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Jan Boman: The region of interest problem in Tomography

Time: Wed 2022-06-08 13.15 - 14.15

Location: Kräftriket, House 6, Room 306 and Zoom

Video link: Meeting ID: 692 1892 7142

Participating: Jan Boman (SU)

Abstract

In Computerized Tomography it is often desirable to reconstruct a function $$f (x) = f (x_1, x_2)$$ on a proper subset S of the (compact) support of f , the region of interest, from measurements of the Radon transform $$Rf (L) =∫_L f ds$$ on the set of lines L that intersect S (ds denotes arc length on L). Well known examples show that this is not possible in general. Instead it was believed for a while that (in dimension 2) it was necessary to know the full Radon transform $$Rf$$ for computing (with reasonable accuracy) f in a region of interest. However, beginning around 2000 new methods were invented that made it possible in some cases to reconstruct $$f$$ in a region of interest based on considerably less than complete data. For instance, if J is a segment of a line $$L_0$$ that contains $$L_0 ∩ supp(f)$$ , and $$Rf (L)$$ is known for all lines L that intersect J, then one can compute the (one-dimensional) Hilbert transform of the restriction of f to J. This has led to much recent work on the Region of Interest problem, for instance to study of uniqueness and stability for inverting the so-called truncated Hilbert transform. This is the problem to find a function $$u(t)$$ with support in an interval $$E ⊂ \mathbf{R}$$ from knowledge of its Hilbert transform $$Hu(t)=\frac{1}{\pi}\int_{\mathbf{R}}u(t')\frac{dt'}{t'-t}$$ for t in another interval F ⊂ R. The difficulty of this problem depends on the relative position of the intervals E and F .