regularized iterative image restoration with ringing reduction linear space invariant image restoration algorithms often introduce ringing effects near sharp intensity transitions it is shown that these artifacts are attributable to the regularization of the ill posed image restoration problem two possible methods to reduce the ringing effects in restored images are proposed the first method incorporates deterministic a priori knowledge about the original image into the restoration algorithm the second method locally regulates the severity of the noise magnification and the ringing phenomenon depending on the edge information in the image a regularized iterative image restoration algorithm is proposed in which both ringing reduction methods are included by making use of the theory of the projections onto convex sets and the concept of norms in a weighted hilbert space both the numerical performance and the visual evaluation of the results are improved by the use of ringing reduction

tomographic image reconstruction using the theory of convex projections the method of convex projections is applied to reconstruct an image in computer tomography this appears to the first time that the method has been used to obtain geometry free reconstruction from ray sum data it is shown that a special set of convex projections duplicates the result of the algebraic reconstruction technique art the similarities and differences of these two methods are discussed it is pointed out that use of a priori information enhances the quality of the results especially when partial data have been used in which case art fails simulations and reconstruction of a ct image are also furnished to demonstrate the feasibility of this method

image recovery in computer tomography from partial fan beam data by convex projections for the image recovery process the authors use the convex projections method also known as the method of projections onto convex sets pocs several incomplete data geometries including those associated with limited source travel and beam blocking internal opacities are considered to enable the recovery several prior knowledge constraints including one associated with the directivity of the image vector are used the overall recovery algorithm can be practically implemented by exploiting the toeplitz structure of key operators

interactive optimization of 3d shape and 2d correspondence using multiple geometric constraints via pocs the traditional approach of handling motion tracking and structure from motion sfm independently in successive steps exhibits inherent limitations in terms of achievable precision and incorporation of prior geometric constraints about the scene this paper proposes a projections onto convex sets pocs framework for iterative refinement of the measurement matrix in the well known factorization method to incorporate multiple geometric constraints about the scene thereby improving the accuracy of both 2d feature point tracking and 3d structure estimates regularities in the scene such as points on line and plane and parallel lines and planes that can be interactively identified and marked at each pocs iteration enforce rank and parallelism constraints on appropriately defined local measurement matrices one for each constraint the pocs framework allows for the integration of the information in each of these local measurement matrices into a single measurement matrix that is closest to the initial observed measurement matrix in frobenius norm which is then factored in the usual manner experimental results demonstrate that the proposed interactive pocs framework consistently improves both 2d correspondences and 3d shape motion estimates and similar results cannot be achieved by enforcing these constraints as either post or preprocessing

convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections solving a convex set theoretic image recovery problem amounts to finding a point in the intersection of closed and convex sets in a hilbert space the projection onto convex sets pocs algorithm in which an initial estimate is sequentially projected onto the individual sets according to a periodic schedule has been the most prevalent tool to solve such problems nonetheless pocs has several shortcomings it converges slowly it is ill suited for implementation on parallel processors and it requires the computation of exact projections at each iteration we propose a general parallel projection method emopsp that overcomes these shortcomings at each iteration of emopsp a convex combination of subgradient projections onto some of the sets is formed and the update is obtained via relaxation the relaxation parameter may vary over an iteration dependent extrapolated range that extends beyond the interval 0 2 used in conventional projection methods emopsp not only generalizes existing projection based schemes but it also converges very efficiently thanks to its extrapolated relaxations theoretical convergence results are presented as well as numerical simulations

the eigenvalues of matrices that occur in certain interpolation problems the eigenvalues of the matrices that occur in certain finite dimensional interpolation problems are directly related to their well posedness and strongly depend on the distribution of the interpolation knots that is on the sampling set we study this dependency as a function of the sampling set itself and give accurate bounds for the eigenvalues of the interpolation matrices the bounds can be evaluated in as few as four arithmetic operations and therefore they greatly simplify the assessment of sampling sets regarding numerical stability the accuracy and usefulness of the bounds are illustrated with examples

limited angle reconstruction using stabilized algorithms the stability of solutions to the limited angle tomography reconstruction problem obtained by using the projections onto convex sets pocs technique are examined although pocs techniques provide a feasible solution to the reconstruction problem the solution is only one sample from the intersection of the closed convex sets that define the solution space a method for evaluating the ensemble of possible solution waveforms that are in the neighborhood of a solution is presented the ensemble characteristics are used to construct an inverse filter which is then applied to the computed solution the results obtained using this method are less sensitive to noise amplification and are less dependent on both starting data and the number of iterations an estimate of the object dependent extrapolation that is possible using either linear or nonlinear constraints is provided

