fast finite difference approximation for identifying parameters in a two dimensional space fractional nonlocal model with variable diffusivity coefficients in this paper we consider an inverse problem for identifying the fractional derivative indices in a two dimensional space fractional nonlocal model based on a generalization of the two sided riemann liouville formulation with variable diffusivity coefficients first we derive an implicit difference method idm for the direct problem and the stability and convergence of the idm are discussed second for the implementation of the idm we develop a fast bi conjugate gradient stabilized method fbi cgstab that is superior in computational performance to gaussian elimination and attains the same accuracy third we utilize the levenberg marquardt l m regularization technique combined with the armijo rule the popular inexact line search condition to solve the modified nonlinear least squares model associated with the parameter identification finally we carry out numerical tests to verify the accuracy and efficiency of the idm numerical investigations are performed with both accurate data and noisy

pilot assisted and blind joint data detection and channel estimation in partial time jamming we consider a communication scenario in which a message is received in the presence of partial time gaussian jamming and additive white gaussian noise we consider a quasi static channel in which the amplitude and phase are constant over each packet transmission the receiver does not know the amplitude and phase of the incoming signal which symbols are jammed or even the statistics of the jammer such as the jamming power and jamming probability in this scenario the receiver must accurately estimate the parameters of the channel and the jamming to achieve good performance we apply the expectation maximization em algorithm to iteratively approximate the maximum likelihood ml estimator for all of the parameters we find that the overall performance of the em algorithm is very sensitive to the initial estimates so we propose a new initial estimator that offers good performance the em algorithm approach requires pilot symbols to resolve a phase ambiguity thus we also present a blind estimation algorithm to avoid the reduction in overall code rate from the use of pilot symbols

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

the total variation flow in rn the purpose of this chapter is to prove existence and uniqueness of the minimizing total variation flow in n frac partial u partial t div left frac du left du right right in left 0 infty right times mathbb r n 3 1 coupled with the initial condition u 0 x u 0 x text x in mathbb r n 3 2 when u0 l loc 1 l n

analytical and numerical solutions of the unsteady 2d flow of mhd fractional maxwell fluid induced by variable pressure gradient abstract the nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations this paper studies the analytical and numerical methods of unsteady 2d flow of magnetohydrodynamic mhd fractional maxwell fluid in a rectangular pipe driven by variable pressure gradient the governing equation is formulated with caputo time dependent fractional derivatives whose orders are distributed in interval 0 2 a challenge is to firstly obtain the exact solution by combining modified separation of variables method with mikusinski type operational calculus meanwhile the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed different to the most classical works both the stability and convergence analysis of two dimensional multi term time fractional momentum equation are derived based on numerical analysis the results show that the velocity increases with the rise of the fractional parameter and relaxation time while an increase in the values of hartmann number leads to a slower velocity in the rectangular pipe

on line estimation of dynamic shock error models based on the kullback leibler information measure develops two sequential or on line estimation schemes in the time domain for dynamic shock error models which are special cases of errors in variables models the author s approach utilizes a state space representation of the model kalman filtering techniques and on line algorithms the first on line algorithm is based on the expectation maximization algorithm and uses a recursive gauss newton scheme to maximize the kullback leibler information measure the second on line algorithm the author proposes is a gradient based scheme and uses stochastic approximations to maximize the log likelihood in comparison to the off line maximum likelihood estimation scheme used in ghosh 1989 the author s on line algorithms have significantly reduced computational costs and negligible memory requirements simulations illustrate the satisfactory performance of the algorithms in estimating errors in variables systems with parameters that vary slowly with time or undergo infrequent jump changes

image segmentation using some piecewise constant level set methods with mbo type of projection in this work we are trying to propose fast algorithms for mumford shah image segmentation using some recently proposed piecewise constant level set methods pclsm two variants of the pclsm will be considered in this work the first variant which we call the binary level set method needs a level set function which only takes values 1 to identify the regions the second variant only needs to use one piecewise constant level set function to identify arbitrary number of regions for the mumford shah image segmentation model with these new level set methods one needs to minimize some smooth energy functionals under some constrains a penalty method will be used to deal with the constraint aos additive operator splitting and mos multiplicative operator splitting schemes will be used to solve the euler lagrange equations for the minimization problems by doing this we obtain some algorithms which are essentially applying the mbo scheme for our segmentation models advantages and disadvantages are discussed for the proposed schemes

unconditionally stable explicit algorithms for nonlinear fluid dynamics problems abstract this paper describes novel explicit algorithms that are unconditionally stable the algorithms are applied to some 1d convection and diffusion problems including nonlinear problems algorithms such as these are of particular interest for massively parallel computers where one is trying to minimize communications while at the same time maintain the stability properties normally associated with implicit schemes it is shown how these stable algorithms can be applied in higher spatial dimensions and how they can be extended to problems defined on unstructured meshes

parameter estimation of superimposed signals using the em algorithm a computationally efficient algorithm for parameter estimation of superimposed signals based on the two step iterative em estimate and maximize with an e step and an m step algorithm is developed the idea is to decompose the observed data into their signal components and then to estimate the parameters of each signal component separately the algorithm iterates back and forth using the current parameter estimates to decompose the observed data better and thus increase the likelihood of the next parameter estimates the application of the algorithm to the multipath time delay and multiple source location estimation problems is considered

analytical solutions of multi term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid this paper derives analytical solutions for a class of new multi term fractional order partial differential equations which include the terms for spatial diffusion time fractional diffusion multi term and reaction these models can be used to describe the nonlinear relationship between the shear stress and shear rate of generalized viscoelastic oldroyd b fluid and burgers fluid by using a modified separation of variables method the governing fractional order partial differential equations are transformed into a set of fractional order ordinary differential equations mikusiski type operational calculus is then employed to obtain the exact solutions of the linear fractional ordinary differential equations with constant coefficients the solutions are expressed in terms of multivariate mittag leffler functions different situations for the unsteady flows of generalized oldroyd b fluid and burgers fluid due to a moving plate are considered via examples integral representations of the solutions are presented it is shown that the presented results reduce to the corresponding results for classical navierstokes oldroyd b maxwell and second grade fluids as special cases

