seismic data restoration via data driven tight frame restoration interpolation of missing traces plays a crucial role in the seismic data processing pipeline efficient restoration methods have been proposed based on sparse signal representation in a transform domain such as fourier wavelet curvelet and shearlet transforms most existing methods are based on transforms with a fixed basis we considered an adaptive sparse transform for restoration of data with complex structures in particular we evaluated a data driven tight frame based sparse regularization method for seismic data restoration the main idea of the data driven tight frame tf is to adaptively learn a set of framelet filters from the currently interpolated data under which the data can be more sparsely represented hence the sparsity promoting l1 norm spl1 minimization methods can produce better restoration results by using the learned filters a split inexact uzawa algorithm which can be viewed as a generalization of the alternating direction of multiplier method admm was applied to solve the presented spl1 model numerical tests were performed on synthetic and real seismic data for restoration of randomly missing traces over a regular data grid our experiments showed that our proposed method obtains the state of the art restoration results in comparison with the traditional fourier based projection onto convex sets the tight frame based method and the recent shearlet regularization admm method

thresholding estimators for linear inverse problems and deconvolutions thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded for signals in a set sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence deconvolutions with kernels having a fourier transform which vanishes at high frequencies are examples of unstable inverse problems where a thresholding in a wavelet basis is a suboptimal estimator a new mirror wavelet basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals this thresholding estimator is used to restore blurred satellite images

blind motion deblurring using multiple images recovery of degraded images due to motion blurring is a challenging problem in digital imaging most existing techniques on blind deblurring are not capable of removing complex motion blurring from the blurred images of complex structures one promising approach is to recover the clear image using multiple images captured for the scene however in practice it is observed that such a multi frame approach can recover a high quality clear image of the scene only after multiple blurred image frames are accurately aligned during pre processing which is a very challenging task even with user interactions in this paper by exploring the sparsity of the motion blur kernel and the clear image under certain domains we propose an alternative iteration approach to simultaneously identify the blur kernels of given blurred images and restore a clear image our proposed approach is not only robust to image formation noises but is also robust to the alignment errors among multiple images a modified version of linearized bregman iteration is then developed to efficiently solve the resulting minimization problem experiments show that our proposed algorithm is capable of accurately estimating the blur kernels of complex camera motions with minimal requirements on the accuracy of image alignment as a result our method is capable of automatically recovering a high quality clear image from multiple blurred images

deconvolution by thresholding in mirror wavelet bases the deconvolution of signals is studied with thresholding estimators that decompose signals in an orthonormal basis and threshold the resulting coefficients a general criterion is established to choose the orthonormal basis in order to minimize the estimation risk wavelet bases are highly sub optimal to restore signals and images blurred by a low pass filter whose transfer function vanishes at high frequencies a new orthonormal basis called mirror wavelet basis is constructed to minimize the risk for such deconvolutions an application to the restoration of satellite images is shown

linearized bregman iterations for compressed sensing finding a solution of a linear equation au f with various minimization properties arises from many applications one such application is compressed sensing where an efficient and robust to noise algorithm to find a minimal l 1 norm solution is needed this means that the algorithm should be tailored for large scale and completely dense matrices a while au and a t u can be computed by fast transforms and the solution we seek is sparse recently a simple and fast algorithm based on linearized bregman iteration was proposed in 28 32 for this purpose this paper is to analyze the convergence of linearized bregman iterations and the minimization properties of their limit based on our analysis here we derive also a new algorithm that is proven to be convergent with a rate furthermore the new algorithm is simple and fast in approximating a minimal l 1 norm solution of au f as shown by numerical simulations hence it can be used as another choice of an efficient tool in compressed sensing

new criteria for exponential stability of nonlinear difference systems with time varying delay general nonlinear time varying difference systems with time varying delay are considered some new explicit criteria for global exponential stability are given two examples are given to illustrate the obtained results

wavelet frame based surface reconstruction from unorganized points applications of wavelet frames to image restoration problems e g image deblurring and inpainting have been successful due to their redundancy and capability of sparsely approximating piecewise smooth functions like images see e g 13 14 8 however wavelet frames have not yet been used for surface reconstruction problems recently in 6 connections between one of the wavelet frame based image restoration model 8 28 48 and variational models e g the rof model 45 were rigorously established such connections not only grant new insights to wavelet frame based image restorations it also case a geometric explanation to wavelet frame based approaches this leads us to a wavelet frame based model as well as a fast algorithm to reconstruct implicit surfaces from unorganized point sets in r 3 we will demonstrate the effectiveness of the proposed model using several commonly used examples

