innovation rate sampling of pulse streams with application to ultrasound imaging signals comprised of a stream of short pulses appear in many applications including bioimaging and radar the recent finite rate of innovation framework has paved the way to low rate sampling of such pulses by noticing that only a small number of parameters per unit time are needed to fully describe these signals unfortunately for high rates of innovation existing sampling schemes are numerically unstable in this paper we propose a general sampling approach which leads to stable recovery even in the presence of many pulses we begin by deriving a condition on the sampling kernel which allows perfect reconstruction of periodic streams from the minimal number of samples we then design a compactly supported class of filters satisfying this condition the periodic solution is extended to finite and infinite streams and is shown to be numerically stable even for a large number of pulses high noise robustness is also demonstrated when the delays are sufficiently separated finally we process ultrasound imaging data using our techniques and show that substantial rate reduction with respect to traditional ultrasound sampling schemes can be achieved

ince s limits for confluent and double confluent heun equations we find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation this is also known as confluent heun equation one solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable z while the other is given by a series of modified bessel functions and converges for z z0 where z0 denotes a regular singularity for short the preceding limit is called ince s limit after ince who have used the same procedure to get the mathieu equations from the whittaker hill ones we find as well that when z0 tends to zero the ince limit of the generalized spheroidal wave equation turns out to be the ince limit of a double confluent heun equation for which solutions are provided finally we show that the schrodinger equation for inverse fourth and sixth power potentials reduces to peculiar cases of the double confluent heun equation and its ince s limit respectively

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

comment on dirac equation in the background of the nutku helicoid metric j math phys 48 092301 2007 the dirac equation written on the boundary of the nutku helicoid space consists of a system of ordinary differential equations we tried to analyze this system and we found that it has a higher singularity than those of the heun equations which give the solutions of the dirac equation in the bulk we also lose an independent integral of motion on the boundary this facts explain why we could not find the solution of the system on the boundary in terms of known functions we make the stability analysis of the helicoid and catenoid cases and end up with an appendix which gives a new example wherein one encounters a form of the heun equation

2d finite rate of innovation reconstruction method for step edge and polygon signals in the presence of noise the finite rate of innovation fri principle is developed for sampling a class of non bandlimited signals that have a finite number of degrees of freedom per unit of time i e signals with fri this sampling scheme is later extended to three classes of sampling kernels with compact support and applied to the step edge reconstruction problem by treating the image row by row in this paper we regard step edges as 2d fri signals and reconstruct them block by block the step edge parameters are obtained from the 2d moments of a given image block experimentally our technique can reconstruct the edge more precisely and track the cramer rao bounds crbs closely with a signal to noise ratio snr larger than 4 db on synthetic step edge images experiments on real images show that our proposed method can reconstruct the step edges under practical conditions i e in the presence of various types of noise and using a real sampling kernel the results on locating the corners of data matrix barcodes using our method also outperform some state of the art barcode decoders

compressed sensing with basis mismatch performance bounds and sparse based estimator compressed sensing cs is a promising emerging domain that outperforms the classical limit of the shannon sampling theory if the measurement vector can be approximated as the linear combination of few basis vectors extracted from a redundant dictionary matrix unfortunately in realistic scenario the knowledge of this basis or equivalently of the entire dictionary is often uncertain i e corrupted by a basis mismatch bm error the consequence of the bm problem is that the estimation accuracy in terms of bayesian mean square error bmse of popular sparse based estimators collapses even if the support is perfectly estimated and in the high signal to noise ratio snr regime this saturation effect considerably limits the effective viability of these estimation schemes in the first part of this work the bayesian cramer rao bound bcrb is derived for cs model with unstructured bm we show that the bcrb foresees the saturation effect of the estimation accuracy of standard sparse based estimators as for instance the omp cosamp or the bp in addition we provide an approximation of this bmse threshold in the second part and in the context of the structured bm model a new estimation scheme called bias correction estimator bice is proposed and its statistical properties are studied the bice acts as a post processing estimation layer for any sparse based estimator and mitigates considerably the bm degradation finally the bice i is a blind algorithm i e is unaware of the uncorrupted dictionary matrix ii is generic since it can be associated to any sparse based estimator iii is fast i e the additional computational cost remains low and iv has good statistical properties to illustrate our results and propositions the bice is applied in the challenging context of the compressive sampling of non bandlimited impulsive signals

analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near ideal estimation performance in various asymptotic frameworks however image processing practice has shown that universal thresholding methods are outperformed by simple bayesian estimators assuming independent wavelet coefficients and heavy tailed priors such as generalized gaussian distributions ggds in this paper we investigate various connections between shrinkage methods and maximum a posteriori map estimation using such priors in particular we state a simple condition under which map estimates are sparse we also introduce a new family of complexity priors based upon rissanen s universal prior on integers one particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length mdl principle we develop analytical expressions for the shrinkage rules implied by ggd and complexity priors this allows us to show the equivalence between universal hard thresholding map estimation using a very heavy tailed ggd and mdl estimation using one of the new complexity priors theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis specifications of the prior a basic concern in image processing applications

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

generalized arnoldi tikhonov method for sparse reconstruction this paper introduces two new algorithms belonging to the class of arnoldi tikhonov regularization methods which are particularly appropriate for sparse reconstruction the main idea is to consider suitable adaptively defined regularization matrices that allow the usual 2 norm regularization term to approximate a more general regularization term expressed in the p norm p geq 1 the regularization matrix can be updated both at each step and after some iterations have been performed leading to two different approaches the first one is based on the idea of the iteratively reweighted least squares method and can be obtained considering flexible krylov subspaces the second one is based on restarting the arnoldi algorithm numerical examples are given in order to show the effectiveness of these new methods and comparisons with some other already existing algorithms are made

