The Funk–Radon transform for hyperplane sections through a common point

The Funk–Radon transform, also known as the spherical Radon transform, assigns to a function on the sphere its mean values along all great circles. Since its invention by Paul Funk in 1911, the Funk–Radon transform has been generalized to other families of circles as well as to higher dimensions. We are particularly interested in the following generalization: we consider the intersections of the sphere with hyperplanes containing a common point inside the sphere. If this point is the origin, this is the same as the aforementioned Funk–Radon transform. We give an injectivity result and a range characterization of this generalized Radon transform by finding a relation with the classical Funk–Radon transform.

Inversion Formula for a Radon-Type Transform Arising in Photoacoustic Tomography with Circular Integrating Detectors

This paper is devoted to a Radon-type transform arising in a version of Photoacoustic Tomography that uses integrating circular detectors. The Radon-type transform that arises can be decomposed into the known Radon-type transforms: the spherical Radon transform and the sectional Radon transform. An inversion formula is obtained by combining existing inversion formulas for the above two Radon-type transforms.

Image reconstruction from radially incomplete spherical Radon data

We study inversion of the spherical Radon transform with centres on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Our result is valid when the support of the image function is inside the data acquisition sphere, outside that sphere, as well as on both sides of the sphere. Furthermore, in addition to the uniqueness result, our method of proof provides reconstruction formulas for all those cases. We present a robust computational algorithm and demonstrate its accuracy and efficiency on several numerical examples.