fixed points of uniformly lipschitzian mappings abstract two fixed point theorems for uniformly lipschitzian mappings in metric spaces due respectively to e lifsic and to t c lim and h k xu are compared within the framework of the so called cat 0 spaces it is shown that both results apply in this setting and that lifsic s theorem gives a sharper result also a new property is introduced that yields a fixed point theorem for uniformly lipschitzian mappings in a class of hyperconvex spaces a class which includes those possessing property p of lim and xu

convergence of an iterative method for total variation denoising in total variation denoising one attempts to remove noise from a signal or image by solving a nonlinear minimization problem involving a total variation criterion several approaches based on this idea have recently been shown to be very effective particularly for denoising functions with discontinuities this paper analyzes the convergence of an iterative method for solving such problems the iterative method involves a lagged diffusivity approach in which a sequence of linear diffusion problems are solved global convergence in a finite dimensional setting is established and local convergence properties including rates and their dependence on various parameters are examined

inequalities in metric spaces with applications abstract analogues of the parallelogram identity and the cn inequality of bruhat and tits in uniformly convex metric spaces are established as an application of the new inequalities we prove two fixed point results for single valued and multi valued lipschitzian mappings

a parallel proximal algorithm for anisotropic total variation minimization total variation tv is a one of the most popular regularizers for stabilizing the solution of ill posed inverse problems this paper proposes a novel proximal gradient algorithm for minimizing tv regularized least squares cost functionals unlike traditional methods that require nested iterations for computing the proximal step of tv our algorithm approximates the latter with several simple proximals that have closed form solutions we theoretically prove that the proposed parallel proximal method achieves the tv solution with arbitrarily high precision at a global rate of converge that is equivalent to the fast proximal gradient methods the results in this paper have the potential to enhance the applicability of tv for solving very large scale imaging inverse problems

cat k spaces weak convergence and fixed points abstract in this paper we show that some of the recent results on fixed point for cat 0 spaces still hold true for cat 1 spaces and so for any cat k space under natural boundedness conditions we also introduce a new notion of convergence in geodesic spaces which is related to the convergence and applied to study some aspects on the geometry of cat 0 spaces at this point two recently posed questions in w a kirk b panyanak a concept of convergence in geodesic spaces nonlinear anal 68 12 2008 3689 3696 are answered in the negative the work finishes with the study of the lifsic characteristic and property p of lim xu to derive fixed point results for uniformly lipschitzian mappings in cat k spaces a conjecture raised in s dhompongsa w a kirk b sims fixed points of uniformly lipschitzian mappings nonlinear anal 65 2006 762 772 on the lifsic characteristic function of cat k spaces is solved in the positive

basic sequences embeddings and the uniqueness of the symmetric structure in unitary matrix spaces abstract for every symmetric sequence space e we denote by ce the associated unitary matrix space we show that every basic sequence in ce has a subsequence which embeds in l2 e this reduces many problems about ce to the analogous problems on e we also study the asymptotic behavior of the embeddings of one unitary sequence space into another and the problem of the uniqueness of the symmetric structure of unitary matrix spaces

efficient multiplicative noise removal method using isotropic second order total variation to overcome and reduce the undesirable staircase effect commonly met in the total variation tv regularization based multiplicative noise removal methods a novel multiplicative noise removal model based on isotropic second order total variation isotv is proposed under the maximum a posteriori map framework under the spectral decomposition framework the isotv is first transformed into an equivalent formulation as a novel weighted l 1 l 2 mixed norm of the second order image derivatives then an efficient alternating iterative algorithm is designed to solve the proposed model finally we prove in detail the convergence of the proposed algorithm a set of experiments on both standard and medical images show that the proposed isotv method yields state of the art results both in terms of peak signal to noise ratio psnr and image perception quality specifically the proposed isotv method can better reduce the staircase effect and preserve image edges more sharpness with medical applications

