2013/02/26 by Reema Al‐Aifari, Reema Al-Aifari, Al-Aifari, Reema +2
Mathematics · Medicine · #Mathematical Analysis and Transform Methods #Medical Imaging Techniques and Applications #Numerical methods in inverse problems #math.CA #math.FA #math.SP
paper · pdf · doi:10.48550/arxiv.1302.6295
24 pages, revised version
arxiv created 2013/11/26 · arxiv updated 2013/11/28
We study a restriction of the Hilbert transform as an operator HT from L2(a2,a4) to L2(a1,a3) for real numbers a1 < a2 < a3 < a4. The operator HT arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection (DBP). There, the reconstruction requires recovering a family of one-dimensional functions f supported on compact intervals [a2,a4] from its Hilbert transform measured on intervals [a1,a3] that might only overlap, but not cover [a2,a4]. We show that the inversion of HT is ill-posed, which is why we investigate the spectral properties of HT. We relate the operator HT 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 HT, which then implies that the spectrum of HT^* HT is discrete. Furthermore, we express the singular value decomposition of HT in terms of the solutions to the Sturm-Liouville problem. The singular values of HT accumulate at both 0 and 1, implying that HT is not a compact operator. We conclude by illustrating the properties obtained for HT numerically.