Abstract. We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In R 3 it maps a function to its surface integrals over circular cones, and in R 2 it maps a function to its integrals along two rays with a common vertex. Such transforms appear in various mathematical models arising in medical imaging, nuclear industry and homeland security. This paper contains new results about inversion of conical Radon transform with fixed opening angle and vertical central axis in R 2 and R 3 . New simple explicit inversion formulae are presented in these cases. Numerical simulations were performed to demonstrate the efficiency of the suggested algorithm in 2D.

The representation of a function by its circular Radon transform (CRT) and various related problems arise in many areas of mathematics, physics and imaging science. There has been a substantial spike of interest towards these problems in the last decade mainly due to the connection between the CRT and mathematical models of several emerging medical imaging modalities. This paper contains some new results about the existence and uniqueness of the representation of a function by its circular Radon transform with partial data. A new inversion formula is presented in the case of the circular acquisition geometry for both interior and exterior problems when the Radon transform is known for only a part of all possible radii. The results are not only interesting as original mathematical discoveries, but can also be useful for applications, e.g. in medical imaging.

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Since Compton cameras were introduced in the use of single photon emission computed tomography, various types of conical Radon transforms, which integrate the emission distribution over circular cones, have been studied. Most of previous works did not address the attenuation factor, which may lead to significant degradation of image quality. In this paper, we consider the problem of recovering an unknown function from conical projections affected by a known constant attenuation coefficient called an attenuated conical Radon transform.In the case of a fixed opening angle and vertical central axis, new explicit inversion formula is derived. Two-dimensional numerical simulations were performed to demonstrate the efficiency of the suggested algorithm.

We study inversion of the spherical Radon transform with centers 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 based on our inversion formula and demonstrate its accuracy and efficiency on several numerical examples