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

sampling streams of pulses with unknown shapes this paper develops a theory for sampling and perfectly reconstructing streams of short pulses of unknown shapes in the continuous time domain the single pulse is modelled as the delayed version of a wavelet sparse signal which is normally not band limited as the delay can be an arbitrary real number it is difficult to develop an exact sampling result for this type of signals we manage to achieve the exact reconstruction of the pulses by using only the knowledge of the fourier transform of the signal at specific frequencies we further introduce a multichannel acquisition system that uses a new family of compact support sampling kernels for extracting the fourier information from the samples the shape of the kernel is independent of the wavelet basis in which the pulse is sparse and hence the same acquisition system can be used with pulses that are sparse on different wavelet bases by exploiting the fact that pulses have short duration and that the sampling kernels have compact support we finally propose a local and sequential algorithm to reconstruct streaming pulses from the samples

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

signal enhancement a composite property mapping algorithm a signal enhancement algorithm is developed that seeks to recover a signal from noise contaminated distorted measurements made on that signal this object is achieved by utilizing a set of properties which the signal is known or is hypothesized as possessing the measured signal is modified to the smallest degree necessary to sequentially possess each of the individual properties conditions for the algorithm s convergence are established in which the primary requirement is that a composite property mapping be closed this is a relatively unrestricted condition in comparison to that required of most existing signal enhancement algorithms

fixed points of generalized approximate message passing with arbitrary matrices the estimation of a random vector with independent components passed through a linear transform followed by a componentwise possibly nonlinear output map arises in a range of applications approximate message passing amp methods based on gaussian approximations of loopy belief propagation have recently attracted considerable attention for such problems for large random transforms these methods exhibit fast convergence and admit precise analytic characterizations with testable conditions for optimality even for certain non convex problem instances however the behavior of amp under general transforms is not fully understood in this paper we consider the generalized amp gamp algorithm and relate the method to more common optimization techniques this analysis enables a precise characterization of the gamp algorithm fixed points that applies to arbitrary transforms in particular we show that the fixed points of the so called max sum gamp algorithm for map estimation are critical points of a constrained maximization of the posterior density the fixed points of the sum product gamp algorithm for estimation of the posterior marginals can be interpreted as critical points of a certain free energy

cardinal exponential splines part ii think analog act digital by interpreting the green function reproduction property of exponential splines in signal processing terms we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts the link is that the latter are the b spline coefficients of the former which happen to be exponential splines motivated by this observation we introduce an extended family of cardinal splines the generalized e splines to generalize the concept for all convolution operators with rational transfer functions we construct the corresponding compactly supported b spline basis functions which are characterized by their poles and zeros thereby establishing an interesting connection with analog filter design techniques we investigate the properties of these new b splines and present the corresponding signal processing calculus which allows us to perform continuous time operations such as convolution differential operators and modulation by simple application of the discrete version of these operators in the b spline domain in particular we show how the formalism can be used to obtain exact discrete implementations of analog filters finally we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real world analog to digital a to d and d to a conversion systems

a generalized accelerated proximal gradient approach for total variation based image restoration this paper proposes a generalized accelerated proximal gradient gapg approach for solving total variation tv based image restoration problems the gapg algorithm generalizes the original apg algorithm by replacing the lipschitz constant with an appropriate positive definite matrix resulting in faster convergence for tv based image restoration problems we further introduce two auxiliary variables that approximate the partial derivatives constraints on the variables can easily be imposed without modifying the algorithm much and the tv regularization can be either isotropic or anisotropic as compared with the recently developed apg based methods for tv based image restoration i e monotone version of the two step iterative shrinkage thresholding algorithm mtwist and monotone version of the fast ist algorithm mfista our gapg is much simpler as it does not require to solve an image denoising subproblem moreover the convergence rate of o k 2 is maintained by our gapg where k is the number of iterations the cost of each iteration in gapg is also lower as a result in our experiments our gapg approach can be much faster than mtwist and mfista the experiments also verify that our gapg converges faster than the original apg and mtwist when they solve identical problems

fri sampling on structured nonuniform grids application to super resolved optical imaging the annihilating filter forms the core of many state of the art techniques for finite rate of innovation fri signal reconstruction and spectral estimation in its standard manifestation the annihilating filter is setup in the discrete time domain with uniform sampling and is used to estimate the parameters of a signal that is expressible as a sum of exponentials in this paper we consider annihilation in the continuous domain using translation operators codenamed opera and show that fri signals can be reconstructed using measurements taken on a structured nonuniform sampling sns grid the parameters are estimable uniquely under certain conditions the transition from uniform sampling to sns comes with specific advantages the measurements could be made in bursts from high signal to noise ratio snr regions which directly enhances parameter estimation accuracy the sampling interval could be made larger than in the uniform case which leads to a reduction in the mean squared error mse and closely spaced exponential parameters could be resolved better we present theoretical developments related to the sns grid and demonstrate accurate parameter estimation monte carlo performance analysis in the presence of noise shows that the opera with the sns approach yields a significant decrease in mse up to 10 to 20 db for certain snr ranges compared with the uniform sampling scenario considering an application to frequency domain optical coherence tomography we show that the proposed methodology results in higher resolution and higher reconstruction accuracy than the standard techniques

graph cut based discrete valued image reconstruction efficient graph cut methods have been used with great success for labeling and denoising problems occurring in computer vision unfortunately the presence of linear image mappings has prevented the use of these techniques in most discrete amplitude image reconstruction problems in this paper we develop a graph cut based framework for the direct solution of discrete amplitude linear image reconstruction problems cast as regularized energy function minimizations we first analyze the structure of discrete linear inverse problem cost functions to show that the obstacle to the application of graph cut methods to their solution is the variable mixing caused by the presence of the linear sensing operator we then propose to use a surrogate energy functional that overcomes the challenges imposed by the sensing operator yet can be utilized efficiently in existing graph cut frameworks we use this surrogate energy functional to devise a monotonic iterative algorithm for the solution of discrete valued inverse problems we first provide experiments using local convolutional operators and show the robustness of the proposed technique to noise and stability to changes in regularization parameter then we focus on nonlocal tomographic examples where we consider limited angle data problems we compare our technique with state of the art discrete and continuous image reconstruction techniques experiments show that the proposed method outperforms state of the art techniques in challenging scenarios involving discrete valued unknowns

