Inertial gradient method for fluorescence molecular tomography

Abstract

Image reconstruction in fluorescence molecular tomography involves seeking stable and meaningful solutions via the inversion of a highly under-determined and severely ill-posed linear mapping. An attractive scheme consists of minimizing a convex objective function that includes a quadratic error term added to a convex and nonsmooth sparsity-promoting regularizer. Choosing [Formula: see text]-norm as a particular case of a vast class of nonsmooth convex regularizers, our paper proposes a low per-iteration complexity gradient-based first-order optimization algorithm for the [Formula: see text]-regularized least squares inverse problem of image reconstruction. Our algorithm relies on a combination of two ideas applied to the nonsmooth convex objective function: Moreau–Yosida regularization and inertial dynamics-based acceleration. We also incorporate into our algorithm a gradient-based adaptive restart strategy to further enhance the practical performance. Extensive numerical experiments illustrate that in several representative test cases (covering different depths of small fluorescent inclusions, different noise levels and different separation distances between small fluorescent inclusions), our algorithm can significantly outperform three state-of-the-art algorithms in terms of CPU time taken by reconstruction, despite almost the same reconstructed images produced by each of the four algorithms.