Functional Liftings of Vectorial Variational Problems with Laplacian Regularization
For researchers in image processing and optimization, this work provides a theoretical framework to handle vectorial data with higher-order regularization, though it is incremental as it extends existing lifting techniques.
The paper proposes a convex relaxation for variational problems with Laplacian-based second-order regularization, extending functional lifting to vectorial data and higher-order regularization. It proves connections to sublabel-accurate multilabeling and demonstrates applicability to 2D image registration.
We propose a functional lifting-based convex relaxation of variational problems with Laplacian-based second-order regularization. The approach rests on ideas from the calibration method as well as from sublabel-accurate continuous multilabeling approaches, and makes these approaches amenable for variational problems with vectorial data and higher-order regularization, as is common in image processing applications. We motivate the approach in the function space setting and prove that, in the special case of absolute Laplacian regularization, it encompasses the discretization-first sublabel-accurate continuous multilabeling approach as a special case. We present a mathematical connection between the lifted and original functional and discuss possible interpretations of minimizers in the lifted function space. Finally, we exemplarily apply the proposed approach to 2D image registration problems.