sampling moments and reconstructing signals of finite rate of innovation shannon meets strang fix consider the problem of sampling signals which are not bandlimited but still have a finite number of degrees of freedom per unit of time such as for example nonuniform splines or piecewise polynomials and call the number of degrees of freedom per unit of time the rate of innovation classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited recently it was shown that by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation it is possible to reconstruct such signals uniquely these sampling schemes however use kernels with infinite support and this leads to complex and potentially unstable reconstruction algorithms in this paper we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm the class of kernels that we can use is very rich and includes functions satisfying strang fix conditions exponential splines and functions with rational fourier transform this last class of kernels is quite general and includes for instance any linear electric circuit we thus show with an example how to estimate a signal of finite rate of innovation at the output of an rc circuit the case of noisy measurements is also analyzed and we present a novel algorithm that reduces the effect of noise by oversampling

morphological filters part i their set theoretic analysis and relations to linear shift invariant filters this paper examines the set theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations which involve supremum infimum operations and additions binary signals are classified as sets and multilevel signals as functions two set theoretic representations of signals are reviewed filters are classified as set processing sp or function processing fp conditions are provided for certain fp filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as sp filters the basic morphological operations of set erosion dilation opening and closing are related to minkowski set operations and are used to construct fp morphological filters emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions the latter case allows the definition of morphological convolutions and correlations toward this goal various properties of fp morphological filters are also examined linear shift invariant filters due to their translation invariance are uniquely characterized by their kernel which is a special collection of input signals increasing linear filters are represented as the supremum of erosions by their kernel functions if the filters are also discrete and have a finite extent impulse response they can be represented as the supremum of erosions only by their minimal with respect to a signal ordering kernel functions stable linear filters can be represented as the sum of at most two weighted suprema of erosions these results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal shaping strategies

moms maximal order interpolation of minimal support we consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly supported basis function spl phi z we first give the expression for the cases of spl phi that have minimal support for a given accuracy also known as approximation order this class of functions which we call maximal order minimal support functions moms is made of linear combinations of the b spline of the same order and of its derivatives we provide an explicit form of the moms that maximizes the approximation accuracy when the step size is small enough we compute the sampling gain obtained by using these optimal basis functions over the splines of the same order we show that it is already substantial for small orders and that it further increases with the approximation order l when l is large this sampling gain becomes linear more specifically its exact asymptotic expression is 2 spl pi e l since the optimal functions are continuous but not differentiable for even orders and even only piecewise continuous for odd orders our result implies that regularity has little to do with approximating performance these theoretical findings are corroborated by experimental evidence that involves compounded rotations of images

sampling piecewise sinusoidal signals with finite rate of innovation methods we consider the problem of sampling piecewise sinusoidal signals classical sampling theory does not enable perfect reconstruction of such signals since they are not band limited however they can be characterized by a finite number of parameters namely the frequency amplitude and phase of the sinusoids and the location of the discontinuities in this paper we show that under certain hypotheses on the sampling kernel it is possible to perfectly recover the parameters that define the piecewise sinusoidal signal from its sampled version in particular we show that at least theoretically it is possible to recover piecewise sine waves with arbitrarily high frequencies and arbitrarily close switching points extensions of the method are also presented such as the recovery of combinations of piecewise sine waves and polynomials finally we study the effect of noise and present a robust reconstruction algorithm that is stable down to snr levels of 7 db

nonlocally centralized sparse representation for image restoration sparse representation models code an image patch as a linear combination of a few atoms chosen out from an over complete dictionary and they have shown promising results in various image restoration applications however due to the degradation of the observed image e g noisy blurred and or down sampled the sparse representations by conventional models may not be accurate enough for a faithful reconstruction of the original image to improve the performance of sparse representation based image restoration in this paper the concept of sparse coding noise is introduced and the goal of image restoration turns to how to suppress the sparse coding noise to this end we exploit the image nonlocal self similarity to obtain good estimates of the sparse coding coefficients of the original image and then centralize the sparse coding coefficients of the observed image to those estimates the so called nonlocally centralized sparse representation ncsr model is as simple as the standard sparse representation model while our extensive experiments on various types of image restoration problems including denoising deblurring and super resolution validate the generality and state of the art performance of the proposed ncsr algorithm

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

image denoising using scale mixtures of gaussians in the wavelet domain we describe a method for removing noise from digital images based on a statistical model of the coefficients of an overcomplete multiscale oriented basis neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables a gaussian vector and a hidden positive scalar multiplier the latter modulates the local variance of the coefficients in the neighborhood and is thus able to account for the empirically observed correlation between the coefficient amplitudes under this model the bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable we demonstrate through simulations with images contaminated by additive white gaussian noise that the performance of this method substantially surpasses that of previously published methods both visually and in terms of mean squared error

examples of heun and mathieu functions as solutions of wave equations in curved spaces we give examples of where the heun function exists as solutions of wave equations encountered in general relativity as a new example we find that while the dirac equation written in the background of nutku helicoid metric yields mathieu functions as its solutions in four spacetime dimensions the trivial generalization to five dimensions results in the double confluent heun function we reduce this solution to the mathieu function with some transformations

fri sampling with arbitrary kernels this paper addresses the problem of sampling non bandlimited signals within the finite rate of innovation fri setting we had previously shown that by using sampling kernels whose integer span contains specific exponentials generalized strang fix conditions it is possible to devise non iterative fast reconstruction algorithms from very low rate samples yet the accuracy and sensitivity to noise of these algorithms is highly dependent on these exponential reproducing kernels actually on the exponentials that they reproduce hence our first contribution here is to provide clear guidelines on how to choose the sampling kernels optimally in such a way that the reconstruction quality is maximized in the presence of noise the optimality of these kernels is validated by comparing with cramer rao s lower bounds crb our second contribution is to relax the exact exponential reproduction requirement instead we demonstrate that arbitrary sampling kernels can reproduce the best exponentials within quite a high accuracy in general and that applying the exact fri algorithms in this approximate context results in near optimal reconstruction accuracy for practical noise levels essentially we propose a universal extension of the fri approach to arbitrary sampling kernels numerical results checked against the crb validate the various contributions of the paper and in particular outline the ability of arbitrary sampling kernels to be used in fri algorithms

