# Lifting methods for manifold-valued variational problems

## TL;DR

This paper reviews lifting methods that transform complex manifold-valued variational problems into convex problems in higher-dimensional spaces, enabling efficient solutions and extending techniques to general convex regularization.

## Contribution

It introduces a refined finite element discretization framework for manifold-valued lifting methods and generalizes total variation regularization techniques.

## Key findings

- Extended lifting methods to manifold-valued problems.
- Supported general convex regularization in the lifted framework.
- Facilitated global optimization of complex variational problems.

## Abstract

Lifting methods allow to transform hard variational problems such as segmentation and optical flow estimation into convex problems in a suitable higher-dimensional space. The lifted models can then be efficiently solved to a global optimum, which allows to find approximate global minimizers of the original problem. Recently, these techniques have also been applied to problems with values in a manifold. We provide a review of such methods in a refined framework based on a finite element discretization of the range, which extends the concept of sublabel-accurate lifting to manifolds. We also generalize existing methods for total variation regularization to support general convex regularization.