generalized higher degree total variation hdtv regularization we introduce a family of novel image regularization penalties called generalized higher degree total variation hdtv these penalties further extend our previously introduced hdtv penalties which generalize the popular total variation tv penalty to incorporate higher degree image derivatives we show that many of the proposed second degree extensions of tv are special cases or are closely approximated by a generalized hdtv penalty additionally we propose a novel fast alternating minimization algorithm for solving image recovery problems with hdtv and generalized hdtv regularization the new algorithm enjoys a tenfold speed up compared with the iteratively reweighted majorize minimize algorithm proposed in a previous paper numerical experiments on 3d magnetic resonance images and 3d microscopy images show that hdtv and generalized hdtv improve the image quality significantly compared with tv

banach saks property in marcinkiewicz spaces abstract we catalogue all marcinkiewicz function and sequence spaces with the banach saks property and present necessary and sufficient conditions for a wide subclass of spaces to possess the p banach saks property 1 p we apply our results to several open problems

the banach saks p property we obtain general theorems which enable the calculation of the banach saks index set for rearrangement invariant function spaces in terms of their boyd indices for many important classes of rearrangement invariant spaces the index set is non trivial if and only if the boyd indices of the underlying space are non trivial although the latter equivalence does not hold in full generality we also study banach saks index sets defined for weakly null sequences of identically distributed independent random variables and their connections with the classical banach saks index set

subspace structure of lorentz lp q spaces and strictly singular operators abstract we study subspaces of lorentz l p q spaces and provide an easy to check characterization of strictly singular operators defined on these spaces as an application we obtain stability under duality for the class of strictly singular operators on l p q spaces extending a theorem of l weis for operators on l p spaces

on uniformly lipschitzian multivalued mappings in banach and metric spaces abstract let x d be a metric space a mapping t x x is said to be uniformly lipschitzian if there exists a constant k such that d t n x t n y k d x y for all x y x and n 1 it is known that such mappings always have fixed points in certain metric spaces for k 1 provided k is sufficiently near 1 these spaces include uniformly convex metric and banach spaces as well as metric spaces having lifsic characteristic greater than 1 a uniformly lipschitzian concept for multivalued mappings is introduced in this paper and multivalued analogues of these results are obtained

mathematical models for local nontexture inpaintings dedicated to stanley osher on the occasion of his 60th birthday abstract inspired by the recent work of bertalmio et al on digital inpaintings siggraph 2000 we develop general mathematical models for local inpaintings of nontexture images on smooth regions inpaintings are connected to the harmonic and biharmonic extensions and inpainting orders are analyzed for inpaintings involving the recovery of edges we study a variational model that is closely connected to the classical total variation tv denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 other models are also discussed based on the mumford shah regularity comm pure appl math xlii 1989 pp 577 685 and curvature driven diffusions cdd of chan and shen j visual comm image rep 12 2001 the broad applications of the inpainting models are demonstrated through restoring scratched old photos disocclusion in vision analysis text removal digital zooming and edge based image coding

on uniformly convex functions abstract in this paper we study uniformly convex functions and uniformly convex functions at a point giving some properties and characterizations of them further we give some examples and applications of these types of functions

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

on some lp inequalities in best approximation theory h x jxp p ax p is strictly increasing in the interval p j o thus h x 0 has a unique solution x p note that x l 2 1 x p is strictly increasing in p and x 0 is the unique solution of the equation p 2 xp p l f2 1 in the interval 0 6 x 1 1 1 in 11 we proved the inequality

phase retrieval through regularization for seismic problems abstractthe most advanced data processing imaging algorithms aim at optimizing a suitably defined cost function consisting of a data misfit term posed in the data domain and a regularization term for strong phase errors in the data however such algorithms fail to achieve an acceptable solution because they force us to match the incorrect phase information to remedy this deficiency we replaced the time domain misfit term by a fourier magnitude fidelity term to recover the desired signal from just the amplitude spectrum of the observations such a problem is known as phase retrieval and it is a subject of interest for those who cannot measure the phase information about the system under study the proposed phase retrieval problem is solved by a fast iterative shrinkage thresholding algorithm and it is used to tackle two common phase problems arising in seismic processing data recovery in the presence of residual static shifts and deconvolution with a missing wavelet phase under sparsity constrain