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

interpreting translation invariant wavelet shrinkage as a new image smoothing scale space coifman and donoho 1995 suggested translation invariant wavelet shrinkage as a way to remove noise from images basically their technique applies wavelet shrinkage to a two dimensional 2 d version of the semi discrete wavelet representation of mallat and zhong 1992 coifman and donoho also showed how the method could be implemented in o nlog n operations where there are n pixels in this paper we provide a mathematical framework for iterated translation invariant wavelet shrinkage and show using a theorem of kato and masuda 1978 that with orthogonal wavelets it is equivalent to gradient descent in l sub 2 i along the semi norm for the besov space b sub 1 sup 1 l sub 1 i which in turn can be interpreted as a new nonlinear wavelet based image smoothing scale space unlike many other scale spaces the characterization is not in terms of a nonlinear partial differential equation

l minimization for wavelet frame based image restoration the theory of tight wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems the success of wavelet frame based models including balanced approach 18 7 and analysis based approach 11 31 50 is due to their capability of sparsely approximating piecewise smooth functions like images motivated by the balanced approach and analysis based approach we shall propose a wavelet frame based l0 minimization model where the l0 norm of the frame coefficients is penalized we adapt the penalty decomposition pd method of 40 to solve the proposed optimization problem some convergence analysis of the adapted pd method will also be provided numerical results showed that the proposed model solved by the pd method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images

towards artifact free characterization of surface topography using complex wavelets and total variation minimization a hybrid method based on dual tree complex wavelet transform and total variation minimization is proposed for erasure of undesirable artifacts that arise in the existing wavelet based methods in surface metrology the complex wavelet transform provides approximate shift invariance and good directional selectivity and attempts to solve the weakness of real discrete wavelet methods reconstruct the complex wavelet coefficients using a total variation minimization principle to eliminate the wavelet shape artifacts and the pseudo gibbs artifacts near the discontinuities which are caused when thresholding small wavelet coefficients by replacing these thresholded complex wavelet coefficients by optimal values that minimize the total variation the method performs a close artifact free surface characterization numerical experiments using a series of typical engineering and bioengineering surfaces demonstrate the remarkable potential of the methodology

proximal thresholding algorithm for minimization over orthonormal bases the notion of soft thresholding plays a central role in problems from various areas of applied mathematics in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis using convex analytical tools we extend this notion to that of proximal thresholding and investigate its properties providing in particular several characterizations of such thresholders we then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems and we establish the strong convergence of a proximal thresholding algorithm to solve it numerical applications to signal recovery are demonstrated

recursive parameter estimation using incomplete data abstract stochastic approximation procedures are considered for the estimation of parameters using incomplete data one procedure is stated and illustrated which often leads to asymptotically efficient estimators others are developed which although possibly not optimal in the above sense will be very much easier to apply this will be particularly advantageous when quick recursive estimates are required examples are given and a link is made between one of the sub optimal methods and the em algorithm author

blind multiuser detection in uplink cdma with multipath fading a sequential em approach we consider joint channel estimation and data detection in uplink asynchronous code division multiple access systems employing aperiodic long spreading sequences in the presence of unknown multipath fading since maximum likelihood ml sequence estimation is too complex to perform multiuser receivers are proposed based on the sequential expectation maximization em algorithm with the prior knowledge of only the signature waveforms the delays and the second order statistics of the fading channel the receivers sequentially estimate the channel using the sequential em algorithm moreover the snapshot estimates of each path are tracked by linear minimum mean squared error filters the user data are detected by a ml sequence detector given the channel estimates the proposed receivers that use the exact expressions have a computational complexity o 2 sup k per bit where k is the number of users using the em algorithm we derive low complexity approximations which have a computational complexity of o k sup 2 per bit simulation results demonstrate that the proposed receivers offer substantial performance gains over conventional pilot symbol assisted techniques and achieve a performance close to the known channel bounds furthermore the proposed receivers even outperform the single user rake receiver with nyquist pilot insertion rate in a single user environment

nonlinear scale space with spatially varying stopping time a general scale space algorithm is presented for denoising signals and images with spatially varying dominant scales the process is formulated as a partial differential equation with spatially varying time the proposed adaptivity is semi local and is in conjunction with the classical gradient based diffusion coefficient designed to preserve edges the new algorithm aims at maximizing a local snr measure of the denoised image it is based on a generalization of a global stopping time criterion presented recently by the author and colleagues most notably the method works well also for partially textured images and outperforms any selection of a global stopping time given an estimate of the noise variance the procedure is automatic and can be applied well to most natural images

sequential algorithms for parameter estimation based on the kullback leibler information measure methods of stochastic approximation are used to convert iterative algorithms for maximizing the kullback leibler information measure into sequential algorithms special attention is given to the case of incomplete data and several algorithms are presented to deal with situations of this kind the application of these algorithms to the identification of finite impulse response systems is considered

iterative regularization and nonlinear inverse scale space applied to wavelet based denoising in this paper we generalize the iterative regularization method and the inverse scale space method recently developed for total variation tv based image restoration to wavelet based image restoration this continues our earlier joint work with others where we applied these techniques to variational based image restoration obtaining significant improvement over the rudin osher fatemi tv based restoration here we apply these techniques to soft shrinkage and obtain the somewhat surprising result that a the iterative procedure applied to soft shrinkage gives firm shrinkage and converges to hard shrinkage and b that these procedures enhance the noise removal capability both theoretically in the sense of generalized bregman distance and for some examples experimentally in terms of the signal to noise ratio leaving less signal in the residual

wavelet domain image restoration with adaptive edge preserving regularization in this paper we consider a wavelet based edge preserving regularization scheme for use in linear image restoration problems our efforts build on a collection of mathematical results indicating that wavelets are especially useful for representing functions that contain discontinuities i e edges in two dimensions or jumps in one dimension we interpret the resulting theory in a statistical signal processing framework and obtain a highly flexible framework for adapting the degree of regularization to the local structure of the underlying image in particular we are able to adapt quite easily to scale varying and orientation varying features in the image while simultaneously retaining the edge preservation properties of the regularizer we demonstrate a half quadratic algorithm for obtaining the restorations from observed data

