Absfrucf- We present a Markov random field model which allows realistic edge modeling while providing stable maximum a posteriori MAP solutions. The proposed model, which we refer to as a generalized Gaussian Markov random field (GGMRF), is named for its similarity to the generalized Gaussian distribution used in robust detection and estimation. The model satisifies several desirable analytical and computational properties for MAP estimation, including continuous dependence of the estimate on the data, invariance of the character of solutions to scaling of data, and a solution which lies at the unique global mini-mum of the U posteriori log-likeihood function. The GGMRF is demonstrated to be useful for image reconstruction in low-dosage transmission tomography. I.

This paper examines the relationship between wavelet-based image processing algorithms and variational problems. Algorithms are derived as exact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following problem: given an image F defined on a square I, minimize over all g in the Besov space B 1 1 (L1 (I)) the functional #F - g# 2 L 2 (I) + ##g# B 1 1 (L 1 (I)) .Weusethetheoryof nonlinear wavelet image compression in L2 (I) to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i.i.d., mean zero, Gaussian noise. A new signal-tonoise ratio, which we claim more accurately reflects the visual perception of noise in images, arises in this derivation. We present extensive computations that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image F:thelarge...

Abstract 1 Over the past ten years there has been considerable interest in statistically optimal reconstruction of image cross-sections from tomographic data. In particular, a variety of such algorithms have been proposed for maximum a posteriori (MAP) reconstruction from emission tomographic data. While MAP estimation requires the solution of an optimization problem, most existing reconstruction algorithms take an indirect approach based on the expectation maximization (EM) algorithm. In this paper we propose a new approach to statistically optimal image reconstruction based on direct optimization of the MAP criterion. The key to this direct optimization approach is greedy pixel-wise computations known as iterative coordinate decent (ICD). We show that the ICD iterations require approximately the same amount of computation per iteration as EM based approaches, but the new method converges much more rapidly (in our experiments typically 5 iterations). Other advantages of the ICD method are that it is easily applied to MAP estimation of transmission tomograms, and typical convex constraints, such as positivity, are simply incorporated.

Abstract 1 Markov random fields (MRF) have been widely used to model images in Bayesian frameworks for image reconstruction and restoration. Typically, these MRF models have parameters that allow the prior model to be adjusted for best performance. However, optimal estimation of these parameters (sometimes referred to as hyperparameters) is difficult in practice for two reasons: 1) Direct parameter estimation for MRF’s is known to be mathematically and numerically challenging. 2) Parameters can not be directly estimated because the true image cross-section is unavailable. In this paper, we propose a computationally efficient scheme to address both these difficulties for a general class of MRF models, and we derive specific methods of parameter estimation for the MRF model known as a generalized Gaussian MRF (GGMRF). The first section of the paper derives methods of direct estimation of scale and shape parameters for a general continuously valued MRF. For the GGMRF case, we show that the ML estimate of the scale parameter, σ, has a simple closed form solution, and we present an efficient scheme for computing the ML estimate of the shape parameter, p, by an off-line numerical computation of the dependence of the partition function on p.

In recent years many authors have introduced certain nonlinear algorithms for data compression, noise removal, and image reconstruction. These methods include...

We represent the standard ramp filter operator of the filtered back-projection (FBP) reconstruction in different bases composed of Haar and Daubechies compactly supported wavelets. The resulting multiscale representation of the ramp filter matrix operator is approximately diagonal. The accuracy of this diagonal approximation becomes better as wavelets with larger number of vanishing moments are used. This wavelet-based representation enables us to formulate a multiscale tomographic reconstruction technique wherein the object is reconstructed at multiple scales or resolutions. A complete reconstruction is obtained by combining the reconstructions at different scales. Our multiscale reconstruction technique has the same computational complexity as the FBP reconstruction method. It differs from other multiscale reconstruction techniques in that 1) the object is defined through a multiscale transformation of the projection domain, and 2) we explicitly account for noise in the projection da...

A fast and exact algorithm for total variation minimization Un algorithme rapide et exact de la minimisation de la variation totale

.................................................................viii CHAPTER 1 INTRODUCTION ..............................................1 CHAPTER 2 PRELIMINARIES ..............................................5 2.1 Definitions and Notations ..............................................5 2.2 Theorems and Inequalities..............................................6 CHAPTER 3 WAVELETS AND BESOV SPACES ............................9 3.1 Biorthogonal Wavelets .................................................9 3.2 Besov Spaces .........................................................15 CHAPTER 4 EMBEDDING, INTERPOLATION, AND DUALITY BETWEEN BESOV SPACES .............................................................18 4.1 Embedding ...........................................................18 4.2 Interpolation..........................................................21 4.3 Duality ...............................................................23 CHAPTER 5 LINEAR HOMOGENEOUS EQUATION...

Statistical methods of discrete tomographic reconstruction pose new problems both in stochastic modeling to define an optimal reconstruction, and in optimization to find that reconstruction. Multiscale models have succeeded in improving representation of structure of varying scale in imagery, a chronic problem for common Markov random fields. This chapter shows that associated multiscale methods of optimization also avoid local minima of the log a posteriori probability better than single-resolution techniques. These methods are applied here to both segmentation/reconstruction of the unknown cross-sections, and estimation of unknown parameters represented by the discrete levels. 1.1 Introduction The reconstruction of images from projections is important in a variety of problems including tasks in medical imaging and non-destructive testing. Perhaps, the reconstruction technique most frequently used in commercial applications is convolution backprojection (CBP) [1]. While CBP...

Abstract. We bound the difference between the solution to the continuous Rudin–Osher–Fatemi image smoothing model and the solutions to various finite-difference approximations to this model. These bounds apply to “typical ” images, i.e., images with edges or with fractal structure. These are the first bounds on the error in numerical methods for ROF smoothing. Key words. Total variation, bounded variation, variational problems, finite-difference methods, image processing. AMS subject classifications. 65N06, 65N12, 94A08 1. Introduction. Image