applications of mixed martingale hardy spaces in fourier analysis abstract in this paper we consider the martingale hardy spaces defined with the help of the mixed l p norm using atomic decomposition and doob s inequality some applications will be shown in fourier analysis such as the uniform boundedness of the partial sums of the walsh fourier series on l p 1 p and the boundedness of the fejer maximal operator from h p to l p 1 2 p as a consequence of the boundedness we get some almost everywhere and norm convergence results

coupling image restoration and segmentation a generalized linear model bregman perspective we introduce a new class of data fitting energies that couple image segmentation with image restoration these functionals model the image intensity using the statistical framework of generalized linear models by duality we establish an information theoretic interpretation using bregman divergences we demonstrate how this formulation couples in a principled way image restoration tasks such as denoising deblurring deconvolution and inpainting with segmentation we present an alternating minimization algorithm to solve the resulting composite photometric geometric inverse problem we use fisher scoring to solve the photometric problem and to provide asymptotic uncertainty estimates we derive the shape gradient of our data fitting energy and investigate convex relaxation for the geometric problem we introduce a new alternating split bregman strategy to solve the resulting convex problem and present experiments and comparisons on both synthetic and real world images

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

riesz type kernels over the ring of integers of a local field abstract letodenote the ring of integers of a local field in this note we prove an approximation theorem for the riesz type kernels n n 1 overo the proof requires a sharp estimate of thedirichlet kerneldn x onp 1 o which may also have independent interest as a consequence we solve the local field analog of 1 concerning rates of convergence of thefejer sumon the walsh system our approach has potential applications to other operators such as cesaro means and abel poisson means of type i

image segmentation by variational methods mumford and shah functional and the discrete approximations in this paper we discuss the links between mumford and shah s variational problem for signal and image segmentation based on an energy functional of a continuous grey level function and the numerical algorithms proposed to solve it these numerical approaches are based on a discrete functional we recall that in one dimension this discrete functional is asymptotically equivalent to the continuous functional this can be summarized in a gamma convergence result we show that the same result holds in dimension two provided that the continuous energy is adapted to the anisotropy of the discrete approaches we display a few experimental results in dimensions one and two

segmentation of digitized dermatoscopic images by two dimensional color clustering a color based segmentation scheme applied to dermatoscopic images is proposed the rgb image is processed in the l u v color space a 2d histogram is computed with the two principal components and then smoothed with a gaussian low pass filter the maxima location and a set of features are computed from the histogram contour lines these features are the number of enclosed pixels the surface of the base and the height of the maximum they allow for the selection of valid clusters which determine the number of classes the image is then segmented using a modified version of the fuzzy c means fcm clustering technique that takes into account the cluster orientation finally the segmented image is cleaned using mathematical morphology the region borders are smoothed and small components are removed

martingale hardy spaces for 0 p 1 the relation between the usually considered martingale hardy spaces is examined it will be shown that if the stochastic basis is regular then all these hardy spaces are equivalent an atomic description of the spacesh andp p will be given for 0 p 1 and with the help of this we shall consider the duals of these spaces moreover the dual space ofvmo p generated by the dual space ofh p will be determined in the case when all algebras are generated by countably many atoms furthermore we shall show in this case that the dual ofvmo 1 ish 1

wavelet based segmentation on the sphere abstract segmentation a useful powerful technique in pattern recognition is the process of identifying object outlines within images there are a number of efficient algorithms for segmentation in euclidean space that depend on the variational approach and partial differential equation modelling wavelets have been used successfully in various problems in image processing including segmentation inpainting noise removal super resolution image restoration and many others wavelets on the sphere have been developed to solve such problems for data defined on the sphere which arise in numerous fields such as cosmology and geophysics in this work we propose a wavelet based method to segment images on the sphere accounting for the underlying geometry of spherical data our method is a direct extension of the tight frame based segmentation method used to automatically identify tube like structures such as blood vessels in medical imaging it is compatible with any arbitrary type of wavelet frame defined on the sphere such as axisymmetric wavelets directional wavelets curvelets and hybrid wavelet constructions such an approach allows the desirable properties of wavelets to be naturally inherited in the segmentation process in particular directional wavelets and curvelets which were designed to efficiently capture directional signal content provide additional advantages in segmenting images containing prominent directional and curvilinear features we present several numerical experiments applying our wavelet based segmentation method as well as the common k means method on real world spherical images including an earth topographic map a light probe image solar data sets and spherical retina images these experiments demonstrate the superiority of our method and show that it is capable of segmenting different kinds of spherical images including those with prominent directional features moreover our algorithm is efficient with convergence usually within a few iterations

