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.     

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.     

Maximum likelihood estimation with side information of a 1-Ddiscrete layered medium from its noisy impulse reflection response

We consider the problem of computing the maximum likelihood estimates of the reflection coefficients of a discrete 1-D layered medium from noisy observations of its impulse reflection response. We have side information in that a known subset of the reflection coefficients are known to be zero; this knowledge could come from either a priori knowledge of a homogeneous subregion inside the scattering medium or from a thresholding operation in which noisy reconstructed reflection coefficients with absolute values below a threshold are known to be zero. Our procedure converges in one or two iterations, each of which requires only setting up and solving a small system of linear equations and running the Levinson algorithm. Numerical examples are provided that demonstrate not only the operation of the algorithm but also that the side information improves the reconstruction of unconstrained reflection coefficients as well as constrained ones due to the nonlinearity of the inverse scattering problem     

Levinson-like and Schur-like fast algorithms for solvingblock-slanted Toeplitz systems of equations arising in wavelet-basedsolution of integral equations

The Wiener-Hopf integral equation of linear least-squares estimation of a wide-sense stationary random process and the Krein integral equation of one-dimensional (1-D) inverse scattering are Fredholm equations with symmetric Toeplitz kernels. They are transformed using a wavelet-based Galerkin method into a symmetric “block-slanted Toeplitz (BST)” system of equations. Levinson-like and Schur-like fast algorithms are developed for solving the symmetric BST system of equations. The significance of these algorithms is as follows. If the kernel of the integral equation is not a Calderon-Zygmund operator, the wavelet transform may not sparsify it. The kernel of the Krein and Wiener-Hopf integral equations does not, in general, satisfy the Calderon-Zygmund conditions. As a result, application of the wavelet transform to the integral equation does not yield a sparse system matrix. There is, therefore, a need for fast algorithms that directly exploit the (symmetric block-slanted Toeplitz) structure of the system matrix and do not rely on sparsity. The first such O(n²) algorithms, viz., a Levinson-like algorithm and a Schur (1917) like algorithm, are presented. These algorithms are also applied to the factorization of the BST system matrix. The Levinson-like algorithm also yields a test for positive definiteness of the BST system matrix. The results obtained are directly applicable to the problem of constrained deconvolution of a nonstationary signal, where the locations of the smooth regions of the signal being deconvolved are known a priori     

A multidimensional nonlinear edge-preserving filter for magneticresonance 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.     

A Kalman filtering approach to stochastic global andregion-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.     

On geometric sequences of reflection coefficients and Gaussianautocorrelations

The author shows that by interpreting the lattice filter as a discrete transmission line, the problem of determining the reflection coefficients associated with a Gaussian autocorrelation can be solved easily using the Schur algorithm. These reflection coefficients have been shown to be in geometric progression; it is claimed that this has been done in a much simpler and more enlightening manner than in the presentation of G. Jacoriti and G. Scarano (see ibid., vol.75, no.7, p.960-961, 1987). The geometric progression of reflection coefficients leads to a stationarity property of the discrete transmission line, which accounts for the striking simplicity of the expressions for the waves traveling in the line     

Time-frequency distribution inversion of the Radon transform [imagereconstruction]

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.     

Relaxation procedure for phase retrieval of nonnegative signals

The paper presents an iterative procedure called the relaxation of autocorrelation equations (RAE) for solving the phase retrieval problem for nonnegative signals. First, the phase retrieval problem is formulated in the spatial domain as a set of polynomial equations with autocorrelations as known data and signal values as unknowns. Then, the RAE procedure solves these equations by recognizing one unknown at a time. While other unknowns are held constant at previously estimated values, a single unknown is varied inside the nonnegative region to globally minimize the sum of squared residuals of the equations with respect to the unknown. In every iteration, this procedure is repeated for each signal value. Since the sum of squared residuals is nonincreasing, the algorithm will either converge to a solution or stagnate; ways to overcome stagnation are suggested. The key feature of the RAE procedure is that unlike iterative transform algorithms, it allows direct control over bounding values of the signal at all times. Several numerical examples illustrate the RAE procedure