an efficient nonconvex regularization for wavelet frame and total variation based image restoration in order to improve the quality of the image many works impose explicit priors on the solution to regularize the ill posed inverse problem in this paper we propose a hybrid variational model which takes advantages of the wavelet tight frame model and the total variation model for image restoration the core of the method is a new nonconvex penalty function that is designed for efficient minimization by means of the firm thresholding and soft shrinkage operations we address the proposed optimization problem by converting it to a constrained problem with variable splitting and using the alternating direction method of multipliers numerical examples for image restoration are given to show that the proposed method outperforms some existing methods in terms of the peak signal to noise ratio and structural similarity index

image restoration using total variation with overlapping group sparsity image restoration is one of the most fundamental issues in imaging science total variation regularization is widely used in image restoration problems for its capability to preserve edges in the literature however it is also well known for producing staircase artifacts in this work we extend the total variation with overlapping group sparsity which we previously developed for one dimension signal processing to image restoration a convex cost function is given and an efficient algorithm is proposed for solving the corresponding minimization problem in the experiments we compare our method with several state of the art methods the results illustrate the efficiency and effectiveness of the proposed method in terms of psnr and computing time

the classification problem for nonatomic weak lp spaces abstract the aim of the paper is to study the isomorphic structure of the weak l p space l p when is a purely nonatomic measure space using maharam s classification of measure algebras it is shown that every such l p is isomorphic to a weak l p space defined on a weighted direct sum of product measure spaces of the type 2 several isomorphic invariants are then obtained in particular it is found that there is a notable difference between the case 1 p 2 and the case where 2 p applying the methods developed we obtain an isomorphic classification of the purely nonatomic weak l p spaces in a special case

fixed points of uniformly lipschitzian mappings in spaces with uniformly normal structure dans un espace de banach x on etudie quand reflexivite et structure normale sont suffisantes pour assurer pour de bon 1 la propriete de point fixe pour des auto applications uniformement lipschitziennes

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

deblurring poisson noisy images by total variation with overlapping group sparsity deblurring poisson noisy images has recently been subject of an increasingly amount of works in various applications such as astronomical imaging fluorescent confocal microscopy imaging single particle emission computed tomography spect and positron emission tomography pet many works promote the introduction of an explicit prior on the solution to regularize the ill posed inverse problem for improving the quality of the images in this paper we consider using the total variation with overlapping group sparsity as a prior information the proposed method can avoid staircase effect and preserve edges in the restored images after converting the proposed model to a constrained problem by variable splitting we solve the corresponding problem with the alternating direction method of multipliers admm numerical examples for deblurring poisson noisy images are given to show that the proposed method outperforms some existing methods in terms of the signal to noise ratio relative error and structural similarity index

poisson image reconstruction with hessian schatten norm regularization poisson inverse problems arise in many modern imaging applications including biomedical and astronomical ones the main challenge is to obtain an estimate of the underlying image from a set of measurements degraded by a linear operator and further corrupted by poisson noise in this paper we propose an efficient framework for poisson image reconstruction under a regularization approach which depends on matrix valued regularization operators in particular the employed regularizers involve the hessian as the regularization operator and schatten matrix norms as the potential functions for the solution of the problem we propose two optimization algorithms that are specifically tailored to the poisson nature of the noise these algorithms are based on an augmented lagrangian formulation of the problem and correspond to two variants of the alternating direction method of multipliers further we derive a link that relates the proximal map of an lp norm with the proximal map of a schatten matrix norm of order p this link plays a key role in the development of one of the proposed algorithms finally we provide experimental results on natural and biological images for the task of poisson image deblurring and demonstrate the practical relevance and effectiveness of the proposed framework

