Li Zongyu, Lange Kenneth, Fessler Jeffrey A
Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122.
Departments of Computational Medicine, Human Genetics, and Statistics, University of California, Los Angeles, CA 90095.
IEEE Trans Comput Imaging. 2022;8:838-850. doi: 10.1109/tci.2022.3209936. Epub 2022 Oct 5.
This paper discusses phase retrieval algorithms for maximum likelihood (ML) estimation from measurements following independent Poisson distributions in very low-count regimes, e.g., 0.25 photon per pixel. To maximize the log-likelihood of the Poisson ML model, we propose a modified Wirtinger flow (WF) algorithm using a step size based on the observed Fisher information. This approach eliminates all parameter tuning except the number of iterations. We also propose a novel curvature for majorize-minimize (MM) algorithms with a quadratic majorizer. We show theoretically that our proposed curvature is sharper than the curvature derived from the supremum of the second derivative of the Poisson ML cost function. We compare the proposed algorithms (WF, MM) with existing optimization methods, including WF using other step-size schemes, quasi-Newton methods such as LBFGS and alternating direction method of multipliers (ADMM) algorithms, under a variety of experimental settings. Simulation experiments with a random Gaussian matrix, a canonical DFT matrix, a masked DFT matrix and an empirical transmission matrix demonstrate the following. 1) As expected, algorithms based on the Poisson ML model consistently produce higher quality reconstructions than algorithms derived from Gaussian noise ML models when applied to low-count data. Furthermore, incorporating regularizers, such as corner-rounded anisotropic total variation (TV) that exploit the assumed properties of the latent image, can further improve the reconstruction quality. 2) For unregularized cases, our proposed WF algorithm with Fisher information for step size converges faster (in terms of cost function and PSNR vs. time) than other WF methods, e.g., WF with empirical step size, backtracking line search, and optimal step size for the Gaussian noise model; it also converges faster than the LBFGS quasi-Newton method. 3) In regularized cases, our proposed WF algorithm converges faster than WF with backtracking line search, LBFGS, MM and ADMM.
本文讨论了在极低计数情况下（例如，每像素0.25个光子），基于独立泊松分布测量进行最大似然（ML）估计的相位恢复算法。为了最大化泊松ML模型的对数似然，我们提出了一种改进的Wirtinger流（WF）算法，该算法使用基于观测Fisher信息的步长。这种方法消除了除迭代次数之外的所有参数调整。我们还为具有二次主元化器的主元化-最小化（MM）算法提出了一种新颖的曲率。我们从理论上表明，我们提出的曲率比从泊松ML代价函数二阶导数的上确界导出的曲率更尖锐。我们在各种实验设置下，将所提出的算法（WF、MM）与现有的优化方法进行了比较，包括使用其他步长方案的WF、拟牛顿方法（如LBFGS）和乘子交替方向法（ADMM）算法。使用随机高斯矩阵、规范DFT矩阵、掩码DFT矩阵和经验传输矩阵进行的模拟实验表明了以下几点。1）正如预期的那样，当应用于低计数数据时，基于泊松ML模型的算法始终比从高斯噪声ML模型导出的算法产生更高质量的重建。此外，纳入正则化器，如利用潜在图像假定属性的角舍入各向异性总变分（TV），可以进一步提高重建质量。2）对于未正则化的情况，我们提出的具有Fisher信息步长的WF算法比其他WF方法收敛得更快（就代价函数和PSNR与时间而言），例如具有经验步长、回溯线搜索和高斯噪声模型最优步长的WF；它也比LBFGS拟牛顿方法收敛得更快。3）在正则化的情况下，我们提出的WF算法比具有回溯线搜索的WF、LBFGS、MM和ADMM收敛得更快。
