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Variational Networks: An Optimal Control Approach to Early Stopping Variational Methods for Image Restoration.
Journal of Mathematical Imaging and Vision ( IF 2 ) Pub Date : 2020-03-11 , DOI: 10.1007/s10851-019-00926-8 Alexander Effland 1 , Erich Kobler 1 , Karl Kunisch 2 , Thomas Pock 1
Journal of Mathematical Imaging and Vision ( IF 2 ) Pub Date : 2020-03-11 , DOI: 10.1007/s10851-019-00926-8 Alexander Effland 1 , Erich Kobler 1 , Karl Kunisch 2 , Thomas Pock 1
Affiliation
We investigate a well-known phenomenon of variational approaches in image processing, where typically the best image quality is achieved when the gradient flow process is stopped before converging to a stationary point. This paradox originates from a tradeoff between optimization and modeling errors of the underlying variational model and holds true even if deep learning methods are used to learn highly expressive regularizers from data. In this paper, we take advantage of this paradox and introduce an optimal stopping time into the gradient flow process, which in turn is learned from data by means of an optimal control approach. After a time discretization, we obtain variational networks, which can be interpreted as a particular type of recurrent neural networks. The learned variational networks achieve competitive results for image denoising and image deblurring on a standard benchmark data set. One of the key theoretical results is the development of first- and second-order conditions to verify optimal stopping time. A nonlinear spectral analysis of the gradient of the learned regularizer gives enlightening insights into the different regularization properties.
中文翻译:
变异网络:一种用于图像复原的早期停止变异方法的最优控制方法。
我们研究了图像处理中一种众所周知的变分方法现象,通常当梯度流过程在收敛到固定点之前停止时,可以获得最佳图像质量。此悖论源于基础变分模型的优化和建模误差之间的折衷,即使采用深度学习方法从数据中学习高度表达性的正则化方法也是如此。在本文中,我们利用了这一悖论,并在梯度流过程中引入了最佳的停止时间,然后通过最佳控制方法从数据中学习了最佳的停止时间。经过时间离散后,我们获得了变分网络,可以将其解释为递归神经网络的一种特殊类型。所学习的变分网络在标准基准数据集上实现图像去噪和图像去模糊的竞争性结果。关键的理论结果之一是开发一阶和二阶条件以验证最佳停止时间。对所学习的正则化器的梯度进行非线性频谱分析,可以洞悉不同的正则化属性。
更新日期:2020-03-11
中文翻译:
变异网络:一种用于图像复原的早期停止变异方法的最优控制方法。
我们研究了图像处理中一种众所周知的变分方法现象,通常当梯度流过程在收敛到固定点之前停止时,可以获得最佳图像质量。此悖论源于基础变分模型的优化和建模误差之间的折衷,即使采用深度学习方法从数据中学习高度表达性的正则化方法也是如此。在本文中,我们利用了这一悖论,并在梯度流过程中引入了最佳的停止时间,然后通过最佳控制方法从数据中学习了最佳的停止时间。经过时间离散后,我们获得了变分网络,可以将其解释为递归神经网络的一种特殊类型。所学习的变分网络在标准基准数据集上实现图像去噪和图像去模糊的竞争性结果。关键的理论结果之一是开发一阶和二阶条件以验证最佳停止时间。对所学习的正则化器的梯度进行非线性频谱分析,可以洞悉不同的正则化属性。