regularized color clustering in medical image database a regularized color clustering algorithm is proposed to solve the color clustering problem in medical image database by incorporating both measures of cluster separability and cluster compactness regularized color clustering allows the automatic extraction of significant color groups with varying populations experimental results in different color spaces show that the regularized color clustering gives superior results in extracting significant distinct abnormal color clusters without significant increases in cluster compactness furthermore results of color clustering in different color spaces show that the luv color space is more suitable for color clustering methods for selecting the regularization constants have also been suggested

estimates for n widths of multiplier operators of multiple walsh series abstract estimates for kolmogorov and gelfand n widths of multiplier operators of multiple walsh series are obtained upper and lower bounds are established for n widths of general multiplier operators these results are applied to get upper and lower bounds for n widths of specific multiplier operators which generate sets of finitely and infinitely differentiable functions in the dyadic sense it is shown that these estimates have order sharp in various important cases

deblurring gaussian blur using a wavelet array transform deblurring the gaussian blur which is a fundamental problem of signal analysis has defined a satisfactory solution the principal reason for the difficulty is the inherently ill conditioned blur matrix which poses a challenge to its stable inversion most of the literature is concerned with a solution to the problem in restricted domains and this solution is in many cases characterised by inversions that are not stable we propose a multiscale inversion method based on wavelet arrays which is applicable to a wide class of images and show that the inversion is stable with respect to noise both in the blurred signal and in the blur variance we include for illustration the result of such a deblurring scheme as applied to a natural image

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

a generalization for fourier transforms of a theorem due to marcinkiewicz it is shown that the maximal operator of the marcinkiewicz means of a tempered distribution is bounded from hp r2 to lp r2 for all p0 p and consequently is of weak type 1 1 where p0 1 as a consequence we obtain a generalization for fourier transforms of a summability result due to marcinkiewicz and zhizhiashvili more exactly the marcinkiewicz means of a function f l1 r2 converge a e to the function in question moreover we prove that the marcinkiewicz means are uniformly bounded on the spaces hp r2 and so they converge in the norm p0 p similar results for the riesz transforms are also given

marcinkiewicz fejer means of d dimensional walsh fourier series abstract the boundedness of marcinkiewicz maximal operator for d dimensional walsh fourier series is studied from the martingale hardy lorentz space h p q into the lorentz space l p q

histogram based segmentation in a perceptually uniform color space in this work we present a segmentation algorithm for color images that uses the watershed algorithm to segment either the two dimensional 2 d or the three dimensional 3 d color histogram of an image for compliance with the way humans perceive color this segmentation has to take place in a perceptually uniform color space like the luv space to avoid oversegmentation the watershed algorithm has to be applied to a smoothed histogram

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

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 the equivalence of soft wavelet shrinkage total variation diffusion total variation regularization and sides soft wavelet shrinkage total variation tv diffusion tv regularization and a dynamical system called sides are four useful techniques for discontinuity preserving denoising of signals and images in this paper we investigate under which circumstances these methods are equivalent in the one dimensional case first we prove that haar wavelet shrinkage on a single scale is equivalent to a single step of space discrete tv diffusion or regularization of two pixel pairs in the translationally invariant case we show that applying cycle spinning to haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of tv diffusion we prove that space discrete tv diffusion and tv regularization are identical and that they are also equivalent to the sides system when a specific force function is chosen afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the laplacian pyramid of the signal we analyze possibilities to avoid gibbs like artifacts for multiscale haar wavelet shrinkage by scaling the thresholds finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and pde variational approaches these methods are based on iterated shift invariant wavelet shrinkage at multiple scales with scaled thresholds

cesaro summability of double walsh fourier series we introduce quasi local operators these include operators of calder6n zygmund type a hybrid hardy space ho of functions of two variables and we obtain sufficient conditions for a quasi local maximal operator to be of weak type 1 as an application we show that cesatro means of the double walsh fourier series of a function f converge a e when f belongs to ho we also obtain the dyadic analogue of a summability result of marcienkiewicz and zygmund valid for all f e l1 provided summability takes place in some positive cone

regularization theory in image restoration the stabilizing functional approach several aspects of the application of regularization theory in image restoration are presented this is accomplished by extending the applicability of the stabilizing functional approach to 2 d ill posed inverse problems inverse restoration is formulated as the constrained minimization of a stabilizing functional the choice of a particular quadratic functional to be minimized is related to the a priori knowledge regarding the original object through a formulation of image restoration as a maximum a posteriori estimation problem this formulation is based on image representation by certain stochastic partial differential equation image models the analytical study and computational treatment of the resulting optimization problem are subsequently presented as a result a variety of regularizing filters and iterative regularizing algorithms are proposed a relationship between the regularized solutions proposed and optimal wiener estimation is identified the filters and algorithms proposed are evaluated through several experimental results

