aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

an adaptive level set method for nondifferentiable constrained image recovery the formulation of a wide variety of image recovery problems leads to the minimization of a convex objective over a convex set representing the constraints derived from a priori knowledge and consistency with the observed signals in previous years nondifferentiable objectives have become popular due in part to their ability to capture certain features such as sharp edges they also arise naturally in minimax inconsistent set theoretic recovery problems at the same time the issue of developing reliable numerical algorithms to solve such convex programs in the context of image recovery applications has received little attention we address this issue and propose an adaptive level set method for nondifferentiable constrained image recovery the asymptotic properties of the method are analyzed and its implementation is discussed numerical experiments illustrate applications to total variation and minimax set theoretic image restoration and denoising problems

total variation structured total least squares method for image restoration in this paper we study the total variation structured total least squares method for image restoration in the image restoration problem the point spread function is corrupted by errors in the model we study the objective function by minimizing two variables the restored image and the estimated error of the point spread function the proposed objective function consists of the data fitting term containing these two variables the magnitude of error and the total variation regularization of the restored image by making use of the structure of the objective function an efficient alternating minimization scheme is developed to solve the proposed model numerical examples are also presented to demonstrate the effectiveness of the proposed model and the efficiency of the numerical scheme

a multi scale vectorial l τ tv framework for color image restoration a general multi scale vectorial total variation model with spatially adapted regularization parameter for color image restoration is introduced in this paper this total variation model contains an l data fidelity for any c 1 2 the use of a spatial dependent regularization parameter improves the reconstruction of features in the image as well as an adequate smoothing for the homogeneous parts the automated adaptation of this regularization parameter is made according to local statistical characteristics of the noise which contaminates the image the corresponding multiscale vectorial total variation model is solved by fenchel duality and inexact semismooth newton techniques numerical results are presented for the cases 1 and 2 which reconstruct images contaminated with salt and pepper noise and gaussian noise respectively

analytical aspects of spatially adapted total variation regularisation abstract in this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem motivated by its application in signal recovery tasks we study in depth the relationship between the weight function and the creation of new discontinuities in the solution a partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed we prove that the weighted total variation minimisation problem is well posed even in the case of vanishing weight function despite the lack of coercivity this is based on the fact that the total variation of the solution is bounded by the total variation of the data a result that it also shown here finally the relationship to the corresponding weighted fidelity problem is explored showing that the two problems can produce completely different solutions even for very simple data functions

an algorithm for the minimization of mixed l sub 1 and l sub 2 norms with application to bayesian estimation the regularizing functional approach is widely used in many estimation problems in practice the solution is defined as one minimum point of a suitable functional the main part of which accounts for the underlying physical model whereas the regularizing part represents some prior information about the unknowns in the bayesian interpretation one has a maximum a posteriori map estimator in which the main and regularizing parts are represented respectively by likelihood and prior distributions when either the prior or likelihood is a laplace distribution and the other is a gaussian distribution one is led to consider functionals that include both absolute and square norms the authors present a characterization of the minimum points of such functionals together with a descent type algorithm for numerical computations the results of monte carlo simulations are also reported

factorized banded inverse preconditioners for matrices with toeplitz structure in this paper we study factorized banded inverse preconditioners for matrices with toeplitz structure we show that if a toeplitz matrix t has certain off diagonal decay property then the factorized banded inverse preconditioner approximates t 1 accurately and the spectra of these preconditioned matrices are clustered around 1 in nonlinear image restoration applications toeplitz related systems of the form i t d t are required to solve where d is a positive nonconstant diagonal matrix we construct inverse preconditioners for such matrices numerical results show that the performance of our proposed preconditioners are superior to that of circulant preconditioners a two dimensional nonlinear image restoration example is also presented to demonstrate the effectiveness of the proposed preconditioner

maximal flow through a domain in place of flows on a discrete network we study flows described by a vector field x y in a plane domain the analogue of the capacity constraint is c x y and the strength of sources and sinks is n f on the boundary and div f in the interior we show that the largest the maximal flow is determined by the minimal cut as in the discrete case the dual problem has a 0 1 solution given by the characteristic function of the minimal cut for the continuous problem the argument depends on the coarea formula for functions of bounded variation

remarks on the thickness and outerthickness of a graph abstract the thickness of a graph is the minimum number of planar subgraphs into which the graph can be decomposed the thickness of complete bipartite graphs km n is known for almost all values of m and n in this paper we solve the thickness of complete bipartite graphs for unknown cases m the outerthickness of a graph is the minimum number of outerplanar subgraphs into which the graph can be decomposed we give lower and upper bounds for outerthickness in the terms of the minimum and maximum degree of a graph let be the minimum degree and the maximum degree of a graph g we show that the following bounds hold for outerthickness 4 o g 2 we also discuss the possibility of determining the upper bound for outerthickness using the number of edges of a given graph

on row relaxation methods for large constrained least squares problems this paper addresses the question of how to construct a row relaxation method for solving large unstructured linear least squares problems with or without linear constraints the proposed approach combines the herman lent hurwitz scheme for solving regularized least squares problems with the lent censor hildreth method for solving linear constraints however numerical experiments show that the herman lent hurwitz scheme has difficulty reaching a least squares solution this difficulty is resolved by applying the riley golub iterative improvement process