nonlinear approximation and the space bv r2 given a function 2 0 q 0 l 2 and a real number t 0 let u f t infg bv g 11 ll vgig where the infimum is taken over all functions g g bv of bounded variation on this and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation as well as in digital image processing techniques for finding minimizers g for u f t based on variational calculus and nonlinear partial differential equations have been put forward by several authors dms ro ms cl the main disadvantage of these approaches is that they are numerically intensive on the other hand it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems for example the above problem with bv replaced by the besov space b l i see e g cdll however since bv has no simple description in terms of wavelet coefficients it is not clear that minimizers for u f t can be realized in this way we shall show in this paper that simple methods based on haar thresholding provide near minimizers for u f t our analysis of this extremal problem brings forward many interesting relations between haar decompositions and the space bv 1 introduction nonlinear approximation has recently played an impor tant role in several problems of image processing including compression noise removal and feature extraction we have in mind techniques such as wavelet compression djl wavelet shrinkage or thresholding djkp1 wavelet packets cw and greedy algorithms mz dt there has also been an impressive contribution of techniques based on variational calculus and nonlinear partial dif ferential equations see e g dms ro ms cl especially to the problems of noise removal and image segmentation the common point between these two approaches is their ability to adapt to the composite nature of images edge tex tures and smooth regions should be treated adaptively a requirement which is certainly not fulfilled by the classical linear filtering techniques

an unstructured mesh control volume method for two dimensional space fractional diffusion equations with variable coefficients on convex domains in this paper we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains which to the best of our knowledge is a new contribution to the literature firstly we present the finite volume scheme for the two dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements secondly we explore the property of the stiffness matrix generated by the integral of the space fractional derivative we find that the stiffness matrix is sparse and not regular therefore we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system which is more efficient than using the gaussian elimination method finally we present several examples to verify our method in which we make a comparison of our method with the finite element method for solving a riesz space fractional diffusion equation on a circular domain the numerical results demonstrate that our method can reduce cpu time significantly while retaining the same accuracy and approximation property as the finite element method the numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains

the cauchy problem for discrete time fractional evolution equations abstract this paper is devoted to the study of discrete time fractional evolution equations involving the riemann liouville like difference operator based on the relationship between c 0 semigroups and a distinguished class of sequences of operators we show the structure of the solutions for the inhomogeneous cauchy problem of abstract fractional difference equations further we establish two criteria for the existence and uniqueness of solutions for the semilinear cauchy problem some examples are also provided to illustrate our main results

close form solution and parameter selection for convex minimization based edge preserving smoothing this work presents a new approach for the analysis of convex minimization based edge preserving image smoothing and the parameter selection therein the global solution that is the response of a convex smoothing model to the ideal step edge is derived in close form by analyzing the close form solution insights are drawn into how the optimal solution responds to edges in the data and how the parameter values affect resultant edges in the solution based on this a scheme is proposed for selecting parameters to achieve desirable responses at edges the theoretic results are substantiated by experiments

a stochastic model for thermal transport of nanofluid in porous media derivation and applications abstract in this paper a stochastic thermal transport model is developed for nanofluid flowing through porous media this model incorporates the influences of nanoparticle migration on convective heat transfer of the colloidal solution we show that levy flight movement patterns of nanoparticles result in the derived model using fractional derivative for the diffusion term the new thermal transport model is then applied to the mixed convective problem which is solved using finite difference method numerical results show that the smaller values of levy index lead to larger nusselt numbers thus the occurrence of long jumps for nanoparticles increases the heat transport of nanofluids the effects of other involved physical parameters are also presented and discussed

space alternating generalized expectation maximization algorithm the expectation maximization em method can facilitate maximizing likelihood functions that arise in statistical estimation problems in the classical em paradigm one iteratively maximizes the conditional log likelihood of a single unobservable complete data space rather than maximizing the intractable likelihood function for the measured or incomplete data em algorithms update all parameters simultaneously which has two drawbacks 1 slow convergence and 2 difficult maximization steps due to coupling when smoothness penalties are used the paper describes the space alternating generalized em sage method which updates the parameters sequentially by alternating between several small hidden data spaces defined by the algorithm designer the authors prove that the sequence of estimates monotonically increases the penalized likelihood objective derive asymptotic convergence rates and provide sufficient conditions for monotone convergence in norm two signal processing applications illustrate the method estimation of superimposed signals in gaussian noise and image reconstruction from poisson measurements in both applications the sage algorithms easily accommodate smoothness penalties and converge faster than the em algorithms

parallel in time multigrid for space time finite element approximations of two dimensional space fractional diffusion equations abstract the paper investigates a non intrusive parallel time integration with multigrid for space fractional diffusion equations in two spatial dimensions which is discretized by the space time finite element method to propagate solutions we develop a multigrid reduction in time mgrit algorithm with time dependent time grid propagators and provide its two level convergence theory under the assumptions of the stability and simultaneous diagonalizability on time grid propagators numerical results show that the proposed method possesses the saturation error order theoretical results of the two level variant deliver good predictions for our model problems and significant speedups of the mgrit can be achieved when compared to the two level variant with f relaxation an equivalent version of the parareal algorithm and the sequential time stepping approach

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

a tunable finite difference method for fractional differential equations with non smooth solutions in this work a finite difference method of tunable accuracy for fractional differential equations fdes with end point singularities is developed modified weighted shifted grunwald letnikov wsgl formulas are proposed to approximate the left and right riemann liouville fractional operators which show better accuracy than the original wsgl formulas due to the use of the correction terms finite difference schemes are constructed to solve two fractional boundary value problems and a space fractional allen cahn equation even if the singularity of the considered fdes is unknown satisfactory numerical solutions can still be obtained by suitably tuning the correction terms various numerical examples are presented to verify the effectiveness of the present method and comparisons with other known methods are also made that demonstrate higher accuracy of the current method

