Inversion of the Spherical Mean Transform with Sources on a Hyperplane

2009/10/08 by Aleksei Beltukov, Beltukov, Aleksei · 3 citations
Mathematics · Engineering · #Numerical methods in inverse problems #Calibration and Measurement Techniques #Photoacoustic and Ultrasonic Imaging

paper · pdf · doi:10.48550/arxiv.0910.1380

Abstract

The object of this study is an integral operator S which averages functions in the Euclidean upper half-space ℝ₊ⁿ over the half-spheres centered on the topological boundary ∂ ℝ₊ⁿ. By generalizing Norton's approach to the inversion of arc means in the upper half-plane, we intertwine S with a convolution operator P. The latter integrates functions in ℝⁿ over the translates of a paraboloid of revolution. Our main result is a set of inversion formulas for P and 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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• Inversion of the elliptical Radon transform arising in migration imaging using the regular Radon transform
• The spherical mean Radon transform with centers on cylindrical surfaces
• Recovering Finite Parametric Distributions and Functions Using the\n Spherical Mean Transform

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