variational phase imaging using the transport of intensity equation we introduce a variational phase retrieval algorithm for the imaging of transparent objects our formalism is based on the transport of intensity equation tie which relates the phase of an optical field to the variation of its intensity along the direction of propagation tie practically requires one to record a set of defocus images to measure the variation of intensity we first investigate the effect of the defocus distance on the retrieved phase map based on our analysis we propose a weighted phase reconstruction algorithm yielding a phase map that minimizes a convex functional the method is nonlinear and combines different ranges of spatial frequencies depending on the defocus value of the measurements in a regularized fashion the minimization task is solved iteratively via the alternating direction method of multipliers our simulations outperform commonly used linear and nonlinear tie solvers we also illustrate and validate our method on real microscopy data of hela cells

a general framework for regularized similarity based image restoration any image can be represented as a function defined on a weighted graph in which the underlying structure of the image is encoded in kernel similarity and associated laplacian matrices in this paper we develop an iterative graph based framework for image restoration based on a new definition of the normalized graph laplacian we propose a cost function which consists of a new data fidelity term and regularization term derived from the specific definition of the normalized graph laplacian the normalizing coefficients used in the definition of the laplacian and associated regularization term are obtained using fast symmetry preserving matrix balancing this results in some desired spectral properties for the normalized laplacian such as being symmetric positive semidefinite and returning zero vector when applied to a constant image our algorithm comprises of outer and inner iterations where in each outer iteration the similarity weights are recomputed using the previous estimate and the updated objective function is minimized using inner conjugate gradient iterations this procedure improves the performance of the algorithm for image deblurring where we do not have access to a good initial estimate of the underlying image in addition the specific form of the cost function allows us to render the spectral analysis for the solutions of the corresponding linear equations in addition the proposed approach is general in the sense that we have shown its effectiveness for different restoration problems including deblurring denoising and sharpening experimental results verify the effectiveness of the proposed algorithm on both synthetic and real examples

iterative reweighted total generalized variation based poisson noise removal model total generalized variation tgv regularization model is one of the most effective methods for denoising and eliminating staircase effect however the tgv regularization model tends to blur edges as the existence of high order derivative in order to avoid the staircase effect while alleviating the edge blurring an iterative reweighted tgv based poisson noise removal model is presented under the assumption that each pixel of noisy image follows a poisson distribution the weight function incorporated in the tgv regularization term is derived from the expectation maximization em algorithm we design a new iterative weighted primal dual algorithm which is an improvement of the classic iterative reweighted algorithm numerical experimental results show the better performance of our model in removing noise effectively while preserving edges and eliminating staircase effect

isomorphic classification of lp q spaces abstract we prove that for all 1 p q p q p 1 p 2 and p 2 q 2 the banach space l p q does not isomorphically embed into the space l p q 0 1 we also prove under the same conditions that the space l p q 0 does not isomorphically embed into the space l p q 0 1 l p q

fixed point theorems for uniformly lipschitzian semigroups in uniformly convex spaces for all x y in c and each n 1 2 goebel and kirk 2 first studied these mappings and they proved that such t has a fixed point if c is a closed bounded convex subset of a uniformly convex banach space x and k k k is the unique solution of the equation 1 a l k k 1 with 6 being the modulus of convexity of x lifschitz s showed that in a hilbert space i a uniformly k lipschitzian mapping t with k c 21j2 has a fixed point lim 6 7 and lim et al 8 extended goebel kirk s theorem in lp spaces recently downing and ray l and ishihara and takahashi 3 verified that lifschitz s theorem is valid for uniformly lipschitzian semigroups in hifbert spaces the purpose of this paper is to prove fixed point theorems for a uniformly k lipschitzian semigroup in uniformly convex banach spaces which is left reversible or for which the space of right uniformly continuous functions on the semigroup has a left invariant mean the results generalize those of downing and ray l lau 4 lim 6 7 and ishihara and takahashi 3

