coincides with a classical blind deconvolution method the proposed algorithms are tested experimentally and compared with existing blind deconvolution methods

bayesian restoration using a new nonstationary edge preserving image prior in this paper we propose a class of image restoration algorithms based on the bayesian approach and a new hierarchical spatially adaptive image prior the proposed prior has the following two desirable features first it models the local image discontinuities in different directions with a model which is continuous valued thus it preserves edges and generalizes the on off binary line process idea used in previous image priors within the context of markov random fields mrfs second it is gaussian in nature and provides estimates that are easy to compute using this new hierarchical prior two restoration algorithms are derived the first is based on the maximum a posteriori principle and the second on the bayesian methodology numerical experiments are presented that compare the proposed algorithms among themselves and with previous stationary and non stationary mrf based with line process algorithms these experiments demonstrate the advantages of the proposed prior

a new sure approach to image denoising interscale orthonormal wavelet thresholding this paper introduces a new approach to orthonormal wavelet image denoising instead of postulating a statistical model for the wavelet coefficients we directly parametrize the denoising process as a sum of elementary nonlinear processes with unknown weights we then minimize an estimate of the mean square error between the clean image and the denoised one the key point is that we have at our disposal a very accurate statistically unbiased mse estimate stein s unbiased risk estimate that depends on the noisy image alone not on the clean one like the mse this estimate is quadratic in the unknown weights and its minimization amounts to solving a linear system of equations the existence of this a priori estimate makes it unnecessary to devise a specific statistical model for the wavelet coefficients instead and contrary to the custom in the literature these coefficients are not considered random any more we describe an interscale orthonormal wavelet thresholding algorithm based on this new approach and show its near optimal performance both regarding quality and cpu requirement by comparing it with the results of three state of the art nonredundant denoising algorithms on a large set of test images an interesting fallout of this study is the development of a new group delay based parent child prediction in a wavelet dyadic tree

relaxed trajectories of integrodifferential equations and optimal control on banach space relaxation optimal control for a class of integrodifferential equation associated with fractional power operator is investigated by gronwall lemma with singularity and delay the existence of mild solutions of integrodifferential equations is proved introducing finitely additive measure control we convexificate original control systems and obtain corresponding relaxed control systems approximation results and equivalent relationship between control differential equations and integrodifferential inclusion are given existence of relaxed optimal control is verified and one relaxed optimal control is just an original one under generalized cesari conditions

existence theory and the maximum principle for relaxed infinite dimensional optimal control problems existence theorems are considered for relaxed optimal control problems described by semilinear systems in banach spaces relaxed controls are used whose values are finitely additive probability measures this class of relaxed controls does not require special assumptions such as compactness on the control set under suitable conditions relaxed trajectories coincide with those obtained from differential inclusions existence theorems for relaxed controls are obtained that apply to distributed parameter systems described by semilinear parabolic and wave equations as well as a version of pontryagin s maximum principle for relaxed optimal control problems

variational bayesian image restoration with a product of spatially weighted total variation image priors in this paper a new image prior is introduced and used in image restoration this prior is based on products of spatially weighted total variations tv these spatial weights provide this prior with the flexibility to better capture local image features than previous tv based priors bayesian inference is used for image restoration with this prior via the variational approximation the proposed restoration algorithm is fully automatic in the sense that all necessary parameters are estimated from the data and is faster than previous similar algorithms numerical experiments are shown which demonstrate that image restoration based on this prior compares favorably with previous state of the art restoration algorithms

optimal control for a class of strongly nonlinear impulsive equations in banach spaces abstract in this paper we consider the optimal control problem of system governed by a class of strongly nonlinear impulsive evolution equations based on the existence of strongly nonlinear impulsive evolution equations which contain nonlinear monotone operators and nonmonotone perturbations we prove the existence of optimal controls for the general lagrange problem the necessary conditions for optimality are also presented