mmse whitening and subspace whitening this correspondence develops a linear whitening transformation that minimizes the mean squared error mse between the original and whitened data i e one that results in a white output that is as close as possible to the input in an mse sense when the covariance matrix of the data is not invertible the whitening transformation is designed to optimally whiten the data on a subspace in which it is contained the optimal whitening transformation is developed both for the case of finite length data vectors and infinite length signals

fresh fri based single image super resolution algorithm in this paper we consider the problem of single image super resolution and propose a novel algorithm that outperforms state of the art methods without the need of learning patches pairs from external data sets we achieve this by modeling images and more precisely lines of images as piecewise smooth functions and propose a resolution enhancement method for this type of functions the method makes use of the theory of sampling signals with finite rate of innovation fri and combines it with traditional linear reconstruction methods we combine the two reconstructions by leveraging from the multi resolution analysis in wavelet theory and show how an fri reconstruction and a linear reconstruction can be fused using filter banks we then apply this method along vertical horizontal and diagonal directions in an image to obtain a single image super resolution algorithm we also propose a further improvement of the method based on learning from the errors of our super resolution result at lower resolution levels simulation results show that our method outperforms state of the art algorithms under different blurring kernels

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

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

generalized conjugate gradient methods for l1 regularized convex quadratic programming with finite convergence the conjugate gradient cg method is an efficient iterative method for solving large scale strongly convex quadratic programming qp in this paper we propose some generalized cg gcg methods for solving the l1 regularized possibly not strongly convex qp that terminate at an optimal solution in a finite number of iterations at each iteration our methods first identify a face of an orthant and then either perform an exact line search along the direction of the negative projected minimum norm subgradient of the objective function or execute a cg subroutine that conducts a sequence of cg iterations until a cg iterate crosses the boundary of this face or an approximate minimizer of over this face or a subface is found we determine which type of step should be taken by comparing the magnitude of some components of the minimum norm subgradient of the objective function to that of its rest components our analysis on finite convergence of these methods makes use of an error bound result and some key pro

single image super resolution via linear mapping of interpolated self examples this paper presents a novel example based single image superresolution procedure that upscales to high resolution hr a given low resolution lr input image without relying on an external dictionary of image examples the dictionary instead is built from the lr input image itself by generating a double pyramid of recursively scaled and subsequently interpolated images from which self examples are extracted the upscaling procedure is multipass i e the output image is constructed by means of gradual increases and consists in learning special linear mapping functions on this double pyramid as many as the number of patches in the current image to upscale more precisely for each lr patch similar self examples are found and because of them a linear function is learned to directly map it into its hr version iterative back projection is also employed to ensure consistency at each pass of the procedure extensive experiments and comparisons with other state of the art methods based both on external and internal dictionaries show that our algorithm can produce visually pleasant upscalings with sharp edges and well reconstructed details moreover when considering objective metrics such as peak signal to noise ratio and structural similarity our method turns out to give the best performance

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed

a representation theory for morphological image and signal processing a unifying theory for many concepts and operations encountered in or related to morphological image and signal analysis is presented the unification requires a set theoretic methodology where signals are modeled as sets systems signal transformations are viewed as set mappings and translational invariant systems are uniquely characterized by special collections of input signals this approach leads to a general representation theory in which any translation invariant increasing upper semicontinuous system can be presented exactly as a minimal nonlinear superposition of morphological erosions or dilations the theory is used to analyze some special cases of image signal analysis systems such as morphological filters median and order statistic filters linear filters and shape recognition transforms although the developed theory is algebraic its prototype operations are well suited for shape analysis hence the results also apply to systems that extract information about the geometrical structure of signals

solving variational inequality problems via smoothing nonsmooth reformulations it has long been known that variational inequality problems can be reformulated as nonsmooth equations recently locally high order convergent newton methods for nonsmooth equations have been well established via the concept of semismoothness when the constraint set of the variational inequality problem is a rectangle several locally convergent newton methods for the reformulated nonsmooth equations can also be globalized in this paper our main aim is to provide globally and locally high order convergent newton methods for solving variational inequality problems with general constraints to achieve this we first prove via convolution that these nonsmooth equations can be well approximated by smooth equations which have desirable properties for the design of newton methods we then reformulate the variational inequality problems as equivalent smoothing nonsmooth equations and apply newton type methods to solve the latter systems and so the variational inequality problems stronger convergence results have been obtained

estimating signals with finite rate of innovation from noisy samples a stochastic algorithm as an example of the recently introduced concept of rate of innovation signals that are linear combinations of a finite number of diracs per unit time can be acquired by linear filtering followed by uniform sampling however in reality samples are rarely noiseless in this paper we introduce a novel stochastic algorithm to reconstruct a signal with finite rate of innovation from its noisy samples even though variants of this problem have been approached previously satisfactory solutions are only available for certain classes of sampling kernels for example kernels that satisfy the strang fix condition in this paper we consider the infinite support gaussian kernel which does not satisfy the strang fix condition other classes of kernels can be employed our algorithm is based on gibbs sampling a markov chain monte carlo method extensive numerical simulations demonstrate the accuracy and robustness of our algorithm

sampling signals with finite rate of innovation the authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation examples of signals with a finite rate of innovation include streams of diracs e g the poisson process nonuniform splines and piecewise polynomials even though these signals are not bandlimited we show that they can be sampled uniformly at or above the rate of innovation using an appropriate kernel and then be perfectly reconstructed thus we prove sampling theorems for classes of signals and kernels that generalize the classic bandlimited and sinc kernel case in particular we show how to sample and reconstruct periodic and finite length streams of diracs nonuniform splines and piecewise polynomials using sinc and gaussian kernels for infinite length signals with finite local rate of innovation we show local sampling and reconstruction based on spline kernels the key in all constructions is to identify the innovative part of a signal e g time instants and weights of diracs using an annihilating or locator filter a device well known in spectral analysis and error correction coding this leads to standard computational procedures for solving the sampling problem which we show through experimental results applications of these new sampling results can be found in signal processing communications systems and biological systems