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

optimal estimation of deterioration from diagnostic image sequence estimation of mechanical structure damage can greatly benefit from the knowledge that the damage accumulates irreversibly over time this paper formulates a problem of estimation of a pixel wise monotonic increasing or decreasing time series of images from noisy blurred image data our formulation includes temporal monotonicity constraints and a spatial regularization penalty we cast the estimation problem as a large scale quadratic programming qp optimization and describe an efficient interior point method for solving this problem the method exploits the special structure of the qp and scales well to problems with more than a million of decision variables and constraints the proposed estimation approach performs well for simulated data we demonstrate an application of the approach to diagnostic images obtained in structural health monitoring experiments and show that it provides a good estimate of the damage accumulation trend while suppressing spatial and temporal noises

a multilevel algorithm for simultaneously denoising and deblurring images in this paper we develop a fast multilevel algorithm for simultaneously denoising and deblurring images under the total variation regularization although much effort has been devoted to developing fast algorithms for the numerical solution and the denoising problem was satisfactorily solved fast algorithms for the combined denoising and deblurring model remain to be a challenge recently several successful studies of approximating this model and subsequently finding fast algorithms were conducted which have partially solved this problem the aim of this paper is to generalize a fast multilevel denoising method to solving the minimization model for simultaneously denoising and deblurring our new idea is to overcome the complexity issue by a detailed study of the structured matrices that are associated with the blurring operator a fast algorithm can then be obtained for directly solving the variational model supporting numerical experiments on gray scale images are presented

preconditioning regularized least squares problems arising from high resolution image reconstruction from low resolution frames in this paper we study the problem of reconstructing a high resolution image from multiple undersampled shifted degraded frames with subpixel displacement errors from multisensors preconditioned conjugate gradient methods with cosine transform based preconditioners and incomplete factorization based preconditioners are applied to solve this image reconstruction problem numerical examples are given to demonstrate the efficiency of these preconditioners we find that cosine transform based preconditioners are effective when the number of shifted low resolution frames are large but are less effective when the number is small however incomplete factorization based preconditioners work quite well independent of the number of shifted low resolution frames

a nonconvex l1 l1 l2 model for image restoration with impulse noise abstract in this paper we propose a model for image restoration with blur and impulse noise it is composed of l 1 data fitting term and a nonconvex regularization term which is the weighted difference of l 1 norm and l 2 norm based on wavelet frame the combined model is difficult to solve by the corresponding euler lagrangian equations here we solve it by alternating direction method of multipliers admm we describe the detailed process of the algorithm and establish the convergence of the algorithm the experimental outcomes on different blurs and different impulse noises demonstrate that the proposed approach is better than those existing methods in terms of standard signal to noise ratio psnr relative error reerr and visual quality

parameter selection for total variation based image restoration using discrepancy principle there are two key issues in successfully solving the image restoration problem 1 estimation of the regularization parameter that balances data fidelity with the regularity of the solution and 2 development of efficient numerical techniques for computing the solution in this paper we derive a fast algorithm that simultaneously estimates the regularization parameter and restores the image the new approach is based on the total variation tv regularized strategy and morozov s discrepancy principle the tv norm is represented by the dual formulation that changes the minimization problem into a minimax problem a proximal point method is developed to compute the saddle point of the minimax problem by adjusting the regularization parameter adaptively in each iteration the solution is guaranteed to satisfy the discrepancy principle we will give the convergence proof of our algorithm and numerically show that it is better than some state of the art methods in terms of both speed and accuracy

discrete time neural network for fast solving large linear l_ 1 estimation problems and its application to image restoration there is growing interest in solving linear l1 estimation problems for sparsity of the solution and robustness against non gaussian noise this paper proposes a discrete time neural network which can calculate large linear l1 estimation problems fast the proposed neural network has a fixed computational step length and is proved to be globally convergent to an optimal solution then the proposed neural network is efficiently applied to image restoration numerical results show that the proposed neural network is not only efficient in solving degenerate problems resulting from the nonunique solutions of the linear l1 estimation problems but also needs much less computational time than the related algorithms in solving both linear l1 estimation and image restoration problems

total bounded variation regularization as a bilaterally constrained optimization problem it is demonstrated that the predual for problems with total bounded variation regularization terms can be expressed as a bilaterally constrained optimization problem existence of a lagrange multiplier and an optimality system are established this allows us to utilize efficient optimization methods developed for problems with box constraints in the context of bounded variation formulations here in particular the primal dual active set method considered as a semismooth newton method is analyzed and superlinear convergence is proved as a by product we obtain that the lagrange multiplier associated with the box constraints acts as an edge detector numerical results for image denoising and zooming resizing show the efficiency of the new approach

a note on halton s conjecture the thickness of a graph g is the minimum number of planar graphs whose union is g halton conjectured that any graph of maximum degree d has thickness at most d 2 4 we disprove the conjecture by showing graphs of thickness d 2 for any d 5

a morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets this paper introduces a discrete scheme for mean curvature motion using a morphological image processing approach an axiomatic approach of image processing and the mean curvature partial differential equation pde are briefly presented then the properties of the proposed scheme are studied in particular consistency and convergence are proved the applications of mean curvature motion in image denoising and form evolution are developed and experiences are presented

a new approach to constrained total least squares image restoration abstract recently there has been a growing interest and progress in using total least squares tls methods for solving blind deconvolution problems arising in image restoration here the true image is to be estimated using only partial information about the blurring operator or point spread function psf which is subject to error and noise in this paper we present a new iterative regularized and constrained tls image restoration algorithm neumann boundary conditions are used to reduce the boundary artifacts that normally occur in restoration processes preliminary numerical tests are reported on some simulated optical imaging problems in order to illustrate the effectiveness of the approach as well as the fast convergence of our iterative scheme

an o m log n time algorithm for the maximal planar subgraph problem based on a new version of the hopcroft and tarjan planarity testing algorithm this paper develops an o m log n time algorithm to find a maximal planar subgraph

on heuristics for determining the thickness of a graph we present an empirical analysis of some heuristics for the graph thickness problem i e decomposing a graph into the minimum number of planar subgraphs the problem has applications in database systems e g the layout of e r diagrams the heuristics are based on some algorithms for finding a maximal planar subgraph of a nonplanar graph empirical results show that a randomized heuristic is a practical tool for solving this problem we also improve upon a previously derived upper bound for the thickness of a graph with respect to its size

hyperbolic householder transformations a class of transformation matrices analogous to the householder matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least squares problems these matrices which we term hyperbolic householder are shown to effect deletion or simultaneous addition and deletion of data with much less sensitivity to rounding errors than for techniques based on normal equations when the addition deletion sets are large this numerical robustness is obtained at the expense of only a modest increase in computations and when only a relatively small fraction of the data set is modified there is a decrease in required computations two applications to signal processing problems are considered first these transformations are used to obtain a square root algorithm for windowed recursive least squares filtering second the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array

extensions of hildreth s row action method for quadratic programming an extended version of hildreth s iterative quadratic programming algorithm is presented geometrically interpreted and proved to produce a sequence of iterates that i converges to the solution and ii has an important intermediate optimality property this extended hildreth algorithm is cast into a new form which more pronouncedly brings out its primal dual nature the application of the algorithm may be governed by an index sequence which is more general than a cyclic sequence namely by an almost cyclic control and a sequence of relaxation parameters is incorporated without ruining convergence the algorithm is a row action method which is particularly suitable for handling large or huge and sparse systems

minimizers of cost functions involving nonsmooth data fidelity terms application to the processing of outliers we present a theoretical study of the recovery of an unknown vector x in mathbb r p such as a signal or an image from noisy data y in mathbb r q by minimizing with respect to x a regularized cost function cal f x y psi x y alpha phi x where psi is a data fidelity term phi is a smooth regularization term and alpha 0 is a parameter typically psi x y ax y 2 where a is a linear operator the data fidelity terms psi involved in regularized cost functions are generally smooth functions only a few papers make an exception to this and they consider restricted situations nonsmooth data fidelity terms are avoided in image processing in spite of this we consider both smooth and nonsmooth data fidelity terms our goal is to capture essential features exhibited by the local minimizers of regularized cost functions in relation to the smoothness of the data fidelity term in order to fix the context of our study we consider psi x y sum i psi a i tx y i where a i t are the rows of a and psi is cal c m on mathbb r setminus 0 we show that if psi 1 0 psi 1 0 then typical data y give rise to local minimizers hat x of cal f y which fit exactly a certain number of the data entries there is a possibly large set hat h of indexes such that a i t hat x y i for every i in hat h in contrast if psi is smooth on mathbb r for almost every y the local minimizers of cal f y do not fit any entry of y thus the possibility that a local minimizer fits some data entries is due to the nonsmoothness of the data fidelity term this is a strong mathematical property which is useful in practice by way of application we construct a cost function allowing aberrant data outliers to be detected and to be selectively smoothed our numerical experiments advocate the use of nonsmooth data fidelity terms in regularized cost functions for special purposes in image and signal processing

an efficient tvl1 algorithm for deblurring multichannel images corrupted by impulsive noise we extend the alternating minimization algorithm recently proposed in y wang j yang w yin and y zhang siam j imag sci 1 2008 pp 248 272 j yang w yin y zhang and y wang siam j imag sci 2 2009 pp 569 592 to the case of recovering blurry multichannel color images corrupted by impulsive rather than gaussian noise the algorithm minimizes the sum of a multichannel extension of total variation and a data fidelity term measured in the ell 1 norm and is applicable to both salt and pepper and random valued impulsive noise we derive the algorithm by applying the well known quadratic penalty function technique and prove attractive convergence properties including finite convergence for some variables and q linear convergence rate under periodic boundary conditions the main computational requirements of the algorithm are fast fourier transforms and a low complexity gaussian elimination procedure numerical results on images with different blurs and impulsive noise are presented to demonstrate the efficiency of the algorithm in addition it is numerically compared to the least absolute deviation method h y fu m k ng m nikolova and j l barlow siam j sci comput 27 2006 pp 1881 1902 and the two phase method j f cai r chan and m nikolova aims j inverse problems and imaging 2 2008 pp 187 204 for recovering grayscale images we also present results of recovering multichannel images

