The sub-Riemannian X-ray Transform on H-type groups: Fourier-Slice Theorems and Injectivity sets

Abstract: We continue the development of X-ray tomography in sub-Riemannian geometry. Using the Fourier Transform adapted to the group structure, we generalize the Fourier Slice Theorem to the class of H-type groups. The Fourier Slice Theorem expresses the X-ray transform as the composition of a Fourier restriction and multiplication operator. We compute the spectral resolution for the resulting operator-valued multiplier, which we use to show that an integrable function on an H-type group is determined by its integrals over sub-Riemannian geodesics. We also relate the support of the X-ray transform $If$ to the support of $f$ in frequency space. These results given explicit answers to the injectivity question in a class of geometries with an abundance of conjugate points.