Mohammad-Djafari Ali
Laboratoire des Signaux et Système, CNRS, CentraleSupélec-University Paris Saclay, 91192 Gif-sur-Yvette, France.
International Science Consulting and Training (ISCT), 91440 Bures-sur-Yvette, France.
Entropy (Basel). 2021 Dec 13;23(12):1673. doi: 10.3390/e23121673.
Classical methods for inverse problems are mainly based on regularization theory, in particular those, that are based on optimization of a criterion with two parts: a data-model matching and a regularization term. Different choices for these two terms and a great number of optimization algorithms have been proposed. When these two terms are distance or divergence measures, they can have a Bayesian Maximum A Posteriori (MAP) interpretation where these two terms correspond to the likelihood and prior-probability models, respectively. The Bayesian approach gives more flexibility in choosing these terms and, in particular, the prior term via hierarchical models and hidden variables. However, the Bayesian computations can become very heavy computationally. The machine learning (ML) methods such as classification, clustering, segmentation, and regression, based on neural networks (NN) and particularly convolutional NN, deep NN, physics-informed neural networks, etc. can become helpful to obtain approximate practical solutions to inverse problems. In this tutorial article, particular examples of image denoising, image restoration, and computed-tomography (CT) image reconstruction will illustrate this cooperation between ML and inversion.
经典的反问题方法主要基于正则化理论，特别是那些基于对包含两部分的准则进行优化的方法：数据-模型匹配项和正则化项。针对这两项已经提出了不同的选择以及大量的优化算法。当这两项是距离或散度度量时，它们可以有贝叶斯最大后验（MAP）解释，其中这两项分别对应于似然模型和先验概率模型。贝叶斯方法在选择这些项时，特别是通过层次模型和隐藏变量选择先验项时，具有更大的灵活性。然而，贝叶斯计算在计算上可能会变得非常繁重。基于神经网络（NN），特别是卷积神经网络、深度神经网络、物理信息神经网络等的机器学习（ML）方法，如分类、聚类、分割和回归，可能有助于获得反问题的近似实际解决方案。在本教程文章中，图像去噪、图像恢复和计算机断层扫描（CT）图像重建的具体示例将说明ML与反演之间的这种协作。