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

smoothing projected gradient method and its application to stochastic linear complementarity problems a smoothing projected gradient spg method is proposed for the minimization problem on a closed convex set where the objective function is locally lipschitz continuous but nonconvex nondifferentiable we show that any accumulation point generated by the spg method is a stationary point associated with the smoothing function used in the method which is a clarke stationary point in many applications we apply the spg method to the stochastic linear complementarity problem slcp and image restoration problems we study the stationary point defined by the directional derivative and provide necessary and sufficient conditions for a local minimizer of the expected residual minimization erm formulation of slcp preliminary numerical experiments using the spg method for solving randomly generated slcp and image restoration problems of large sizes show that the spg method is promising

digital filters as absolute norm regularizers a variational formulation of the digital filtering problem based on a certain regularizing functional is presented in particular the special case where the regularizing functional is defined by means of l sub 1 norm is considered and a characterization of its minima is presented it is shown that for a certain class of signals one solution to the regularization problem can be obtained by means of a suitable recursive median filter some examples and a possible extension of the variational approach are also discussed

image restoration by convex projections using adaptive constraints and the l sub 1 norm a new projection method based on the row action projection rap algorithm for image restoration is presented the new implementation is computationally attractive and facilitates local adaption of projection operators the local mean and the least l sub 1 norm set of solutions are used as constraints computer simulations illustrate the new methods to be very competitive in restoring missing spectral components of a degraded image

parallel application of block iterative methods in medical imaging and radiation therapy some row action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems recently developed block iterative versions of such special purpose methods enable parallel computation when the underlying problem is appropriately decomposed this opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning all in their fully discretized modelling approach since there is more than one way of deriving block iterative versions of any row action method the choice has to be made with reference to the underlying real world problem

quasi newton approach to nonnegative image restorations abstract image restoration or deblurring is the process of attempting to correct for degradation in a recorded image typically the blurring system is assumed to be linear and spatially invariant and fast fourier transform fft based schemes result in efficient computational image restoration methods however real images have properties that cannot always be handled by linear methods in particular an image consists of positive light intensities and thus a nonnegativity constraint should be enforced this constraint and other ways of incorporating a priori information have been suggested in various applications and can lead to substantial improvements in the reconstructions nevertheless such constraints are rarely implemented because they lead to nonlinear problems which require demanding computations we suggest efficient implementations for three nonnegatively constrained restorations schemes constrained least squares maximum likelihood and maximum entropy we show that with a certain parameterization and using a quasi newton scheme these methods are very similar in addition our formulation reveals a connection between our approach for maximum likelihood and the expectation maximization em method used extensively by astronomers numerical experiments illustrate that our approach is superior to em both in terms of accuracy and efficiency

a genetic algorithm for determining the thickness of a graph abstract the thickness of a graph is the minimum number of planar subgraphs into which the graph can be decomposed determining the thickness of a given graph is known to be an np complete problem this paper discusses the possibility of determining the thickness of a graph using a genetic algorithm our tests show that the genetic approach outperforms the earlier heuristic algorithms reported in the literature

stabilized hyperbolic householder transformations a modification of the hyperbolic householder scheme is introduced which is demonstrably stable theoretically according to an established stability criterion and which exhibits superior numerical behavior in simulations the modified transform scheme effects downdating by applying conventional orthonormal rather than hyperbolic householder transformations to the data the latter have preferable numerical properties however the construction of these orthonormal operators itself requires hyperbolic computations thus the proposed method is in some sense half hyperbolic and half orthonormal there is no computational penalty incurred with these stabilized hyperbolic householder transforms they enjoy an operation count identical to their conventional counterparts

regularized constrained total least squares image restoration in this paper the problem of restoring an image distorted by a linear space invariant lsi point spread function psf that is not exactly known is formulated as the solution of a perturbed set of linear equations the regularized constrained total least squares rctls method is used to solve this set of equations using the diagonalization properties of the discrete fourier transform dft for circulant matrices the rctls estimate is computed in the dft domain this significantly reduces the computational cost of this approach and makes its implementation possible even for large images an error analysis of the rctls estimate based on the mean squared error mse criterion is performed to verify its superiority over the constrained total least squares ctls estimate numerical experiments for different errors in the psf are performed to test the rctls estimator objective and visual comparisons are presented with the linear minimum mean squared error lmmse and the regularized least squares rls estimator our experiments show that the rctls estimator reduces significantly ringing artifacts around edges as compared to the two other approaches

a property of the minimum vectors of a regularizing functional defined by means of the absolute norm we consider a regularizing functional defined by means of the l sub 1 norm where the regularization is obtained using first differences as is well known such a functional can be put in relation with recursive median filters of appropriate window length we show that at least one of the minima is reached at a vector whose components have values over the same discrete set of the given signal this suggests a simple method to refine the approximate solution to the regularization problem which can be obtained with recursive median filters of increasing order we also report an example of application where the refinement method is employed for a signal detection problem

iterative solution of linear programs by perturbing a linear program to a quadratic program it is possible to solve the latter in its dual variable space by iterative techniques such as successive over relaxation sor methods this provides a solution to the original linear program