closed loop restoration approach to blurry images based on machine learning and feedback optimization blind image deconvolution bid aims to remove or reduce the degradations that have occurred during the acquisition or processing it is a challenging ill posed problem due to a lack of enough information in degraded image for unambiguous recovery of both point spread function psf and clear image although recently many powerful algorithms appeared however it is still an active research area due to the diversity of degraded images as well as degradations closed loop control systems are characterized with their powerful ability to stabilize the behavior response and overcome external disturbances by designing an effective feedback optimization in this paper we employed feedback control to enhance the stability of bid by driving the current estimation quality of psf to the desired level without manually selecting restoration parameters and using an effective combination of machine learning with feedback optimization the foremost challenge when designing a feedback structure is to construct or choose a suitable performance metric as a controlled index and a feedback information our proposed quality metric is based on the blur assessment of deconvolved patches to identify the best psf and computing its relative quality the kalman filter based extremum seeking approach is employed to find the optimum value of controlled variable to find better restoration parameters learning algorithms such as multilayer perceptron and bagged decision trees are used to estimate the generic psf support size instead of trial and error methods the problem is modeled as a combination of pattern classification and regression using multiple training features including noise metrics blur metrics and low level statistics multi objective genetic algorithm is used to find key patches from multiple saliency maps which enhance performance and save extra computation by avoiding ineffectual regions of the image the proposed scheme is shown to outperform corresponding open loop schemes which often fails or needs many assumptions regarding images and thus resulting in sub optimal results

reiterated ergodic algebras and applications we redefine the homogenization algebras without requiring the separability assumption we show that this enables one to treat more complicated homogenization problems than those solved by the previous theory in particular we exhibit an example of algebra which contrary to the algebra of almost periodic functions induces no homogenization algebra we prove some general compactness results which are then applied to the resolution of some homogenization problems related to the generalized reynolds type equations and to some nonlinear hyperbolic equations

a study of methods of choosing the smoothing parameter in image restoration by regularization the method of regularization is portrayed as providing a compromise between fidelity to the data and smoothness with the tradeoff being determined by a scalar smoothing parameter various ways of choosing this parameter are discussed in the case of quadratic regularization criteria they are compared algebraically and their statistical properties are comparatively assessed from the results of all extensive simulation study based on simple images

generalized sure for exponential families applications to regularization stein s unbiased risk estimate sure was proposed by stein for the independent identically distributed i i d gaussian model in order to derive estimates that dominate least squares ls recently the sure criterion has been employed in a variety of denoising problems for choosing regularization parameters that minimize an estimate of the mean squared error mse however its use has been limited to the i i d case which precludes many important applications in this paper we begin by deriving a sure counterpart for general not necessarily i i d distributions from the exponential family this enables extending the sure design technique to a much broader class of problems based on this generalization we suggest a new method for choosing regularization parameters in penalized ls estimators we then demonstrate its superior performance over the conventional generalized cross validation and discrepancy approaches in the context of image deblurring and deconvolution the sure technique can also be used to design estimates without predefining their structure however allowing for too many free parameters impairs the estimate s performance to address this inherent tradeoff we propose a regularized sure objective and demonstrate its use in the context of wavelet denoising

optimal spline smoothing of fmri time series by generalized cross validation linear parametric regression models of fmri time series have correlated residuals one approach to address this problem is to condition the autocorrelation structure by temporal smoothing smoothing splines with the degree of smoothing selected by generalized cross validation gcv spline provide a method to find an optimal smoother for an fmri time series the purpose of this study was to determine if gcv spline of fmri time series yields unbiased variance estimates of linear regression model parameters gcv spline was evaluated with a real fmri data set and bias of the variance estimator was computed for simulated time series with autocorrelation structures derived from fmri data this study only considered fmri experimental designs of boxcar type the results from the real data suggest that gcv spline determines appropriate amounts of smoothing the simulations show that the variance estimates are on average unbiased the unbiased variance estimates come at some cost to signal detection efficiency this study demonstrates that gcv spline is an appropriate method for smoothing fmri time series

interpolation sets and the size of quotients of function spaces on a locally compact group the research of the second author was partially supported by the spanish ministry of science including feder funds grant mtm2011 23118 and fundacio caixa castello bancaixa grant number p1 1b2014 35

