A Layer Stripping Fast Algorithm for the Two-Dimensional Direct Current Inverse Resistivity Problem

The inverse problem of reconstructing the resistivity of the earth, varying both laterally and with depth, from direct current measurements is considered. The problem is formulated as a multidimensional inverse scattering problem and solved using a layer stripping algorithm. This algorithm recursively reconstructs the resistivity and electrical potential on horizontal planes of increasing depth by downward continuation. This is the first exact solution to the inverse resistivity problem for resistivity varying laterally as well as with depth. The algorithm is an extension of an algorithm proposed by Levy for resistivity varying in one dimension.     

An asymmetric discrete-time approach for the design and analysis ofperiodic waveguide gratings

A discrete-time approach is introduced for the analysis of periodic waveguide gratings with gain (or loss) extending concepts developed for transfer matrix and Gel'fand-Levitan-Marchenko (GLM) inverse scattering techniques. The periodic waveguide grating with gain (or loss) is modeled as a lossy layered dielectric that allows for a digital signal processing (DSP) formulation of the forward and inverse scattering problem. It is shown that the DSP forward scattering formulation as an asymmetric two-component wave system is equivalent to the impedance matching matrix method. A numerical example is presented to emphasize this result. The DSP formulation is an exact discrete design, not just an approximation to a continuous design, and includes all multiple reflections, transmission scattering losses, and absorption effects. A comparison of the continuous GLM, discrete GLM, and discrete Krein inverse problem formulations for a medium with gain (or loss) is presented. The discrete lossy formulations generalize previous lossless results and are found from two different types of reflection data. Since slab gratings are discrete (not continuous) structures, the integral equations used to describe the continuous inverse problem are shown to become matrix equations. Thus, our result enables fast algorithms to be used to solve the inverse problem. A fast algorithm is presented allowing for the complete reconstruction of the grating parameters from its two-sided response in a recursive (slab by slab) fashion     

Two-dimensional linear prediction and spectral estimation on apolar raster

A zero-mean homogeneous random field is defined on a discrete polar raster. Given sample values inside a disk of finite radius, the authors wish to estimate the field's power spectral density using linear prediction. Issues arising include estimation of covariance lags and extendibility of a finite set of lag estimates into a positive semidefinite covariance extension (required for a meaningful spectral density). The authors give a generalized autocorrelation procedure that guarantees positive semidefinite covariance estimates. It first interpolates the data using Gaussians, computes its Radon transform, and applies familiar 1D techniques to each slice. Some numerical examples are provided to justify the validity of the proposed procedure. The authors also propose a correlation-matching covariance extension procedure that uses the Radon transform to extend a given set of covariance lags to the entire plane, when this is possible, and discuss circumstances for which this is impossible     

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.     

Partitioning algorithms for 1-D and 2-D discrete phase-retrievalproblems with disconnected support

The discrete phase-retrieval problem with disconnected support is to reconstruct a discrete-time signal whose support is the union of disjoint intervals from the magnitude of its discrete Fourier transform. We use the Good-Thomas (1958) FFT mapping to transform the one-dimensional (1-D) version of this problem into a two-dimensional (2-D) discrete phase-retrieval problem. We then solve the latter problem by partitioning it into a set of 1-D phase-retrieval problems. The discrete and modulated Radon transforms are used to formulate two coupled 1-D problems, the solution to which specifies solutions to the other 1-D problems. This effectively partitions the original disconnected support 1-D problem into smaller 1-D problems, which may be solved in parallel. Small amounts of noise in the data can be rejected by using more than two coupled 1-D problems, The 2-D phase-retrieval problem with disconnected support is also considered. Numerical examples are provided, including comparison with the hybrid I/O algorithm     

Split versions of the Levinson-like and Schur-like fast algorithmsfor solving block-slanted-Toeplitz systems of equations

In Joshi and Yagle (1998) the Fredholm equations of one-dimensional (1-D) inverse scattering and LLS estimation were transformed via the orthonormal wavelet transform into a series of symmetric “block-slanted-Toeplitz” (BST) systems of equations. Levinson-like and Schur-like fast algorithms were presented for solving the BST systems. Here, we present split versions of the Levinson-like and Schur-like fast algorithms. The significance of these split algorithms is as follows. Although the Levinson-like and Schur-like fast algorithms reduce the complexity of solving the BST systems from O(n³) to O(n²), there still exists an inherent redundancy in these algorithms in the case where the BST system matrices have centrosymmetric blocks. This situation arises when a symmetric wavelet basis function (like the Littlewood-Paley) is used in the problem transformation. This redundancy is exploited here to derive the split Levinson-like and split Schur-like fast algorithms. These split algorithms reduce the number of multiplications required at each iteration by a factor of two, as compared with the Levinson-like and Schur-like algorithms     

Two methods for Toeplitz-plus-Hankel approximation to a datacovariance matrix

Recently, fast algorithms have been developed for computing the optimal linear least squares prediction filters for nonstationary random processes (fields) whose covariances have (block) Toeplitz-Hankel form. If the covariance of the random process (field) must be estimated from the data, the following problem is presented: given a data covariance matrix, computer from the available data, find the Toeplitz-plus-Hankel matrix closest to this matrix in some sense. The authors give two procedures for computing the Toeplitz-plus-Hankel matrix that minimizes the Hilbert-Schmidt norm of the difference between the two matrices. The first approach projects the data covariance matrix onto the subspace of Toeplitz-plus-Hankel matrices, for which basis functions can be computed using a Gram-Schmidt orthonormalization. The second approach projects onto the subspace of symmetric Toeplitz plus skew-persymmetric Hankel matrices, resulting in a much simpler algorithm. The extension to block Toeplitz-plus-Hankel data covariance matrix approximation is also addressed     

A layer-recursive formula for the medium filter for volumeconductor problems in radially symmetric layered media [bioelectricityapplication]

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.     

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.     

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.