Contemporary Mathematics Volume 278, 2001 Local tomography and related problems Carlos A. Berenstein Medical tomography is one of the most visible recent contributions of Mathe- matics to the general well-being. CT and MRI are now almost routine diagnostic tools. These two imaging methods and other tomographic instrumentation have also significant scientific and industrial applications, quite often unexpected. The problems discussed below were inspired by work on a prototype instrumentation to measure space plasma originally conceived with M. Coplan and J. Moore (see [ZCMB] for a 2-d version, the full 3-d case was done jointly with M. Shahshahani.) Let us start by recalling the definition of the Radon transform we use and its relationship to the wave operator. (The basic references for the Radon transform are [Hel, Na, KS], each of them has a different, complementary, point of view.) First, consider the original case studied by Radon, the Radon transform in the Euclidean plane. (Although in his original paper [R], he also briefly discusses the inversion formula for the 3-d case.) Let w = (cos 0, sin 0) E 8 1 , p E 1R, thus the equation X · W = p represents the equation of a line £ perpendicular to W and of signed distance to the origin equal to p. For a function f sufficiently nice, for instance, continuous of compact support, one defines the Radon transform of f evaluated in this line as given by Rwf(p) := Rf(w,p) := J f(x)ds = 1: f(xo + tw.L)dt, where x 0 is any fixed point in£, w.L = (- sin 0, cos 0) is the rotate of w by ~. Clearly, Rf is defined in the family of all lines in the plane and depends only on the line£ and not in its equation. In particular, it satisfies Rf(-w, -p) = Rf(w,p). One can see that the definition of the Radon transform can be easily extended to other classes offunctions, e.g. S(1R2 ), L 1 (1R2 ), and, more interestingly for the usual applications like CT, f bounded and compactly supported, sufficiently smooth except along some curves, where f has jump singularities. The case of 3-d, applicable to MRI, is similar with the exception that the same equation x · w = p represents a 2-d plane in the space and ds has to be interpreted as the area measure along this hyperplane. Although the discussion is more general, we are considering n = 2 or 3 in what 2000 Mathematics Subject Classification. Primary 44Al2, Secondary 92C55. This lecture reflects past and ongoing research with a number of collaborators, Enrico Casa- dio Tarabusi, Roger Gay, Boris Rubin, and David Walnut, among them. The author's research has been partly supported by NSF grants DMS-9622249 and DMS-0070044. © 2001 AmPrican Math'mat.ical Soci'ty 3
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