An analytical inversion formula for the overdetermined spherical section transform

Abstract This paper addresses the inversion of the overdetermined spherical section transform, which integrates a function defined on the unit sphere in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> over the full set of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> -dimensional subspheres obtained by intersecting the unit sphere with hyperplanes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> . The inversion of the spherical section transform and its variants have applications in various tomographic imaging techniques, including thermoacoustic and photoacoustic tomography, as well as Compton camera imaging. We derive analytic inversion formulas for the overdetermined spherical section transform in n -dimensions and demonstrate the stability of inversion in three dimensions.