new methods of controlled total variation reduction for digital functions we consider the functional of total variation for functions of either one or two discrete variables we introduce a new principle for construction of evolution processes that gradually reduce the total variation of a function the method is derived from analysis of relaxation of the functional along the directions given by the haar wavelets an approach that does not require regularization of either the functional or its singular gradient we further present a rigorous analysis of the analytical properties of the resulting evolution schemes the purely discrete theory grows out of universal observations about the relation between the total variation and the haar functions an additional advantage of the method is that it can be directly implemented as an effective numerical procedure moreover it can be intertwined in a natural way with a nonlinear phase space filtering which further increases control of the deformation of a signal finally we report that the algorithms can be used as remarkable tools for signal processing and image enhancement

splines in higher order tv regularization splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces in this paper we show in a strictly discrete setting that splines of degree m 1 solve also a minimization problem with quadratic data term and m th order total variation tv regularization term in contrast to problems with quadratic regularization terms involving m th order derivatives the spline knots are not known in advance but depend on the input data and the regularization parameter more precisely the spline knots are determined by the contact points of the m th discrete antiderivative of the solution with the tube of width 2 around the m th discrete antiderivative of the input data we point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the sobolev space w 2 0 m from this point of view the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines

combined curvelet shrinkage and nonlinear anisotropic diffusion in this paper a diffusion based curvelet shrinkage is proposed for discontinuity preserving denoising using a combination of a new tight frame of curvelets with a nonlinear diffusion scheme in order to suppress the pseudo gibbs and curvelet like artifacts the conventional shrinkage results are further processed by a projected total variation diffusion in which only the insignificant curvelet coefficients or high frequency part of the signal are changed by use of a constrained projection numerical experiments from piecewise smooth to textured images show good performances of the proposed method to recover the shape of edges and important detailed components in comparison to some existing methods

on wavelet fundamental solutions to the heat equation heatlets abstract we present an application of wavelet theory in partial differential equations we study the wavelet fundamental solutions to the heat equation the heat evolution of an initial wavelet state is called a heatlet like wavelets for the l2 space heatlets are atomic heat evolutions in the sense that any general heat evolution can be assembled from a heatlet according to some simple rules we study the basic properties and algorithms of heatlets and related functions

reconstruction of wavelet coefficients using total variation minimization we propose a model to reconstruct wavelet coefficients using a total variation minimization algorithm the approach is motivated by wavelet signal denoising methods where thresholding small wavelet coefficients leads to pseudo gibbs artifacts by replacing these thresholded coefficients by values minimizing the total variation our method performs a nearly artifact free signal denoising in this paper we detail the algorithm based on a subgradient descent combining a projection on a linear space the convergence of the algorithm is established and numerical experiments are reported

staircasing in semidiscrete stabilised inverse linear diffusion algorithms we consider a semidiscrete model problem for the approximation of stabilised inverse linear diffusion processes the work is motivated by an important observation on fully discrete schemes concerning the so called staircasing phenomenon when sharpening monotone data profiles fully discrete methods generally introduce stepfunction type solutions reminiscent of staircases in this work we show by an analysis of dynamical systems in corresponding semidiscrete formulations that already the semidiscrete numerical model contains the relevant information on the occurrence of staircasing numerical experiments confirm and complement the theoretical results

on line estimation of hidden markov model parameters based on the kullback leibler information measure sequential or online hidden markov model hmm signal processing schemes are derived and their performance is illustrated by simulation the online algorithms are sequential expectation maximization em schemes and are derived by using stochastic approximations to maximize the kullback leibler information measure the schemes can be implemented either as filters or fixed lag or sawtooth lag smoothers they yield estimates of the hmm parameters including transition probabilities markov state levels and noise variance in contrast to the offline em algorithm baum welch scheme which uses the fixed interval forward backward scheme the online schemes have significantly reduced memory requirements and improved convergence and they can estimate hmm parameters that vary slowly with time or undergo infrequent jump changes similar techniques are used to derive online schemes for extracting finite state markov chains imbedded in a mixture of white gaussian noise wgn and deterministic signals of known functional form with unknown parameters

nonlinear diffusion in laplacian pyramid domain for ultrasonic speckle reduction a new speckle reduction method i e laplacian pyramid based nonlinear diffusion lpnd is proposed for medical ultrasound imaging with this method speckle is removed by nonlinear diffusion filtering of bandpass ultrasound images in laplacian pyramid domain for nonlinear diffusion in each pyramid layer a gradient threshold is automatically determined by a variation of median absolute deviation mad estimator the performance of the proposed lpnd method has been compared with that of other speckle reduction methods including the recently proposed speckle reducing anisotropic diffusion srad and nonlinear coherent diffusion ncd in simulation and phantom studies an average gain of 1 55 db and 1 34 db in contrast to noise ratio was obtained compared to srad and ncd respectively the visual comparison of despeckled in vivo ultrasound images from liver and carotid artery shows that the proposed lpnd method could effectively preserve edges and detailed structures while thoroughly suppressing speckle these preliminary results indicate that the proposed speckle reduction method could improve image quality and the visibility of small structures and fine details in medical ultrasound imaging

generalised nonlocal image smoothing we propose a discrete variational approach for image smoothing consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures defined on image patches one of such dissimilarity measures is the weighted l 2 distance between patches in such a case we derive an iterative neighbourhood filter that induces a new similarity measure in the photometric domain it can be regarded as an extended patch similarity measure that evaluates not only the patch similarity of two chosen pixels but also the similarity of their corresponding neighbours this leads to a more robust smoothing process since the pixels selected for averaging are more coherent with the local image structure by slightly modifying the way the similarities are computed we obtain two related filters the nl means filter of buades et al siam multiscale model simul 4 2 490 530 2005b and the nds filter of mrazek et al geometric properties for incomplete data computational imaging and vision vol 31 pp 335 352 springer dordrecht 2006 in fact the proposed approach can be considered as a generalisation of the latter filter to the space of patches we also provide novel insights into relations of the nds filter with diffusion regularisation methods as well as with some recently proposed graph regularisation techniques we evaluate our method for the task of denoising greyscale and colour images degraded with gaussian and salt and pepper noise demonstrating that it compares very well to other more sophisticated approaches

