STABILITY OF THE INTERIOR PROBLEM FOR POLYNOMIAL REGION OF INTEREST

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.

Figure 1

The setup for finding a bound for $g_k^{\prime }(p)$.

Figure 2

Contour of integration Γ.

Figure 3

Analytic continuations of φ_k(r) from the interval I_δ.

Figure 4

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Figure 7

Conformal mappings taking initial semi-disk D⁺ to upper half-plane. The boldface lines indicate the images of (δ, R₁ − δ), while the dotted lines indicate the images of ∂D⁺ \ (δ, R₁ − δ).

Figure 8

$S_{\delta }^{″}$ and the maximum argument on the semicircle.

Figure 9

A plot of φ_c(x) for three different values of c