Abstract.In this paper we demonstrate that the framework of nonlinear spectral decompositions based on total variation (TV) regularization is very well suited for image fusion as well as more general image manipulation tasks. The well-localized and edge-preserving spectral TV decomposition allows to select frequencies of a certain image to transfer particular features, such as wrinkles in a face, from one image to another. We illustrate the effectiveness of the proposed approach in several numerical experiments, including a comparison to the competing techniques of Poisson image editing, linear osmosis, wavelet fusion and Laplacian pyramid fusion. We conclude that the proposed spectral TV image decomposition framework is a valuable tool for semi-and fullyautomatic image editing and fusion.

Nonlinear variational methods have become very powerful tools for many image processing tasks. Recently a new line of research has emerged, dealing with nonlinear eigenfunctions induced by convex functionals. This has provided new insights and better theoretical understanding of convex regularization and introduced new processing methods. However, the theory of nonlinear eigenvalue problems is still at its infancy. We present a new flow that can generate nonlinear eigenfunctions of the form T (u) = λu, where T (u) is a nonlinear operator and λ ∈ R is the eigenvalue. We develop the theory where T (u) is a subgradient element of a regularizing one-homogeneous functional, such as total-variation (TV) or total-generalized-variation (TGV). We introduce two flows: a forward flow and an inverse flow; for which the steady state solution is a nonlinear eigenfunction. The forward flow monotonically smooths the solution (with respect to the regularizer) and simultaneously increases the L 2 norm. The inverse flow has the opposite characteristics. For both flows, the steady state depends on the initial condition, thus different initial conditions yield different eigenfunctions. This enables a deeper investigation into the space of nonlinear eigenfunctions, allowing to produce numerically diverse examples, which may be unknown yet. In addition we suggest an indicator to measure the affinity of a function to an eigenfunction and relate it to pseudo-eigenfunctions in the linear case.

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Image demosaicing is an important step in the image processing pipeline for digital cameras, and it is one of the many tasks within the field of image restoration. A well-known characteristic of natural images is that most patches are smooth, while high-content patches like textures or repetitive patterns are much rarer, which results in a long-tailed distribution. This distribution can create an inductive bias when training machine learning algorithms for image restoration tasks and for image demosaicing in particular. There have been many different approaches to address this challenge, such as utilizing specific losses or designing special network architectures. What makes our work is unique in that it tackles the problem from a training protocol perspective. Our proposed training regime consists of two key steps. The first step is a data-mining stage where sub-categories are created and then refined through an elimination process to only retain the most helpful subcategories. The second step is a cyclic training process where the neural network is trained on both the mined subcategories and the original dataset. We have conducted various experiments to demonstrate the effectiveness of our training method for the image demosaicing task. Our results show that this method outperforms standard training across a range of architecture sizes and types, including CNNs and Transformers. Moreover, we are able to achieve state-of-the-art results with a significantly smaller neural network, compared to previous state-of-the-art methods.

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