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Simultaneous activity and attenuation emission tomography as a nonlinear ill‐posed problem
Author(s) -
Dicken Volker
Publication year - 1998
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19980781518
Subject(s) - tikhonov regularization , attenuation , single photon emission computed tomography , tomography , inverse problem , nonlinear system , regularization (linguistics) , emission computed tomography , spect imaging , algorithm , mathematics , radon transform , iterative reconstruction , physics , computer science , mathematical analysis , positron emission tomography , nuclear medicine , optics , artificial intelligence , medicine , quantum mechanics
In Single Photon Emission Computed Tomography (SPECT) one is interested in reconstructing the activity distribution f of some radiopharmaceutical. Alas the data gathered suffer from attenuation described by the tissue density μ. We only have noisy sample values of the Attenuated‐Radon‐Transform\documentclass{article}\pagestyle{empty}\begin{document}$$ A(f,\mu)(\omega,s) = \mathop \smallint \nolimits_{ - \infty }^\infty f(s\omega ^ \bot + t\omega)\exp (- \mathop \smallint \nolimits_t^\infty \mu (s\omega ^ \bot + \tau \omega)d\tau)dt $$\end{document} (which is nonlinear in μ) per imaged slice at hand. Traditional theory for SPECT reconstruction treats μ as a known parameter. In practical applications however μ is not known, but crudely estimated or neglected. We try to develop an algorithm that approximates both f and μ from SPECT data alone, in order to obtain quantitatively accurate SPECT images. This is done using Tikhonov regularization techniques developed for non‐linear parameter estimation problems in differential equations and an adapted Gauß‐Newton‐CG minimization algorithm.