implementation of the regularized structured total least squares algorithms for blind image deblurring abstract the structured total least squares stls problem has been introduced to handle problems involving structured matrices corrupted by noise often the problem is ill posed recently regularization has been proposed in the stls framework to solve ill posed blind deconvolution problems encountered in image deblurring when both the image and the blurring function have uncertainty the kernel of the regularized stls rstls problem is a least squares problem involving block toeplitz toeplitz block matrices in this paper an algorithm is described to solve this problem based on a particular implementation of the generalized schur algorithm gsa it is shown that this new implementation improves the computational efficiency of the straightforward implementation of gsa from o n 2 5 to o n 2 where n is the number of pixels in the image

cubic rings and the exceptional jordan algebra in a previous paper eg we described an integral structure j e on the exceptional jordan algebra of hermitian 3 3 matrices over the cayley octonions here we use modular forms and niemeier s classification of even unimodular lattices of rank 24 to further investigate j and the integral even lattice j0 ze in j specifically we study ring embeddings of totally real cubic rings a into j which send the identity of a to e and we give a new proof of r borcherds s result that j0 is characterized as a euclidean lattice by its rank type discriminant and minimal norm

a duality based splitting method for ell 1 tv image restoration with automatic regularization parameter choice a novel splitting method is presented for the ell 1 tv restoration of degraded images subject to impulsive noise the functional is split into an ell 2 tv denoising part and an ell 1 ell 2 deblurring part the dual problem of the relaxed functional is smooth with convex constraints and can be solved efficiently by applying an arrow hurwicz type algorithm to the augmented lagrangian formulation the regularization parameter is chosen automatically based on a balancing principle the accuracy the fast convergence and the robustness of the algorithm and the use of the parameter choice rule are illustrated on some benchmark images and compared with an existing method

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

nonideal sampling and regularization theory shannon s sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some shift invariant space which may or may not be bandlimited in this paper we present some further justification for this type of representation while addressing the issue of the specification of the best reconstruction space we consider a realistic setting where a multidimensional signal is prefiltered prior to sampling and the samples are corrupted by additive noise we adopt a variational approach to the reconstruction problem and minimize a data fidelity term subject to a tikhonov like continuous domain l2 regularization to obtain the continuous space solution we present theoretical justification for the minimization of this cost functional and show that the globally minimal continuous space solution belongs to a shift invariant space generated by a function generalized b spline that is generally not bandlimited when the sampling is ideal we recover some of the classical smoothing spline estimators the optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a b spline like basis to make the scheme practical we specify the generating functions corresponding to the most popular families of regularization operators derivatives iterated laplacian as well as a new generalized one that leads to a new brand of matern splines we conclude the paper by proposing a stochastic interpretation of the reconstruction algorithm and establishing an equivalence with the minimax and minimum mean square error mmse wiener solutions of the generalized sampling problem

formulation and solution of structured total least norm problems for parameter estimation the total least squares tls method is a generalization of the least squares ls method for solving overdetermined sets of linear equations ax spl ap b the tls method minimizes spl par e r spl par sub f where r b a e x so that b r spl isin range a e given a spl isin c sup m spl times n with m spl ges n and b spl isin c sup m spl times 1 the most common tls algorithm is based on the singular value decomposition svd of a b however the svd based methods may not be appropriate when the matrix a has a special structure since they do not preserve the structure previously a new problem formulation known as structured total least norm stln and the algorithm for computing the stln solution have been developed the stln method preserves the special structure of a or a b and can minimize the error in the discrete l sub p norm where p 1 2 or spl infin in this paper the stln problem formulation is generalized for computing the solution of stln problems with multiple right hand sides ax spl ap b it is shown that these problems can be converted to ordinary stln problems with one right hand side in addition the method is shown to converge to the optimal solution in certain model reduction problems furthermore the application of the stln method to various parameter estimation problems is studied in which the computed correction matrix applied to a or a b keeps the same toeplitz structure as the data matrix a of a b respectively in particular the l sub 2 norm stln method is compared with the ls and tls methods in deconvolution transfer function modeling and linear prediction problems

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

an adaptive fixed point proximity algorithm for solving total variation denoising models abstract we study an adaptive fixed point proximity algorithm to solve the total variation denoising model the objective function is a sum of two convex functions and one of them is composed by an affine transformation which is usually a regularization term by decoupling and splitting the problem is changed into two subproblems due to the nonsmooth and nondifferentiability of the subproblem we solve its proximity minimization problem instead of the original one to overcome the staircase effect during the process of denoising an adaptive criterion on proximity parameters is put forward at last we apply the improved algorithm to solve the isotropic total variation denoising model the numerical results are given to illustrate the efficiency of the algorithm

d4 modular forms in this paper we study modular forms on two simply connected groups of type d 4 over q one group g s is a globally split group of type d 4 viewed as the group of isotopies of the split rational octonions the other g c is the isotopy group of the rational nonsplit octonions we study automorphic forms on g s in analogy to the work of gross gan and savin on g 2 namely we study automorphic forms whose component at infinity corresponds to a quaternionic discrete series representation we study automorphic forms on g c using gross s formalism of algebraic modular forms finally we follow work of gan savin gross rallis and others to study an exceptional theta correspondence connecting modular forms on g c and g s this can be thought of as an octonionic generalization of the jacquet langlands correspondence

