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2012•Transactions of the American Mathematical SocietyOpen access

A generalization of the Mader-Helgason inversion formulas for Radon transforms

Y. A. Antipov, Boris Rubin

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Abstract

In 1927, Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on $\mathbb {R}^n$. These formulas differ from the original ones by Radon and seem to be forgotten. We generalize Mader’s formulas to totally geodesic Radon transforms in any dimension on arbitrary constant curvature space. Another new interesting inversion formula for the $k$-plane transform was presented in the recent book “Integral geometry and Radon transform” by S. Helgason. We extend this formula to arbitrary constant curvature space. The paper combines tools of integral geometry and complex analysis.

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In 1927, Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on $\mathbb {R}^n$. These formulas differ from the original ones by Radon and seem to be forgotten. We generalize Mader’s formulas to totally geodesic Radon transforms in any dimension on arbitrary constant curvature space. Another new interesting inversion formula for the $k$-plane transform was presented in the recent book “Integral geometry and Radon transform” by S. Helgason. We extend this formula to arbitrary constant curvature space. The paper combines tools of integral geometry and complex analysis.

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Available abstract

In 1927, Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on $\mathbb {R}^n$. These formulas differ from the original ones by Radon and seem to be forgotten. We generalize Mader’s formulas to totally geodesic Radon transforms in any dimension on arbitrary constant curvature space. Another new interesting inversion formula for the $k$-plane transform was presented in the recent book “Integral geometry and Radon transform” by S. Helgason. We extend this formula to arbitrary constant curvature space. The paper combines tools of integral geometry and complex analysis.

Key concepts: Integral geometry, Mathematics, Radon transform, Hyperplane, Inversion (geology), Geodesic, Curvature, Radon

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