We present numerical minimization schemes based on the Chambolle—Pock primal dual algorithm. Within this framework, we compare various regularization terms, including non-negativity constraints, H 1 -regularization and total variation regularization. Compared to standard quadratic Tikhonov regularization, TV-regularization is demonstrated to increase the reconstruction quality from conical Radon data. In this paper, we consider the inverse problem for the cone beam transform of vector fields. This transform maps a vector field to its line integrals along rays coming out from points on a given source trajectory.

We proved that the solenoidal part of the field can be reconstructed, if the trajectory satisfies the Kirillov—Tuy condition of order 2.

## Radon transform

It means that every plane, intersecting the support of the field, intersects the trajectory twice. An exact inversion formula is presented. A reconstruction algorithm, based on the obtained formula is developed. Numerical experiments confirm reliability of the algorithm. The elliptic Radon transform eRT integrates functions over ellipses in 2D and ellipsoids of revolution in 3D.

It thus serves as a model for linearized seismic imaging under the common offset scanning geometry where sources and receivers are offset by a constant vector. As an inversion formula of eRT is unknown we propose certain imaging operators generalized backprojection operators which allow to reconstruct some singularities of the searched-for reflectivity function from seismic measurements. We calculate and analyze the principal symbols of these imaging operators as pseudo-differential operators to understand how they map, emphasize or de-emphasize singularities. We use this information to develop local reconstruction operators that reconstruct relatively independently of depth and offset.

Numerical examples illustrate the theoretical findings.

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This paper describes a coplanar non invasive non destructive capacitive imaging device. We first introduce a mathematical model for its output, and discuss some of its theoretical capabilities. We show that the data obtained from this device can be interpreted as a weighted Radon transform of the electrical permittivity of the measured object near its surface.

The quality of the images leads us to expect that excellent results can be delivered by ad hoc optimized inversion formulas. There are also interesting, yet unexplored, theoretical questions on imaging that this sensor will allow us to test. We investigate the inverse source problem for the wave equation, arising in photo- and thermoacoustic tomography. There exist quite a few theoretically exact inversion formulas explicitly expressing the solution of this problem in terms of the measured data, under the assumption of the constant and known speed of sound.

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However, almost all of these formulas require data to be measured either on an unbounded surface, or on a closed surface completely surrounding the object. This is too restrictive for practical applications. The alternative approach we present, under certain restriction on geometry, yields a theoretically exact reconstruction of the standard Radon projections of the source from the data measured on a finite open surface.

In addition, this technique reduces the time interval where the data should be known. In general, our method requires a pre-computation of densities of certain single-layer potentials. However, in the case of a truncated circular or spherical acquisition surface, these densities are easily obtained analytically, which leads to fully explicit asymptotically fast algorithms. We test these algorithms in a series of numerical simulations.

Broken ray transforms BRTs are typically considered to be reciprocal, meaning that the transform is independent of the direction in which a photon travels along a given broken ray.

However, if the photon can change its energy or be absorbed and re-radiated at a different frequency at the vertex of the ray, then reciprocity is lost. In optics, non-reciprocal BRTs are applicable to imaging problems with fluorescent contrast agents. In the case of x-ray imaging, problems with single Compton scattering also give rise to non-reciprocal BRTs. In this paper, we focus on tomographic optical fluorescence imaging and show that, by reversing the path of a photon and using the non-reciprocity of the data function, we can reconstruct simultaneously and independently all optical properties of the medium the intrinsic attenuation coefficients at the excitation and the fluorescence frequency and the concentration of the contrast agent.

Our results are also applicable to inverting BRTs that arise due to single Compton scattering.

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To find out more, see our Privacy and Cookies policy. Close this notification. Inverse Problems. Scope Relevant topics that would be considered for inclusion in this special issue include: Mathematical models of imaging techniques using generalized Radon transforms Applications to real-world problems Broken-ray, V-line, conical, star, American football and other related transforms Exact and approximate inversion formulas in various geometries Reconstruction algorithms and analysis of artifacts Theoretical results such as injectivity, support theorems, range description of the transforms, which play an important role in imaging applications Potential authors are invited to contact one of the editors to discuss suitability prior to submission.

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### The Radon Transform and Medical Imaging

Thanks for telling us about the problem. Return to Book Page. Of value to mathematicians, physicists, and engineers, this excellent introduction to Radon transform covers both theory and applications, with a rich array of examples and literature that forms a valuable reference. This edition is a revised and updated version by the author of his pioneering work. Get A Copy. Paperback , pages. More Details Original Title. Other Editions 1.

## Fundamentals of the Radon Transform | SpringerLink

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