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.

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...

Abstract—Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The well-known sparsity of the wavelet coefficients of real-world images is modeled by heavy-tailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of infinite) combination of Gaussian densities. This class includes, among others, the generalized Gaussian, the Jeffreys, and the Gaussian mixture priors. Necessary and sufficient conditions are stated under which the prior induced by a thresholding/shrinking denoising rule is a GSM. This result is then used to show that the prior induced by the “nonnegative garrote ” thresholding/shrinking rule, herein termed the garrote prior, is a GSM. To compute the maximum a posteriori estimate, we propose a new generalized expectation maximization (GEM) algorithm, where the missing variables are the scale factors of the GSM densities. The maximization step of the underlying expectation maximization algorithm is replaced with a linear stationary second-order iterative method. The result is a GEM algorithm of ( log) computational complexity. In a series of benchmark tests, the proposed approach outperforms or performs similarly to state-of-the art methods, demanding comparable (in some cases, much less) computational complexity. Index Terms—Bayesian, deconvolution, expectation maximization (EM), generalized expectation maximization (GEM), Gaussian scale mixtures (GSM), heavy-tailed priors, wavelet. I.

In this paper, we propose an iterative algorithm for enhancing the resolution of monochrome and color image sequences. Various approaches toward motion estimation are investigated and compared. Improving the spatial resolution of an image sequence critically depends upon the accuracy of the motion estimator. The problem is complicated by the fact that the motion field is prone to significant errors since the original high-resolution images are not available. Improved motion estimates may be obtained by using a more robust and accurate motion estimator, such as a pel-recursive scheme instead of block matching. In processing color image sequences, there is the added advantage of having more flexibility in how the final motion estimates are obtained, and further improvement in the accuracy of the motion field is therefore possible. This is because there are three different intensity fields (channels) conveying the same motion information. In this paper, the choice of which motion estimator to use versus how the final estimates are obtained is weighed to see which issue is more critical in improving the estimated high-resolution sequences. Toward this end, an iterative algorithm is proposed, and two sets of experiments are presented. First, several different experiments using the same motion estimator but three different data fusion approaches to merge the individual motion fields were performed. Second, estimated high-resolution images using the block matching estimator were compared to those obtained by employing a pel recursive scheme. Experiments were performed on a real color image sequence, and performance was measured by the peak signal to noise ratio (PSNR). Index Terms---High-resolution video, motion compensation, resolution enhancement. I.

The notion of a multiresolution support is introduced. This is a sequence of boolean images, related to significant pixels at each of a number of resolution levels. The multiresolution support is then used for noise suppression, in the context of image filtering, or iterative image restoration. Algorithmic details, and a range of practical examples, illustrate this approach.

In applications that demand highly detailed images, it is often not feasible or sometimes possible to acquire images of such high resolution by just using hardware (high precision optics and charge coupled devices (CCDs)). Instead, image processing approaches can be used to construct a high resolution image from multiple, degraded, low resolution images. It is assumed that the low resolution images have been subsampled (thus introducing aliasing) and displaced by sub-pixel shifts with respect to a reference frame. Therefore, the problem can be divided into three sub-problems: registration (estimating the shifts), restoration, and interpolation. None of the methods which appeared in the literature solve the registration and restoration sub-problems simultaneously. This is sub-optimal, since the registration and restoration steps are inter-dependent. Based on previous restoration and identification work using the Expectation-Maximization (EM) algorithm, the proposed approach estimates th...

In recent years, wavelet-based algorithms have been successful in different signal processing tasks. The wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients. Discontinuities often carry relevant signal information, and therefore, they represent a critical part to analyze. In this paper, we study the dependency across scales of the wavelet coefficients generated by discontinuities. We start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual (see Theorem 1 in Section II). We then introduce the notion of footprints, which are scale space vectors that model discontinuities in piecewise polynomial signals exactly. We show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints. This also leads to efficient approximation of piecewise smooth functions. Finally, we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising, compression, and (nonblind) deconvolution. In the case of compression, we also prove that at high rates, footprint-based algorithms attain optimal performance (see Theorem 3 in Section V).

Abstract—Despite its practical importance in image processing and computer vision, blind blur identification and blind image restoration have so far been addressed under restrictive assumptions such as all-pole stationary image models blurred by zero- or minimum-phase point-spread functions. Relying upon diversity (availability of a sufficient number of multiple blurred images), we develop blind FIR blur identification and order determination schemes. Apart from a minimal persistence of excitation condition (also present with nonblind setups), the inaccessible input image is allowed to be deterministic or random and of unknown color or distribution. With the blurs satisfying a certain co-primeness condition in addition, we establish existence and uniqueness results which guarantee that single-input/multiple-output FIR blurred images can be restored blindly, though perfectly in the absence of noise, using linear FIR filters. Results of simulations employing the blind order determination, blind blur identification, and blind image restoration algorithms are presented. When the SNR is high, direct image restoration is found to yield better results than indirect image restoration which employs the estimated blurs. In low SNR, indirect image restoration performs well while the direct restoration results vary with the delay but improve with larger equalizer orders. Index Terms—Blind blur estimation, blind image restoration, multichannel image restoration. I.

this paper image restoration applications, where multiple distorted versions of the same original image are available, are considered. A general adaptive restoration algorithm is derived on the basis of a set theoretic regularization technique. The adaptivity of the algorithm is introduced in two ways: (a) by a constraint ,operator which incorporates properties of the response of the hu- man visual system into the restoration process and (b) by a weight matrix which assigns greater importance for the deconvolution process to areas of high spatial activity than to areas of low spatial activity. Different degrees of trust are assigned to the various distorted images depending on the amounts of noise. The proposed algorithm is general and can be used for any type of linear distortion and constraint operators. It can also be used to restore signals other than images. Experimental results obtained by an iterative implementation of the proposed algorithms are pre- sented. c 1990 Academic Press, Inc

In this paper, learning-based algorithms for image restoration and blind image restoration are proposed. Such algorithms deviate from the traditional approaches in this area, by utilizing priors that are learned from similar images. Original images and their degraded versions by the known degradation operator (restoration problem) are utilized for designing the VQ codebooks. The codevectors are designed using the blurred images. For each such vector, the high frequency information obtained from the original images is also available. During restoration, the high frequency information of a given degraded image is estimated from its low frequency information based on the codebooks. For the blind restoration problem, a number of codebooks are designed corresponding to various versions of the blurring function. Given a noisy and blurred image, one of the codebooks is chosen based on a similarity measure, therefore providing the identification of the blur. To make the restoration process computationally efficient, the Principal Component Analysis (PCA) and VQ-Nearest Neighborhood approaches are utilized. Simulation results are presented to demonstrate the effectiveness of the proposed algorithms.