framelet based blind motion deblurring from a single image how to recover a clear image from a single motion blurred image has long been a challenging open problem in digital imaging in this paper we focus on how to recover a motion blurred image due to camera shake a regularization based approach is proposed to remove motion blurring from the image by regularizing the sparsity of both the original image and the motion blur kernel under tight wavelet frame systems furthermore an adapted version of the split bregman method is proposed to efficiently solve the resulting minimization problem the experiments on both synthesized images and real images show that our algorithm can effectively remove complex motion blurring from natural images without requiring any prior information of the motion blur kernel

accelerating l1 l2 deblurring using wavelet expansions of operators abstract image deblurring is a fundamental problem in imaging usually solved with computationally intensive optimization procedures the goal of this paper is to provide new efficient strategies to reduce computing times for simple deblurring models regularized using orthogonal wavelet transforms we show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis the proposed methodology consists of three ingredients i a sparse approximation of the blur operator in wavelet bases ii a diagonal preconditioner and iii an implementation on massively parallel architectures combining the three ingredients leads to acceleration factors ranging from 4 to 250 on a typical workstation for instance a 1024 1024 image can be deblurred in 0 15 s

infinite eigenvalue assignment for singular systems abstract in this paper the infinite eigenvalue assignment problem for singular systems is studied necessary and sufficient conditions are presented under which there exists a state feedback such that the closed loop system is regular and has only infinite eigenvalues the main result is proved constructively based on some simple numerical algorithms these numerical algorithms consist of an orthogonal reduction to an upper block hessenberg form and a simple linear recursion deduced from 2 2 givens transformations

alternating direction method of multipliers for linear inverse problems in this paper we propose an iterative method using alternating direction method of multipliers admm strategy to solve linear inverse problems in hilbert spaces with a general convex penalty term when the data is given exactly we give a convergence analysis of our admm algorithm without assuming the existence of a lagrange multiplier in case the data contains noise we show that our method is a regularization method as long as it is terminated by a suitable stopping rule various numerical simulations are performed to test the efficiency of the method

wavelet frame based scene reconstruction from range data abstract how to reconstruct the scene a visible surface from a set of scattered noisy and possibly sparse range data is a challenging problem in robotic navigation and computer graphics as most real scenes can be modeled by piecewise smooth surfaces traditional surface fitting techniques e g smoothing spline generally can not preserve sharp discontinuities of surfaces based on sparse approximation of piecewise smooth functions in frame domain we propose a new tight frame based formulation for reconstructing a piecewise smooth surface from a sparse range data set which is robust to both additive noise and outliers furthermore the resulting minimization problem from our formulation can be efficiently solved by the split bregman method 1 2 the numerical experiments show that the proposed approach is capable of reconstructing a piecewise smooth surface with sharp edges from sparse range data corrupted with noise and outliers

exponential ultimate boundedness of nonlinear stochastic difference systems with time varying delays in this paper we discuss the boundedness of nonlinear stochastic difference systems with time varying delays several criteria on exponential ultimate boundedness in mean square are derived examples are also discussed to illustrate the effectiveness of the obtained results

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

simple stability criteria for nonlinear time varying discrete systems abstract we present two sufficient conditions for asymptotic stability of nonlinear time varying discrete systems our main results generalize mori s result for linear discrete time invariant systems to nonlinear time varying systems

a nonstationary accelerating alternating direction method for frame based poissonian image deblurring abstract poissonian image deblurring problem has been an attractive research subject in many areas of imaging science and engineering in order to obtain accurate restorations one has to run several times to manually select a good value of a regularization parameter in many models in this paper we first propose an optimization model by combining a new data fitting term with a frame based analysis regularizer and a nonnegative constraint a discrepancy principle based nonstationary accelerating alternating direction method called dp naadm is then proposed to solve the optimization model without estimating the regularization parameter numerical results show that dp naadm provides competitive restoration results compared to the well known poisson image deconvolution by augmented lagrangian method called pidal also dp naadm provides better restorations in terms of both accuracy and visual quality than the discrepancy principle based pidal method

restoration of chopped and nodded images by framelets in infrared astronomy an observed image from a chop and nod process can be considered as the result of passing the original image through a high pass filter here we propose a restoration algorithm which builds up a tight framelet system that has the high pass filter as one of the framelet filters our approach reduces the solution of restoration problem to that of recovering the missing coefficients of the original image in the tight framelet decomposition the framelet approach provides a natural setting to apply various sophisticated framelet denoising schemes to remove the noise without reducing the intensity of major stars in the image a proof of the convergence of the algorithm based on convex analysis is also provided simulated and real images are tested to illustrate the efficiency of our method over the projected landweber method

deconvolution a wavelet frame approach this paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches which has already appeared in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b as wavelet frame based high resolution image reconstruction methods we first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation which completes the formulation given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b this formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties these missing coefficients are recovered iteratively together with a built in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating this approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b however an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers this paper is to establish the theoretical foundation of this wavelet frame approach in particular a proof of convergence an analysis of the stability of algorithms and a study of the minimization property of solutions are given