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

a multiscale algorithm for image segmentation by variational method most segmentation algorithms are composed of several procedures split and merge small region elimination boundary smoothing each depending on several parameters the introduction of an energy to minimize leads to a drastic reduction of these parameters the authors prove that the most simple segmentation tool the region merging algorithm made according to the simplest energy is enough to compute a local energy minimum belonging to a compact class and to achieve the job of most of the tools mentioned above the authors explain why merging in a variational framework leads to a fast multiscale multichannel algorithm with a pyramidal structure the obtained algorithm is o n ln n where n is the number of pixels of the picture this fast algorithm is applied to make grey level and texture segmentation and experimental results are shown

salt and pepper noise removal by median type noise detectors and detail preserving regularization this paper proposes a two phase scheme for removing salt and pepper impulse noise in the first phase an adaptive median filter is used to identify pixels which are likely to be contaminated by noise noise candidates in the second phase the image is restored using a specialized regularization method that applies only to those selected noise candidates in terms of edge preservation and noise suppression our restored images show a significant improvement compared to those restored by using just nonlinear filters or regularization methods only our scheme can remove salt and pepper noise with a noise level as high as 90

tongue image matching using color content this paper addresses the problem of color image matching in medical diagnosis the color matching of tongue images in different color spaces with different metrics have been investigated and is reported in this article two new metrics namely sorted metric and probabilistic combined metric are proposed existing distance measurements in coordinate space do not satisfy the reflexivity axiom that means they are not the valid metrics to overcome this limitation the sorted metric in coordinate space is proposed in coordinate space to improve the matching performance a probabilistic combined metric is proposed based on the theory of combining classifier these metrics are applied for the matching of tongue color images and the results are encouraging

linearized proximal alternating minimization algorithm for motion deblurring by nonlocal regularization abstract non blind motion deblurring problems are highly ill posed and so it is quite difficult to find the original sharp and clean image to handle ill posedness of the motion deblurring problem we use nonlocal total variation abbreviated as tv regularization approaches nonlocal tv can restore periodic textures and local geometric information better than local tv but since nonlocal tv requires weighted difference between pixels in the whole image it demands much more computational resources than local tv by using the linearization of the fidelity term and the proximal function our proposed algorithm does not require any inversion of blurring operator and nonlocal operator therefore the proposed algorithm is very efficient for motion deblurring problems we compare the numerical performance of our proposed algorithm with that of several state of the art algorithms for deblurring problems our numerical results show that the proposed method is faster and more robust than state of the art algorithms on motion deblurring problems

herz spaces and restricted summability of fourier transforms and fourier series a general summability method the so called summability is considered for multi dimensional fourier transforms and fourier series a new inequality for the hardy littlewood maximal function is verified it is proved that if the fourier transform of is in a herz space then the restricted maximal operator of the means of a distribution is of weak type 1 1 provided that the supremum in the maximal operator is taken over a cone like set from this it follows that t f f over a cone like set a e for all f l1 r d moreover

marcinkiewicz summability of multi dimensional fourier transforms and fourier series abstract a generalization of marcinkiewicz summability of multi dimensional fourier transforms and fourier series is investigated with the help of a continuous function under some weak conditions on we show that the maximal operator of the marcinkiewicz means of a tempered distribution is bounded from h p x d to l p x d for all d d p and consequently is of weak type 1 1 where 0 1 is depending only on and x r or x t as a consequence we obtain a generalization of a summability result due to marcinkiewicz and zhizhiashvili for d dimensional fourier transforms and fourier series more exactly the marcinkiewicz means of a function f l 1 x d converge a e to f moreover we prove that the marcinkiewicz means are uniformly bounded on the spaces h p x d and so they converge in norm d d p similar results are shown for conjugate functions some special cases of the marcinkiewicz summation are considered such as the fejer cesaro weierstrass picar bessel de la vallee poussin rogosinski and riesz summations

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

pointwise convergence of multiple trigonometric series abstract this is devoted to the study of pointwise convergence of the rectangular partial sums of multiple trigonometric series with coefficients satisfying certain conditions involving finite order differences three types of such conditions are posed convergence may be understood both in the unrestricted sense and in the restricted sense we prove that if the first arithmetic mean of an n fold trigonometric series of the above type converges then so does its rectangular partial sum as a corollary we obtain that any multiple fourier series of the above type converges restrictedly almost everywhere these results extend those of chen chen and hsieh moricz and moricz and waterman

