Inversion of the Spherical Mean Transform with Sources on a Hyperplane

Details Inversion of the Spherical Mean Transform with Sources on a Hyperplane Inversion of the Spherical Mean Transform with Sources on a Hyperplane

2009-10-08

arxiv.org/abs/0910.1380v1

Mathematics

Classical Analysis and ODEs

29 pages, 4 figures

Scientific paper

The object of this study is an integral operator $\mathcal{S}$ which averages functions in the Euclidean upper half-space $\mathbb{R}_{+}^{n}$ over the half-spheres centered on the topological boundary $\partial \mathbb{R}_{+}^{n}$. By generalizing Norton's approach to the inversion of arc means in the upper half-plane, we intertwine $\mathcal{S}$ with a convolution operator $\mathcal{P}$. The latter integrates functions in $\mathbb{R}^{n}$ over the translates of a paraboloid of revolution. Our main result is a set of inversion formulas for $\mathcal{P}$ and $\mathcal{S}$ derived using a combination of Fourier analysis and classical Radon theory. These formulas appear to be new and are suitable for practical reconstructions.

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