forward fourier wavelet regularized deconvolution for ill conditioned systems we propose an efficient hybrid fourier wavelet regularized deconvolution forward algorithm that performs noise regularization via scalar shrinkage in both the fourier and wavelet domains the fourier shrinkage exploits the fourier transform s economical representation of the colored noise inherent in deconvolution whereas the wavelet shrinkage exploits the wavelet domain s economical representation of piecewise smooth signals and images we derive the optimal balance between the amount of fourier and wavelet regularization by optimizing an approximate mean squared error mse metric and find that signals with more economical wavelet representations require less fourier shrinkage forward is applicable to all ill conditioned deconvolution problems unlike the purely wavelet based wavelet vaguelette deconvolution wvd moreover its estimate features minimal ringing unlike the purely fourier based wiener deconvolution even in problems for which the wvd was designed we prove that forward s mse decays with the optimal wvd rate as the number of samples increases further we demonstrate that over a wide range of practical sample lengths forward improves on wvd s performance

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

paley wiener characterization of kernels for finite rate of innovation sampling exact reconstruction of finite rate of innovation signals can be achieved by employing customized sampling kernels that satisfy certain frequency domain properties we impose compact support in time as an additional constraint considering frequency domain reconstruction we derive conditions for admissible sampling kernels and corresponding sampling rates our constructive kernel design methodology is based on the paley wiener theorem for compactly supported functions the new kernels satisfy generalized strang fix conditions and have specific polynomial modulated exponential reproducing properties unlike exponential splines which have a support that is directly proportional to the number of exponentials they can generate the proposed kernels have a support that is independent of that number to analyze noise robustness we consider a special member of the class that has a sum of modulated splines sms form in the time domain and optimize its parameters to minimize the noise variance the sum of sincs sos kernel reported in the literature is an instance of this construction in noise robustness analysis sms kernels show improvement in mean squared error mse compared with the state of the art alternatives in continuous time noise the improvement in mse is about 2 db for low signal to noise ratio snr and 7 db for high snr in the case of discrete white gaussian noise the mse is lower by as much as 25 db by using a higher order sms kernel compared with the sos kernel for snrs in the range of 10 15 db

exact sampling results for some classes of parametric nonbandlimited 2 d signals we present sampling results for certain classes of two dimensional 2 d signals that are not bandlimited but have a parametric representation with a finite number of degrees of freedom while there are many such parametric signals it is often difficult to propose practical sampling schemes therefore we will concentrate on those classes for which we are able to give exact sampling algorithms and reconstruction formulas we analyze in detail a set of 2 d diracs and extend the results to more complex objects such as lines and polygons unlike most multidimensional sampling schemes the methods we propose perfectly reconstruct such signals from a finite number of samples in the noiseless case some of the techniques we use are already encountered in the context of harmonic retrieval and error correction coding in particular singular value decomposition svd based methods and the annihilating filter approach are both explored as inherent parts of the developed algorithms potentials and limitations of the algorithms in the noisy case are also pointed out applications of our results can be found in astronomical signal processing image processing and in some classes of identification problems

implementation of linear digital filters based on morphological representation theory recent advances in mathematical morphology have led to a representation theory that covers increasing linear translation invariant ilti filters as a subclass of morphological filters the representation is based on supremum minimum and addition operations and does not require any multiplications however this representation has not been practical because its essence is to evaluate the supremum of a set with an infinite number of elements based on the previously developed morphological representation we present a finite max min representation for ilti filters the new representation does not require the use of multipliers and is practical in the sense that it consists of a finite number of maximum minimum and addition operations the proposed max min representation is modified to cover fir linear shift invariant filters in general the modified representation requires only one multiplication per output sample and has lower computational complexity than the max min representation other related topics such as duality roundoff noise and implementation are also discussed

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

sampling schemes for multidimensional signals with finite rate of innovation in this paper we consider the problem of sampling signals that are nonband limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation streams of diracs and piecewise polynomials are the examples of such signals and thus are known as signals with finite rate of innovation fri we know that the classical band limited sine sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band limited however the recent results on fri sampling suggest that it is possible to sample and perfectly reconstruct such nonband limited signals using a rich class of kernels in this paper we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials satisfy strang fix conditions in fact the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples using these moments and the annihilating filter method prony s method the innovative part of the signal and therefore the signal itself is perfectly reconstructed in particular we present local directional derivatives based and global complex moments based radon transform based sampling schemes for classes of fri signals such as sets of diracs bilevel and planar polygons quadrature domains e g circles ellipses and cardioids 2d polynomials with polygonal boundaries and n dimensional diracs and convex polytopes this work has been explored in a promising way in super resolution algorithms and distributed compression and might find its applications in photogrammetry computer graphics and machine vision

sampling curves with finite rate of innovation in this paper we extend the theory of sampling signals with finite rate of innovation fri to a specific class of two dimensional curves which are defined implicitly as the zeros of a mask function here the mask function has a parametric representation as a weighted summation of a finite number of complex exponentials and therefore has finite rate of innovation an associated edge image which is discontinuous on the predefined parametric curve is proved to satisfy a set of linear annihilation equations we show that it is possible to reconstruct the parameters of the curve i e to detect the exact edge positions in the continuous domain based on the annihilation equations robust reconstruction algorithms are also developed to cope with scenarios with model mismatch moreover the annihilation equations that characterize the curve are linear constraints that can be easily exploited in optimization problems for further image processing e g image up sampling we demonstrate one potential application of the annihilation algorithm with examples in edge preserving interpolation experimental results with both synthetic curves as well as edges of natural images clearly show the effectiveness of the annihilation constraint in preserving sharp edges and improving snrs

