Maximum Likelihood Estimation of Regularization Parameters in High-Dimensional Inverse Problems: An Empirical Bayesian Approach Part I: Methodology and Experiments
Maximum Likelihood Estimation of Regularization Parameters in High-Dimensional Inverse Problems: An Empirical Bayesian Approach Part I: Methodology and Experiments
复制标题
高维逆问题中正则化参数的最大似然估计：经验贝叶斯方法第一部分：方法和实验
DOI：
10.1137/20m1339829
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Alain Durmus
中科院分区：
文献类型：
--
作者：
A. F. Vidal;Valentin De Bortoli;M. Pereyra;Alain Durmus
许多成像问题需要求解病态或不适定的逆问题。成像方法通常通过正则化估计问题使其适定性来解决这个困难。这通常需要设置所谓的正则化参数的值，这些参数控制强制执行的正则化的量。众所周知，这些参数很难事先设定，并且可能对恢复的估计值产生巨大影响。在这项工作中，我们提出了一个一般的经验贝叶斯方法设置正则化参数的成像问题是凸w.r.t.未知的形象。我们的方法校准正则化参数直接从观测数据的最大边际似然估计，并可以同时估计多个正则化参数。此外，所提出的算法使用相同的基本运营商作为近端优化算法，即梯度和近端运营商，因此，它是直接适用于目前通过使用近端优化技术解决的问题。我们的方法证明了一系列的实验和比较与文献中的替代方法。所考虑的实验包括图像去噪，非盲图像反卷积，和高光谱解混，使用合成和分析先验涉及L1，全变差，全变差和L1，和总广义变差伪范数。在配套论文arXiv：2008.05793中给出了所提出的方法的详细理论分析。
Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularising the estimation problem to make it well-posed. This often requires setting the value of the so-called regularisation parameters that control the amount of regularisation enforced. These parameters are notoriously difficult to set a priori, and can have a dramatic impact on the recovered estimates. In this work, we propose a general empirical Bayesian method for setting regularisation parameters in imaging problems that are convex w.r.t. the unknown image. Our method calibrates regularisation parameters directly from the observed data by maximum marginal likelihood estimation, and can simultaneously estimate multiple regularisation parameters. Furthermore, the proposed algorithm uses the same basic operators as proximal optimisation algorithms, namely gradient and proximal operators, and it is therefore straightforward to apply to problems that are currently solved by using proximal optimisation techniques. Our methodology is demonstrated with a range of experiments and comparisons with alternative approaches from the literature. The considered experiments include image denoising, non-blind image deconvolution, and hyperspectral unmixing, using synthesis and analysis priors involving the L1, total-variation, total-variation and L1, and total-generalised-variation pseudo-norms. A detailed theoretical analysis of the proposed method is presented in the companion paper arXiv:2008.05793.
高维反问题中正则化参数的最大似然估计：经验贝叶斯方法。
影响因子：
2.1
作者：
De Bortoli V
通讯作者：
De Bortoli V
来自少量噪声数据的稀疏重建：具有广义伽马超先验的分层贝叶斯模型分析
影响因子：
2.1
作者：
Calvetti, Daniela;Pragliola, Monica;Somersalo, Erkki;Strang, Alexander
通讯作者：
Strang, Alexander
影响因子：
14.2
作者：
Arridge, Simon;Maass, Peter;Schonlieb, Carola-Bibiane
通讯作者：
Schonlieb, Carola-Bibiane