optimization by simulated annealing an experimental evaluation part ii graph coloring and number partitioning this is the second in a series of three papers that empirically examine the competitiveness of simulated annealing in certain well studied domains of combinatorial optimization simulated annealing is a randomized technique proposed by s kirkpatrick c d gelatt and m p vecchi for improving local optimization algorithms here we report on experiments at adapting simulated annealing to graph coloring and number partitioning two problems for which local optimization had not previously been thought suitable for graph coloring we report on three simulated annealing schemes all of which can dominate traditional techniques for certain types of graphs at least when large amounts of computing time are available for number partitioning simulated annealing is not competitive with the differencing algorithm of n karmarkar and r m karp except on relatively small instances moreover if running time is taken into account natural annealing schemes cannot even outperform multiple random runs of the local optimization algorithms on which they are based in sharp contrast to the observed performance of annealing on other problems

the convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations this paper reviews the convergence properties of linear stationary iterative processes for solving large unstructured systems of linear equations special attention is given to the case where the system to be solved is singular and possibly inconsistent the review starts by investigating the iteration bf x k 1 h bf x k bf f k 0 1 2 cdots when the system i h bf x bf f is inconsistent in this case the sequence left bf x k right always diverges nevertheless when h is convergent there exists a vector bf q in text null i h such that the sequence hat bf x k bf x k k bf q converges the limit of this sequence hat bf x solves a certain least squares problem and the sequence of residual vectors bf r k i h bf x k bf f converges to i h hat bf x bf f furthermore if h is similar to a real diagonal matrix then the corresponding norms of bf r k and hat bf x k hat bf x decr

on regularized least norm problems this paper investigates the regularized least norm problem text minimize f mathbf x varepsilon s mathbf x s s a mathbf x mathbf b p where varepsilon is a positive constant 1 s infty and 1 p infty let mathbf x varepsilon denote the solution that corresponds to a given value of varepsilon and let mathbf x denote the minimum l s norm solution of the unregularized least l p norm problem it is shown that mathbf x varepsilon is a continuous function of varepsilon mathbf x varepsilon s leqq mathbf x s lim varepsilon to infty mathbf x varepsilon mathbf 0 and lim varepsilon to 0 mathbf x varepsilon mathbf x furthermore if the system a mathbf x mathbf b is solvable then there exists a positive constant delta such that mathbf x varepsilon mathbf x for all varepsilon in 0 delta the question of whether mathbf x varepsilon mat

anisotropic second and fourth order diffusion models based on convolutional virtual electric field for image denoising this paper proposes novel second order and fourth order anisotropic diffusion models for image denoising these two models are based on convolutional virtual electric field convef the proposed second order anisotropic diffusion model was introduced by introducing the convef into the perona malik p m equation and the fourth order one is coined by combining convef and the you kaveh y k equation the employment of the convef model allows to improve the estimation of high order derivatives and makes the proposed methods more robust to noise in addition the proposed convef based y k model is an anisotropic diffusion equation which can avoid the over smoothness of edges and provide better edge protection capability except for better denoising some experiments are conducted to demonstrate the efficiency of the proposed methods

fast nonconvex nonsmooth minimization methods for image restoration and reconstruction nonconvex nonsmooth regularization has advantages over convex regularization for restoring images with neat edges however its practical interest used to be limited by the difficulty of the computational stage which requires a nonconvex nonsmooth minimization in this paper we deal with nonconvex nonsmooth minimization methods for image restoration and reconstruction our theoretical results show that the solution of the nonconvex nonsmooth minimization problem is composed of constant regions surrounded by closed contours and neat edges the main goal of this paper is to develop fast minimization algorithms to solve the nonconvex nonsmooth minimization problem our experimental results show that the effectiveness and efficiency of the proposed algorithms

on the thickness of graphs of given degree abstract the results presented here refer to the determination of the thickness of a graph that is the minimum number of planar subgraphs into which the graph can be decomposed a useful general preliminary result obtained here is theorem 8 a planar graph has a planar representation for any arbitrary placement of its nodes in the plane it is then proved that if we have positive integers d and t such that any graph of degree at most d has thickness at most t the following hold theorem 9 any graph of degree d has thickness at most t roof d 1 d theorem 10 any graph of degree d can always be embedded in a regular graph g of any degree d corollary 5 any graph of degree d has thickness at most roof d 2 theorem 12 with d and t defined as above we have d 4t 2 corollary 6 if t 2 then d 6 we further conjecture that indeed the thickness of any graph of degree not exceeding 6 is never more than 2 and that more generally t roof d 2 4 since the design and fabrication of vlsi computer chips is essentially a concrete representation of the planar decomposition of a graph all these results are of direct practical interest

