Spectral and Asymptotic Analysis for the Truncated Hilbert Transform Arising in Limited Data Tomography

Reema Al-Aifari^* and Alexander Katsevich

In Computerized Tomography a 2D or 3D object is reconstructed from projection data (Radon transform data) from multiple directions. When the X-ray beams are sufficiently wide to fully embrace the object and when the beams from a sufficiently dense set of directions around the object can be used, this problem and its solution are well understood. When the data are more limited the image reconstruction problem becomes much more challenging; leading to configurations where only a subregion of the object is illuminated from all angles. \newline In this talk we consider a limited data problem in 2D Computerized Tomography which - with a result by Gelfand and Graev - gives rise to 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 $a_1 < a_2 < a_3 < a_4$. We present the framework of tomographic reconstruction from limited data and the method of differentiated back-projection (DBP) which leads to the operator $H_T$. The reconstruction from the DBP method requires recovering a family of 1D functions $f$ supported on compact intervals $[a_2,a_4]$ from its Hilbert transform measured on intervals $[a_1,a_3]$ that might only overlap, but not cover $[a_2,a_4]$. \newline We relate the operator $H_T$ to a self-adjoint \textit{two-interval Sturm-Liouville problem}, for which the spectrum is discrete. The Sturm-Liouville operator is found to commute with $H_T$, which allows to express the singular value decomposition of $H_T$ in terms of the solutions to the Sturm-Liouville problem. We then find that $0$ and $1$ are the two accumulation points of the singular values of $H_T$. Thus, $H_T$ is not compact but its inversion is ill-posed. \newline With these results, we then address the question of the rate of convergence of the singular values by considering the corresponding Sturm-Liouville problem. The WKB method allows to approximate solutions to the Sturm-Liouville problem for large eigenvalues and away from the singular points $a_1, a_2, a_3, a_4$. Close to these singularities, the eigenfunctions can be approximated by Bessel functions. Analyzing the behavior of the eigenfunctions near the singularities then allows to estimate the convergence rate of the singular values of $H_T$. We conclude by illustrating the properties obtained for the SVD of $H_T$ numerically.

Mathematics Subject Classification: 47A52 47A10

Keywords: tomographic reconstruction; limited data; ill-posed; spectral analysis; asymptotic analysis

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