nested conjugate gradient algorithm with nested preconditioning for non linear image restoration we develop a novel optimization algorithm which we call nested non linear conjugate gradient cg algorithm nncg for image restoration based on quadratic data fitting and smooth non quadratic regularization the algorithm is constructed as a nesting of two conjugate gradient iterations the outer iteration is constructed as a preconditioned non linear cg algorithm the preconditioning is performed by the inner cg iteration that is linear the inner cg iteration which performs preconditioning for outer cg iteration itself is accelerated by an another fft based non iterative preconditioner we prove that the method converges to a stationary point for both convex and non convex regularization functionals we demonstrate experimentally that proposed method outperforms the well known majorization minimization method used for convex regularization and a non convex inertial proximal method for non convex regularization functional

fft based preconditioners for toeplitz block least squares problems discretized two dimensional deconvolution problems arising e g in image restoration and seismic tomography can be formulated as least squares computations min b tx 2 where t is often a large scale rectangular toeplitz block matrix the authors consider solving such block least squares problems by the preconditioned conjugate gradient algorithm using square nonsingular circulant block and related preconditioners constructed from the blocks of the rectangular matrix t preconditioning with such matrices allows efficient implementation using the one dimensional or two dimensional fast fourier transform fft two block preconditioners related to those proposed by t chan and j olkin for square nonsingular toeplitz block systems are derived and analyzed it is shown that for important classes of t the singular values of the preconditioned matrix are clustered around one this extends the authors earlier work on preconditioners for toeplitz least squares iterations for one dimensiona

convergence theorems for nonexpansive semigroups in cat 0 spaces abstract let c be a convex and compact subset of a cat 0 space x in this paper we consider the following iterative scheme for a one parameter nonexpansive semigroup t t t 0 on c x 1 c x n 1 t t n x n 1 x n where 0 1 and t n 0 and we prove that under certain conditions x n converges to a common fixed point of the semigroup t t t 0

strong convergence of a general iteration scheme in cat 0 spaces abstract we introduce and study strong convergence of a general iteration scheme for a finite family of asymptotically quasi nonexpansive maps in convex metric spaces and c a t 0 spaces the new iteration scheme includes modified mann and ishikawa iterations the three step iteration scheme of xu and noor and the scheme of khan domlo and fukhar ud din as special cases in banach spaces our results are refinements and generalizations of several recent results from the current literature

majorization minimization algorithms for wavelet based image restoration standard formulations of image signal deconvolution under wavelet based priors regularizers lead to very high dimensional optimization problems involving the following difficulties the non gaussian heavy tailed wavelet priors lead to objective functions which are nonquadratic usually nondifferentiable and sometimes even nonconvex the presence of the convolution operator destroys the separability which underlies the simplicity of wavelet based denoising this paper presents a unified view of several recently proposed algorithms for handling this class of optimization problems placing them in a common majorization minimization mm framework one of the classes of algorithms considered when using quadratic bounds on nondifferentiable log priors shares the infamous singularity issue si of iteratively re weighted least squares irls algorithms the possibility of having to handle infinite weights which may cause both numerical and convergence issues in this paper we prove several new results which strongly support the claim that the si does not compromise the usefulness of this class of algorithms exploiting the unified mm perspective we introduce a new algorithm resulting from using bounds for nonconvex regularizers the experiments confirm the superior performance of this method when compared to the one based on quadratic majorization finally an experimental comparison of the several algorithms reveals their relative merits for different standard types of scenarios

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

design of steerable filters for feature detection using canny like criteria we propose a general approach for the design of 2d feature detectors from a class of steerable functions based on the optimization of a canny like criterion in contrast with previous computational designs our approach is truly 2d and provides filters that have closed form expressions it also yields operators that have a better orientation selectivity than the classical gradient or hessian based detectors we illustrate the method with the design of operators for edge and ridge detection we present some experimental results that demonstrate the performance improvement of these new feature detectors we propose computationally efficient local optimization algorithms for the estimation of feature orientation we also introduce the notion of shape adaptable feature detection and use it for the detection of image corners

total variation and high order total variation adaptive model for restoring blurred images with cauchy noise abstract in this paper we propose a novel model to restore an image corrupted by blur and cauchy noise the model is composed of a data fidelity term and two regularization terms including total variation and high order total variation total variation provides well preserved edge features but suffers from staircase effects in smooth regions whereas high order total variation can alleviate staircase effects moreover we introduce a strategy for adaptively selecting regularization parameters we develop an efficient alternating minimization algorithm for solving the proposed model numerical examples suggest that the proposed method has the advantages of better preserving edges and reducing staircase effects

