Alfred K. Louis

Mathematical problems of computerized tomography

The data measured in computerized tomography; e.g., the X-ray attenuation in X-ray tomography or the resonance phenomena in nuclear magnetic resonance tomography, have to be processed to produce the pictures on which the diagnostic evaluation of the physician is based. This process consists of the solution of the following mathematical problem. The data depend on the searched-for distribution and this dependence can be described as an integral transform. To produce the final picture amounts to the inversion of the integral transform. This paper is concerned with the description of the integral transforms modeling the different techniques in computerized tomography. Among other things, the following questions are treated. Which numerical problems do we have to encounter in inverting the transforms; e.g., what accuracy in the reconstruction can we expect in dependence on the accuracy of the data. To what extent is a distribution determined by a finite number of measurements. Is it possible to recover the distribution reliably if the data are incomplete.

Nonlinear iterative methods for linear ill-posed problems in Banach spaces

We introduce and discuss nonlinear iterative methods to recover the minimum-norm solution of the operator equation Ax = y in Banach spaces X, Y, where A is a continuous linear operator from X to Y. The methods are nonlinear due to the use of duality mappings which reflect the geometrical aspects of the underlying spaces. The space X is required to be smooth and uniformly convex, whereas Y can be an arbitrary Banach space. The case of exact as well as approximate and disturbed data and operator are taken into consideration and we prove the strong convergence of the sequence of the iterates.

A mollifier method for linear operator equations of the first kind

An inversion method for the solution of ill-posed linear problems is presented. It is based on the idea of computing a mollified version of the searched-for solution and the approximate inverse operator is computed with exactly given quantities. The method is compared with known methods such as the Tikhonov-Phillips and Backus-Gilbert methods. Numerical tests verify the advantages, which are: no additional or artificial discretisation of the solution is needed, locally varying point-spread functions are easily realised, a simple change of the regularisation parameter with regard of a posteriori parameter strategies is implemented and a straightforward interpretation of the regularised solution is possible. When the approximation inversion operator is computed the solution can be computed by parallel processing.

Local Tomographic Methods in Sonar

Tomographic methods are described that will reconstruct object boundaries in shallow water using sonar data. The basic ideas involve microlocal analysis, and they are valid under weak assumptions even if the data do not correspond exactly to our model.

Incomplete data problems in X-ray computerized tomography

SummaryIn the present paper we study truncated projections for the fanbeam geometry in computerized tomography. First we derive consistency conditions for the divergent beam transform. Then we study a singular value decomposition for the case where only the interior rays in the fan are provided, as for example in region-of-interest tomography. We show that the high angular frequency components of the searched-for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition.

Incomplete data problems in X-ray computerized tomography. I. Singular value decomposition of the limited angle transform

SummaryThe reconstruction of an object from its x-ray scans is achieved by the inverse Radon transform of the measured data. For fast algorithms and stable inversion the directions of the x rays have to be equally distributed. In the present paper we study the intrinsic problems arising when the directions are restricted to a limited range by computing the singular value decomposition of the Radon transform for the limited angle problem. Stability considerations show that parts of the spectrum cannot be reconstructed and the irrecoverable functions are characterized.

Medical imaging: state of the art and future development

The aim of medical imaging is to provide in an non - invasive way morphological information about a human patient. The information is obtained by performing an ” experiment ” where the interaction of a source of radiation anf the tissue under consideration is measured. From the measured data the desired information has to be computed, hence we face an inverse problem. It is always ill - posed in the sense that small errors in the data can be amplified to large changes in the reconstruction. For developing efficient and stable software we have to study the mathematical model; i. e., the description of the experiment based on physical and engeneering knowledge. In optimal situations it is possible to derive” inversion formulas” which relate in a constructive way the data to the searched - for information. Reconstruction algorithms can be found by discretisizing these formulas. But of course we have to perform a stability analysis in order to design the software such that the influence of the data noise is reduced as much as possible. If such inversion formulas are unknown or cannot be discretisized in an accurate way direct discretization and iterative methods are used for the computation.

A unified approach to regularization methods for linear ill-posed problems

The aim of this paper is to study regularization methods for linear ill-posed problems. Linear methods are Tikhonov-Phillips methods, iterative methods, truncated singular value decomposition, Backus-Gilbert-type methods and approximate inverse, for example. The first three are generally studied as filter methods where a special filter for the singular value decomposition can be computed. In the other methods mentioned the regularization is achieved by either smoothing the data or the solution. More general is the approximate inverse introduced by Louis (1996 Inverse Problems 12 175-90). Here we show that all these methods can be viewed either as smoothing the pseudo-inverse or equivalently as first smoothing the data and then applying the pseudo-inverse. The smoothing of the data or of the pseudo-inverse has to be at least of the order of the smoothing of the operator in the problem to be solved. Conditions for the order-optimality of the methods are given.

Contour reconstruction in 3-D X-ray CT

The authors derives an algorithm for reconstructing contours of an object from 3D cone beam X-ray data. By reducing the amount of the searched-for information, contours, or density jumps instead of the densities themselves, the authors are able to develop fast algorithms for data incomplete with respect to both the movement of the X-ray source and the detector reading. The method is related to local or Lambda tomography. Numerical simulations show the efficiency of the algorithm.

An iterative regularization method for the solution of the split feasibility problem in Banach spaces

The split feasibility problem (SFP) consists of finding a common point in the intersection of finitely many convex sets, where some of the sets arise by imposing convex constraints in the range of linear operators. We are concerned with its solution in Banach spaces. To this end we generalize the CQ algorithm of Byrne with Bregman and metric projections to obtain an iterative solution method. In case the sets projected onto are contaminated with noise we show that a discrepancy principle renders this algorithm a regularization method. We measure the distance between convex sets by local versions of the Hausdorff distance, which in contrast to the standard Hausdorff distance allow us to measure the distance between unbounded sets. Hereby we prove a uniform continuity result for both kind of projections. The performance of the algorithm is demonstrated with some numerical experiments.