algebraic structure of semigroup compactifications pym s and veech s theorems and strongly prime points abstract the spectrum of an admissible subalgebra a g of luc g g the algebra of right uniformly continuous functions on a locally compact group g constitutes a semigroup compactification g a of g in this paper we analyze the algebraic behaviour of those points of g a that lie in the closure of a g sets sets whose characteristic function can be approximated by functions in a g this analysis provides a common ground for far reaching generalizations of veech s property the action of g on g luc g is free and pym s local structure theorem this approach is linked to the concept of translation compact set recently developed by the authors and leads to characterizations of strongly prime points in g a points that do not belong to the closure of g g where g g a g all these results will be applied to show that in many of the most important algebras left invariant means of a g when such means are present are supported in the closure of g g

a sure approach for digital signal image deconvolution problems in this paper we are interested in the classical problem of restoring data degraded by a convolution and the addition of a white gaussian noise the originality of the proposed approach is twofold first we formulate the restoration problem as a nonlinear estimation problem leading to the minimization of a criterion derived from stein s unbiased quadratic risk estimate secondly the deconvolution procedure is performed using any analysis and synthesis frames that can be overcomplete or not new theoretical results concerning the calculation of the variance of the stein s risk estimate are also provided in this work simulations carried out on natural images show the good performance of our method with respect to conventional wavelet based restoration methods

regularity for solutions of nonlinear delay evolution equations abstract the existence of solutions for a class of strongly nonlinear delay evolution equations is studied the regularity and a variation of solutions of the given equations are also given

automatic parameter selection for denoising algorithms using a no reference measure of image content across the field of inverse problems in image and video processing nearly all algorithms have various parameters which need to be set in order to yield good results in practice usually the choice of such parameters is made empirically with trial and error if no ground truth reference is available some analytical methods such as cross validation and stein s unbiased risk estimate sure have been successfully used to set such parameters however these methods tend to be strongly reliant on restrictive assumptions on the noise and also computationally heavy in this paper we propose a no reference metric q which is based upon singular value decomposition of local image gradient matrix and provides a quantitative measure of true image content i e sharpness and contrast as manifested in visually salient geometric features such as edges in the presence of noise and other disturbances this measure 1 is easy to compute 2 reacts reasonably to both blur and random noise and 3 works well even when the noise is not gaussian the proposed measure is used to automatically and effectively set the parameters of two leading image denoising algorithms ample simulated and real data experiments support our claims furthermore tests using the tid2008 database show that this measure correlates well with subjective quality evaluations for both blur and noise distortions

higher degree total variation hdtv regularization for image recovery we introduce novel image regularization penalties to overcome the practical problems associated with the classical total variation tv scheme motivated by novel reinterpretations of the classical tv regularizer we derive two families of functionals involving higher degree partial image derivatives we term these families as isotropic and anisotropic higher degree tv hdtv penalties respectively the isotropic penalty is the mixed norm of the directional image derivatives while the anisotropic penalty is the separable norm of directional derivatives these functionals inherit the desirable properties of standard tv schemes such as invariance to rotations and translations preservation of discontinuities and convexity the use of mixed norms in isotropic penalties encourages the joint sparsity of the directional derivatives at each pixel thus encouraging isotropic smoothing in contrast the fully separable norm in the anisotropic penalty ensures the preservation of discontinuities while continuing to smooth along the line like features this scheme thus enhances the linenlike image characteristics analogous to standard tv we also introduce efficient majorize minimize algorithms to solve the resulting optimization problems the numerical comparison of the proposed scheme with classical tv penalty current second degree methods and wavelet algorithms clearly demonstrate the performance improvement specifically the proposed algorithms minimize the staircase and ringing artifacts that are common with tv and wavelet schemes while better preserving the singularities we also observe that anisotropic hdtv penalty provides consistently improved reconstructions compared with the isotropic hdtv penalty

