Functional Lifting for Variational Problems with Higher-Order Regularization
Variational approaches are an established paradigm in the field of image processing. The non-convexity of the functional can be addressed by functional lifting and convex relaxation techniques, which aim to solve a convex approximation of the original energy on a larger space. However, so far these approaches have been limited to first-order , gradient-based regularizers such as the total variation . In this work, we propose a way to extend functional lifting to a second-order regularizer derived from the Laplacian. We prove that it can be represented efficiently and thus allows numerical optimization. We experimentally demonstrate the usefulness on a synthetic convex denoising problem and on synthetic as well as real-world image registration problems.
This work was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—394737018 “Functional Lifting 2.0: Efficient Convexifications for Imaging and Vision”. We would like to thank Emanuel Laude and Thomas Möllenhoff for providing their library prost, which was used to solve the saddle-point formulation of the problems.
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