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Variational networks: an optimal control approach to early stopping variational methods for image restoration. (English) Zbl 1434.68626
Summary: 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.
MSC:
68U10 Computing methodologies for image processing
34H05 Control problems involving ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
65L05 Numerical methods for initial value problems
68T07 Artificial neural networks and deep learning
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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