A Kalman filtering approach to stochastic global and region-of-interest tomography.

We define two forms of stochastic tomography. In global tomography, the goal is to reconstruct an object from noisy observations of all of its projections. In region-of-interest (ROI) tomography, the goal is to reconstruct a small portion of an object (an ROI) from noisy observations of its projections densely sampled in and near the ROI and sparsely sampled away from the ROI. We solve both problems by expanding the object and its projections in a circular harmonic (Fourier) series in the angular variable so that the Radon transform becomes Abel transforms of integer orders applied to the harmonics. The algorithm has three major components. First, we fit state-space models to each order of Abel transform and thus represent the Radon transform operation as a parallel bank of systems, each of which computes the appropriate Abel transform of a circular harmonic. A variable transformation here allows either the global or ROI problem to be solved. Second, the object harmonics are modeled as a Brownian branch. This is a two-point boundary value system, which is Markovianized into a form suitable for the Kalman filter. Finally, a parallel bank of Kalman smoothing filters independently estimates each circular harmonic from the noisy projection data. Numerical examples illustrate the proposed procedure.

Reconstruction from projections under time-frequency constraints.

Low-pass filtering computed tomography (CT) images to reduce noise may smooth or modify image features which are very important to the physician. Image features are often more easily identified and processed in the time-frequency plane. The authors use time-frequency distributions for spatially varying filtering of noisy CT images, constraining time-frequency representation coefficients of the projection data or of the reconstructed image to be zero in certain regions of the time-frequency plane. The authors consider two different applications: 1) filtering the projection data and then performing image reconstruction; and 2) filtering the reconstructed image directly. Criteria minimized, subject to constraints, may be either a deterministic minimum weighted perturbation of the given projection data or a stochastic minimum mean-square error in colored Gaussian noise. Results show improvement over processing the image with a linear spatially invariant filter.

A multidimensional nonlinear edge-preserving filter for magnetic resonance image restoration.

The paper presents a multidimensional nonlinear edge-preserving filter for restoration and enhancement of magnetic resonance images (MRI). The filter uses both interframe (parametric or temporal) and intraframe (spatial) information to filter the additive noise from an MRI scene sequence. It combines the approximate maximum likelihood (equivalently, least squares) estimate of the interframe pixels, using MRI signal models, with a trimmed spatial smoothing algorithm, using a Euclidean distance discriminator to preserve partial volume and edge information. (Partial volume information is generated from voxels containing a mixture of different tissues.) Since the filter's structure is parallel, its implementation on a parallel processing computer is straightforward. Details of the filter implementation for a sequence of four multiple spin-echo images is explained, and the effects of filter parameters (neighborhood size and threshold value) on the computation time and performance of the filter is discussed. The filter is applied to MRI simulation and brain studies, serving as a preprocessing procedure for the eigenimage filter. (The eigenimage filter generates a composite image in which a feature of interest is segmented from the surrounding interfering features.) It outperforms conventional pre and post-processing filters, including spatial smoothing, low-pass filtering with a Gaussian kernel, median filtering, and combined vector median with average filtering.

Region-of-interest tomography using exponential radial sampling.

The authors combine several ideas, including nonuniform sampling and circular harmonic expansions, into a new procedure for reconstructing a small region of interest (ROI) of an image from a set of its projections that are densely sampled in the ROI and coarsely sampled outside the ROI. Specifically, the radial sampling density of both the projections and the reconstructed image decreases exponentially with increasing distance from the ROI. The problem and data are reminiscent of the recently formulated local tomography problem; however, the authors' algorithm reconstructs the ROI of the image itself, not the filtered version of it obtained using local tomography. The new algorithm has the added advantages of speed (it can be implemented entirely using the FFT) and parallelizability (each image harmonic is computed independently). Numerical examples compare the new algorithm to filtered backprojection.

Optimization of MRI protocols and pulse sequence parameters for eigenimage filtering.

The eigenimage filter generates a composite image in which a desired feature is segmented from interfering features. The signal-to-noise ratio (SNR) of the eigenimage equals its contrast-to-noise ratio (CNR) and is directly proportional to the dissimilarity between the desired and interfering features. Since image gray levels are analytical functions of magnetic resonance imaging (MRI) parameters, it is possible to maximize this dissimilarity by optimizing these parameters. For optimization, the authors consider four MRI pulse sequences: multiple spin-echo (MSE); spin-echo (SE); inversion recovery (IR); and gradient-echo (GE). The authors use the mathematical expressions for MRI signals along with intrinsic tissue parameters to express the objective function (normalized SNR of the eigenimage) in terms of MRI parameters. The objective function along with a set of diagnostic or instrumental constraints define a multidimensional nonlinear constrained optimization problem, which the authors solve by the fixed point approach. The optimization technique is demonstrated through its application to phantom and brain images. The authors show that the optimal pulse sequence parameters for a sequence of four MSE and one IR images almost doubles the smallest normalized SNR of the brain eigenimages, as compared to the conventional brain protocol.

