The three main tools in the single image restoration theory are the maximum likelihood (ML) estimator, the maximum a posteriori probability (MAP) estimator, and the set theoretic approach using projection onto convex sets (POCS). This paper utilizes the above known tools to propose a unified methodology toward the more complicated problem of superresolution restoration. In the superresolution restoration problem, an improved resolution image is restored from several geometrically warped, blurred, noisy and downsampled measured images. The superresolution restoration problem is modeled and analyzed from the ML, the MAP, and POCS points of view, yielding a generalization of the known superresolution restoration methods. The proposed restoration approach is general but assumes explicit knowledge of the linear space- and time-variant blur, the (additive Gaussian) noise, the different measured resolutions, and the (smooth) motion characteristics. A hybrid method combining the simplicity of the ML and the incorporation of nonellipsoid constraints is presented, giving improved restoration performance, compared with the ML and the POCS approaches. The hybrid method is shown to converge to the unique optimal solution of a new definition of the optimization problem. Superresolution restoration from motionless measurements is also discussed. Simulations demonstrate the power of the proposed methodology.

In this paper we analyze the performance of an iterative algorithm, similar to the discrete Paponiis-Gerchberg algorithm, and which can be used to recover missing samples in finite-length records of band-limited data. No assumptions are made regarding the distribution of the missing samples, in contrast with the often studied extrapolation problem, in which the known samples are grouped together. Indeed, it is possible to regard the observed signal as a sampled version of the original one, and to interpret the reconstruction result studied herein as a sampling result. We show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data. We give upper and lower bounds for the error as a function of the number of iterations, together with the signals for which the bounds are attained. Also, we analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix. From this analysis we infer the optimum value of the relaxation constant. We also point out, among all sampling sets with the same density, those for which the convergence rate of the recovery algorithm is maximum or minimum. For low-pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer. The worst convergence rates generally occur when the missing samples are contiguous.

In this paper we present a new image recovery algorithm to remove, in addition to blocking, ringing artifacts from compressed images and video. This new algorithm is based on the theory of projections onto convex sets (POCS). A new family of directional smoothness constraint sets is defined based on line processes modeling of the image edge structure. The definition of these smoothness sets also takes into account the fact that the visibility of compression artifacts in an image is spatially-varying. To overcome the numerical difficulty in computing the projections onto these sets, a divide-and-conquer (DAC) strategy is introduced. According to this strategy new smoothness sets are derived such that their projections are easier to compute. The effectiveness of the proposed algorithm is demonstrated through numerical experiments using MPEG-based coders-decoders (codecs). I Introduction At the present time transform-based compression is among the most popular and it is widely used for b...

We present a statistical characterization of texture images in the context of an overcomplete complex wavelet transform. The characterization is based on empirical observations of statistical regularities in such images, and parameterized by (1) the local autocorrelation of the coefficients in each subband; (2) both the local auto-correlation and cross-correlation of coefficient magnitudes at other orientations and spatial scales; and (3) the first few moments of the image pixel histogram. We develop an efficient algorithm for synthesizing random images subject to these constraints using alternated projections, and demonstrate its effectiveness on a wide range of synthetic and natural textures. In particular, we show that many important structural elements in textures (e.g., edges, repeated patterns or alternated patches of simpler texture), can be captured through joint second order statistics of the coefficient magnitudes. We also show the flexibility of the representation, by applying to a variety...

We present a universal statistical model for texture images in the context of an overcomplete complex wavelet transform. The model is parameterized by a set of statistics computed on pairs of coefficients corresponding to basis functions at adjacent spatial locations, orientations, and scales. We develop an efficient algorithm for synthesizing random images subject to these constraints, by iteratively projecting onto the set of images satisfying each constraint, and we use this to test the perceptual validity of the model. In particular, we demonstrate the necessity of subgroups of the parameter set by showing examples of texture synthesis that fail when those parameters are removed from the set. We also demonstrate the power of our model by successfully synthesizing examples drawn from a diverse collection of artificial and natural textures.

The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily on the pioneering work of Gerchberg, Saxton, and Fienup. Despite the widespread use of the algorithms proposed by these three researchers, current mathematical theory cannot explain their remarkable success. Nevertheless, great insight can be gained into the behavior, the shortcomings, and the performance of these algorithms from their possible counterparts in convex optimization theory. An important step in this direction was made two decades ago when the error reduction algorithm was identified as a nonconvex alternating projection algorithm. The purpose of this paper is to formulate the phase retrieval problem with mathematical care and to establish new connections between well established numerical phase retrieval schemes and classical convex optimization methods. Specifically, it is shown that Fienup’s basic inputoutput algorithm corresponds to Dykstra’s algorithm, and that Fienup’s hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. This work provides a theoretical framework to better understand and, potentially, improve existing phase recovery algorithms. 1 1

In this paper the problem of restoring an image distorted by a linear space-invariant (LSI) point-spread function (psf) which 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 psf errors are performed to test the RCTLS estimator for this problem. Objective and visual comparisons are presented with the linear minimum mean-squared-error (LMMSE) and the re...

Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the Douglas-Rachford algorithm for monotone operator-splitting, is obtained under general conditions. Applications to non-Gaussian image denoising in a tight frame are also demonstrated. Index Terms — Convex optimization, denoising, Douglas-Rachford, frame, nondifferentiable optimization, Poisson noise,

We present a statistical characterization of texture images in the context of an overcomplete complex wavelet transform. The characterization is based on empirical observations of statistical regularities in such images, and parameterized by (1) the local auto-correlation of the coefficients in each subband; (2) both the local auto-correlation and cross-correlation of coefficient magnitudes at other orientations and spatial scales; and (3) the first few moments of the image pixel histogram. We develop an efficient algorithm for synthesizing random images subject to these constraints using alternated projections, and demonstrate its effectiveness on a wide range of synthetic and natural textures. We also show the flexibility of the representation, by applying to a variety of tasks which can be viewed as constrained image synthesis problems. Vision is arguably our most important sensory system, judging from both the ubiquity of visual forms of communication, and the large proportion of ...

this paper, we consider an image to be a sample of a random eld that is parameterized by a small set of statistics. In a very high-dimensional space such as that of all digitized images, the samples of such a random eld lie close to the hypersurface of images sharing the same sample statistics, and we can approximate the probability density as a uniform distribution over this hypersurface [13]. Thus, assuming the random eld parameters are known, the statistical description is replaced by a deterministic one. In this context, an image corrupted by noise is an N-dimensional vector (N the number of pixels) that has been displaced from its original position to a point outside of its associated JP is supported by a fellowship from the Programa Nacional de FPI (Spanish Government). EPS is supported by NSF CAREER grant MIP-9796040.