edge preserving pet image reconstruction using trust optimization transfer iterative image reconstruction for positron emission tomography can improve image quality by using spatial regularization the most commonly used quadratic penalty often oversmoothes sharp edges and fine features in reconstructed images while nonquadratic penalties can preserve edges and achieve higher contrast recovery existing optimization algorithms such as the expectation maximization em and preconditioned conjugate gradient pcg algorithms work well for the quadratic penalty but are less efficient for high curvature or nonsmooth edge preserving regularizations this paper proposes a new algorithm to accelerate edge preserving image reconstruction by using two strategies trust surrogate and optimization transfer descent trust surrogate approximates the original penalty by a smoother function at each iteration but guarantees the algorithm to descend monotonically optimization transfer descent accelerates a conventional optimization transfer algorithm by using conjugate gradient and line search results of computer simulations and real 3 d data show that the proposed algorithm converges much faster than the conventional em and pcg for smooth edge preserving regularization and can also be more efficient than the current state of art algorithms for the nonsmooth ell 1 regularization

equidistributed random variables in lp q abstract we consider linear combinations of independent identically distributed random variables in l p q in fact we provide several norm inequalities for sums from a larger class of equidistributed random variables

iterative image restoration combining total variation minimization and a second order functional a noise removal technique using partial differential equations pdes is proposed here it combines the total variational tv filter with a fourth order pde filter the combined technique is able to preserve edges and at the same time avoid the staircase effect in smooth regions a weighting function is used in an iterative way to combine the solutions of the tv filter and the fourth order filter numerical experiments confirm that the new method is able to use less restrictive time step than the fourth order filter numerical examples using images with objects consisting of edge flat and intermediate regions illustrate advantages of the proposed model

robustness improvement of hyperspectral image unmixing by spatial second order regularization the acquisition of hundreds of images of a scene each at a different wavelength is known as hyperspectral imaging this high amount of data allows the extraction of much more information from hyperspectral images compared with conventional color images the forward looking imaging approach emerged from remote sensing but is still not very widespread in industrial and other practical applications spectral unmixing in particular aims at the determination of the components present in a scene as well as the abundance to which each component contributes this information is valuable for instance when discrimination tasks are to be performed involving not only spectral but also spatial information was found to have the potential to improve the unmixing results several publications use spatial first order regularization closely related to the total variation approach to incorporate this spatial information like in classical image processing this approach favors piecewise constant pixel transitions this is why it was proposed in the literature to use second order regularization instead of first order to approach piecewise linear transitions therefore we introduce hessian based regularization to hyperspectral unmixing and propose an algorithm to calculate the regularized result we use simulated data and images measured in our laboratory to show that both the first and second order approaches share many properties and produce similar results the second order approach however is more robust and thus more accurate in finding the minimum both methods smoothen the images in the case of supervised unmixing i e the component spectra are known beforehand and enhance unsupervised unmixing when the spectra are not known

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

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

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples

hessian schatten norm regularization for linear inverse problems we introduce a novel family of invariant convex and non quadratic functionals that we employ to derive regularized solutions of ill posed linear inverse imaging problems the proposed regularizers involve the schatten norms of the hessian matrix which are computed at every pixel of the image they can be viewed as second order extensions of the popular total variation tv semi norm since they satisfy the same invariance properties meanwhile by taking advantage of second order derivatives they avoid the staircase effect a common artifact of tv based reconstructions and perform well for a wide range of applications to solve the corresponding optimization problems we propose an algorithm that is based on a primal dual formulation a fundamental ingredient of this algorithm is the projection of matrices onto schatten norm balls of arbitrary radius this operation is performed efficiently based on a direct link we provide between vector projections onto norm balls and matrix projections onto schatten norm balls finally we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p