Inversion and Symmetries of the Star Transform

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along "star-shaped" trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula and discuss its stability properties. As an unexpected bonus of our approach, we prove a conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials

Inversion of the star transform

We define the star transform as a generalization of the broken ray transform introduced by us in previous work. The advantages of using the star transform include the possibility to reconstruct the absorption and the scattering coefficients of the medium separately and simultaneously (from the same data) and the possibility to utilize scattered radiation which, in the case of the conventional X-ray tomography, is discarded. In this paper, we derive the star transform from physical principles, discuss its mathematical properties and analyze numerical stability of inversion. In particular, it is shown that stable inversion of the star transform can be obtained only for configurations involving odd number of rays. Several computationally-efficient inversion algorithms are derived and tested numerically.Comment: Accepted to Inverse Problems in this for

Inversion of the Broken Ray Transform

The broken ray transform (BRT) is an integral of a function along a union of two rays with a common vertex. Consider an X-ray beam scanning an object of interest. The ray undergoes attenuation and scatters in all directions inside the object. This phenomena may happen repeatedly until the photons either exit the object or are completely absorbed. In our work we assume the single scattering approximation when the intensity of the rays scattered more than once is negligibly small. Among all paths that the scattered rays travel inside the object we pick the one that is a union of two segments with one common scattering point. The intensity of the ray which traveled this path and exited the object can be measured by a collimated detector. The collimated detector is able to measure the intensity of X-rays from the selected direction. The logarithm of such a measurement is the broken ray transform of the attenuation coefficient plus the logarithm of the scattering coefficient at the scattering point (vertex) and a known function of the scattering angle. In this work we consider the reconstruction of X-ray attenuation coefficient distribution in a plane from the measurements on two or three collimated detector arrays. We derive an exact local reconstruction formula for three flat collimated detectors or three curved or pin-hole collimated detectors. We obtain a range condition for the case of three curved or pin-hole detectors and provide a special case of the range condition for three flat detectors. We generalize the reconstruction formula to four and more detectors and find an optimal set of parameters that minimize noise in the reconstruction. We introduce a more accurate scattering model which takes into account energy shifts due to the Compton effect, derive an exact reconstruction formula and develop an iterative reconstruction method for the energy-dependent case. To solve the problem we assume that the radiation source is monoenergetic and the dependence of the attenuation coefficient on energy is linear on an energy interval from the minimal to the maximal scattered energy. %initial radiation energy. We find the parameters of the linear dependence of the attenuation on energy as a function of a point in the reconstruction plane