Iterative methods for statistically-based reconstruction from projections are computationally costly relative to convolution backprojection, but allow useful image reconstruction from sparse and noisy data. We present a method for Bayesian reconstruction which relies on updates of single pixel values, rather than the entire image, at each iteration. 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. We demonstrate that Bayesian segmentation using GS iteration produces useful estimates at much lower signal-to-noise ratios than required for continuously valued reconstruction. This paper includes analysis of the convergence properties of gradient ascent and GS for reconstruction from integral projections, and simulations of both maximum-likelihood and maximum a posteriori cases.

The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformly-spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

Abstruct-The problem of reconstructing a multidimensional field from noisy, limited projection measurements is approached using an object-based stochastic field model. Objects within a cross section are characterized by a fiite-dimensional set of parameters, which are estimated directly from limited, noisy projection measurements using maximum likelihood estimation. In Part I, the computational structure and performance of the ML estimation procedure are investigated for the problem of locating a single object in a deterministic background; simulations are also presented. In Part 11, the issue of robustness to modeling errors is addressed. PART I PERFORMANCE ANALYSIS

This paper examines the problem of reconstructing a voxelized representation of 3D space from a series of images. An iterative algorithm is used to find the scene model which jointly explains all the observed images by determining which region of space is responsible for each of the observations. The current approach formulates the problem as one of optimization over estimates of these responsibilities. The process converges to a distribution of responsibility which accurately reflects the constraints provided by the observations, the positions and shape of both solid and transparent objects, and the uncertainty which remains. Reconstruction is robust, and gracefully represents regions of space in which there is little certainty about the exact structure due to limited, non-existent, or contradicting data. Rendered images of voxel spaces recovered from synthetic and real observation images are shown. To Appear: International Conference on Computer Vision, 1999 1 Introduction Given in...

Algebraic reconstruction methods, such as the algebraic reconstruction technique (ART) and the related simultaneous ART (SART), reconstruct a two--dimensional (2-D) or three--dimensional (3-D) object from its X-ray projections. The algebraic methods have, in certain scenarios, many advantages over the more popular Filtered Backprojection approaches and have also recently been shown to perform well for 3-D cone-beam reconstruction. However, so far the slow speed of these iterative methods have prohibited their routine use in clinical applications. In this paper, we address this shortcoming and investigate the utility of widely available 2-D texture mapping graphics hardware for the purpose of accelerating the 3-D algebraic reconstruction. We find that this hardware allows 3-D cone-beam reconstructions to be obtained at almost interactive speeds, with speed-ups of over 50 with respect to implementations that only use general-purpose CPUs. However, we also find that the reconstruction quality is rather sensitive to the resolution of the framebuffer, and to address this critical issue we propose a scheme that extends the precision of a given framebuffer by 4 bits, using the color channels. With this extension, a 12-bit framebuffer delivers useful reconstructions for 0.5% tissue contrast, while an 8-bit framebuffer requires 4%. Since graphics hardware generates an entire image for each volume projection, it is most appropriately used with an algebraic reconstruction method that performs volume correction at that granularity as well, such as SART or SIRT. We chose SART for its faster convergence properties. Index Terms---Algebraic reconstruction technique (ART), computed tomography, cone beam reconstruction, hardware acceleration, simultaneous algebraic reconstruction techni...

Abstract. We describe an optimization problem arising in reconstructing 3D medical images from Positron Emission Tomography (PET). A mathematical model of the problem, based on the Maximum Likelihood principle is posed as a problem of minimizing a convex function of several millions variables over the standard simplex. To solve a problem of these characteristics, we develop and implement a new algorithm, Ordered Subsets Mirror Descent, and demonstrate, theoretically and computationally, that it is well suited for solving the PET reconstruction problem. Key words: positron emission tomography, maximum likelihood, image reconstruction, convex optimization, mirror descent. 1

In this dissertation we explore the use of wavelets in certain linear inverse problems with discrete, noisy data. We observe discrete samples of a process y(u) = (Kf)(u)+ z(u), where K is a linear operator, z is a noise process, and f is a function we wish to recover from the data. In the problems that we consider, the inverse of K, K \Gamma1 , either does not exist or is poorly behaved. Such problems are termed ill-posed i.e., ones in which small changes in the data may lead to large changes in the recovered version of f . Our methods are most effective for problems where the operator K is homogeneous with respect to dilations, such as integration, fractional integration, convolution, and the Radon transform. The theoretical framework in which we work is that of Donoho's (1992) WaveletVaguelette Decomposition (WVD). The WVD uses wavelets and vaguelettes (almost wavelets) to decompose the operator K. Although this formally resembles the Singular Value Decomposition (SVD), the use o...

Fourier-based reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourier-based reprojection methods. We apply a min-max interpolation method for the nonuniform fast Fourier transform (NUFFT) to minimize the interpolation errors. Numerical results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic reprojection and backprojection than conventional interpolation methods.

Emission computed tomography (ECT) is a technology for medical imaging whose importance is increasing rapidly. There is a growing appreciation for the value of the functional (as opposed to anatomical) information that is provided by ECT and there are significant advancements taking place, both in the instrumentation for data collection, and in the computer methods for generating images from the measured data. These computer methods are designed to solve the inverse problem known as “image reconstruction from projections.” This paper uses the various models of the data collection process as the framework for presenting an overview of the wide variety of methods that have been developed for image reconstruction in the major subfields of ECT, which are positron emission tomography (PET) and single-photon emission computed tomography (SPECT). The overall sequence of the major sections in the paper, and the presentation within each major section, both proceed from the more realistic and general models to those that are idealized and application specific. For most of the topics, the description proceeds from the three-dimensional case to the two-dimensional case. The paper presents a broad overview of algorithms for PET and SPECT, giving references to the literature where these algorithms and their applications are described in more detail.

The sampled Radon transform of a 2D function can be represented as a continuous linear map A : L 2(\Omega\Gamma ! R N , where (Au) j = hu; / j i and / j is the characteristic function of a strip through \Omega approximating the set of line integrals in the sample. The image reconstruction problem is: given a vector b 2 R N , find an image (or density function) u(x; y) such that Au = b. In general there are infinitely many solutions; we seek the solution with minimal 2-norm, which leads to a matrix equation Bw = b, where B is a square dense matrix with several convenient properties. We analyze the use of Gauss-Seidel iteration applied to the problem, observing that while the iteration formally converges, there exists a near null space into which the error vectors migrate, after which the iteration stalls. The null space and near null space of B are characterized in order to develop a multilevel scheme. Based on the principles of the Multilevel Projection Method (PML), this scheme l...