edge preserving wavelet thresholding for image denoising in this paper we consider a general setting for wavelet based image denoising methods in fact in both deterministic regularization methods and stochastic maximum a posteriori estimations the denoised image email protected is obtained by minimizing a functional which is the sum of a data fidelity term and a regularization term that enforces a roughness penalty on the solution the latter is usually defined as a sum of potentials which are functions of a derivative of the image by considering particular families of dyadic wavelets we propose the use of new potential functions which allows us to preserve and restore important image features such as edges and smooth regions during the wavelet denoising process numerical results are presented showing the optimal performance of the denoising algorithm obtained

parameters estimation for a new anomalous thermal diffusion model in layered media in this paper we study an inverse problem of parameters estimation for a new time fractional heat conduction model in multilayered medium in the anomalous thermal diffusion model we consider the fractional derivative boundary conditions and the conduction obeys modified fourier law with riemannliouville fractional operator of different order in each layer for the direct problem we construct an effective finite difference scheme by using the balance method to deal with the discontinuity interface for the inverse problem we apply the nonlinear conjugate gradient ncg method with different conjugated coefficients to simultaneously identify the fractional exponent in each layer finally we use experimental data to verify the effectiveness of the proposed technique in which the jacobian matrix is achieved by a derivative free approach we analyze the sensitivity coefficients and the convergence behaviors of the ncg algorithm the simulation results confirm that the fractional heat conduction model with estimated parameters gives a more accurate fitting than the classical counterpart and the ncg method is a feasible and effective technique for the inverse problem of parameters estimation in fractional model

locally adaptive regression splines in this paper least squares penalized regression estimates withtotal variation penalities are considered it is shown that theseestimators are least squares splines with locally data adaptive placed knotpoints algorithms and asymptotic properties are discussed

a finite volume method for two dimensional riemann liouville space fractional diffusion equation and its efficient implementation abstract we develop a finite volume method based on crank nicolson time discretization for the two dimensional nonsymmetric riemann liouville space fractional diffusion equation stability and convergence are then carefully discussed we prove that the finite volume scheme is unconditionally stable and convergent with second order accuracy in time and min 1 1 order in space with respect to a weighted discrete norm here 0 1 are the space fractional order indexes in x and y directions respectively furthermore we rewrite the finite volume scheme into a matrix form and develop a matrix free preconditioned fast krylov subspace iterative method which only requires storage of o n and computational cost of o n log n per iteration without losing any accuracy compared to the direct gaussian elimination method here n is the total number of spatial unknowns consequently the fast finite volume method is particularly suitable for large scale modeling and simulation numerical experiments verify the theoretical results and show strong potential of the fast method

nonlinear evolution equations as fast and exact solvers of estimation problems we develop computationally efficient procedures for solving certain restoration problems in one dimension including the one dimensional 1 d discrete versions of the total variation regularized problem introduced by sauer and bouman and the constrained total variation minimization problem introduced by rudin et al the procedures are exact and have time complexity o nlogn and space complexity o n where n is the number of data samples they are based on a simple nonlinear diffusion equation proposed by pollak et al and related to the perona malik equation a probabilistic interpretation for this diffusion equation in 1 d is provided by showing that it produces optimal solutions to a sequence of estimation problems we extend our methods to two dimensions where they no longer have similar optimality properties however we experimentally demonstrate their effectiveness for image restoration

nonlinear wavelet image processing variational problems compression and noise removal through wavelet shrinkage this paper examines the relationship between wavelet based image processing algorithms and variational problems algorithms are derived as exact or approximate minimizers of variational problems in particular we show that wavelet shrinkage can be considered the exact minimizer of the following problem given an image f defined on a square i minimize over all g in the besov space b11 l1 i the functional f g l2 i 2 g b11 l1 i we use the theory of nonlinear wavelet image compression in l2 i to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i i d mean zero gaussian noise a new signal to noise ratio snr which we claim more accurately reflects the visual perception of noise in images arises in this derivation we present extensive computations that support the hypothesis that near optimal shrinkage parameters can be derived if one knows or can estimate only two parameters about an image f the largest for which f bq lq i 1 q 2 1 2 and the norm f bq lq i both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than donoho and johnstone s visushrink procedure an example suggests however that donoho and johnstone s 1994 1995 1996 sureshrink method which uses a different shrinkage parameter for each dyadic level achieves a lower error than our procedure

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

joint parameter estimation and symbol detection for linear or nonlinear unknown channels we present an iterative method for joint channel parameter estimation and symbol selection via the baum welch algorithm or equivalently the expectation maximization em algorithm channel parameters including noise variance are estimated using a maximum likelihood criterion the markovian properties of the channel state sequence enable us to calculate the required likelihood using a forward backward algorithm the calculated likelihood functions can easily give optimum decisions on information symbols which minimize the symbol error probability the proposed receiver can be used for both linear and nonlinear channels it improves the system throughput by making saving in the transmission of known symbols usually employed for channel identification simulation results which show fast convergence are presented

fine granularity and spatially adaptive regularization for projection based image deblurring this paper studies two classes of regularization strategies to achieve an improved tradeoff between image recovery and noise suppression in projection based image deblurring the first is based on a simple fact that r times landweber iteration leads to a fixed level of regularization which allows us to achieve fine granularity control of projection based iterative deblurring by varying the value r the regularization behavior is explained by using the theory of lagrangian multiplier for variational schemes the second class of regularization strategy is based on the observation that various regularized filters can be viewed as nonexpansive mappings in the metric space a deeper understanding about different regularization filters can be gained by probing into their asymptotic behavior the fixed point of nonexpansive mappings by making an analogy to the states of matter in statistical physics we can observe that different image structures smooth regions regular edges and textures correspond to different fixed points of nonexpansive mappings when the temperature regularization parameter varies such an analogy motivates us to propose a deterministic annealing based approach toward spatial adaptation in projection based image deblurring significant performance improvements over the current state of the art schemes have been observed in our experiments which substantiates the effectiveness of the proposed regularization strategies

estimation of optimal pde based denoising in the snr sense this paper is concerned with finding the best partial differential equation based denoising process out of a set of possible ones we focus on either finding the proper weight of the fidelity term in the energy minimization formulation or on determining the optimal stopping time of a nonlinear diffusion process a necessary condition for achieving maximal snr is stated based on the covariance of the noise and the residual part we provide two practical alternatives for estimating this condition by observing that the filtering of the image and the noise can be approximated by a decoupling technique with respect to the weight or time parameters our automatic algorithm obtains quite accurate results on a variety of synthetic and natural images including piecewise smooth and textured ones we assume that the statistics of the noise were previously estimated no a priori knowledge regarding the characteristics of the clean image is required a theoretical analysis is carried out where several snr performance bounds are established for the optimal strategy and for a widely used method wherein the variance of the residual part equals the variance of the noise

