Volume 34, Issue 6
Analysis of a Numerical Method for Radiative Transfer Equation Based Bioluminescence Tomography

Rongfang Gong, Joseph Eichholz, Xiaoliang Cheng & Weimin Han

J. Comp. Math., 34 (2016), pp. 648-670.

Published online: 2016-12

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  • Abstract

In the bioluminescence tomography (BLT) problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem is ill-posed and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter to balance the accuracy and stability of approximate solutions. In this paper, a parameter-dependent coupled complex boundary method (CCBM) based Tikhonov regularization is applied to the BLT problem governed by the radiative transfer equation (RTE). By properly adjusting the parameter in the Robin boundary condition, we achieve one important property: the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy. The discrete-ordinate finite-element method is used to compute numerical solutions. Numerical results are provided to illustrate the performance of the proposed method.

  • Keywords

Bioluminescence tomography radiative transfer equation Tikhonov regularization coupled complex boundary method convergence

  • AMS Subject Headings

92C55 65F22 80M10.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

grf _ath@nuaa.edu.cn (Rongfang Gong)

eichholz@rose-hulman.edu (Joseph Eichholz)

xiaoliangcheng@zju.edu.cn (Xiaoliang Cheng)

weimin-han@uiowa.edu (Weimin Han)

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@Article{JCM-34-648, author = {Gong , Rongfang and Eichholz , Joseph and Cheng , Xiaoliang and Han , Weimin }, title = {Analysis of a Numerical Method for Radiative Transfer Equation Based Bioluminescence Tomography}, journal = {Journal of Computational Mathematics}, year = {2016}, volume = {34}, number = {6}, pages = {648--670}, abstract = { In the bioluminescence tomography (BLT) problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem is ill-posed and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter to balance the accuracy and stability of approximate solutions. In this paper, a parameter-dependent coupled complex boundary method (CCBM) based Tikhonov regularization is applied to the BLT problem governed by the radiative transfer equation (RTE). By properly adjusting the parameter in the Robin boundary condition, we achieve one important property: the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy. The discrete-ordinate finite-element method is used to compute numerical solutions. Numerical results are provided to illustrate the performance of the proposed method.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1607-m2016-0515}, url = {http://global-sci.org/intro/article_detail/jcm/9818.html} }
TY - JOUR T1 - Analysis of a Numerical Method for Radiative Transfer Equation Based Bioluminescence Tomography AU - Gong , Rongfang AU - Eichholz , Joseph AU - Cheng , Xiaoliang AU - Han , Weimin JO - Journal of Computational Mathematics VL - 6 SP - 648 EP - 670 PY - 2016 DA - 2016/12 SN - 34 DO - http://doi.org/10.4208/jcm.1607-m2016-0515 UR - https://global-sci.org/intro/article_detail/jcm/9818.html KW - Bioluminescence tomography KW - radiative transfer equation KW - Tikhonov regularization KW - coupled complex boundary method KW - convergence AB - In the bioluminescence tomography (BLT) problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem is ill-posed and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter to balance the accuracy and stability of approximate solutions. In this paper, a parameter-dependent coupled complex boundary method (CCBM) based Tikhonov regularization is applied to the BLT problem governed by the radiative transfer equation (RTE). By properly adjusting the parameter in the Robin boundary condition, we achieve one important property: the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy. The discrete-ordinate finite-element method is used to compute numerical solutions. Numerical results are provided to illustrate the performance of the proposed method.
Rongfang Gong , Joseph Eichholz , XiaoliangCheng & Weimin Han. (2020). Analysis of a Numerical Method for Radiative Transfer Equation Based Bioluminescence Tomography. Journal of Computational Mathematics. 34 (6). 648-670. doi:10.4208/jcm.1607-m2016-0515
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