a row relaxation method for large l1 problems abstract this paper presents a row relaxation method for solving the regularized l 1 problem minimize 1 2 e x 2 2 a x b 1 it is shown that the dual of this problem has the form minimize 1 2 a t y 2 2 e b t y subject to y 1 and if y solves the dual then a t y e solves the primal the fact that the dual variables have simple bounds enables us to apply a wide range of methods the method derived in this paper is a row relaxation method that resembles kaczmarz s method this feature makes it suitable for solving problems in which a is large sparse and unstructured another advantage is that the method is easily adapted to handle linear constraints the paper introduces an iterative improvement technique that shifts the limit of the iterative process toward a solution of the unregularized problem minimize a x b 1

a simple simulated annealing algorithm for the maximum clique problem abstract simulated annealing technique has mostly been used to solve various optimization and learning problems and it is well known that the maximum clique problem is one of the most studied np hard optimization problems owing to its numerous applications in this note a simple simulated annealing algorithm for the maximum clique problem is proposed and tested on all 80 dimacs maximum clique instances although it is simple the proposed simulated annealing algorithm is efficient on most of the dimacs maximum clique instances the simulation results show that the proposed simulated annealing algorithm outperforms a recent efficient simulated annealing algorithm proposed by xu and ma and the solutions obtained by the proposed simulated annealing algorithm have the equal quality with those obtained by a recent trust region heuristic algorithm of stanislav busygin

a simultaneous iterative method for computing projections on polyhedra a simultaneous version of hildreth s iterative algorithm for norm minimization over linear inequalities is presented proofs are given showing that the algorithm converges from any starting point to its projection on the linear constraints set in the feasible case and to the nearest least squares solution in the general case

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

efficient minimization methods of mixed l 2 l 1 and l 1 l 1 norms for image restoration image restoration problems are often solved by finding the minimizer of a suitable objective function usually this function consists of a data fitting term and a regularization term for the least squares solution both the data fitting and the regularization terms are in the ell 2 norm in this paper we consider the least absolute deviation lad solution and the least mixed norm lmn solution for the lad solution both the data fitting and the regularization terms are in the ell 1 norm for the lmn solution the regularization term is in the ell 1 norm but the data fitting term is in the ell 2 norm since images often have nonnegative intensity values the proposed algorithms provide the option of taking into account the nonnegativity constraint the lmn and lad solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods at each iteration of the interior point method a structured linear system must be solved the preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems experimental results are used to demonstrate the effectiveness of our approach we also show the quality of the restored images using the minimization of mixed ell 2 ell 1 and ell1 ell 1 norms is better than that using only the ell 2 norm

recursive median filters of increasing order a variational approach for a certain class of signals it is known that the recursive median filters provide a minimum of a related regularizing functional defined by means of the l sub 1 norm we generalize such results and we show that a good approximation to the minima of such a functional can be obtained for any signal by means of successive applications of recursive median filters of increasing order

iterative schemes for the least 2 norm solution of piecewise linear programs abstract let f r n be a convex polyhedral function we show how to find the normal minimizer of f and the associated lagrange multipliers by computing x arg min x f x x 2 2 approximately for a sequence of 0 via any relaxation method applied to the corresponding dual problems our schemes generalize those of managasarian and de leone for solving very large sparse linear programs

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

structured total least squares for color image restoration the problem of 3 x 3 color mixing image restoration is considered the blurring matrices as well as the observed image are contaminated by noise therefore the total least squares tls method is employed to restore the original image since the blurring matrices are also structured we apply the structured total least squares stls method j b rosen h park and j glick siam j matrix anal appl 17 1996 pp 110 126 the blurring matrices are generally ill conditioned thus tikhonov s regularization is used to stabilize the solution since neumann boundary conditions are used in the restoration process the discrete cosine transform dct based preconditioner is effective for the linear systems encountered in our stls algorithm

fast deconvolution with approximated psf by rstls with antireflective boundary conditions the problem of reconstructing signals and images from degraded ones is considered in this paper the latter problem is formulated as a linear system whose coefficient matrix models the unknown point spread function and the right hand side represents the observed image moreover the coefficient matrix is very ill conditioned requiring an additional regularization term different boundary conditions can be proposed in this paper antireflective boundary conditions are considered since both sides of the linear system have uncertainties and the coefficient matrix is highly structured the regularized structured total least squares approach seems to be the more appropriate one to compute an approximation of the true signal image with the latter approach the original problem is formulated as an highly nonconvex one and seldom can the global minimum be computed it is shown that regularized structured total least squares problems for antireflective boundary conditions can be decomposed into single variable subproblems by a discrete sine transform such subproblems are then transformed into one dimensional unimodal real valued minimization problems which can be solved globally some numerical examples show the effectiveness of the proposed approach

the adventures of a simple algorithm abstract this paper presents analyzes and tests a row action method for solving the regularized linear programming problem minimize 1 2 x 2 2 c t x subject to a x b where is a given positive constant the proposed method is based on the observation that the dual of this problem has the form maximize b t y 1 2 a t y c 2 2 subject to y 0 and if y solves the dual then the point x a t y c provides the unique solution of the primal maximizing the dual objective function by changing one variable at a time results in an effective row action method which is suitable for solving large sparse problems one aim of this paper is to clarify the convergence properties of the proposed scheme let y k denote the estimated dual solution at the end of the k th iteration and let x k a t y k c denote the corresponding primal estimate it is proved that the sequence x k converges to x while the sequence y k converges to a point y that solves the dual the only assumption which is needed in order to establish these claims is that the feasible region is not empty yet perhaps the more striking features of the algorithm are related to temporary situations in which it attempts to solve an inconsistent system in such cases the sequence y k obeys the rule y k u k k v where u k is a fastly converging sequence and v is a fixed vector that satisfies a t v 0 and b t v 0 so the sequence x k is almost unchanged for many iterations the paper ends with numerical experiments that illustrate the effects of this phenomenon and the close links with kaczmarz s sor method