Acceleration and filtering in the generalized Landweber iteration using a variable shaping matrix.

The generalized Landweber iteration with a variable shaping matrix is used to solve the large linear system of equations arising in the image reconstruction problem of emission tomography. The method is based on the property that once a spatial frequency image component is almost recovered within in in the generalized Landweber iteration, this component will still stay within in during subsequent iterations with a different shaping matrix, as long as this shaping matrix satisfies the convergence criterion for the component. Two different shaping matrices are used: the first recovers low-frequency image components; and the second may be used either to accelerate the reconstruction of high-frequency image components, or to attenuate these components to filter the image. The variable shaping matrix gives results similar to truncated inverse filtering, but requires much less computation and memory, since it does not rely on the singular value decomposition.

Time-frequency distribution inversion of the Radon transform [image reconstruction].

In using filtered backprojection to compute the inverse Radon transform, the ramp filter amplifies noise. Spatially invariant noise filters reduce resolution. It is desirable to filter noise where projections have no local high-frequency components. Using the short-time Fourier transform, the authors apply a time-frequency mask filter that zeroes out projections where local signal energy is below a threshold. Results show improvement over reconstructions using spatially invariant smoothing filters.

A fast algorithm for backprojection with linear interpolation.

In the filtered backprojection procedure for image reconstruction from projections, backprojection dominates the computation time. A simple algorithm that reduces the number of multiplications in linear interpolation and backprojection stage by 50%, with a small increase in the number of additions, is proposed. The algorithm performs the interpolation and backprojection of four views together. Examples of implementation are given and extension to interpolation of more than four views is discussed.

A comparative analysis of several transformations for enhancement and segmentation of magnetic resonance image scene sequences.

The performance of the eigenimage filter is compared with those of several other filters as applied to magnetic resonance image (MRI) scene sequences for image enhancement and segmentation. Comparisons are made with principal component analysis, matched, modified-matched, maximum contrast, target point, ratio, log-ratio, and angle image filters. Signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), segmentation of a desired feature (SDF), and correction for partial volume averaging effects (CPV) are used as performance measures. For comparison, analytical expressions for SNRs and CNRs of filtered images are derived, and CPV by a linear filter is studied. Properties of filters are illustrated through their applications to simulated and acquired MRI sequences of a phantom study and a clinical case; advantages and weaknesses are discussed. The conclusion is that the eigenimage filter is the optimal linear filter that achieves SDF and CPV simultaneously.

Acceleration of Landweber-type algorithms by suppression of projection on the maximum singular vector.

A procedure that speeds up convergence during the initial stage (the first 100 forward and backward projections) of Landweber-type algorithms, for iterative image reconstruction for positron emission tomography (PET), which include the Landweber, generalized Landweber, and steepest descent algorithms, is discussed. The procedure first identifies the singular vector associated with the maximum singular value of the PET system matrix, and then suppresses projection of the data on this singular vector after a single Landweber iteration. It is shown that typical PET system matrices have a significant gap between their two largest singular values; hence, this suppression allows larger gains in subsequent iterations, speeding up convergence by roughly a factor of three.

A layer-recursive formula for the medium filter for volume conductor problems in radially symmetric layered media.

Simplified algorithms for the computation of the filter coefficients used in solutions of the forward and inverse volume conductor problems in a multilayered cylindrical geometry are derived. The new algorithms are layer-recursive, as opposed to previous algorithms which were specific for the structure studied. The new algorithms not only eliminate the need to derive algebraically cumbersome filter expressions, but also speed up their numerical evaluation.

Numerical study of multigrid implementations of some iterative image reconstruction algorithms.

The numerical behavior of multigrid implementations of the Landweber, generalized Landweber, ART, and MLEM iterative image reconstruction algorithms is investigated. Comparisons between these algorithms, and with their single-grid implementations, are made on two small-scale synthetic PET systems, for phantom objects exhibiting different characteristics, and on one full-scale synthetic system, for a Shepp-Logan phantom. The authors also show analytically the effects of noise and initial condition on the generalized Landweber iteration, and note how to choose the shaping operator to filter out noise in the data, or to enhance features of interest in the reconstructed image. Original contributions include (1) numerical studies of the convergence rates of single-grid and multigrid implementations of the Landweber, generalized Landweber, ART, and MLEM iterations and (2) effects of noise and initial condition on the generalized Landweber iteration, with procedures for filtering out noise or enhancing image features.