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Tikhonov, A.N. (1963) Solution of Incorrectly Formulated Problems and the Regularization Method. Soviet Mathematics Doklady, 4, 1035-1038.

has been cited by the following article:

  • TITLE: An Alternative Model to Radon Transform for Gamma Ray Emission Tomography

    AUTHORS: Christian Jeanguillaume, Richard Simonnet, Christine Cavaro-Menard, Abdelha Douiri, Ines Bouali, Jean Jacques Loeb

    KEYWORDS: Radon Transform, Tomography, SPECT, Large and Long Hole Collimators, Inverse Problems

    JOURNAL NAME: Advances in Molecular Imaging, Vol.7 No.2, April 30, 2017

    ABSTRACT: The Radon transform fits badly Single Photon Emission Tomography (SPECT). However, Thin Holes Collimator (THC) and Radon model are widely used. The CACAO project has been proposed to enhance the quality of SPECT images. CACAO is a short hand notation for computer aided collimation tomography. The main idea of this project is to use collimators with much larger holes to increase the sensitivity, and slightly longer holes to increase the spatial resolution. The acquisition sequence includes a translation. The Radon projection is replaced by a 2D sum. A dedicated reconstruction algorithm has been developed. If the physical advantage of the project in terms of sensitivity and spatial resolution is generally admitted, a question remains unanswered: Would the ill-posedness of the inverse problem ruin the quality of the reconstructed images? In this article, a representation of the 2D direct problem matrix is derived. This allows us to compare the two inverse problems (CACAO versus THC). The condition number was used for this comparison. We studied the variation of these condition numbers with several parameters. For a proper set of parameters, the CACAO inverse problem may appear easier to solve and more accurately than the THC one.