Frank Natterer

Inversion of the attenuated Radon transform

We derive an exact inversion formula for the attenuated Radon transform. The formula is closely related to Novikovs inversion formula, but our derivation is completely different. We also give an implementation of the inversion formula very similar to the filtered backprojection algorithm of x-ray tomography and present numerical results.

Error bounds for tikhonov regularization in hilbert scales

For an ill-posed problem Af = g we consider Tikhonov reguiariration: Minimize where is the norm in a Hilbert scale (Hs). Assuming that the norm is equivalent to the norm for some a > O and f e Hq for some q > O , we show that the accuracy of Tikhonov regularization is asymptotically best possible, provided that ω is chosen optimally and p ≥ (q - a)/2.

A propagation-backpropagation method for ultrasound tomography

Ultrasound tomography is modelled by the inverse problem of a 2D Helmholtz equation at fixed frequency with plane-wave irradiation. It is assumed that the field is measured outside the support of the unknown potential f for finitely many incident waves. Starting out from an initial guess f0 for f we propagate the measured field through the object f0 to yield a computed held whose difference to the measurements is in turn backpropagated. The backpropagated field is used to update f0. The propagation as well as the backpropagation are done by a finite difference marching scheme. The whole process is carried out in a single-step fashion, i.e. the updating is done immediately after backpropagating a single wave. It is very similar to the well known ART method in X-ray tomography, with the projection and backprojection step replaced by propagation and backpropagation.

Determination of tissue attenuation in emission tomography of optically dense media

In emission tomography of optically dense media one is faced with the problem of determining the attenuation distribution which must be taken into account when setting up the equations from which the activity distribution is computed. We describe a method for computing the attenuation distribution from the emission data. We assume that the true attenuation distribution is approximately an affine distortion of a known prototype attenuation distribution.

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.

Regularisierung schlecht gestellter Probleme durch Projektionsverfahren

SummaryThe numerical solution of ill-posed problems by projection methods is considered. Regularization is carried out simply by choosing an optimal discretization parameter. It is shown by asymptotic estimates and by numerical examples that this kind of regularization is as efficient as the method of Tikhonov and Phillips.

Sampling in fan beam tomography

In fan beam tomography, functions in

Computerized Tomography with Unknown Sources

\mathbb{R}^2

Über die punktweise Konvergenz Finiter Elemente

are constructed from integrals along straight lines emanating from a finite number of sources sitting on a circle around the reconstruction region. Using a sampling theorem for periodic functions and asymptotic estimates for the Fourier transform of the fan beam transform, the exact sampling conditions are found for standard fan beam scanning necessary to obtain a certain resolution. New efficient sampling schemes and corresponding reconstruction algorithms are also found. These sampling schemes need significantly less data to obtain a certain resolution than a standard fan beam geometry. The correctness of the sampling conditions for standard fan beam scanning and the superiority of the new schemes are shown by computer simulations.

Mathematical Methods in Tomography

We give a reconstruction method for the attenuation coefficient in a 2D cross section of a 3D object from tomographic data which does not require knowledge of the positions or the intensity of the sources, and the sources may even be inside the object. Such a situation arises in emission computerized tomography. The method is based on consistency conditions in the range of the relevant integral transforms. The paper contains a detailed description of the algorithm and numerical results for computer generated data.