the feasible solution in signal restoration the feasible solution to the signal restoration problem is defined as the one which satisfies all constraints which can be imposed on the true solution a very important set of constraints can be obtained by examining the statistics of the noise these and other constraints can be described as closed convex sets thus projection onto closed convex sets is the numerical method used to obtain a feasible solution examples of this method demonstrate its usefulness in one and two dimensional signal restoration the limitations of the method are discussed

the discrete gerchberg algorithm the discrete version of the gerchberg algorithm for iterative restoration of a time constrained function from only partial knowledge of its spectrum or vice versa is analyzed although convergence is guaranteed eigenvalues close to unity inhibit iteration to the limit identification of these large eigenvalues allowing extrapolation to the limit is described

an iterative algorithm for signal reconstruction from bispectrum the author presents a procedure to reconstruct a minimum or nonminimum phase discrete time signal from its bispectrum the algorithm is iterative and based on using the method of projections onto convex sets pocs in an inner product space in which the vector space consists of real sequences and vector addition is defined in terms of the convolution operation prior information such as an energy bound on the signal can be incorporated into this algorithm an algorithm that employs energy information was found to produce better results in simulation than one that ignores energy constraints

euler implicit explicit iterative scheme for the stationary navier stokes equations in this paper a new uniqueness assumption a2 of the solution for the stationary navier stokes equations is presented under assumption a2 the exponential stability of the solution bar u bar p for the stationary navier stokes equations is proven moreover the euler implicit explicit scheme based on the mixed finite element is applied to solve the stationary navier stokes equations finally the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme

optimal pulse shape design for digital communication systems by projections onto convex sets recent developments in dsp hardware has made it possible to use almost any pulse shape for digital data trans mission this resulted in a search for algorithms capable of constructing pulse shapes matching to the properties of a given channel the projection onto convex sets technique is suitable for the solution of this problem particularly because of its ffexibility in modeling a variety of constraints the use of the technique for the optimal pulse shape design problem is demonstrated through a detailed example from power line communications i introduction ince nyquist s invention of the intersymbol interference isi the optimal pulse shaping problem for digital data transmission over channels with various properties has been investigated by a number of researchers 1 7 the main differences between various digital nyquist pulse design al gorithms published in the literature are the type of the filter fir or iir they choose to generate the pulse shape and the formulation and solution of the optimization problem some of these approaches use general optimization techniques and they can handle additional constraints if needed however addition of each new constraint requires reformulation of the problem and the solution algorithm the projections onto convex sets pocs technique is an optimization method introduced in the 1960 s 8 a number of successful applications of the pocs technique in the signal processing area have been reported in the recent years 9 ll the pocs technique is attractive for optimum pulse shape design particularly because of its modular way of formulating problems with many constraints in this technique the addition of new constraints or removal of some of the existing ones does not require reformulation it is also possible to handle certain inconsistent requirements in the pocs formulation 12 if there are nonintersecting convex sets it is possible to find the vector which is as close as possible to one set of constraints while satisfying the rest

information theoretic image formation the emergent role of information theory in image formation is surveyed unlike the subject of information theoretic communication theory information theoretic imaging is far from a mature subject the possible role of information theory in problems of image formation is to provide a rigorous framework for defining the imaging problem for defining measures of optimality used to form estimates of images for addressing issues associated with the development of algorithms based on these optimality criteria and for quantifying the quality of the approximations the definition of the imaging problem consists of an appropriate model for the data and an appropriate model for the reproduction space which is the space within which image estimates take values each problem statement has an associated optimality criterion that measures the overall quality of an estimate the optimality criteria include maximizing the likelihood function and minimizing mean squared error for stochastic problems and minimizing squared error and discrimination for deterministic problems the development of algorithms is closely tied to the definition of the imaging problem and the associated optimality criterion algorithms with a strong information theoretic motivation are obtained by the method of expectation maximization related alternating minimization algorithms are discussed in quantifying the quality of approximations global and local measures are discussed global measures include the mean squared error and discrimination between an estimate and the truth and probability of error for recognition or hypothesis testing problems local measures include fisher information

an iterative method for the extrapolation of band limited functions abstract we derive an iterative algorithm to extrapolate band limited functions using the operators of fourier transform and projection on convex subsets the algorithm allows the use of a priori knowledge in the extrapolation procedure

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

elliptic lie algebras and tubular algebras abstract this article is to study relations between tubular algebras of ringel and elliptic lie algebras in the sense of saito yoshii using the explicit structure of the derived categories of tubular algebras given by happel ringel we prove that the elliptic lie algebra of type d 4 1 1 e 6 1 1 e 7 1 1 or e 8 1 1 is isomorphic to the ringel hall lie algebra of the root category of the tubular algebra with the same type as a by product of our proof we obtain a chevalley basis of the elliptic lie algebra following indecomposable objects of the root category of the corresponding tubular algebra this can be viewed as an analogue of the frenkel malkin vybornov theorem in which they described a chevalley basis for each untwisted affine kac moody lie algebra by using indecomposable representations of the corresponding affine quiver

a projection operator for the restoration of divergence free vector fields the theory of image restoration by projection onto convex sets can also be applied to the restoration of vector fields these can have properties that restrict them to lie in well defined closed convex sets one of the properties divergence freedom is considered and the theory and numerical implementation of its projection operator are presented the performance of the operator is illustrated by restoring from partial information two simulated divergence free vector fields this projection operator finds an important application in the restoration of velocity fields or optical flows computed from an image sequence when the real velocity field is known a priori to be divergence free

signal recovery from wavelet transform maxima the paper presents an iterative algorithm for signal recovery from discrete time wavelet transform maxima the signal recovery algorithm is developed by using the method of projections onto convex sets convergence of the algorithm is assured

an optimal technique for constraint based image restoration and reconstruction a new technique for finding an optimal feasible solution to the general image reconstruction and restoration problem is described this method allows the use of prior knowledge of the properties of both the solution and any noise present on the data the problem is formulated as the optimization of a cost function over the intersection of a number of convex constraint sets each set being defined as containing those solutions consistent with a particular constraint a duality theorem is then applied to yield a dual problem in which the unknown image is replaced by a model defined in terms of a finite dimensional parameter vector and the kernels of the integral equations relating the data and solution the dual problem may then be solved for the model parameters using a gradient descent algorithm this method serves as an alternative to the primal constrained optimization and projection onto convex sets pocs algorithms problems in which this new approach is appropriate are discussed an example is given for image reconstruction from noisy projection data applying the dual method results in a fast nonlinear algorithm simulation results demonstrate the superiority of the optimal feasible solution over one obtained using a suboptimal approach

direct fourier reconstruction in computer tomography direct fourier transform ft reconstruction of images in computerized tomography ct is not widely used because of the difficulty of precisely interpolating from polar to cartesian samples in this paper an exact interpolation scheme is proposed which in practice can be approached with arbitrary accuracy using well conditioned algorithms several features of the direct ft method are discussed a method that allows angular band limiting of the data before processing to avoid angular aliasing artifacts in the reconstructed image is discussed and experimentally verified the experimental results demonstrate the feasibility of direct ft reconstruction of ct data

halftone to continuous tone conversion of error diffusion coded images considers the problem of reconstructing a continuous tone contone image from its halftoned version where the halftoning process is done by error diffusion the authors present an iterative nonlinear decoding algorithm for halftone to contone conversion and show simulation results that compare the performance of the algorithm to that of conventional linear low pass filtering they find that the new technique results in subjectively superior reconstruction as there is a natural relationship between error diffusion and spl sigma spl delta modulation the reconstruction algorithm can also be applied to the decoding problem for spl sigma spl delta modulators

image restoration by the method of convex projections part 2 applications and numerical results the image restoration theory discussed in a previous paper by youla and webb 1 is applied to a simulated image and the results compared with the well known method known as the gerchberg papoulis algorithm the results show that the method of image restoration by projection onto convex sets by providing a convenient technique for utilizing a priori information performs significantly better than the gerchberg papoulis method

weak and norm convergence of a parallel projection method in hilbert spaces abstract we show that for a parallel projection method for finding the common point of convex sets in a hilbert space under very mild conditions about the variable weights and the variable relaxation parameters the constructed sequence is weakly convergent to a point of the intersection of the sets we also prove norm convergence of the sequence under additional conditions on the convex sets

reconstruction of oversampled band limited signals from spl sigma spl delta encoded binary sequences considers the application of spl sigma spl delta modulators to analog to digital conversion the authors have previously shown that for constant input signals optimal nonlinear decoding can achieve large gains in signal to noise ratio snr over linear decoding the present paper shows a similar result for band limited input signals the new nonlinear decoding algorithm is based on projections onto convex sets pocs and alternates between a time domain operation and a band limitation to find a signal invariant under both the time domain operation results in a quadratic programming problem the band limitation can be based on singular value decomposition of a certain matrix the authors show simulation results for the snr performance of a pocs based decoder and a linear decoder for the single loop double loop and two stage spl sigma spl delta modulators and for a specific fourth order interpolative modulator depending on the modulator and the oversampling ratio improvements in snr of up to 10 20 db can be achieved