two projection methods for regularized total least squares approximation abstract regularized total least squares is a useful approach for solving ill posed overdetermined systems of equations when both the model matrix and the observed data are contaminated by noise a newton based regularized total least squares method was proposed by lee et al 2013 16 but may not be efficient for large scale problems here we consider two projection based algorithms applied to this method for the solution of the large scale problem the first fixes the underlying subspace dimension while the second expands the subspace dynamically during the iterations by employing a generalized krylov subspace expansion experimental results demonstrate that the two projection based algorithms can be successfully applied for the solution of the large scale regularized total least squares problems

a regularized structured total least squares algorithm for high resolution image reconstruction abstract high resolution image reconstruction is an important problem in image processing in general the blurring matrices are ill conditioned and it is necessary to compute a regularized solution moreover error exists not only in the blurred image but also the blurring matrix thus the total least squares method tends to give better results than the ordinary least squares method since the blurring matrices are also structured it is more appropriate to apply structured total least squares stls ng et al int j imaging systems technol 12 2002 35 recently proposed a regularized constrained total least squares rctls algorithm for this problem rctls is essentially a different name for regularized structured total least squares rstls however ng et al solved a problem that approximates the rctls problem the algorithm proposed in this paper solves the exact regularization of the stls problem and it is a faster algorithm also proposed is a preconditioner for the linear systems encountered in our rstls algorithm

the l 1 potts functional for robust jump sparse reconstruction we investigate the nonsmooth and nonconvex l 1 potts functional in discrete and continuous time we show gamma convergence of discrete l 1 potts functionals toward their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer for the discrete l 1 potts problem we introduce an o n 2 time and o n space algorithm to compute an exact minimizer we apply l 1 potts minimization to the problem of recovering piecewise constant signals from noisy measurements f it turns out that the l 1 potts functional has a quite interesting blind deconvolution property in fact we show that mildly blurred jump sparse signals are reconstructed by minimizing the l 1 potts functional furthermore for strongly blurred signals and a known blurring operator we derive an iterative reconstruction algorithm

structural source identification using a generalized tikhonov regularization abstract this paper addresses the problem of identifying mechanical exciting forces from vibration measurements the proposed approach is based on a generalized tikhonov regularization that allows taking into account prior information on the measurement noise as well as on the main characteristics of sources to identify like its sparsity or regularity to solve such a regularization problem efficiently a generalized iteratively reweighted least squares girls algorithm is introduced proposed numerical and experimental validations reveal the crucial role of prior information in the quality of the source identification and the performance of the girls algorithm

a simulated annealing algorithm for determining the thickness of a graph the thickness of a graph is the minimum number of planar subgraphs into which the graph can be decomposed determining the thickness of a given graph is known to be an np complete problem in this paper we introduce a new heuristic algorithm for determining the thickness of a graph our algorithm is based on the simulated annealing optimization scheme we compare the quality of the solutions and running times of our algorithm against previously tested heuristic algorithms we show that the simulated annealing is a fast and efficient method to obtain good approximations for the thickness of a graph we also give a new upper bound for the thickness of complete tripartite graphs whose vertex sets are of equal size

parametric maximum flow algorithms for fast total variation minimization this report studies the global minimization of anisotropically discretized total variation tv energies with an l p in particular l 1 and l 2 fidelity term using parametric maximum flow algorithms to minimize s t cut representations of these energies the tv l 2 model also known as the rudin osher fatemi rof model is suitable for restoring images contaminated by gaussian noise while the tv l 1 model is able to remove impulsive noise from grayscale images and perform multiscale decompositions of them preliminary numerical results on large scale two dimensional ct and three dimensional brain mr images are presented to illustrate the effectiveness of these approaches

a primal dual active set method for non negativity constrained total variation deblurring problems this paper studies image deblurring problems using a total variation based model with a non negativity constraint the addition of the non negativity constraint improves the quality of the solutions but makes the solution process a difficult one the contribution of our work is a fast and robust numerical algorithm to solve the non negatively constrained problem to overcome the nondifferentiability of the total variation norm we formulate the constrained deblurring problem as a primal dual program which is a variant of the formulation proposed by chan golub and mulet for unconstrained problems here dual refers to a combination of the lagrangian and fenchel duals to solve the constrained primal dual program we use a semi smooth newton s method we exploit the relationship between the semi smooth newton s method and the primal dual active set method to achieve considerable simplification of the computations the main advantages of our proposed scheme are no parameters need significant adjustment a standard inverse preconditioner works very well quadratic rate of local convergence theoretical and numerical numerical evidence of global convergence and high accuracy of solving the optimality system the scheme shows robustness of performance over a wide range of parameters a comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme