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BIRS Workshop Lecture Videos

Non-central Funk-Radon transforms Agranovsky, Mark


The classical Funk (Funk-Radon) transform, evaluating integrals of functions on the unit sphere in $\mathbb R^n$ over cross-sections by linear hyperspaces, is well studied. This transform has many applications, in medical imaging ($Q$-ball method in diffusion MRI), geometric tomography (intersection bodies problem). Last years, a similar transform (shifted Funk transform) associated with cross-sections of the unit sphere by $k$-planes passing through a fixed point (center), which is not necessarily the origin, appeared in the focus of researchers. The kernel of a shifted Funk transform with the center inside the unit sphere was described and inversion formulas were obtained. In my talk an universal approach will be discussed which allows to treat the case of arbitrarily located center. In most cases, shifted Funk transform has nontrivial kernel, so that single Funk data is not enough to recover a function on the unit sphere. Hence it is natural to ask when and how functions can be recovered from {\it multiple} Funk data. This question is completely answered for pairs of shifted Funk transforms. We fully describe all geometric configurations of the centers which provide injectivity of the paired transform and, correspondingly, unique recovery of functions on the unit sphere. A corresponding reconstruction procedure is given. The approach relies on the action of a hyperbolic automorphism group of the real ball and a billiard-like dynamics of a self-mapping of the unit sphere, generated by the set of centers. The common kernel of a pair of Funk transforms consists of related automorphic functions and is determined by the type of the above dynamics on the unit sphere.

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