# Functional Liftings of Vectorial Variational Problems with Laplacian Regularization

## TL;DR

This paper introduces a convex relaxation method based on functional lifting for vectorial variational problems with Laplacian regularization, enabling advanced image processing applications like 2D image registration.

## Contribution

It develops a novel convex relaxation framework that extends existing multilabeling approaches to higher-order regularization and vectorial data, with theoretical and practical insights.

## Key findings

- Encompasses sublabel-accurate multilabeling as a special case
- Provides a mathematical connection between lifted and original functionals
- Successfully applied to 2D image registration problems

## Abstract

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.