Abbreviated Journal Title
SIAM J. Math. Anal.
Hilbert transform; spectrum; Sturm-Liouville; limited data; tomography; inverse problems; SPHEROIDAL WAVE-FUNCTIONS; CONE-BEAM CT; IMAGE-RECONSTRUCTION; FOURIER-ANALYSIS; FAN-BEAM; BACKPROJECTION; UNCERTAINTY; PROJECTIONS; Mathematics, Applied
We study a restriction of the Hilbert transform as an operator H-T from L-2(a(2), a(4)) to L-2(a(1), a(3)) for real numbers a(1) < a(2) < a(3) < a(4). The operator H-T arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection. There, the reconstruction requires recovering a family of one-dimensional functions f supported on compact intervals [a(2), a(4)] from its Hilbert transform measured on intervals [a(1), a(3)] that migH(T) only overlap, but not cover [a(2), a(4)]. We show that the inversion of H-T is ill-posed, which is why we investigate the spectral properties of H-T. We relate the operator H-T to a self-adjoint two-interval Sturm-Liouville problem, for which we prove that the spectrum is discrete. The Sturm-Liouville operator is found to commute with H-T, which then implies that the spectrum of (HTHT)-H-* is discrete. Furthermore, we express the singular value decomposition of H-T in terms of the solutions to the Sturm-Liouville problem. The singular values of H-T accumulate at both 0 and 1, implying that H-T is not a compact operator. We conclude by illustrating the properties obtained for H-T numerically.
Siam Journal on Mathematical Analysis
Al-Aifari, Reema and Katsevich, Alexander, "Spectral Analysis of The Truncated Hilbert Transform With Overlap" (2014). Faculty Bibliography 2010s. 4972.