2015/11/05 by Alexander Katsevich, Katsevich, Alexander, Alexander Tovbis +1
Mathematics · Medicine · #34B24 #34L10 #44A12 #44A15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Medical Imaging Techniques and Applications #Numerical methods in inverse problems #math.CA #msc:34B24 #msc:34L10 #msc:44A12 #msc:44A15
paper · pdf · doi:10.48550/arxiv.1511.01967
23 pages, 2 figures
openalex publication_date 2015/11/05 · arxiv created 2015/11/06 · arxiv updated 2015/11/09 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We study the interior problem of tomography. The starting point is the Gelfand-Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the line. Let Ω1 be the interval where f is supported, and Ω2 be the interval where the Hilbert transform of f can be computed using the Gelfand-Graev formula. The equation we study is H1 f=g|Ω2, where H1 is the FHT that integrates over Ω1 and gives the result on Ω2, i.e. H1: L2(Ω1)→ L2(Ω2). In the case of the interior problem the tomographic data are truncated, and Ω1 is no longer a subset of Ω2. In this paper we consider the case when the intervals Ω1=(a1,0) and Ω2=(0,a2) are adjacent. Here a1 < 0 < a2. First we find a differential operator L that commutes with H1. Using the Titchmarsh-Weyl theory, we show that L has only continuous spectrum and obtain two isometric transformations U1, U2, such that U2 H1 U1^* is the multiplication operator with the function σ(λ), λ≥(a12+a22)/8. Here λ is the spectral parameter. Then we show that σ(λ)→0 as λ→∞ exponentially fast. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators U1, U2 as λ→∞. When the intervals are symmetric, i.e. -a1=a2, the operators U1, U2 are obtained explicitly in terms of hypergeometric functions.