Adagrad-optimized variational Bayesian reconstruction with sparsity-adaptive normal-generalized inverse Gaussian prior for fluorescence molecular tomography

Fluorescence molecular tomography (FMT) is a promising medical imaging technology with the ability to quantitatively reconstruct the three-dimensional distribution of fluorescently labeled probes in vivo. However, due to the strong scattering properties of biological tissues, conventional reconstruction methods encounter challenges such as low reconstruction accuracy and high computational complexity. Here, an adaptive online variational Bayesian method based on the normal-generalized inverse Gaussian (NGIG) prior is proposed. This method reduces computational complexity while ensuring that the globally optimal solution is maintained. Specifically, by utilizing variational inference, the optimization of the objective function is converted into a convex optimization problem that minimizes the variational lower bound, effectively reducing the function's complexity. Furthermore, to accurately capture the prior distribution, the NGIG prior is introduced. It imposes probabilistic constraints on the sparsity structure. This approach alleviates the adverse effects caused by overly strict sparsity constraints. In addition, the adaptive gradient algorithm (Adagrad) is employed to dynamically adjust the parameter learning rate, thereby preventing the algorithm from becoming trapped in local optima during the posterior inference process. The effectiveness of the proposed method is validated through numerical simulations and fluorescence source implantation experiments. The results show that the adaptive online variational Bayesian (AOVB)-NGIG method achieves superior performance in both fluorescence source localization and shape recovery. The minimum localization error is 0.243 mm, accompanied by a dice coefficient of 0.889. Meanwhile, the root mean square error and relative intensity error remain relatively low, indicating that the reconstructed results are the closest to the actual light source. These outcomes demonstrate that AOVB-NGIG can reliably reconstruct the spatial characteristics of the fluorescence source with high accuracy. This study is expected to advance the preclinical and clinical applications of FMT in early tumor detection.