Bioluminescence tomography via a shape optimization method based on a complex-valued model
Qianqian Wu, Rongfang Gong, Wei Gong, Ziyi Zhang, Shengfeng Zhu
TL;DR
This work tackles the BLT inverse source problem by recasting it as a shape optimization task that simultaneously recovers source geometry and strength from boundary measurements under a steady-state elliptic diffusion model. A key innovation is decoupling the source support from its intensity using an adjoint-based formulation and a parameter-free, coupled complex boundary method (CCBM) within a level-set framework, enabling topological changes without a priori geometry. The authors establish well-posedness, derive shape derivatives, and develop a two-step algorithm that first evolves the source boundary via level-set descent and then reconstructs intensity with CCBM, achieving robust performance on noisy data and enabling accurate recovery of multiple, closely spaced, or nested sources. Numerical experiments in 2D demonstrate superior geometric accuracy and stability compared with existing approaches, indicating strong potential for practical, noninvasive BLT applications with complex source morphologies.
Abstract
In this study, we investigate the inverse source problem arising in bioluminescence tomography, the objective of which is to reconstruct both the support and the intensity of an internal light source from boundary measurements governed by an elliptic model. A shape optimization framework is developed in which the source intensity and its support are decoupled through first-order optimality conditions. To enhance the stability of the reconstruction, we incorporate a parameter-dependent coupled complex boundary method together with perimeter and volume regularizations. Source support is represented by a level set function, allowing the algorithm to naturally accommodate topological changes and recover multiple, closely spaced, or nested source regions. Theoretical justifications for the proposed formulation and regularization strategy are established, and extensive numerical experiments are performed to assess the reconstruction accuracy for both noise-free and noisy data. The results demonstrate that our method achieves robust and accurate recovery of source geometry and intensity, and exhibits clear advantages over existing approaches.