This paper introduces an expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with low-complexity, expressed in terms of the wavelet coecients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, the resulting criteria are solved approximately or require very demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation oered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. The algorithm alternates between an E-step based on the fast Fourier transform (FFT) and a DWT-based M-step, resulting in an ecient iterative process requiring O(N log N) operations per iteration. Thus, it is the rst image restoration algorithm that optimizes a wavelet-based penalized likelihood criterion and has computational complexity comparable to that of standard wavelet denoising or frequency domain deconvolution methods. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach outperforms several of the best existing methods in benchmark tests, and in some cases is also much less computationally demanding.

This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden

In this paper, we propose the application of the hierarchical Bayesian paradigm to the image restoration problem. We derive expressions for the iterative evaluation of the two hyperparameters applying the evidence and maximum a posteriori (MAP) analysis within the hierarchical Bayesian paradigm. We show analytically that the analysis provided by the evidence approach is more realistic and appropriate than the MAP approach for the image restoration problem. We furthermore study the relationship between the evidence and an iterative approach resulting from the set theoretic regularization approach for estimating the two hyperparameters, or their ratio, defined as the regularization parameter. Finally the proposed algorithms are tested experimentally.

Iterative shrinkage/thresholding (IST) 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 IST 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 IST (TwIST) algorithms, exhibiting much faster convergence rate than IST for ill-conditioned problems. For a vast class of nonquadratic convex regularizers ( 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.

The point-spread function (PSF) of a blurred image is often unknown a priori --- the blur must first be identified from the degraded image data before restoring the image. We introduce generalized cross-validation (GCV) to address the blur identification problem. Motivated by the success of GCV in identifying optimal smoothing parameters for image restoration, we have extended the method to the problem of identifying blur parameters as well. The GCV criterion identifies model parameters for the blur, the image, and the regularization parameter, providing all the information necessary to restore the image. Experiments are presented which show that GCV is capable of yielding good identification results. Furthermore, a comparison of the GCV criterion to maximum likelihood (ML) estimation shows that GCV often outperforms ML in identifying the blur and image model parameters. To appear in IEEE Transactions on Image Processing. This work was supported in part by the Joint Services Electroni...

Many important problems in engineering and science are well-modeled by Poisson processes. In many applications it is of great interest to accurately estimate the intensities underlying observed Poisson data. In particular, this work is motivated by photon-limited imaging problems. This paper studies a new Bayesian approach to Poisson intensity estimation based on the Haar wavelet transform. It is shown that the Haar transform provides a very natural and powerful framework for this problem. Using this framework, a novel multiscale Bayesian prior to model intensity functions is devised. The new prior leads to a simple, Bayesian intensity estimation procedure. Furthermore, we characterize the correlation behavior of the new prior and show that it has 1/f spectral characteristics. The new framework is applied to photon-limited image estimation and its potential to improve nuclear medicine imaging is examined.

Superresolution reconstruction produces a high-resolution image from a set of low-resolution images. Previous iterative methods for superresolution had not adequately addressed the computational and numerical issues for this ill-conditioned and typically underdetermined large scale problem. We propose efficient block circulant preconditioners for solving the Tikhonov-regularized superresolution problem by the conjugate gradient method. We also extend to underdetermined systems the derivation of the generalized cross-validation method for automatic calculation of regularization parameters. Effectiveness of our preconditioners and regularization techniques is demonstrated with superresolution results for a simulated sequence and a forward looking infrared (FLIR) camera image sequence.

The goal of image restoration is to reconstruct the original scene from a de-graded observation. This recovery process is critical to many image processing applications. Although classical linear im-age restoration has been thoroughly studied [ 1, 21, the more difficult problem of blind image restoration has numerous research possibilities. Our objective in this article is to introduce the problem of blind deconvolution for images, provide an overview of the basic principles and methodologies behind the existing algorithms, and examine the current trends and the potential of this difficult signal processing problem. A broad review of blind deconvolution methods for images is given to portray the experience of the authors and of the many other re-searchers in this area. We first introduce the blind deconvolution problem for general signal processing applications. The specific challenges encountered in image related resto-

Abstract—In many image restoration/resolution enhancement applications, the blurring process, i.e., point spread function (PSF) of the imaging system, is not known or is known only to within a set of parameters. We estimate these PSF parameters for this ill-posed class of inverse problem from raw data, along with the regularization parameters required to stabilize the solution, using the generalized cross-validation method (GCV). We propose efficient approximation techniques based on the Lanczos algorithm and Gauss quadrature theory, reducing the computational complexity of the GCV. Data-driven PSF and regularization parameter estimation experiments with synthetic and real image sequences are presented to demonstrate the effectiveness and robustness of our method. Index Terms—Blind restoration, blur identification, generalized cross-validation, quadrature rules, superresolution. I.

A broad range of inverse problems can be abstracted into the problem of minimizing the sum of several convex functions in a Hilbert space. We propose a proximal decomposition algorithm for solving this problem with an arbitrary number of nonsmooth functions and establish its weak convergence. The algorithm fully decomposes the problem in that it involves each function individually via its own proximity operator. A significant improvement over the methods currently in use in the area of inverse problems is that it is not limited to two nonsmooth functions. Numerical applications to signal and image processing problems are demonstrated.