Abstract
Inverse transport theory concerns the reconstruction of the absorption and scattering coefficients in a transport equation from knowledge of the albedo operator, which models all possible boundary measurements. Uniqueness and stability results are well known and are typically obtained for errors of the albedo operator measured in the \begin{document} $L^1$ \end{document} sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in \begin{document} $O(1)$ \end{document} errors of the albedo operator and hence in \begin{document} $O(1)$ \end{document} error predictions on the reconstruction of the coefficients, which are not useful. This paper revisit such stability estimates by introducing a more forgiving metric on the measurements errors, namely the \begin{document} $1-$ \end{document} Wasserstein distances, which penalize blurring or misalignment by an amount proportional to the width of the blurring kernel or to the amount of misalignment. We obtain new stability estimates in this setting. We also consider the effect of errors, still measured in the \begin{document} $1-$ \end{document} Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allows us to consider discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.
Highlights
We consider the inverse problem theory of the following stationary linear transport equation:v · ∇u(x, v) + σ(x, v)u = k(x, v′, v)u(x, v′)dv′, (x, v) ∈ X × VV u(x, v) = g(x, v),(x, v) ∈ Γ−. (1.1)The solution u(x, v) models the density of particles, such as photons, as a function of space x and velocity v
We introduce measurement error on the albedo operator A that are significantly more forgiving and more likely to be met in practical settings
Our choice of the 1−Wasserstein distance in inverse transport theory is based on two observations: (i) it is sufficiently versatile to provide small penalties on a large class of classical errors such as small blurring or misalignment, both at the detector and source levels; and (ii) its structure is quite amenable to relatively simple mathematical analysis and generalizations of results obtained in the setting of L1 errors
Summary
We consider the inverse problem theory of the following stationary (timeindependent) linear transport equation:. A second objective is to consider the setting in which the source term g is replaced by another source term that is close in the same Wasserstein sense This allows us to consider the case of errors in the alignment and design of (e.g.,) laser probes. This type of error, typically neglected in standard stability estimates, is important in practice to model probing signals that are never perfectly known. Our choice of the 1−Wasserstein distance in inverse transport theory is based on two observations: (i) it is sufficiently versatile to provide small penalties on a large class of classical errors such as small blurring or misalignment, both at the detector and source levels; and (ii) its structure is quite amenable to relatively simple mathematical analysis and generalizations of results obtained in the setting of L1 errors. For a selected list of references, see, e.g., [1, 10, 11, 12, 20, 21, 22, 23, 27, 29, 30, 31]
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