a crank nicolson adi galerkin legendre spectral method for the two dimensional riesz space distributed order advection diffusion equation abstract in the paper a crank nicolson alternating direction implicit adi galerkin legendre spectral scheme is presented for the two dimensional riesz space distributed order advection diffusion equation the gauss quadrature has a higher computational accuracy than the mid point quadrature rule which is proposed to approximate the distributed order riesz space derivative so that the considered equation is transformed into a multi term fractional equation moreover the transformed equation is solved by discretizing in space by the adi galerkin legendre spectral scheme and in time using the crank nicolson difference method stability and convergence analysis are verified for the numerical approximation a lot of numerical results are demonstrated to justify the theoretical analysis

iterative multiuser receivers for cdma channels an em based approach maximum likelihood detection for the multiuser code division multiple access cdma channel is prohibitively complex this paper considers new iterative multiuser receivers based on the expectation maximization em algorithm and related more powerful space alternating algorithms the latter algorithms include the sage algorithm and a new missing parameter space alternating algorithm that alternately updates individual parameter components or treats them as probabilistic missing data application of these em based algorithms to the problem of discrete parameter estimation i e data detection in the gaussian multiple access channel leads to a variety of convergent receiver structures that incorporate soft decision feedback for interference cancellation and or sequential updating of iterative bit estimates convergence and performance analyses are based on well known properties of the em algorithm and on numerical simulation

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

a time space spectral method for the time space fractional fokker planck equation and its inverse problem abstract in the paper we consider a time space spectral method to get the numerical solution of time space fractional fokker planck initial boundary value problem the temporal discretization is constructed by jacobi polynomials and the spatial discretization is composed by legendre polynomials moreover we present the stability and convergence analysis strictly the main advantages of the provided method are spectrally accurate in time and space and high computational efficiency in addition we introduce the inverse problem based on the spectral form with high order accuracy of the direct problem for the first time the levenberg marquardt l m method is proposed to estimate the two fractional derivatives and 2 some numerical results presented are consistent with the theoretical analysis

combined complex ridgelet shrinkage and total variation minimization a new algorithm for the characterization of engineering surface topographies with line singularities is proposed it is based on thresholding complex ridgelet coefficients combined with total variation tv minimization the discrete ridgelet transform is designed by first using a discrete radon transform based on the nonequispaced fast fourier transform nfft and then applying a dual tree complex wavelet transform dt cwt the nfft based approach of the radon transform completely avoids linear interpolations of the cartesian to polar grid and requires only o n2 log n arithmetic operations for n by n arrays while its inverse preserves the good reconstruction quality of the filtered backprojection the dt cwt in the second step of the ridgelet transform provides approximate shift invariance on the projections of the radon transform after hard thresholding the ridgelet coefficients they are restored using tv minimization to eliminate the pseudo gibbs artifacts near the discontinuities numerical experiments demonstrate the remarkable ability of the methodology to extract line scratches

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

a maximum likelihood approach to blind multiuser interference cancellation this paper addresses the problem of blind multiple access interference mai and intersymbol interference isi suppression in direct sequence code division multiple access ds cdma systems a novel approach to obtain the coefficients of a linear receiver using the maximum likelihood ml principle is proposed the method is blind because it only exploits the statistical features of the transmitted symbols and gaussian noise in the channel we demonstrate that an adequate linear constraint on these coefficients ensures that the desired user is extracted and the resulting linearly constrained maximum likelihood linear lcmll receiver can be efficiently implemented using the iterative space alternating generalized expectation maximization sage algorithm in order to take advantage of the diversity inherent to multipath channels we also introduce a blind rake multiuser receiver that proceeds in two steps first soft estimates of the desired user transmitted symbols are obtained from each propagation path using a bank of appropriate lcmll receivers afterwards these estimates are adequately combined to enhance the signal to interference and noise ratio sinr computer simulations show that the proposed blind algorithms for multiuser detection are near far resistant and attain convergence using small blocks of data thus outperforming existing linearly constrained minimum variance lcmv blind receivers

efficient nonlocal means for denoising of textural patterns this paper contributes two novel techniques in the context of image restoration by nonlocal filtering first we introduce an efficient implementation of the nonlocal means filter based on arranging the data in a cluster tree the structuring of data allows for a fast and accurate preselection of similar patches in contrast to previous approaches the preselection is based on the same distance measure as used by the filter itself it allows for large speedups especially when the search for similar patches covers the whole image domain i e when the filter is truly nonlocal however also in the windowed version of the filter the cluster tree approach compares favorably to previous techniques in respect of quality versus computational cost second we suggest an iterative version of the filter that is derived from a variational principle and is designed to yield nontrivial steady states it reveals to be particularly useful in order to restore regular textured patterns

sequence estimation in the presence of random parameters via the em algorithm the expectation maximization em algorithm was first introduced in the statistics literature as an iterative procedure that under some conditions produces maximum likelihood hit parameter estimates in this paper we investigate the application of the em algorithm to sequence estimation in the presence of random disturbances and additive white gaussian noise as examples of the use of the em algorithm we look at the random phase and fading channels and show that a formulation of the sequence estimation problem based on the em algorithm can provide a means of obtaining ml sequence estimates a task that has been previously too complex to perform

a signal processing approach to generalized 1 d total variation total variation tv is a powerful method that brings great benefit for edge preserving regularization despite being widely employed in image processing it has restricted applicability for 1 d signal processing since piecewise constant signals form a rather limited model for many applications here we generalize conventional tv in 1 d by extending the derivative operator which is within the regularization term to any linear differential operator this provides flexibility for tailoring the approach to the presence of nontrivial linear systems and for different types of driving signals such as spike like piecewise constant and so on conventional tv remains a special case of this general framework we illustrate the feasibility of the method by considering a nontrivial linear system and different types of driving signals

