锥形Radon变换及其顶点位于球面上且轴垂直于球的性质
The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes
AI总结：
本文研究了顶点位于n维空间球面上的圆锥积分变换的可逆性，发展了基于级数展开和降维到广义阿贝尔型一维积分方程的反演方法，并推导了广义阿贝尔方程的唯一性结果。
摘要：
从圆锥积分恢复函数最近因其在新型医学成像技术（如使用康普顿相机的发射计算机断层摄影）中的相关性而变得重要。本文研究了积分圆锥的顶点被限制在n维空间的球面上且对称轴垂直于球的情况。我们证明了所考虑变换的可逆性，并发展了一种基于级数展开和降维到广义阿贝尔型一维积分方程的反演方法。由于出现的核不满足标准假设，我们还推导了广义阿贝尔方程的唯一性结果，其中核在对角线上有零点。最后，我们展示了如何数值实现我们的反演方法并展示了数值结果。
英文摘要：
Recovering a function from its integrals over circular cones recently gained significance because of its relevance to novel medical imaging technologies such emission tomography using Compton cameras. In this paper we investigate the case where the vertices of the cones of integration are restricted to a sphere in $n$-dimensional space and symmetry axes are orthogonal to the sphere. We show invertibility of the considered transform and develop an inversion method based on series expansion and reduction to a system of one-dimensional integral equations of generalized Abel type. Because the arising kernels do not satisfy standard assumptions, we also develop a uniqueness result for generalized Abel equations where the kernel has zeros on the diagonal. Finally, we demonstrate how to numerically implement our inversion method and present numerical results.
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