convergence of the linearized bregman iteration for l1 norm minimization one of the key steps in compressed sensing is to solve the basis pursuit problem min u n u au f bregman iteration was very successfully used to solve this problem in 40 also a simple and fast iterative algorithm based on linearized bregman iteration was proposed in 40 which is described in detail with numerical simulations in 35 a convergence analysis of the smoothed version of this algorithm was given in 11 the purpose of this paper is to prove that the linearized bregman iteration proposed in 40 for the basis pursuit problem indeed converges

wavelet algorithms for high resolution image reconstruction high resolution image reconstruction refers to the reconstruction of high resolution images from multiple low resolution shifted degraded samples of a true image in this paper we analyze this problem from the wavelet point of view by expressing the true image as a function in cal l bbb r 2 we derive iterative algorithms which recover the function completely in the cal l sense from the given low resolution functions these algorithms decompose the function obtained from the previous iteration into different frequency components in the wavelet transform domain and add them into the new iterate to improve the approximation we apply wavelet packet thresholding methods to denoise the function obtained in the previous step before adding it into the new iterate our numerical results show that the reconstructed images from our wavelet algorithms are better than that from the tikhonov least squares approach extension to super resolution image reconstruction where some of the low resolution images are missing is also considered

wavelet deblurring algorithms for spatially varying blur from high resolution image reconstruction abstract high resolution image reconstruction refers to reconstructing a higher resolution image from multiple low resolution samples of a true image in chan et al wavelet algorithms for high resolution image reconstruction research report cuhk 2000 20 department of mathematics the chinese university of hong kong 2000 we considered the case where there are no displacement errors in the low resolution samples i e the samples are aligned properly and hence the blurring operator is spatially invariant in this paper we consider the case where there are displacement errors in the low resolution samples the resulting blurring operator is spatially varying and is formed by sampling and summing different spatially invariant blurring operators we represent each of these spatially invariant blurring operators by a tensor product of a lowpass filter which associates the corresponding blurring operator with a multiresolution analysis of l 2 r 2 using these filters and their duals we derive an iterative algorithm to solve the problem based on the algorithmic framework of chanet al wavelet algorithms for high resolution image reconstruction research report cuhk 2000 20 department of mathematics the chinese university of hong kong 2000 our algorithm requires a nontrivial modification to the algorithms in chan et al wavelet algorithms for high resolution image reconstruction research report cuhk 2000 20 department of mathematics the chinese university of hong kong 2000 which apply only to spatially invariant blurring operators our numerical examples show that our algorithm gives higher peak signal to noise ratios and lower relative errors than those from the tikhonov least squares approach

affine systems in l2 rd the analysis of the analysis operator abstract discrete affine systems are obtained by applying dilations to a given shift invariant system the complicated structure of the affine system is due first and foremost to the fact that it is not invariant under shifts affine frames carry the additional difficulty that they are global in nature it is the entire interaction between the various dilation levels that determines whether the system is a frame and not the behaviour of the system within one dilation level we completely unravel the structure of the affine system with the aid of two new notions the affine product and a quasi affine system this leads to a characterization of affine frames the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the fourier domain several results such as a general oversampling theorem follow from these characterizations most importantly the affine product can be factored during a multiresolution analysis construction and this leads to a complete characterization of all tight frames that can be constructed by such methods moreover this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions and that the freedom offered by redundancy can be fully exploited in these constructions an

stability analysis of nonlinear retarded difference equations in banach spaces in this paper we give new sufficient conditions for asymptotic stability and instability of nonlinear difference equations with delays in infinite dimensional spaces x k 1 f k x k x k 1 x k r x k x k 0 1 2 where x is a banach space our results are obtained using a general comparison condition on the right hand side function f k which generalizes the stability and instability results obtained by yang miminis naulin and vanegas an application to the stabilizability problem of retarded control systems and illustrative examples of the obtained results are given

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

stability analysis of a general family of nonlinear positive discrete time delay systems in this paper we propose a new approach to analyse the stability of a general family of nonlinear positive discrete time delay systems first we introduce a new class of nonlinear positive discrete time delay systems which generalises some existing discrete time delay systems second through a new technique that relies on the comparison and mathematical induction method we establish explicit criteria for stability and instability of the systems three numerical examples are given to illustrate the feasibility of the obtained results

adaptive wavelet galerkin methods for linear inverse problems we introduce and analyze numerical methods for the treatment of inverse problems based on an adaptive wavelet galerkin discretization these methods combine the theoretical advantages of the wavelet vaguelette decomposition wvd in terms of optimally adapting to the unknown smoothness of the solution together with the numerical simplicity of galerkin methods in a first step we simply combine a thresholding algorithm on the data with a galerkin inversion on a fixed linear space in a second step a more elaborate method performs the inversion by an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed this leads to a significant reduction of the computational cost