Pizzetti formulae and the Radon Transform on the Sphere

2022/03/07 by Alí Guzmán Adán, Adán, Alí Guzmán, Mihaela Vajiac +1
Computer Science · Mathematics · Medicine · #28C10 #33C55 #44A12 #46F10 #58C35 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Medical Image Segmentation Techniques #Medical Imaging Techniques and Applications

paper · pdf · doi:10.48550/arxiv.2203.03472

openalex publication_date 2022/03/07 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we obtain Pizzetti-type formulae on regions of the the unit sphere \mathbbS^m-1 of ℝ^m, and study their applications to the problem of inverting the spherical Radon transform. In particular, we approach integration over (m-2)-dimensional sub-spheres of \mathbbS^m-1, (m-1)-dimensional sub-balls, and over (m-1)-dimensional spherical caps as the action of suitable concentrated delta distributions. In turn, this leads to Pizzetti formulae that express such integrals in terms of the action of SO(m-1)-invariant differential operators. In the last section of the paper, we use some of these expressions to derive the inversion formulae for the Radon transform on \mathbbS^m-1 in a direct way.

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