Fluorescence molecular tomography (FMT) is a non-invasive technique that enables quantitative analysis of pathological processes at the cellular and molecular levels in vivo. The reconstruction of FMT is an ill-posed inverse problem, making it challenging to achieve fast and accurate reconstruction. Regularization methods, such as Tikhonov regularization and sparsity regularization, are typically used to address this issue. Given that tumors are small and sparse compared to the entire imaging domain, sparsity regularization is usually beneficial. The fast iterative shrinkage thresholding algorithm (FISTA) is proposed for theL₁-norm regularization problem and has shown good performance. Classical FISTA employs a linearly increasing search strategy to determine the Lipschitz constant. However, if the proximal gradient condition is satisfied during the initial stages of algorithm iteration, the Lipschitz constant remains unchanged, hindering the convergence of the algorithm. To address this issue, we propose a step-size search method based on a restart strategy, which can provide appropriate Lipschitz constants during the iterations to accelerate the convergence speed of FISTA.

In this study, an adaptive Lipschitz constant is provided at each iteration. The Lipschitz constant is increased by a growth factor containing gradient information. When the Lipschitz constant remains unchanged between two iterations, it may be too large, resulting in a small step size and slow convergence. Therefore, a truncation restart strategy is employed. The initial Lipschitz constant is selected as the current Lipschitz constant. We call this method restart fast iterative shrinkage thresholding algorithm (R-FISTA).

To test the performance of R-FISTA, numerical simulation experiments and in vivo experiments are conducted with both classical FISTA and R-FISTA. In the simulation experiments with different numbers of excitation points, R-FISTA takes less time compared to FISTA (Table 2 and Fig. 2). In addition, different levels (5%, 10%, 15%, 20%, 25%) of noises are considered to test the stability of the method. We find that R-FISTA provides better results according to location error (LE) compared to FISTA (Fig. 3 and Fig. 4). Notably, R-FISTA consumes less reconstruction time compared to FISTA. The real mouse experiment further shows that R-FISTA has a faster convergence speed compared to FISTA, consistent with the simulations (Fig. 2 and Fig. 4). These results demonstrate that R-FISTA accelerates the convergence speed of FISTA.

In this study, we propose a fast reconstruction algorithm for FMT based on FISTA, named R-FISTA. A restart strategy is proposed to search for the step size, providing appropriate Lipschitz constants during the iterations, thereby accelerating the convergence speed of FISTA. Numerical simulation experiments and in vivo experiments have shown that compared with classical FISTA, the R-FISTA algorithm effectively accelerates the reconstruction speed while ensuring high accuracy of FMT reconstruction. This fast reconstruction algorithm makes real-time 3D reconstruction possible. Deep learning has been a hot topic in recent years and has been applied to FMT, such as 3D deep encoder-decoder networks and stacked autoencoder neural networks. However, the explanation and generalization of deep learning methods need further study. Our future work will focus on combining the model with the network to solve the ill-posedness of FMT.