Arnold Lent

Iterative reconstruction algorithms

Abstract In this paper a survey of recent results on iterative reconstruction algorithms is given. These results, many of which have not yet appeared elsewhere, are applicable to a very general formulation of the reconstruction problem based on the series expansion approach. A set of optimization criteria and a number of iterative reconstruction algorithms are stated, together with theorems on the convergence of the algorithms to optimum images. The efficacy of the algorithms is compared to that of the convolution method. In particular, the falseness of the claim that ART and the backprojection method are the same is demonstrated.

ART: Mathematics and applications

Abstract Some properties of the algebraic reconstruction techniques (ART) for reconstructing objects from their projections (e.g. electron micrographs) are discussed. Some generalizations of previously published ART algorithms are given. In particular, ART is extended to handle weighted projection data. An early conjecture about ART is proved for one of the algorithms: it converges to the most uniform solution of the constraint equations provided by the projection data. Other convergence properties of the ART algorithms are discussed and proved. Some new ART algorithms are described. These are believed to converge to optimal reconstructions consistent with the projection data. The importance of choosing the correct ray widths in case of real projection data is demonstrated, and a method for calculating correct ray widths is given. A method is proposed for estimating the optimal number of iterations in a reconstruction. The performance of ART on real data is demonstrated both in the absence of and in the presence of noise.

Iterative algorithms for large partitioned linear systems, with applications to image reconstruction

Abstract We present a unifying framework for a wide class of iterative methods in numerical linear algebra. In particular, the class of algorithms contains Kaczmarzs and Richardsons methods for the regularized weighted least squares problem with weighted norm. The convergence theory for this class of algorithms yields as corollaries the usual convergence conditions for Kaczmarzs and Richardsons methods. The algorithms in the class may be characterized as being group-iterative, and incorporate relaxation matrices, as opposed to a single relaxation parameter. We show that some well-known iterative methods of image reconstruction fall into the class of algorithms under consideration, and are thus covered by the convergence theory. We also describe a novel application to truly three-dimensional image reconstruction.

Relaxation methods for image reconstruction

The problem of recovering an image (a function of two variables) from experimentally available integrals of its grayness over thin strips is of great importance in a large number of scientific areas. An important version of the problem in medicine is that of obtaining the exact density distribution within the human body from X-ray projections. One approach that has been taken to solve this problem consists of translating the available information into a system of linear inequalities. The size and the sparsity of the resulting system (typically, 25,000 inequalities with fewer than 1 percent of the coefficients nonzero) makes methods using successive relaxations computationally attractive, as compared to other ways of solving systems of inequalities. In this paper, it is shown that, for a consistent system of linear inequalities, any sequence of relaxation parameters lying strictly between 0 and 2 generates a sequence of vectors which converges to a solution. Under the same assumptions, for a system of linear equations, the relaxation method converges to the minimum norm solution. Previously proposed techniques are shown to be special cases of our procedure with different choices of relaxation parameters. The practical consequences for image reconstruction of the choice of the relaxation parameters are discussed.

Measurement of spatial distribution of refractive index in tissues by ultrasonic computer assisted tomography

Abstract Computerized tomography is used to calculate two independent images representing distributions of refractive index ( n ) and acoustic attenuation (α) within 1–3 mm thick cross-sections through excised organs, including canine hearts and human breasts. Values of n and α are calculated from profiles of propagation times and amplitudes, respectively, of digitized acoustic pulses obtained by rectilinear transmission scans of the tissue at multiple angles of view. Images of the local speed of ultrasound show high values in regions of muscle, breast parenchyma, medullary carcinoma and connective tissue and show low values in regions of fat. Images of acoustic attenuation show high values in regions of connective tissue and borders of scirrhus carcinoma and low values in regions of fat.

Methods of least squares and SIRT in reconstruction.

Abstract In this paper we show that a particular version of the Simultaneous Iterative Reconstruction Technique (SIRT) proposed by Gilbert in 1972 strongly resembles the Richardson least-squares algorithm. By adopting the adjustable parameters of the general Richardson algorithm, we have been able to produce generalized SIRT algorithms with improved convergence. A particular generalization of the SIRT algorithm, GSIRT, has an adjustable parameter σ and the starting picture ρ0 as input. A value 1 2 for σ and a weighted back-projection for ρ0 produce a stable algorithm. We call the SIRT-like algorithms for the solution of the weighted leastsquares problems LSIRT and present two such algorithms, LSIRT1 and LSIRT2, which have definite computational advantages over SIRT and GSIRT. We have tested these methods on mathematically simulated phantoms and find that the new SIRT methods converge faster than Gilberts SIRT but are more sensitive to noise present in the data. However, the faster convergence rates allow termination before the noise contribution degrades the reconstructed image excessively.

The Dynamic Spatial Reconstructor A Computed Tomography System for High-Speed Simultaneous Scanning of Multiple Cross Sections of the Heart*

A new generation whole-body computed tomography system has been developed to provide accurate visualization and measurement of the vital functions of the heart, lungs, and circulation. This dynamic spatial reconstructor system (DSR) provides stop-action (01-sec), rapidly sequential (60-per-second), synchronous volume (240 simultaneous adjacent 1-mm-thick transaxial sections) reconstructions and display of the full anatomic extents of the internal and external surfaces of the heart throughout successive cardiac cycles, and will permit visualization of the three-dimensional vascular anatomy and circulatory functions in all regions of the body of patients with cardiovascular and other circulatory disabilities.

On the Bayesian approach to image reconstruction

A new iterative method is proposed for finding the optimal Bayesian estimate of an unknown image from its projection data (experimentally obtained integrals of its grayness over thin strips). Convergence of the method is proved and its performance is illustrated. The method compares favorably with previously proposed procedures.

A computer implementation of a bayesian analysis of image reconstruction

We attack the problem of recovering an image (a function of two variables) from experimentally available integrals of its grayness over thin strips. This problem is of great importance in a large number of scientific areas. An important version of the problem in medicine is that of obtaining the exact density distribution within the human body from X-ray projections. This paper proposes an algorithm for finding an optimal Bayesian estimate of an unknown image from the projection data, shows that the algorithm is geometrically convergent, reports on its computer implementation, and demonstrates its performance on a medical problem.

Iterative relaxation methods for image reconstruction

The problem of recovering an image (a function of two variables) from experimentally available integrals of its grayness over thin strips is of great importance in a large number of scientific areas. An important version of the problem in medicine is that of obtaining the exact density distribution within the human body from X-ray projections. One approach that has been taken to solve this problem consists of translating the available information into a system of linear inequalities. The size and the sparsity of the resulting system of inequalities (typically, 25,000 inequalities with less than 1% of the coefficients nonzero) makes methods using successive relaxations computationally attractive. A variety of such methods have been proposed with differing relaxation parameters.