Bilevel optimization for calibrating point spread functions in blind deconvolution

Abstract

Blind deconvolution problems arise in many imaging modalities, where both the underlying point spread function, which parameterizes the convolution operator, and the source image need to be identified. In this work, a novel bilevel optimization approach to blind deconvolution is proposed. The lower-level problem refers to the minimization of a total-variation model, as is typically done in non-blind image deconvolution. The upper-level objective takes into account additional statistical information depending on the particular imaging modality. Bilevel problems of such type are investigated systematically. Analytical properties of the lower-level solution mapping are established based on Robinson's strong regularity condition. Furthermore, several stationarity conditions are derived from the variational geometry induced by the lower-level problem. Numerically, a projected-gradient-type method is employed to obtain a Clarke-type stationary point and its convergence properties are analyzed. We also implement an efficient version of the proposed algorithm and test it through the experiments on point spread function calibration and multiframe blind deconvolution.

Solving inverse problems using data-driven models

“…This set-up can be tailored towards learning the regularization functional , or the data fidelity term , or an appropriate component in a forward operator , e.g. in blind image deconvolution (Hintermüller and Wu 2015). The starting point is to have access to supervised training data that are generated by a -valued random variable .…”

“…A revival of bilevel learning in the context of non-smooth regularizers took place in 2013 with a series of papers: De los Reyes and Schönlieb (2013), Calatroni, De los Reyes and Schönlieb (2014), De los Reyes, Schönlieb and Valkonen (2016, 2017), Calatroni, De los Reyes and Schönlieb (2017), Van Chung, De los Reyes and Schönlieb (2017), Kunisch and Pock (2013), Chen, Pock and Bischof (2012), Chen, Yu and Pock (2015), Chung and Espanol (2017), Hintermüller and Wu (2015), Hintermüller and Rautenberg (2017), Hintermüller, Rautenberg, Wu and Langer (2017), Baus, Nikolova and Steidl (2014) and Schmidt and Roth (2014). A critical theoretical issue is the well-posedness of the learning; another is to derive a characterization of the optimal solutions that can be used in the design of computational methods.…”

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

“…Let us remark that even for finite-dimensional problems, there are few recent references dealing with stationarity conditions and solution algorithms for this type of problems (see, e.g. [18,30,33,34,38]). …”

Modern regularization methods for inverse problems

“…2017, Kobler, Klatzer, Hammernik and Pock 2017, Klatzer et al. 2017), and other works related to image processing (Ochs, Ranftl, Brox and Pock 2015, Hintermüller and Wu 2015).…”