By Gestur Olafsson and Eric Todd Quinto (Editor)
Because their emergence in 1917, tomography and inverse difficulties stay lively and demanding fields that mix natural and utilized arithmetic and supply powerful interaction among different mathematical difficulties and purposes. The utilized facet is healthier identified for scientific and clinical use, specifically, clinical imaging, radiotherapy, and business non-destructive checking out. medical professionals use tomography to work out the interior constitution of the physique or to discover useful info, corresponding to metabolic approaches, noninvasively. Scientists notice defects in items, the topography of the sea flooring, and geological details utilizing X-rays, geophysical measurements, sonar, or different facts. This quantity, in response to the lectures within the brief direction The Radon rework and functions to Inverse difficulties on the American Mathematical Society assembly in Atlanta, GA, January 3-4, 2005, brings jointly articles on mathematical elements of tomography and similar inverse difficulties. The articles disguise introductory fabric, theoretical difficulties, and useful matters in three-D tomography, impedance imaging, neighborhood tomography, wavelet tools, regularization and approximate inverse, sampling, and emission tomography. All contributions are written for a basic viewers, and the authors have integrated references for additional examining.
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Additional resources for The Radon Transform, Inverse Problems, and Tomography
Example text
Cormack, one of the pioneers of tomography: 'In practice one can make only a finite number of measurements with beams of finite width, and the question which arises is how many observations should be made, and how should they be related to each other in order to reconstruct the object' [8]. While Cormack himself proceeded with geometric arguments to discover an efficient data acquisition scheme, we will use two-dimensional sampling theory to address this question, an approach first introduced by Lindgren and Rattey [40, 55].
The corresponding mollifier then is e>y(y) = (2TT)-1/I-2|2/|-1 , forM V. Lakshminarayanan, Three dimensional reconstruction from radiographs and electron micrographs: applications of convolutions instead of Fourier Transforms, Proc. Natl. Acad. Sci. USA 68 (1971), 262-277. Alexander G. Ramrn and Alexander I. Zaslavsky, Singularities of the Radon transform, Bull. Amer. Math. Soc. 25 (1993), 109-115. F. R. Liu, Carlos A. Berenstein, and David Walnut, Wavelet-based multiresolution local tomography, IEEE Trans. Image Processing 6 (1997), 1412-1430. Lawrence A. Shepp and J.