Abstract
Incorporating prior information into inverse problems, e.g. via maximum-a-posteriori estimation, is an important technique for facilitating robust inverse problem solutions. In this paper, we devise two novel approaches for linear inverse problems that permit problem-specific statistical prior selections within the compound Gaussian (CG) class of distributions. The CG class subsumes many commonly used priors in signal and image reconstruction methods including those of sparsity-based approaches. The first method developed is an iterative algorithm, called generalized compound Gaussian least squares (G-CG-LS), that minimizes a regularized least squares objective function where the regularization enforces a CG prior. G-CG-LS is then unrolled, or unfolded, to furnish our second method, which is a novel deep regularized (DR) neural network, called DR-CG-Net, that learns the prior information. A detailed computational theory on convergence properties of G-CG-LS and thorough numerical experiments for DR-CG-Net are provided. Due to the comprehensive nature of the CG prior, these experiments show that our unrolled DR-CG-Net outperforms competitive prior art methods in tomographic imaging and compressive sensing, especially in challenging low-training scenarios.
Abstract (translated)
将先验信息融入反问题中,例如通过最大后验估计,是促进稳健反问题解的重要技术。在本文中,我们提出了两种新的线性反问题方法,允许在复合高斯(CG)分布中进行问题特定的统计先验选择。CG类包括许多在信号和图像重建方法中常用的先验,包括基于稀疏度的方法。我们开发的第一种方法是一个迭代算法,称为一般化复合高斯最小二乘(G-CG-LS)方法,它最小化一个正则化最小二乘目标函数,其中正则化强制执行CG先验。G-CG-LS然后展开或展开,以提供我们的第二种方法,即名为DR-CG-Net的新型深度正则化(DR)神经网络,它学习先验信息。关于G-CG-LS的收敛性质的详细计算理论和DR-CG-Net的深入数值实验都在本文中提供了。由于CG先验的全局性,这些实验表明,我们的未展开DR-CG-Net在断层成像和压缩感知领域优于竞争先驱技术,尤其是在具有挑战性的低训练场景中。
URL
https://arxiv.org/abs/2311.17248
