Generalized stability estimates in inverse ...
Document type :
Compte-rendu et recension critique d'ouvrage
DOI :
Title :
Generalized stability estimates in inverse transport theory
Author(s) :
Bal, Guillaume [Auteur]
Department of Applied Physics and Applied Mathematics [New York] [APAM]
Jollivet, Alexandre [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Department of Applied Physics and Applied Mathematics [New York] [APAM]
Jollivet, Alexandre [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Journal title :
Inverse Problems and Imaging
Pages :
59-90
Publisher :
AIMS American Institute of Mathematical Sciences
Publication date :
2018
ISSN :
1930-8337
English keyword(s) :
Linear transport
inverse problems
stability estimates
Wasserstein distance
inverse problems
stability estimates
Wasserstein distance
HAL domain(s) :
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
English abstract : [en]
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 ...
Show more >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 $L^1$ sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in $O(1)$ errors of the albedo operator and hence in $O(1)$ 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 $1-$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 $1-$Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allow us to consider a discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.Show less >
Show more >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 $L^1$ sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in $O(1)$ errors of the albedo operator and hence in $O(1)$ 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 $1-$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 $1-$Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allow us to consider a discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.Show less >
Language :
Anglais
Popular science :
Non
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