Symmetric star transforms and the algebraic geometry of their dual differential operators

TL;DR

This paper explores the algebraic and geometric properties of the star transform, a generalized Radon transform, linking its invertibility to the symmetry of its dual differential operator and algebraic varieties.

Contribution

It introduces the dual differential operator for the star transform and connects its properties to symmetry groups and algebraic geometry, including Fano varieties and cubic surfaces.

Findings

Symmetric star transforms have dual operators with G-invariant polynomial symbols.

Degenerate symmetry leads to zero sets of elementary symmetric polynomials.

Non-invertible 2D star transforms relate to lines on the Cayley cubic surface.

Abstract

The star transform is a generalized Radon transform mapping a function on $\mathbb{R}^n$ to the function whose value at a point is the integral along a union of rays emanating from the point in a fixed set of directions, called branch vectors. We show that the injectivity and inversion properties of the star transform are connected to its dual differential operator, an object introduced in this paper. We prove that if the set of branch vectors forms a symmetric shape with respect to the action of a finite rotation group $G$, then the symbol of its dual differential operator belongs to the ring of $G$-invariant polynomials. Furthermore, we show that star transforms with degenerate symmetry correspond to linear subspaces contained in the zero set of certain elementary symmetric polynomials, and we investigate the associated real algebraic Fano varieties. In particular, non-invertible star transforms in dimension 2 correspond to certain real lines on the Cayley nodal cubic surface.