Asymptotic behaviour of the singular values for the truncated Hilbert transform

We present new results on the singular value decomposition (SVD) of the truncated Hilbert transform (THT). The THT problem consists in recovering a function f(x) with support on an interval [a₂, a₄] from the knowledge of its Hilbert transform over an interval [a₁, a₃] which overlaps with the support of f, i.e. a₁ <; a₂ <; a₃ <; a₄. This problem has applications in 2D and 3D tomography for the reconstruction of a region of interest using the differential back-projection. Recent work by Al-Aifari and Katsevich demonstrates that the spectrum of the singular values of the THT has two accumulation points in 0 and in 1. For the interior problem, Katsevich and Tovbis have given a characterization of the asymptotic behaviour of the singular values. Building on these results, we derive here the asymptotic behaviour of the singular values of the THT close to 1 and close to 0, and show that the two limits are connected by a simple coordinate transformation. A comparison with the SVD of a discretized version of the problem shows that the asymptotic expressions for the singular values and singular functions are already accurate for small indices.