For centers outside the sphere, the shifted Funk transform reduces, via a Möbius change of variables, to a parallel slice transform, yielding exact injectivity and inversion results.
The Funk-Radon transform for hyperplane sections through a common point
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abstract
The Funk-Radon transform, also known as the spherical Radon transform, assigns to a function on the sphere its mean values along all great circles. Since its invention by Paul Funk in 1911, the Funk-Radon transform has been generalized to other families of circles as well as to higher dimensions. We are particularly interested in the following generalization: we consider the intersections of the sphere with hyperplanes containing a common point inside the sphere. If this point is the origin, this is the same as the aforementioned Funk--Radon transform. We give an injectivity result and a range characterization of this generalized Radon transform by finding a relation with the classical Funk--Radon transform.
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2019 1verdicts
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On Two Families of Funk-Type Transforms
For centers outside the sphere, the shifted Funk transform reduces, via a Möbius change of variables, to a parallel slice transform, yielding exact injectivity and inversion results.