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

a new fuzzy c means method with total variation regularization for segmentation of images with noisy and incomplete data the objective function of the original fuzzy c mean method is modified by a regularizing functional in the form of total variation tv with regard to gradient sparsity and a regularization parameter is used to balance clustering and smoothing an alternating direction method of multipliers in conjunction with the fast discrete cosine transform is used to solve the tv regularized optimization problem the new algorithm is tested on both synthetic and real data and is demonstrated to be effective and robust in treating images with noise and missing data incomplete data

almost everywhere summability of multiple walsh fourier series abstract we prove that for the n dimensional walsh fourier series the maximal operator of the marcinkiewicz means is of weak type 1 1 moreover the marcinkiewicz means nf of the function f l1 converge a e to f as n

cesaro summability of two dimensional walsh fourier series we introduce p quasi local operators and the two dimensional dyadic hardy spaces hp defined by the dyadic squares it is proved that if a sublinear operator t is p quasi local and bounded from l to l then it is also bounded from hp to lp 0 p 1 as an application it is shown that the maximal operator of the cesaro means of a martingale is bounded from hp to lp 1 2 p and is of weak type 1 1 provided that the supremum in the maximal operator is taken over a positive cone so we obtain the dyadic analogue of a summability result with respect to two dimensional trigonometric fourier series due to marcinkievicz and zygmund more exactly the cesaro means of a function f l1 converge a e to the function in question provided again that the limit is taken over a positive cone finally it is verified that if we take the supremum in a cone but for two powers only then the maximal operator of the cesaro means is bounded from hp to lp for every 0 p

finite difference approximation of the mumford shah functional we study the pointwise convergence and the of a family of nonlocal functionals defined in input amstex loadmsbm l 1 roman loc bbb r n to a local functional that depends on the gradient of u and on the set of discontinuity points of u we apply this result to approximate a minimum problem introduced by mumford and shah to study edge detection in computer vision theory 1998 john wiley sons inc

pointwise convergence of double trigonometric series abstract the pointwise convergence problem of the rectangular partial sums of a certain type of double trigonometric series is considered this type of series obeys certain conditions on the finite order differences of its coefficients we prove that if the cesaro sums of the double series converge unrestrictedly then so do its partial sums it is pointed out that the converse of the last statement may not hold for the same kind of double trigonometric series as a corollary it is shown that the double fourier series of the mentioned type converges unrestrictedly almost everywhere generalizations of the above results to the restricted case are also established these results generalize the theorems of chen

variational image segmentation model coupled with image restoration achievements image segmentation and image restoration are two important topics in image processing with a number of important applications in this paper we propose a new multiphase segmentation model by combining image restoration and image segmentation models utilizing aspects of image restoration the proposed segmentation model can effectively and robustly tackle images with a high level of noise or blurriness missing pixels or vector values in particular one of the most important segmentation models the piecewise constant mumford shah model can be extended easily in this way to segment gray and vector valued images corrupted for example by noise blur or information loss after coupling a new data fidelity term which borrowed from the field of image restoration it can be solved efficiently using the alternating minimization algorithm and we prove the convergence of this algorithm with three variables under mild conditions experiments on many synthetic and real world images demonstrate that our method gives better segmentation results in terms of quality and quantity in comparison to other state of the art segmentation models especially for blurry images and those with information loss highlightsprovide a new methodology and a new model for multiphase image segmentation extend the piecewise constant mumford shah model to handle blurry case easily has the ability to segment images with gaussian poisson and impulsive noises effective in segmenting vector valued image and image observed with information loss the convergence of the am algorithm with three variables adopted is proved

lebesgue points of double fourier series and strong summability abstract a general summability method of double fourier series is given with the help of a double sequence k n under some conditions on we show that the marcinkiewicz means of a function f l 1 t 2 converge to f at each modified strong lebesgue point the same holds for a weaker version of lebesgue points for the so called modified lebesgue points of f l p t 2 whenever 1 p the sufficient conditions of are proved for the fejer abel and cesaro summations as an application we give simple proofs for the classical one dimensional strong summability results of hardy and littlewood marcinkiewicz zygmund and gabisoniya and generalize them for strong summability some other special cases of the summation are considered as well such as the weierstrass picard bessel de la vallee poussin rogosinski and riesz summations

cesaro summability of hardy spaces on the ring of integers in a local field abstract let o be the ring of integers in a local field k we solve an open problem due to m h taibleson 1975 math notes vol 15 princeton univ press princeton nj suppose f l1 o does the cesaro means of f converge to f almost everywhere if k has characteristic zero to this end we study the hp lp boundedness of the associated maximal operator to get the corresponding interpolation result on hardy lorentz spaces in particular we obtain that is of weak type 1 1 the proof mainly depends on certain estimates for the oscillatory dirichlet kernels which are refinements of those obtained earlier by the author 1997 j math anal appl 208 528 552

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

cesaro summability of multi dimensional trigonometric fourier series the d dimensional classical hardy spaces hp td are introduced and it is shown that the maximal operator of the riemann sums of a distribution is bounded from hp td to lp t2 d d 1 p and is of weak type 1 1 provided that the supremum in the maximal operator is taken over a positive cone the same is proved for the conjugate riemann sums as a consequence we obtain that every function f l1 td is a e riemann summable to f provided again that the limit is taken over a positive cone