a variational method in image recovery this paper is concerned with a classical denoising and deblurring problem in image recovery our approach is based on a variational method by using the legendre fenchel transform we show how the nonquadratic criterion to be minimized can be split into a sequence of half quadratic problems easier to solve numerically first we prove an existence and uniqueness result and then we describe the algorithm for computing the solution and we give a proof of convergence finally we present some experimental results for synthetic and real images

some qualitative properties for the total variation flow we prove the existence of a finite extinction time for the solutions of the dirichlet problem for the total variation flow for the neumann problem we prove that the solutions reach the average of its initial datum in finite time the asymptotic profile of the solutions of the dirichlet problem is also studied it is shown that the profiles are nonzero solutions of an eigenvalue type problem that seems to be unexplored in the previous literature the propagation of the support is analyzed in the radial case showing a behaviour entirely different to the case of the problem associated with the p laplacian operator finally the study of the radially symmetric case allows us to point out other qualitative properties that are peculiar of this special class of quasilinear equations

an iterative shrinkage approach to total variation image restoration the problem of restoration of digital images from their degraded measurements plays a central role in a multitude of practically important applications a particularly challenging instance of this problem occurs in the case when the degradation phenomenon is modeled by an ill conditioned operator in such a situation the presence of noise makes it impossible to recover a valuable approximation of the image of interest without using some a priori information about its properties such a priori information commonly referred to as simply priors is essential for image restoration rendering it stable and robust to noise moreover using the priors makes the recovered images exhibit some plausible features of their original counterpart particularly if the original image is known to be a piecewise smooth function one of the standard priors used in this case is defined by the rudin osher fatemi model which results in total variation tv based image restoration the current arsenal of algorithms for tv based image restoration is vast in this present paper a different approach to the solution of the problem is proposed based upon the method of iterative shrinkage aka iterated thresholding in the proposed method the tv based image restoration is performed through a recursive application of two simple procedures viz linear filtering and soft thresholding therefore the method can be identified as belonging to the group of first order algorithms which are efficient in dealing with images of relatively large sizes another valuable feature of the proposed method consists in its working directly with the tv functional rather then with its smoothed versions moreover the method provides a single solution for both isotropic and anisotropic definitions of the tv functional thereby establishing a useful connection between the two formulae finally a number of standard examples of image deblurring are demonstrated in which the proposed method can provide restoration results of superior quality as compared to the case of sparse wavelet deconvolution

susan a new approach to low level image processing this paper describes a new approach to low level image processing in particular edge and corner detection and structure preserving noise reduction non linear filtering is used to define which parts of the image are closely related to each individual pixel each pixel has associated with it a local image region which is of similar brightness to that pixel the new feature detectors are based on the minimization of this local image region and the noise reduction method uses this region as the smoothing neighbourhood the resulting methods are accurate noise resistant and fast details of the new feature detectors and of the new noise reduction method are described along with test results

structure texture image decomposition modeling algorithms and parameter selection this paper explores various aspects of the image decomposition problem using modern variational techniques we aim at splitting an original image f into two components u and where u holds the geometrical information and holds the textural information the focus of this paper is to study different energy terms and functional spaces that suit various types of textures our modeling uses the total variation energy for extracting the structural part and one of four of the following norms for the textural part l2 g l1 and a new tunable norm suggested here for the first time based on gabor functions apart from the broad perspective and our suggestions when each model should be used the paper contains three specific novelties first we show that the correlation graph between u and may serve as an efficient tool to select the splitting parameter second we propose a new fast algorithm to solve the tv l1 minimization problem and third we introduce the theory and design tools for the tv gabor model

em based recursive estimation of channel parameters recursive online expectation maximization em algorithm along with stochastic approximation is employed in this paper to estimate unknown time invariant variant parameters the impulse response of a linear system channel is modeled as an unknown deterministic vector process and as a gaussian vector process with unknown stochastic characteristics using these models which are embedded in white or colored gaussian noise different types of recursive least squares rls kalman filtering and smoothing and combined rls and kalman type algorithms are derived directly from the recursive em algorithm the estimation of unknown parameters also generates new recursive algorithms for situations such as additive colored noise modeled by an autoregressive process the recursive em algorithm is shown as a powerful tool which unifies the derivations of many adaptive estimation methods

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

a new method for evaluating the log likelihood gradient the hessian and the fisher information matrix for linear dynamic systems a method is presented for evaluating the log likelihood gradient score the hessian and the fisher information matrix of the parameters of linear dynamic stochastic systems the method incorporates the optimal kalman smoothing equations and is therefore ideal for simultaneous state estimation and parameter identification the result can be used for efficient implementation of gradient based algorithms for maximum likelihood identification of the unknown system parameters and for assessing the mean square estimation accuracy

comparison of em based algorithms for mimo channel estimation iterative channel estimation can improve the channel state information csi with respect to noniterative estimation new iterative channel estimators based on the expectation maximization em algorithm are proposed in this paper a first estimator called the unbiased em uem is designed to unbias the em estimates a second estimator is then put forward which is based on the expectation conditional maximization ecm algorithm and its complexity is lower than that of the em an unbiased ecm uecm estimator is also proposed although the unbiasedness of the uem and uecm estimators is not rigorously proved the use of these names is explained in the paper the new estimators are compared with well known ones such as the em the decision directed dd and the data aided da estimators simulations are reported for a turbo receiver operating over frequency selective multiple input multiple output channels it is shown that the uem channel estimator outperforms the em and that the ecm based estimators are very close to the em based ones

shearlet based total variation diffusion for denoising we propose a shearlet formulation of the total variation tv method for denoising images shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization common approaches in combining wavelet like representations such as curvelets with tv or diffusion methods aim at reducing gibbs type artifacts after obtaining a nearly optimal estimate we show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain we also analyze the performance of a shearlet based diffusion method numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform furthermore the shearlet tv scheme requires far fewer iterations than similar competitors