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

relaxed control for a class of strongly nonlinear impulsive evolution equations relaxed control for a class of strongly nonlinear impulsive evolution equations is investigated existence of solutions of strongly nonlinear impulsive evolution equations is proved and properties of original and relaxed trajectories are discussed the existence of optimal relaxed control and relaxation results are also presented for illustration one example is given

approximable wap and luc interpolation sets abstract extending and unifying concepts extensively used in the literature we introduce the notion of approximable interpolation sets for algebras of functions on locally compact groups especially for weakly almost periodic functions and for uniformly continuous functions we characterize approximable interpolation sets both in combinatorial terms and in terms of the luc and wap compactifications and analyse some of their properties

relaxation of nonlinear impulsive controlled systems on banach spaces relaxation control for a class of semilinear impulsive controlled systems is investigated existence of mild solutions for semilinear impulsive controlled systems is proved by introducing a regular countably additive measure we convexify the original control systems and obtain the corresponding relaxed control systems the existence of optimal relaxed controls and relaxation results is also proved

properties of relaxed trajectories of evolution equations and optimal control this paper presents a density result for a general class of nonlinear infinite dimensional systems that includes an unbounded linear part a result on the existence of optimal controls for a corresponding uncertain system is also proved

a variational approach for bayesian blind image deconvolution in this paper the blind image deconvolution bid problem is addressed using the bayesian framework in order to solve for the proposed bayesian model we present a new methodology based on a variational approximation which has been recently introduced for several machine learning problems and can be viewed as a generalization of the expectation maximization em algorithm this methodology reaps all the benefits of a full bayesian model while bypassing some of its difficulties we present three algorithms that solve the proposed bayesian problem in closed form and can be implemented in the discrete fourier domain this makes them very cost effective even for very large images we demonstrate with numerical experiments that these algorithms yield promising improvements as compared to previous bid algorithms furthermore the proposed methodology is quite general with potential application to other bayesian models for this and other imaging problems

multi wiener sure let deconvolution in this paper we propose a novel deconvolution algorithm based on the minimization of a regularized stein s unbiased risk estimate sure which is a good estimate of the mean squared error we linearly parametrize the deconvolution process by using multiple wiener filters as elementary functions followed by undecimated haar wavelet thresholding due to the quadratic nature of sure and the linear parametrization the deconvolution problem finally boils down to solving a linear system of equations which is very fast and exact the linear coefficients i e the solution of the linear system of equations constitute the best approximation of the optimal processing on the wiener haar threshold basis that we consider in addition the proposed multi wiener sure let approach is applicable for both periodic and symmetric boundary conditions and can thus be used in various practical scenarios the very competitive both in computation time and quality results show that the proposed algorithm which can be interpreted as a kind of nonlinear wiener processing can be used as a basic tool for building more sophisticated deconvolution algorithms

monte carlo sure a black box optimization of regularization parameters for general denoising algorithms we consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white gaussian noise to achieve this we propose to minimize stein s unbiased risk estimate sure which provides a means of assessing the true mean squared error mse purely from the measured data without need for any knowledge about the noise free signal specifically we present a novel monte carlo technique which enables the user to calculate sure for an arbitrary denoising algorithm characterized by some specific parameter setting our method is a black box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form this therefore permits the use of sure for optimization of a wide variety of denoising algorithms we justify our claims by presenting experimental results for sure based optimization of a series of popular image denoising algorithms such as total variation denoising wavelet soft thresholding and wiener filtering smoothing splines in the process we also compare the performance of these methods we demonstrate numerically that sure computed using the new approach accurately predicts the true mse for all the considered algorithms we also show that sure uncovers the optimal values of the parameters in all cases

