Stability of the interior problem with polynomial attenuation in the region of interest

    Authors

    E. Katsevich; A. Katsevich;G. Wang

    Comments

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    Abbreviated Journal Title

    Inverse Probl.

    Keywords

    TRUNCATED HILBERT TRANSFORM; IMAGE-RECONSTRUCTION; LOCAL TOMOGRAPHY; ALGORITHMS; CT; Mathematics, Applied; Physics, Mathematical

    Abstract

    In many practical applications, it is desirable to solve the interior problem of tomography without requiring knowledge of the attenuation function f(a) on an open set within the region of interest (ROI). It was proved recently that the interior problem has a unique solution if f(a) is 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 of f(a) inside the ROI from only the local data. Hence, we consider here only the case of a polynomial, rather than piecewise polynomial, f(a) on the ROI. Assuming that the degree of the polynomial is known, along with some other fairly mild assumptions on f(a), we prove a stability estimate for the interior problem. Additionally, we prove the following general uniqueness result. If there is an open set U on which f(a) is the restriction of a real-analytic function, then f(a) is uniquely determined by only the line integrals through U. It turns out that two known uniqueness theorems are corollaries of this result.

    Journal Title

    Inverse Problems

    Volume

    28

    Issue/Number

    6

    Publication Date

    1-1-2012

    Document Type

    Article

    Language

    English

    First Page

    28

    WOS Identifier

    WOS:000305401300022

    ISSN

    0266-5611

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