Local Reconstructions in Exponential Tomography

✍ Scribed by A.I. Katsevich

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 220 KB
• Volume: 199
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1996.0151

✦ Synopsis

In this paper, pseudolocal and local approaches to the tomographic reconstruction of discontinuities of an unknown function f from its exponential Radon Ž . transform data g , p are developed. A function f is supposed to be piecewise-Ž . continuous and compactly supported. A pseudolocal tomography function f x is d introduced and it is proved that the difference f y f is continuous. Therefore, d locations and values of jumps of f can be recovered from f , the computation of d which is pseudolocal: for the reconstruction of f at a point x one needs to know d Ž . Ž . < < g ,p only for , p satisfying ⌰ и x y p F d. Investigation of the properties of f as d ª 0 is given. Also a local exponential tomography function f Ž . is d ⌳ proposed and it is proved that f Ž . is the result of action on f of an elliptic ⌳ < < Ž Ž. . pseudo-differential operator with the principal symbol . Thus sing supp f s ⌳ Ž . Ž . Ž . sing supp f and, moreover, the asymptotics of f x as x ª S, the discontinuity ⌳ curve of f, are established. These asymptotics allow one to find values of jumps of f across S using local exponential tomography.