Regularization Graphs -- A unified framework for variational regularization of inverse problems
arXiv:2111.03509 · doi:10.1088/1361-6420/ac668d
Abstract
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear operators and convex functionals, assembled by means of operators that can be seen as generalizations of classical infimal convolution operators. This class of functionals exhaustively covers existing regularization approaches and it is flexible enough to craft new ones in a simple and constructive way. We provide well-posedness and convergence results with the proposed class of functionals in a general setting. Further, we consider a bilevel optimization approach to learn optimal weights for such regularization graphs from training data. We demonstrate that this approach is capable of optimizing the structure and the complexity of a regularization graph, allowing, for example, to automatically select a combination of regularizers that is optimal for given training data.
References in corpus (7)
- A combined first and second order variational approach for image reconstruction
- Bilevel parameter learning for higher-order total variation regularisation models
- Higher-order total variation approaches and generalisations
- The Residual Method for Regularizing Ill-Posed Problems
- Infimal convolution of oscillation total generalized variation for the recovery of images with structured texture
- A convex variational model for learning convolutional image atoms from incomplete data
- Adaptive image processing: first order PDE constraint regularizers and a bilevel training scheme