Functional Lifting: Nonlinear Inverse Scale Space Iterations and Neural Fields

Variational models lie at the core of formulating and understanding problems in modern mathematical image processing. Suitable mathematical models that accurately describes the given problem often have undesirable properties, in particular non-convexity. This thesis is built around a functional lifting strategy which embeds non-convex problems into a larger space, so that the lifted problem is convex and that global minimizers of the lifted problem can be mapped to global minimizers of the original problem. The contributions of this thesis are threefold.
We give a thorough introduction to the calibration-based lifting approach. In the literature, this approach is described in a 𝑊1,1 setting. A first contribution of this thesis is the compilation of measure theoretic results in order to extend the approach and related theory to the BV setting. In addition, we also discuss possible discretization and optimization approaches found in the literature.
Calibration-based lifting is then used in order to extend the use case of inverse scale
space iterations. These iterations are related to non-linear spectral representation and filtering techniques, which have proven to give impressive results in different imaging applications. While existing theory is built around the convex deconvolution or denoising problem, our goal is to generalize the results to variational problems with arbitrary, possibly non-convex data term and total variation regularizer.
Finally, we propose a neural őelds based stochastic optimization approach for solving variational problems with non-convex data term and total variation regularizer. By combining calibration-based lifting with the powerful neural fields computational framework, we aim to present a novel stochastic optimization strategy for this difficult class of variational problems.