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, discuss its stability properties and demonstrate its efficiency on numerical examples. As an unexpected bonus of our approach, we prove a long-standing conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials.