Quantitative thermo-acoustic imaging: an exact reconstruction formula

Habib Ammari Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 Rue dʼUlm, 75005 Paris, France Josselin Garnier Laboratoire de Probabilités et Modèles Aléatoires, France Wenjia Jing Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 Rue dʼUlm, 75005 Paris, France Loc Hoang Nguyen Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 Rue dʼUlm, 75005 Paris, France

Analysis of PDEs mathscidoc:2206.03005

Journal of Differential Equations, 254, (3), 1375-1395, 2013.2
This paper aims to mathematically advance the field of quantitative thermo-acoustic imaging. Given several electromagnetic data sets, we establish for the first time an analytical formula for reconstructing the absorption coefficient from thermal energy measurements. Since the formula involves derivatives of the given data up to the third order, it is unstable in the sense that small measurement noises may cause large errors. However, in the presence of measurement noise, the obtained formula, together with a noise regularization technique, provides a good initial guess for the true absorption coefficient. We finally correct the errors by deriving a reconstruction formula based on the least square solution of an optimal control problem and prove that this optimization step reduces the errors occurring and enhances the resolution.
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@inproceedings{habib2013quantitative,
  title={Quantitative thermo-acoustic imaging: an exact reconstruction formula},
  author={Habib Ammari, Josselin Garnier, Wenjia Jing, and Loc Hoang Nguyen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220622155006411535450},
  booktitle={Journal of Differential Equations},
  volume={254},
  number={3},
  pages={1375-1395},
  year={2013},
}
Habib Ammari, Josselin Garnier, Wenjia Jing, and Loc Hoang Nguyen. Quantitative thermo-acoustic imaging: an exact reconstruction formula. 2013. Vol. 254. In Journal of Differential Equations. pp.1375-1395. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220622155006411535450.
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