parameter estimation for blind and non blind deblurring using residual whiteness measures image deblurring id is an ill posed problem typically addressed by using regularization or prior knowledge on the unknown image and also on the blur operator in the blind case id is often formulated as an optimization problem where the objective function includes a data term encouraging the estimated image and blur in blind id to explain the observed data well typically the squared norm of a residual plus a regularizer that penalizes solutions deemed undesirable the performance of this approach depends critically among other things on the relative weight of the regularizer the regularization parameter and on the number of iterations of the algorithm used to address the optimization problem in this paper we propose new criteria for adjusting the regularization parameter and or the number of iterations of id algorithms the rationale is that if the recovered image and blur in blind id is well estimated the residual image is spectrally white contrarily a poorly deblurred image typically exhibits structured artifacts e g ringing oversmoothness yielding residuals that are not spectrally white the proposed criterion is particularly well suited to a recent blind id algorithm that uses continuation i e slowly decreases the regularization parameter along the iterations in this case choosing this parameter and deciding when to stop are one and the same thing our experiments show that the proposed whiteness based criteria yield improvements in snr on average only 0 15 db below those obtained by clairvoyantly stopping the algorithm at the best snr we also illustrate the proposed criteria on non blind id reporting results that are competitive with state of the art criteria such as monte carlo based gsure and projected sure which however are not applicable for blind id

hessian based norm regularization for image restoration with biomedical applications we present nonquadratic hessian based regularization methods that can be effectively used for image restoration problems in a variational framework motivated by the great success of the total variation tv functional we extend it to also include second order differential operators specifically we derive second order regularizers that involve matrix norms of the hessian operator the definition of these functionals is based on an alternative interpretation of tv that relies on mixed norms of directional derivatives we show that the resulting regularizers retain some of the most favorable properties of tv i e convexity homogeneity rotation and translation invariance while dealing effectively with the staircase effect we further develop an efficient minimization scheme for the corresponding objective functions the proposed algorithm is of the iteratively reweighted least square type and results from a majorization minimization approach it relies on a problem specific preconditioned conjugate gradient method which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time we validate the overall proposed regularization framework through deblurring experiments under additive gaussian noise on standard and biomedical images

a convergent least squares regularized blind deconvolution approach the aim of this work is to present a new and efficient optimization method for the solution of blind deconvolution problems with data corrupted by gaussian noise which can be reformulated as a constrained minimization problem whose unknowns are the point spread function psf of the acquisition system and the true image the objective function we consider is the weighted sum of the least squares fit to data discrepancy and possible regularization terms accounting for specific features to be preserved in both the image and the psf the solution of the corresponding minimization problem is addressed by means of a proximal alternating linearized minimization palm algorithm in which the updating procedure is made up of one step of a gradient projection method along the arc and the choice of the parameter identifying the steplength in the descent direction is performed automatically by exploiting the optimality conditions of the problem the resulting approach is a particular case of a general scheme whose convergence to stationary points of the constrained minimization problem has been recently proved the effectiveness of the iterative method is validated in several numerical simulations in image reconstruction problems

weakly almost periodic functions and fourier stieltjes algebras of locally compact groups a noncompact locally compact group g is called an eberlein group if w g b g where w g is the algebra of continuous weakly almost periodic functions on g and b g is the uniform closure of the fourier stieltjes algebra of g we show that if g is a noncompact in group or a noncompact nilpotent group then w g b g contains a linear isometric copy of l in particular g is not an eberlein group on the other hand finite direct products of eucidean motion groups and by a result of w veech noncompact semisimple analytic groups with finite centers are eberlein groups 1 introduction let g be a locally compact group c g the space of bounded continuous complex valued functions on g with the sup norm w g the space of continuous weakly almost periodic w a p functions on g b g the fourier stieltjes algebra of g and b g its uniform closure in c g it is known that b g c w g and if g is compact then c g w g b g therefore in this paper g is assumed to be noncompact in his efforts to characterize w a p functions w f eberlein first raised the question whether for abelian g b g w g w rudin 24 made the initial contribution to this question by showing that b g c w g if g is abelian and contains a discrete subgroup which is not of bounded order and d ramirez 23 by studying discrete abelian groups with bounded orders together with the structure theorem for locally compact abelian groups was able to show that rudin s conclusion holds for all abelian groups see the monographs 4 and 11 for complete accounts of this result in this paper we will extend rudin ramirez result to include many nonabelian groups our main result is the following

