Unraveling X-ray Transforms on the Heisenberg group

Abstract: The classical X-ray Transform maps a function on Euclidean space to a function on the space of lines on this Euclidean space by integrating the function over the given line. Inverting the X-ray transform has wide-ranging applications, including to medical imaging and seismology. Much work has been done to understand this inverse problem in Euclidean space, Euclidean domains, and more generally, for symmetric spaces and Riemannian manifolds with boundary where the lines become geodesics. We formulate a sub-Riemannian version of the X-ray transform on the simplest sub-Riemannnian manifold, the Heisenberg group. Here serious geometric obstructions to classical inverse problems, such as existence of conjugate points, appear generically. With tools adapted to the geometry, such as an operator-valued Fourier Slice Theorem, we prove nonetheless that an integrable function on the Heisenberg group is indeed determined by its line integrals over sub-Riemannian (as well as over its compatible Riemannian and Lorentzian) geodesics.

We also pose an abundance of accessible follow-up questions, standard in the inverse problems community, concerning the sub-Riemannian case, and report progress answering some of them.

analysis of PDEsdifferential geometrymetric geometryoptimization and controlspectral theory

Audience: researchers in the topic

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Sub-Riemannian Seminars

Series comments: The "Sub-Riemannian seminars" are the union of the "Séminaire de géométrie et analyse sous-riemannienne" (held in Paris since 2011) and the "International Sub-Riemannian Seminars", which were born in spring 2020 as a reaction to the COVID-19 pandemic.

The new format will gather every 3 weeks on average, alternating between these types of sessions:

- physical session in Paris (Laboratoire Jacques-Louis Lions), also transmitted online on Zoom.

- fully online session on Zoom.

- special session hosted physically somewhere else, and transmitted online.

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