STABILITY OF THE INTERIOR PROBLEM FOR POLYNOMIAL REGION OF INTEREST.
In many practical applications, it is desirable to solve the interior problem of tomography without requiring knowledge of the attenuation functionfaon an open set within the region of interest (ROI). It was proved recently that the interior problem has a unique solution iffais assumed to be piecewise polynomial on the ROI. In this paper, we tackle the related question of stability. It is well-known that lambda tomography allows one to stably recover the locations and values of the jumps offainside the ROI from only the local data. Hence, we consider here only the case of a polynomial, rather than piecewise polynomial,faon the ROI. Assuming that the degree of the polynomial is known, along with some other fairly mild assumptions onfa, we prove a stability estimate for the interior problem. Additionally, we prove the following general uniqueness result. If there is an open setUon whichfais the restriction of a real-analytic function, thenfais uniquely determined by only the line integrals throughU. It turns out that two known uniqueness theorems are corollaries of this result.
Princeton University, Princeton, NJ 08544Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364Biomedical Imaging Division, VT-WFU School of Biomedical Engineering and Sciences, Virginia Tech, Blacksburg, VA 24061, USA.
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Katsevich
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Wang