weakly almost periodic functions and thin sets in discrete groups a subset e of an infinite discrete group g is called i an rw set if any bounded function on g supported by e is weakly almost periodic ii a weak p sidon set 1 p 2 if on 11 e the lp norm is bounded by a constant times the maximal c norm of i i g iii a t set if xe n e and ex n e are finite whenever x e and iv an ft set if it is a finite union of t sets in this paper we study relationships among these four classes of thin sets we show among other results that a every infinite group g contains an rw set which is not an ft set b countable weak p sidon sets 1 p 4 3 are f t sets

properties of relaxed trajectories for a class of nonlinear evolution equations on a banach space in this paper we consider a class of nonconvex control problems in a banach space which is convexified by introducing measure valued controls we study some topological properties of the set of trajectories of the original system and that of the relaxed system it is shown that under certain reasonable assumptions the set of trajectories of the original system is dense in an appropriate topology in the set of relaxed trajectories as a corollary of this result it follows that the attainable set of the original system is dense in that of the relaxed system these results are directly useful in terminal optimization problems and time optimal control problems for illustration we present two examples in 6

properties of the relaxed trajectories of evolution equations and optimal control this paper considers a class of nonconvex control problems in a banach space governed by a semilinear evolution equation and establishes the existence of admissible pairs then the introduction of measure valued controls convexifies the system and under general assumptions it is shown that the set of trajectories of the original system is dense in an appropriate topology in the set of relaxed trajectories some useful topological properties of the set of relaxed trajectories are also determined furthermore some optimization problems are solved and in many cases the values of the original and relaxed problems are shown to be equal finally another relaxation result is proved for a different class of systems described by a general nonlinear evolution equation

relaxed controls for a class of strongly nonlinear delay evolution equations abstract relaxed controls for a class of strongly nonlinear delay evolution equations are investigated existence of solutions of strongly nonlinear delay equations is proved and properties of original and relaxed trajectories are discussed the existence of optimal relaxed controls and relaxation result are also presented for illustration two examples are given

regularization parameter selection for nonlinear iterative image restoration and mri reconstruction using gcv and sure based methods regularized iterative reconstruction algorithms for imaging inverse problems require selection of appropriate regularization parameter values we focus on the challenging problem of tuning regularization parameters for nonlinear algorithms for the case of additive possibly complex gaussian noise generalized cross validation gcv and weighted mean squared error mse approaches based on stein s unbiased risk estimate sure need the jacobian matrix of the nonlinear reconstruction operator representative of the iterative algorithm with respect to the data we derive the desired jacobian matrix for two types of nonlinear iterative algorithms a fast variant of the standard iterative reweighted least squares method and the contemporary split bregman algorithm both of which can accommodate a wide variety of analysis and synthesis type regularizers the proposed approach iteratively computes two weighted sure type measures predicted sure and projected sure which require knowledge of noise variance 2 and gcv which does not need 2 for these algorithms we apply the methods to image restoration and to magnetic resonance image mri reconstruction using total variation and an analysis type l1 regularization we demonstrate through simulations and experiments with real data that minimizing predicted sure and projected sure consistently lead to near mse optimal reconstructions we also observe that minimizing gcv yields reconstruction results that are near mse optimal for image restoration and slightly suboptimal for mri theoretical derivations in this paper related to jacobian matrix evaluations can be extended in principle to other types of regularizers and reconstruction algorithms

eberlein oligomorphic groups we study the fourier stieltjes algebra of roelcke precompact non archimedean polish groups and give a model theoretic description of the hilbert compactification of these groups we characterize the family of such groups whose fourier stieltjes algebra is dense in the algebra of weakly almost periodic functions those are exactly the automorphism groups of 0 stable 0 categorical structures this analysis is then extended to all semitopological semigroup compactifica tions s of such a group s is hilbert representable if and only if it is an inverse semigroup we also show that every factor of the hilbert compactification is hilbert representable

homogenization of a stochastic nonlinear reaction diffusion equation with a large reaction term the almost periodic framework abstract homogenization of a stochastic nonlinear reaction diffusion equation with a large nonlinear term is considered under a general besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation namely the stochastic nonlinear convection diffusion equation which we explicitly derive in terms of appropriate functionals we study some particular cases such as the periodic framework and many others this is achieved under a suitable generalized concept of convergence for stochastic processes