from local kernel to nonlocal multiple model image denoising we review the evolution of the nonparametric regression modeling in imaging from the local nadaraya watson kernel estimate to the nonlocal means and further to transform domain filtering based on nonlocal block matching the considered methods are classified mainly according to two main features local nonlocal and pointwise multipoint here nonlocal is an alternative to local and multipoint is an alternative to pointwise these alternatives though obvious simplifications allow to impose a fruitful and transparent classification of the basic ideas in the advanced techniques within this framework we introduce a novel single and multiple model transform domain nonlocal approach the block matching and 3 d filtering bm3d algorithm which is currently one of the best performing denoising algorithms is treated as a special case of the latter approach

why simple shrinkage is still relevant for redundant representations shrinkage is a well known and appealing denoising technique introduced originally by donoho and johnstone in 1994 the use of shrinkage for denoising is known to be optimal for gaussian white noise provided that the sparsity on the signal s representation is enforced using a unitary transform still shrinkage is also practiced with nonunitary and even redundant representations typically leading to very satisfactory results in this correspondence we shed some light on this behavior the main argument in this work is that such simple shrinkage could be interpreted as the first iteration of an algorithm that solves the basis pursuit denoising bpdn problem while the desired solution of bpdn is hard to obtain in general we develop a simple iterative procedure for the bpdn minimization that amounts to stepwise shrinkage we demonstrate how the simple shrinkage emerges as the first iteration of this novel algorithm furthermore we show how shrinkage can be iterated turning into an effective algorithm that minimizes the bpdn via simple shrinkage steps in order to further strengthen the denoising effect

revisiting the relationship between adaptive smoothing and anisotropic diffusion with modified filters anisotropic diffusion has been known to be closely related to adaptive smoothing and discretized in a similar manner this paper revisits a fundamental relationship between two approaches it is shown that adaptive smoothing and anisotropic diffusion have different theoretical backgrounds by exploring their characteristics with the perspective of normalization evolution step size and energy flow based on this principle adaptive smoothing is derived from a second order partial differential equation pde not a conventional anisotropic diffusion via the coupling of fick s law with a generalized continuity equation where a source or sink exists which has not been extensively exploited we show that the source or sink is closely related to the asymmetry of energy flow as well as the normalization term of adaptive smoothing it enables us to analyze behaviors of adaptive smoothing such as the maximum principle and stability with a perspective of a pde ultimately this relationship provides new insights into application specific filtering algorithm design by modeling the source or sink in the pde we introduce two specific diffusion filters the robust anisotropic diffusion and the robust coherence enhancing diffusion as novel instantiations which are more robust against the outliers than the conventional filters

a fast accurate approximation method with multigrid solver for two dimensional fractional sub diffusion equation a fast accurate approximation method with multigrid solver is proposed to solve a two dimensional fractional sub diffusion equation using the finite difference discretization of fractional time derivative a block lower triangular toeplitz matrix is obtained where each main diagonal block contains a two dimensional matrix for the laplacian operator our idea is to make use of the block circulant approximation via fast fourier transforms so that the resulting task is to solve a block diagonal system where each diagonal block matrix is the sum of a complex scalar times the identity matrix and a laplacian matrix we show that the accuracy of the approximation scheme is of o because of the special diagonal block structure we employ the multigrid method to solve the resulting linear systems the convergence of the multigrid method is studied numerical examples are presented to illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver an approximation method is proposed to approximate the original block lower triangular toeplitz linear system sufficient condition is given to guarantee a high accuracy of the approximation method multigrid solver is employed to solve the resulting block diagonal linear system multigrid solver is proven to be linear convergence numerical examples illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver

iterative and sequential algorithms for multisensor signal enhancement in problems of enhancing a desired signal in the presence of noise multiple sensor measurements will typically have components from both the signal and the noise sources when the systems that couple the signal and the noise to the sensors are unknown the problem becomes one of joint signal estimation and system identification the authors specifically consider the two sensor signal enhancement problem in which the desired signal is modeled as a gaussian autoregressive ar process the noise is modeled as a white gaussian process and the coupling systems are modeled as linear time invariant finite impulse response fir filters the main approach consists of modeling the observed signals as outputs of a stochastic dynamic linear system and the authors apply the estimate maximize em algorithm for jointly estimating the desired signal the coupling systems and the unknown signal and noise spectral parameters the resulting algorithm can be viewed as the time domain version of the frequency domain approach of feder et al 1989 where instead of the noncausal frequency domain wiener filter the kalman smoother is used this approach leads naturally to a sequential adaptive algorithm by replacing the kalman smoother with the kalman filter and in place of successive iterations on each data block the algorithm proceeds sequentially through the data with exponential weighting applied to allow adaption to nonstationary changes in the structure of the data a computationally efficient implementation of the algorithm is developed an expression for the log likelihood gradient based on the kalman smoother filter output is also developed and used to incorporate efficient gradient based algorithms in the estimation process

adaptive mlsde using the em algorithm the theory of adaptive sequence detection incorporating estimation of channel and related parameters is studied in the context of maximum likelihood ml principles in a general framework based on the expectation and maximization em algorithm a generalized ml sequence detection and estimation gmlsde criterion is derived based on the em approach and it is shown how the per survivor processing and per branch processing methods emerge naturally from gmlsde gmlsde is developed into a real time detection estimation algorithm using the online em algorithm with coupling between estimation and detection by utilizing titterington s 1984 stochastic approximation approach different adaptive ml sequence detection and estimation mlsde algorithms are formulated in a unified manner for different channel models and for different amounts of channel knowledge available at the receiver computer simulation results are presented for differentially encoded quadrature phase shift keying in frequency flat and selective fading channels and comparisons are made among the performances of the various adaptive mlsde algorithms derived earlier

a dictionary learning approach for poisson image deblurring the restoration of images corrupted by blur and poisson noise is a key issue in medical and biological image processing while most existing methods are based on variational models generally derived from a maximum a posteriori map formulation recently sparse representations of images have shown to be efficient approaches for image recovery following this idea we propose in this paper a model containing three terms a patch based sparse representation prior over a learned dictionary the pixel based total variation regularization term and a data fidelity term capturing the statistics of poisson noise the resulting optimization problem can be solved by an alternating minimization technique combined with variable splitting extensive experimental results suggest that in terms of visual quality peak signal to noise ratio value and the method noise the proposed algorithm outperforms state of the art methods