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Computer Tomography and Quantum Mechanics

Part of: Physiological, cellular and medical topics Applications to specific physical systems Integral transforms, operational calculus

Published online by Cambridge University Press:  01 July 2016

Lev B. Klebanov  and
Svetlozar T. Rachev
Lev B. Klebanov*
Affiliation:
Russian Academy of Natural Sciences
Svetlozar T. Rachev*
Affiliation:
University of California, Santa Barbara
*
∗ Postal address: Institute of Mathematical Geology, Russian Academy of Natural Science, 12 Shpalemaya, 191187, Russia.
∗∗ Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara CA 93106-3110, USA.
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Abstract

In this paper we study some topics of interest to specialists in computer tomography. These are the following. (a) The Radon transform and its applications to computer tomography. (b) Problems of computer tomography with partially known data. Estimates of stability will be given for different types of distance in the space of probability distributions. We consider the problem with partially known tomographic data as a stability problem for appropriately chosen distances. This approach allows us to give a solution of the so-called computer tomography paradox. (c) The relation of quantum mechanics to computer tomography. An intriguing method for ‘measuring' wavefunctions by tomographic methods (CAT scans) opens a new approach to various problems in quantum mechanics. Using the method outlined for the solution of the computer tomography paradox, we derive inequalities that estimate the amount of information on the wavefunctions resulting from real CAT scans, i.e. CAT scans based on the finite number of measured marginals (projections) of the Wigner distributions. In conclusion, we propose a new version of the mathematical justification of CAT scans.

Keywords

COMPUTER TOMOGRAPHYMOMENT PROBLEMSQUANTUM MECHANICS

MSC classification

Primary: 92C55: Biomedical imaging and signal processing 44A12: Radon transform
Secondary: 81V99: None of the above, but in this section
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 3 , September 1997 , pp. 595 - 606
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Research supported by the ISF Grant No NXL 300 (Dr Klebanov) and by the Alexander von Humboldt Research Award for Senior U.S. Scientists (Dr Rachev), during his stay at Institute für Mathematische Stochastik, University of Freiburg.

References

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