Ken D. Sauer

A generalized Gaussian image model for edge-preserving MAP estimation

The authors present a Markov random field model which allows realistic edge modeling while providing stable maximum a posterior (MAP) solutions. The model, referred 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 satisfies 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 minimum of the a posteriori log-likelihood function. The GGMRF is demonstrated to be useful for image reconstruction in low-dosage transmission tomography.

A three-dimensional statistical approach to improved image quality for multislice helical CT.

Multislice helical computed tomography scanning offers the advantages of faster acquisition and wide organ coverage for routine clinical diagnostic purposes. However, image reconstruction is faced with the challenges of three-dimensional cone-beam geometry, data completeness issues, and low dosage. Of all available reconstruction methods, statistical iterative reconstruction (IR) techniques appear particularly promising since they provide the flexibility of accurate physical noise modeling and geometric system description. In this paper, we present the application of Bayesian iterative algorithms to real 3D multislice helical data to demonstrate significant image quality improvement over conventional techniques. We also introduce a novel prior distribution designed to provide flexibility in its parameters to fine-tune image quality. Specifically, enhanced image resolution and lower noise have been achieved, concurrently with the reduction of helical cone-beam artifacts, as demonstrated by phantom studies. Clinical results also illustrate the capabilities of the algorithm on real patient data. Although computational load remains a significant challenge for practical development, superior image quality combined with advancements in computing technology make IR techniques a legitimate candidate for future clinical applications.

A local update strategy for iterative reconstruction from projections

A method for Bayesian reconstruction which relies on updates of single pixel values, rather than the entire image, at each iteration is presented. The technique is similar to Gauss-Seidel (GS) iteration for the solution of differential equations on finite grids. The computational cost per iteration of the GS approach is found to be approximately equal to that of gradient methods. For continuously valued images, GS is found to have significantly better convergence at modes representing high spatial frequencies. In addition, GS is well suited to segmentation when the image is constrained to be discretely valued. It is shown that Bayesian segmentation using GS iteration produces useful estimates at much lower signal-to-noise ratios than required for continuously valued reconstruction. The convergence properties of gradient ascent and GS for reconstruction from integral projections are analyzed, and simulations of both maximum-likelihood and maximum a posteriori cases are included. >

A unified approach to statistical tomography using coordinate descent optimization

Over the past years there has been considerable interest in statistically optimal reconstruction of cross-sectional images 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. 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 propose a novel method for computing the ICD updates, which we call ICD/Newton-Raphson. We show that ICD/Newton-Raphson requires approximately the same amount of computation per iteration as EM-based approaches, but the new method converges much more rapidly (in our experiments, typically five to ten iterations). Other advantages of the ICD/Newton-Raphson method are that it is easily applied to MAP estimation of transmission tomograms, and typical convex constraints, such as positivity, are easily incorporated.

Fast Model-Based X-Ray CT Reconstruction Using Spatially Nonhomogeneous ICD Optimization

Recent applications of model-based iterative reconstruction (MBIR) algorithms to multislice helical CT reconstructions have shown that MBIR can greatly improve image quality by increasing resolution as well as reducing noise and some artifacts. However, high computational cost and long reconstruction times remain as a barrier to the use of MBIR in practical applications. Among the various iterative methods that have been studied for MBIR, iterative coordinate descent (ICD) has been found to have relatively low overall computational requirements due to its fast convergence. This paper presents a fast model-based iterative reconstruction algorithm using spatially nonhomogeneous ICD (NH-ICD) optimization. The NH-ICD algorithm speeds up convergence by focusing computation where it is most needed. The NH-ICD algorithm has a mechanism that adaptively selects voxels for update. First, a voxel selection criterion VSC determines the voxels in greatest need of update. Then a voxel selection algorithm VSA selects the order of successive voxel updates based upon the need for repeated updates of some locations, while retaining characteristics for global convergence. In order to speed up each voxel update, we also propose a fast 1-D optimization algorithm that uses a quadratic substitute function to upper bound the local 1-D objective function, so that a closed form solution can be obtained rather than using a computationally expensive line search algorithm. We examine the performance of the proposed algorithm using several clinical data sets of various anatomy. The experimental results show that the proposed method accelerates the reconstructions by roughly a factor of three on average for typical 3-D multislice geometries.

ML parameter estimation for Markov random fields with applications to Bayesian tomography

Markov random fields (MRFs) 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 hyper parameters) is difficult in practice for two reasons: i) direct parameter estimation for MRFs is known to be mathematically and numerically challenging; ii)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 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, sigma, 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.The second section of the paper presents a fast algorithm for computing ML parameter estimates when the true image is unavailable. To do this, we use the expectation maximization(EM) algorithm. We develop a fast simulation method to replace the E-step, and a method to improve parameter estimates when the simulations are terminated prior to convergence.Experimental results indicate that our fast algorithms substantially reduce computation and result in good scale estimates for real tomographic data sets.

Parallel computation of sequential pixel updates in statistical tomographic reconstruction

While Bayesian methods can significantly improve the quality of tomographic reconstructions, they require the solution of large iterative optimization problems. Recent results indicate that the convergence of these optimization problems can be improved by using sequential pixel updates, or Gauss-Seidel iterations. However, Gauss-Seidel iterations may be perceived as less useful when parallel computing architectures are use. We show that for degrees of parallelism of typical practical interest, the Gauss-Seidel iterations updates may be computed in parallel with little loss in convergence speed. In this case, the theoretical speed up of parallel implementations is nearly linear with the number of processors.

Efficient block motion estimation using integral projections

Several efficient techniques have previously been proposed to reduce the computational burden of block matching for motion estimation in video coding. The goal is efficient motion estimation with minimal error in the motion-compensated predicted image. We present a block motion estimation scheme which is based on matching of integral projections of motion blocks with those of the search area in the previous frame. Like many other techniques, ours operates in a sequence of decreasing search radii, but it performs an exhaustive search at each level of the hierarchy. The projection method is much less computationally costly than block matching and has a prediction accuracy of competitive quality with both full block matching and other efficient techniques. Our algorithm also takes advantage of the similarity of motion vectors in adjacent blocks in typical imagery by subsampling the motion vector field. It has the added advantage of allowing parallel computation of vertical and horizontal displacements.

Nonlinear prediction methods for estimation of clique weighting parameters in non-Gaussian image models

NonGaussian Markov image models are effective in the preservation of edge detail in Bayesian formulations of restoration and reconstruction problems. Included in these models are coefficients quantifying the statistical links among pixels in local cliques, which are typically assumed to have an inverse dependence on distance among the corresponding neighboring pixels. Estimation of these coefficients is a nontrivial task for Non Gaussian models. We present rules for coefficient estimation for edge- preserving models which are particularly effective for edge preservation and noise suppression, using a predictive technique analogous to estimation of the weights of optimal weighted median filters.

Bayesian estimation of transmission tomograms using segmentation based optimization

The authors present a method for nondifferentiable optimization in maximum a posteriori estimation of computed transmission tomograms. This problem arises in the application of a Markov random field image model with absolute value potential functions. Even though the required optimization is on a convex function, local optimization methods, which iteratively update pixel values, become trapped on the nondifferentiable edges of the function. An algorithm which circumvents this problem by updating connected groups of pixels formed in an intermediate segmentation step is proposed. Experimental results showed that this approach substantially increased the rate of convergence and the quality of the reconstruction. >