Recovering a function from spherical means in 3D using local data

Rafik Aramyan

doi:10.3934/ipi.2023050

Article Contents

2024, Volume 18, Issue 3: 690-707. Doi: 10.3934/ipi.2023050

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Recovering a function from spherical means in 3D using local data

Rafik Aramyan^{1, 2,}

1.
Institute of Mathematics of NAS of RA, Armenia
2.
Russian-Armenian University, Yerevan, Armenia

Received: October 2022

Revised: November 2023

Early access: November 24, 2023

Published online: November 24, 2023

[The work was supported by the Science Committee of RA, in the frames of the research project 21AG‐1A045].

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Abstract

In this article, we study the spherical mean Radon transform in $ \mathbf R^3 $ with detectors centered on a plane. A novel iterative formula to inverse the spherical mean Radon transform in $ 3D $ is presented. To recover a function from its spherical mean values we use local data which is the benefit of this formula. The inversion of the spherical mean Radon transform is used in mathematical models of radar imaging, thermo- and photo-acoustic tomography, and others.

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Mathematics Subject Classification: Primary: 45Q05, 44A12; Secondary: 65R32.

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Figure 1. An illustration to (12)

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Figure 2. the restriction of $ f $ onto the plane $ y = 3 $

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Figure 3. the approximation of $ f $ obtained by (3) for $ n = 2 $

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Figure 4. the approximation of the piecewise smooth function obtained by (3) for $ n = 2 $

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