iterative sparsification projection fast and robust sparse signal approximation in this paper we address recovery of sparse signals from compressed measurements and sparse signal approximation which have received considerable attention over the last decade first we revisit smoothed l0 sl0 a well known sparse recovery algorithm and give some insights into it that have not been noticed previously specifically we re derive the sl0 algorithm based on proximal methods and using recent results in solving nonconvex problems by proximal algorithms we provide a convergence guarantee for it in addition inspired by this derivation we propose a general family of algorithms which we call iterative sparsification projection isp having sl0 as a special member our algorithmic framework starts with an initial guess for the unknown sparse vector and then iteratively sparsifies it using a fixed threshold followed by projecting the result onto the admissible solution set the threshold is then decreased and the same process is repeated the algorithm terminates when the threshold becomes sufficiently small or another stopping criterion is satisfied we also propose a robust projection to handle the situations with observation noise or model uncertainties our extensive simulations confirm the promising performance of the isp algorithms compared with some well known algorithms

sampling and reconstruction of signals with finite rate of innovation in the presence of noise recently it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals namely certain signals of finite rate of innovation a common feature of such signals is that they have a finite number of degrees of freedom per unit of time and can be reconstructed from a finite number of uniform samples in order to prove sampling theorems vetterli et al considered the case of deterministic noiseless signals and developed algebraic methods that lead to perfect reconstruction however when noise is present many of those schemes can become ill conditioned in this paper we revisit the problem of sampling and reconstruction of signals with finite rate of innovation and propose improved more robust methods that have better numerical conditioning in the presence of noise and yield more accurate reconstruction we analyze in detail a signal made up of a stream of diracs and develop algorithmic tools that will be used as a basis in all constructions while some of the techniques have been already encountered in the spectral estimation framework we further explore preconditioning methods that lead to improved resolution performance in the case when the signal contains closely spaced components for classes of periodic signals such as piecewise polynomials and nonuniform splines we propose novel algebraic approaches that solve the sampling problem in the laplace domain after appropriate windowing building on the results for periodic signals we extend our analysis to finite length signals and develop schemes based on a gaussian kernel which avoid the problem of ill conditioning by proper weighting of the data matrix our methods use structured linear systems and robust algorithmic solutions which we show through simulation results

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

a fast wavelet based reconstruction method for magnetic resonance imaging in this work we exploit the fact that wavelets can represent magnetic resonance images well with relatively few coefficients we use this property to improve magnetic resonance imaging mri reconstructions from undersampled data with arbitrary k space trajectories reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage thresholding algorithm ista which unfortunately converges slowly to make the approach more practical we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k space trajectory we present a mathematical analysis that explains the performance of the algorithms using simulated and in vivo data we show that our nonlinear method is fast as it accelerates ista by almost two orders of magnitude we also show that it remains competitive with tv regularization in terms of image quality

a stable numerical method for inverting shape from moments we derive a stable technique based upon matrix pencils for the reconstruction of or approximation by polygonal shapes from moments we point out that this problem can be considered the dual of 2 d numerical quadrature over polygonal domains an analysis of the sensitivity of the problem is presented along with some numerical examples illustrating the relevant points finally an application to the problem of gravimetry is explored where the shape of a gravitationally anomalous region is to be recovered from measurements of its exterior gravitational field

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

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

multiple frequency dbim twist algorithm for microwave breast imaging a novel distorted born iterative method dbim algorithm is proposed for microwave breast imaging based on the two step iterative shrinkage thresholding method we show that this implementation is more flexible and robust than using traditional krylov subspace methods such as the cgls as solvers of the ill posed linear problem this paper presents several strategies to increase the algorithm s robustness a hybrid multifrequency approach to achieve an optimal tradeoff between imaging accuracy and reconstruction stability a new approach to estimate the average breast tissues properties based on sampling along their range of possible values and running a few dbim iterations to find the minimum error and finally a new regularization strategy for the dbim method based on the l 1 norm and the pareto curve we present reconstruction examples which illustrate the benefits of these optimization strategies which have resulted in a dbim algorithm that outperforms our previous implementations for microwave breast imaging

polynomial spline signal approximations filter design and asymptotic equivalence with shannon s sampling theorem the least squares polynomial spline approximation of a signal g t in l sub 2 r is obtained by projecting g t on s sup n r the space of polynomial splines of order n it is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g t with a b spline of order n more specifically the coefficients of the b spline interpolation of order 2n 1 of the sampled filtered sequence are identical to the coefficients of the least squares approximation of g t of order n it is shown that this approximation can be obtained from a succession of three basic operations prefiltering sampling and postfiltering which confirms the parallel with the classical sampling reconstruction procedure for bandlimited signals the frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift invariant basis functions of s sup n r the standard expansion in terms of b spline coefficients a representation in terms of sampled signal values and a representation using orthogonal basis functions

a family of second order methods for convex ell _1 l1 regularized optimization this paper is concerned with the minimization of an objective that is the sum of a convex function f and an ell 1 l1 regularization term our interest is in active set methods that incorporate second order information about the function f to accelerate convergence we describe a semismooth newton framework that can be used to generate a variety of second order methods including block active set methods orthant based methods and a second order iterative soft thresholding method the paper proposes a new active set method that performs multiple changes in the active manifold estimate at every iteration and employs a mechanism for correcting these estimates when needed this corrective mechanism is also evaluated in an orthant based method numerical tests comparing the performance of three active set methods are presented

ellipse fitting using the finite rate of innovation sampling principle standard approaches for ellipse fitting are based on the minimization of algebraic or geometric distance between the given data and a template ellipse when the data are noisy and come from a partial ellipse the state of the art methods tend to produce biased ellipses we rely on the sampling structure of the underlying signal and show that the x and y coordinate functions of an ellipse are finite rate of innovation fri signals and that their parameters are estimable from partial data we consider both uniform and nonuniform sampling scenarios in the presence of noise and show that the data can be modeled as a sum of random amplitude modulated complex exponentials a low pass filter is used to suppress noise and approximate the data as a sum of weighted complex exponentials the annihilating filter used in fri approaches is applied to estimate the sampling interval in the closed form we perform experiments on simulated and real data and assess both objective and subjective performances in comparison with the state of the art ellipse fitting methods the proposed method produces ellipses with lesser bias furthermore the mean squared error is lesser by about 2 to 10 db we show the applications of ellipse fitting in iris images starting from partial edge contours and to free hand ellipses drawn on a touch screen tablet

