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 ℓ1-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 ℓ1-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.

中文翻译：

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荧光分子断层扫描的惯性梯度法

荧光分子断层扫描中的图像重建涉及通过高度欠定和严重不适定的线性映射的反演来寻求稳定和有意义的解决方案。一个有吸引力的方案包括最小化一个凸目标函数，该函数包括一个添加到凸和非平滑稀疏促进正则化器的二次误差项。选择ℓ1-norm 作为一大类非光滑凸正则化器的特例，我们的论文提出了一种低每次迭代复杂度的基于梯度的一阶优化算法ℓ1- 图像重建的正则化最小二乘逆问题。我们的算法依赖于应用于非光滑凸目标函数的两种思想的组合：Moreau-Yosida 正则化和基于惯性动力学的加速。我们还在我们的算法中加入了基于梯度的自适应重启策略，以进一步提高实际性能。大量的数值实验表明，在几个具有代表性的测试案例中（涵盖不同深度的小荧光夹杂物、不同的噪声水平和不同的小荧光夹杂物之间的分离距离），我们的算法在 CPU 方面可以显着优于三种最先进的算法尽管四种算法中的每一种都产生了几乎相同的重建图像，但重建所花费的时间。