School of Electronics and Information Engineering, Soochow University, Suzhou 215021, China.
Biomed Eng Online. 2010 May 20;9:20. doi: 10.1186/1475-925X-9-20.
The inverse problem of fluorescent molecular tomography (FMT) often involves complex large-scale matrix operations, which may lead to unacceptable computational errors and complexity. In this research, a tree structured Schur complement decomposition strategy is proposed to accelerate the reconstruction process and reduce the computational complexity. Additionally, an adaptive regularization scheme is developed to improve the ill-posedness of the inverse problem.
The global system is decomposed level by level with the Schur complement system along two paths in the tree structure. The resultant subsystems are solved in combination with the biconjugate gradient method. The mesh for the inverse problem is generated incorporating the prior information. During the reconstruction, the regularization parameters are adaptive not only to the spatial variations but also to the variations of the objective function to tackle the ill-posed nature of the inverse problem.
Simulation results demonstrate that the strategy of the tree structured Schur complement decomposition obviously outperforms the previous methods, such as the conventional Conjugate-Gradient (CG) and the Schur CG methods, in both reconstruction accuracy and speed. As compared with the Tikhonov regularization method, the adaptive regularization scheme can significantly improve ill-posedness of the inverse problem.
The methods proposed in this paper can significantly improve the reconstructed image quality of FMT and accelerate the reconstruction process.
荧光分子断层扫描（FMT）的逆问题通常涉及复杂的大规模矩阵运算，这可能导致不可接受的计算误差和复杂性。在这项研究中，提出了一种树状 Schur 补分解策略，以加速重建过程并降低计算复杂度。此外，还开发了一种自适应正则化方案来改善逆问题的不适定性。
通过树结构中的 Schur 补系统沿着两条路径对全局系统进行分层分解。将所得子系统与双共轭梯度法相结合求解。逆问题的网格是结合先验信息生成的。在重建过程中，正则化参数不仅对空间变化而且对目标函数的变化都是自适应的，以解决逆问题的不适定性。
仿真结果表明，树状 Schur 补分解策略在重建准确性和速度方面明显优于传统的共轭梯度（CG）和 Schur CG 方法等先前方法。与 Tikhonov 正则化方法相比，自适应正则化方案可以显著改善逆问题的不适定性。
本文提出的方法可以显著提高 FMT 的重建图像质量并加速重建过程。