shape analysis and reduction of the morphological basis for digital moving average filters a digital moving average filter with nonnegative mask weights that sum to unity is a gray scale morphological filter as such it possesses a representation as a maximum of erosions over its morphological basis the present paper investigates the special properties of the digital moving average morphological basis these properties being for the most part combinatoric in nature shapes of digital structuring elements are related to the filter weights by means of shape classes this is accomplished by defining a characteristic of a signal relative to the filter weights and then characterizing basis elements by means of the characteristic the notion of shape is then extended to a more general class of morphological filters the concept or a morphological quasi average is introduced and the power of such filters to approximate morphological filters is addressed the effect of basis reduction is studied for a rather wide class of morphological filters the aim being to quantify the degree to which a reduce

curl constrained gradient estimation for image recovery from highly incomplete spectral data in this paper we introduce new gradient based methods for image recovery from a small collection of spectral coefficients of the fourier transform which is of particular interest for several scanning technologies such as magnetic resonance imaging since gradients of a medical image are much more sparse or compressible than the corresponding image classical ell 1 minimization methods have been used to recover these relative differences the image values can then be obtained by integration algorithms imposing boundary constraints compared with classical gradient recovery methods we propose two new techniques that improve reconstruction first we cast the gradient recovery problem as a compressed sensing problem taking into account that the curl of the gradient field should be zero second inspired by the emerging field of signal processing on graphs we formulate the gradient recovery problem as an inverse problem on graphs iteratively reweighted ell 1 recovery methods are proposed to recover these relative differences and the structure of the similarity graph once the gradient field is estimated the image is recovered from the compressed fourier measurements using least squares estimation numerical experiments show that the proposed approach outperforms the state of the art image recovery methods

a sampling framework for solving physics driven inverse source problems partial differential equations are central to describing many physical phenomena in many applications these phenomena are observed through a sensor network with the aim of inferring their underlying properties leveraging from certain results in sampling and approximation theory we present a new framework for solving a class of inverse source problems for physical fields governed by linear partial differential equations specifically we demonstrate that the unknown field sources can be recovered from a sequence of so called generalized measurements by using multidimensional frequency estimation techniques next we show that for physics driven fields this sequence of generalized measurements can be estimated by computing a linear weighted sum of the sensor measurements whereby the exact weights of the sums correspond to those that reproduce multidimensional exponentials when used to linearly combine translates of a particular prototype function related to the green s function of our underlying field explicit formulas are then derived for the sequence of weights which map sensor samples to the exact sequence of generalized measurements when the green s function satisfies the generalized strang fix condition otherwise the same mapping yields a close approximation of the generalized measurements based on this new framework we develop practical noise robust sensor network strategies for solving the inverse source problem and then present numerical simulation results to verify their performance

incremental proximal methods for large scale convex optimization we consider the minimization of a sum sum i 1 mf i x consisting of a large number of convex component functions f i for this problem incremental methods consisting of gradient or subgradient iterations applied to single components have proved very effective we propose new incremental methods consisting of proximal iterations applied to single components as well as combinations of gradient subgradient and proximal iterations we provide a convergence and rate of convergence analysis of a variety of such methods including some that involve randomization in the selection of components we also discuss applications in a few contexts including signal processing and inference machine learning

an alternating direction algorithm for total variation reconstruction of distributed parameters augmented lagrangian variational formulations and alternating optimization have been adopted to solve distributed parameter estimation problems the alternating direction method of multipliers admm is one of such formulations optimization methods very recently the number of applications of the admm or variants of it to solve inverse problems in image and signal processing has increased at an exponential rate the reason for this interest is that admm decomposes a difficult optimization problem into a sequence of much simpler problems in this paper we use the admm to reconstruct piecewise smooth distributed parameters of elliptical partial differential equations from noisy and linear blurred observations of the underlying field the distributed parameters are estimated by solving an inverse problem with total variation tv regularization the proposed instance of the admm solves in each iteration an and a decoupled optimization problems an operator splitting is used to simplify the treatment of the tv regularizer avoiding its smooth approximation and yielding a simple yet effective admm reconstruction method compared with previously proposed approaches the competitiveness of the proposed method with respect to the state of the art is illustrated in simulated 1 d and 2 d elliptical equation problems which are representative of many real applications

towards generalized fri sampling with an application to source resolution in radioastronomy it is a classic problem to estimate continuous time sparse signals like point sources in a direction of arrival problem or pulses in a time of flight measurement the earliest occurrence is the estimation of sinusoids in time series using prony s method this is at the root of a substantial line of work on high resolution spectral estimation the estimation of continuous time sparse signals from discrete time samples is the goal of the sampling theory for finite rate of innovation fri signals both spectral estimation and fri sampling usually assume uniform sampling but not all measurements are obtained uniformly as exemplified by a concrete radioastronomy problem we set out to solve thus we develop the theory and algorithm to reconstruct sparse signals typically sum of sinusoids from nonuniform samples we achieve this by identifying a linear transformation that relates the unknown uniform samples of sinusoids to the given measurements these uniform samples are known to satisfy the annihilation equations a valid solution is then obtained by solving a constrained minimization such that the reconstructed signal is consistent with the given measurements and satisfies the annihilation constraint thanks to this new approach we unify a variety of fri based methods we demonstrate the versatility and robustness of the proposed approach with five fri reconstruction problems namely dirac reconstructions with irregular time or fourier domain samples fri curve reconstructions dirac reconstructions on the sphere and point source reconstructions in radioastronomy the proposed algorithm improves substantially over state of the art methods and is able to reconstruct point sources accurately from irregularly sampled fourier measurements under severe noise conditions

