星变换的反演与对称性
Inversion and Symmetries of the Star Transform
摘要：
星变换是一种广义Radon变换，它将一个二元函数映射到其沿着“星形”轨迹的积分，该轨迹由从一个公共顶点发出的有限数量的射线组成。这种算子出现在基于基本粒子散射的各种成像模态的数学模型中。本文对星变换的逆变换进行了全面的研究。给出了星变换可逆的充要条件，引入了一个新的反演公式，并讨论了它的稳定性。作为一个意想不到的奖金，我们的方法，我们证明了一个猜想，从代数几何的零集的基本对称多项式。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
