To recover a sharp image from its blurry observation is the problem known as image deblurring. It frequently arises in imaging sciences and technologies, including optical, medical, and astronomical applications, and is crucial for allowing to detect important features and patterns such as those of a distant planet or some microscopic tissue. Mathematically, image deblurring is intimately connected to backward diffusion processes (e.g., inverting the heat equation), which are notoriously unstable. As inverse problem solvers, deblurring models therefore crucially depend upon proper regularizers or conditioners that help secure stability, often at the necessary cost of losing certain highfrequency details in the original images. Such regularization techniques can ensure the existence, uniqueness, or stability of deblurred images. The present work follows closely the general framework described in our recent monograph [18], but also contains more updated views and approaches to image deblurring, including, e.g., more discussion on stochastic signals, the Bayesian/Tikhonov approach to Wiener filtering, and the iterated-shrinkage algorithm of Daubechies et al. [30,31] for wavelet-based deblurring. The work thus contributes to the development of generic, systematic, and unified frameworks in contemporary image processing.

We propose a new way to iteratively solve large scale ill-posed problems by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain iteratively approximations to the Tikhonov L-curve and its corner. Monitoring the change of the approximate L-curves allows us to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired solution can be reconstructed with a low number of iterations. We apply the technique to an idealized image reconstruction problem in positron emission tomography. Keywords: Tikhonov regularization, multiobjective optimization, ill-posed, L-curve, envelope, preconditioned conjugate gradients 1991 MSC Classification: primary 65F10, secondary 65R30, 90C29 1 Introduction Many problem in applied mathematics lead to models of the form F (x) = y + ffl; where x is an unknown vector of parameters, often restricted to a subset\Omega ae IR n (e.g., by nonnegativity con...

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem, which can be regularized within a Bayesian context by using an a priori model of the reconstructed solution. Since real satellite data show spatially variant characteristics, we propose here to use an inhomogeneous model. We use the Maximum Likelihood Estimator (MLE) to estimate its parameters and we demonstrate that the MLE computed on the corrupted image is not suitable for image deconvolution, because it is not robust to noise. Then we show that the estimation is correct only if it is made from the original image. As this image is unknown, we need to compute an approximation of su#ciently good quality to provide useful estimation results.

Pattern analysis of naturally synthesized images is crucial for a number of important fields including image processing, computer vision, artificial intelligence, and computer graphics. Benefited from several important works in existence, the current research note proposes a novel free-boundary variational model for segmented image decomposition. As an inverse problem solver, the new model outputs not only the boundaries of individual objects as achieved by the Mumford-Shah model (Comm. Pure Applied Math., 42:577-685, 1989), but also a structure decomposition comprising a smooth (or cartoonish) component, an oscillatory component (or texture), and a square-integrable residue (or noise). Motivations and justifications from vision research are emphasized, and some preliminary mathematical analysis is given.

In a number of applications, certain computations to be done with a given matrix are performed by replacing this matrix by its best low rank approximation. This has the effect of yielding the most relevant part of the desired solution while discarding noise and redundancies. One such application is that of regularization where the righthand side of the original linear system is noisy or inaccurate while the coefficient matrix is very ill-conditioned. Solving such linear systems accurately is counter-productive as the noise tends to be amplified. A common remedy is to compute the pseudo-inverse solution in which the inverses of the smallest singular values are replaced by zeros or small quantities. A similar procedure is also used in methods related to Principal Component Analysis, such as in Latent Semantic Indexing in information retrieval. Here the low-rank approximation to the original matrix is used to analyze similarities with a given query vector. This paper presents a few conjugate-gradient like methods to provide solutions to these two types of problems by iterative procedures which utilize only matrix-vector products.

Tikhonov regularization with a modified total variation regularization functional is used to recover an image from noisy, blurred data. This approach is appropriate for image processing in that it does not place a priori smoothness conditions on the solution image. An efficient algorithm is presented for the discretized problem which combines a fixed point iteration to handle nonlinearity with an effective preconditioned conjugate gradient iteration for large linear systems. Reconstructions and convergence results are presented for an application to satellite image reconstruction. Keywords--- Image reconstruction, conjugate gradient, preconditioner, total variation, deconvolution, cell-centered finite differences, fixed point iteration, regularization I. Introduction T HE problem of recovering an image from noisy, blurred data is fundamentally that of solving a Fredholm first kind integral equation whose kernel is smooth and non-degenerate. This problem is ill-posed in that small pe...

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem, which can be regularized within a Bayesian context by using an a priori model of the reconstructed solution. Homogeneous regularization models do not provide sufficiently satisfactory results, since real satellite data show spatially variant characteristics. We propose

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem. Direct inversion leads to unacceptable noise amplification. Usually, the problem is either regularized during the inversion process, or the noise is filtered after deconvolution and decomposition in the wavelet transform domain. Herein, we have developed the second solution, by thresholding the coefficients of a new complex wavelet packet transform

In this paper, the regularized solutions of an ill--conditioned system of linear equations are computed for several values of the regularization parameter . Then, these solutions are extrapolated at = 0 by various vector rational extrapolations techniques built for that purpose. These techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition. Numerical results illustrate the effectiveness of the procedures. 1

Maximum entropy (maxent) approach, formally equivalent to maximum likelihood, is a widely used density-estimation method. When input datasets are small, maxent is likely to overfit. Overfitting can be eliminated by various smoothing techniques, such as regularization and constraint relaxation, but theory explaining their properties is often missing or needs to be derived for each case separately. In this dissertation, we propose a unified treatment for a large and general class of smoothing techniques. We provide fully general guarantees on their statistical performance and propose optimization algorithms with complete convergence proofs. As special cases, we can easily derive performance guarantees for many known regularization types including L1 and L2-squared regularization. Furthermore, our general approach enables us to derive entirely new regularization functions with superior statistical guarantees. The new regularization functions use information about the structure of the feature space, incorporate information about sample selection bias, and combine information across several related density-estimation tasks. We propose algorithms solving a large and general subclass of generalized maxent problems, including all