the 192 solutions of the heun equation a machine generated list of 192 local solutions of the heun equation is given they are analogous to kummer s 24 solutions of the gauss hypergeometric equation since the two equations are canonical fuchsian differential equations on the riemann sphere with four and three singular points respectively tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the coxeter group d n each of the 192 expressions is labeled by an element of d 4 of the 192 24 are equivalent expressions for the local heun function hl and it is shown that the resulting order 24 group of transformations of hl is isomorphic to the symmetric group s 4 the isomorphism encodes each transformation as a permutation of an abstract four element set not identical to the set of singular points

sampling and exact reconstruction of pulses with variable width recent sampling results enable the reconstruction of signals composed of streams of fixed shaped pulses these results have found applications in topics as varied as channel estimation biomedical imaging and radio astronomy however in many real signals the pulse shapes vary throughout the signal in this paper we show how to sample and perfectly reconstruct lorentzian pulses with variable width in the noiseless case perfect recovery is guaranteed by a set of theorems in addition we verify that our algorithm is robust to model mismatch and noise this allows us to apply the technique to two practical applications electrocardiogram ecg compression and bidirectional reflectance distribution function brdf sampling ecg signals are one dimensional but the brdf is a higher dimensional signal which is more naturally expressed in a spherical coordinate system this motivated us to extend the theory to the 2d and spherical cases experiments on real data demonstrate the viability of the proposed model for ecg acquisition and compression as well as the efficient representation and low rate sampling of specular brdfs

image super resolution via sparse representation this paper presents a new approach to single image superresolution based upon sparse signal representation research on image statistics suggests that image patches can be well represented as a sparse linear combination of elements from an appropriately chosen over complete dictionary inspired by this observation we seek a sparse representation for each patch of the low resolution input and then use the coefficients of this representation to generate the high resolution output theoretical results from compressed sensing suggest that under mild conditions the sparse representation can be correctly recovered from the downsampled signals by jointly training two dictionaries for the low and high resolution image patches we can enforce the similarity of sparse representations between the low resolution and high resolution image patch pair with respect to their own dictionaries therefore the sparse representation of a low resolution image patch can be applied with the high resolution image patch dictionary to generate a high resolution image patch the learned dictionary pair is a more compact representation of the patch pairs compared to previous approaches which simply sample a large amount of image patch pairs reducing the computational cost substantially the effectiveness of such a sparsity prior is demonstrated for both general image super resolution sr and the special case of face hallucination in both cases our algorithm generates high resolution images that are competitive or even superior in quality to images produced by other similar sr methods in addition the local sparse modeling of our approach is naturally robust to noise and therefore the proposed algorithm can handle sr with noisy inputs in a more unified framework

recovery of sparse translation invariant signals with continuous basis pursuit we consider the problem of decomposing a signal into a linear combination of features each a continuously translated version of one of a small set of elementary features although these constituents are drawn from a continuous family most current signal decomposition methods rely on a finite dictionary of discrete examples selected from this family e g shifted copies of a set of basic waveforms and apply sparse optimization methods to select and solve for the relevant coefficients here we generate a dictionary that includes auxiliary interpolation functions that approximate translates of features via adjustment of their coefficients we formulate a constrained convex optimization problem in which the full set of dictionary coefficients represents a linear approximation of the signal the auxiliary coefficients are constrained so as to only represent translated features and sparsity is imposed on the primary coefficients using an l1 penalty the basis pursuit denoising bp method may be seen as a special case in which the auxiliary interpolation functions are omitted and we thus refer to our methodology as continuous basis pursuit cbp we develop two implementations of cbp for a one dimensional translation invariant source one using a first order taylor approximation and another using a form of trigonometric spline we examine the tradeoff between sparsity and signal reconstruction accuracy in these methods demonstrating empirically that trigonometric cbp substantially outperforms taylor cbp which in turn offers substantial gains over ordinary bp in addition the cbp bases can generally achieve equally good or better approximations with much coarser sampling than bp leading to a reduction in dictionary dimensionality

on reducing the heun equation to the hypergeometric equation abstract the reductions of the heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified heun to hypergeometric reductions are similar to classical hypergeometric identities but the conditions for the existence of a reduction involve features of the heun equation that the hypergeometric equation does not possess namely its cross ratio and accessory parameters the reductions include quadratic and cubic transformations which may be performed only if the singular points of the heun equation form a harmonic or an equianharmonic quadruple respectively and several higher degree transformations this result corrects and extends a theorem in a previous paper which found only the quadratic transformations siam j math anal 10 3 1979 655

on polynomial solutions of the heun equation by making use of a recently developed method to solve linear differential equations of arbitrary order we find a wide class of polynomial solutions to the heun equation we construct the series solution to the heun equation before identifying the polynomial solutions the heun equation extended by the addition of a term x is also amenable for polynomial solutions

exponential wavelet iterative shrinkage thresholding algorithm for compressed sensing magnetic resonance imaging it is beneficial for both hospitals and patients to accelerate mri scanning recently a new fast mri technique based on cs was proposed however the reconstruction quality and computation time of cs mri did not meet the standard of clinical use therefore we proposed a novel algorithm based on three successful components the sparsity of ewt the rapidness of fista and the excellent tuning in sista the proposed method was dubbed exponential wavelet iterative shrinkage threshold algorithm ewista experiments over four kinds of mr images brain ankle knee and adhd indicated that the proposed ewista showed better reconstruction performance than the state of the art algorithms such as fcsa ista fista sista and ewt ista moreover ewista was faster than ista and ewt ista but slightly slower than fcsa fista and sista

a fast image recovery algorithm based on splitting deblurring and denoising in this paper we employ a popular splitting strategy to design a fast iterative algorithm for image restoration we divide the algorithm into two steps i e deblurring step and denoising step in the deblurring step fourier transform is employed for image deblurring under the periodic boundary condition in the denoising step we use a simple and fast method called fast iterative shrinkage thresholding algorithm fista to reduce image noise in addition we also give the convergence analysis for the proposed method visual and quantitative results demonstrate the proposed algorithm applied to l 1 regularization model and total variation tv regularization model is a faster algorithm and keeps image details well