# Sublabel-Accurate Relaxation of Nonconvex Energies

arXiv: https://arxiv.org/abs/1512.01383 · HTML: https://mlsanity.com/paper/1512.01383/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2015.

## Built on by (most cited first, 4 of 4)

- [VA-DepthNet: A Variational Approach to Single Image Depth Prediction](https://mlsanity.com/paper/2302.06556/) (2023)
- [Parameterized Temperature Scaling for Boosting the Expressive Power in Post-Hoc Uncertainty Calibration](https://mlsanity.com/paper/2102.12182/) (2021)
- [Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields](https://mlsanity.com/paper/2107.06028/) (2021)
- [Efficient and Flexible Sublabel-Accurate Energy Minimization](https://mlsanity.com/paper/2206.09596/) (2022)

## Abstract (arXiv)

We propose a novel spatially continuous framework for convex relaxations based on functional lifting. Our method can be interpreted as a sublabel-accurate solution to multilabel problems. We show that previously proposed functional lifting methods optimize an energy which is linear between two labels and hence require (often infinitely) many labels for a faithful approximation. In contrast, the proposed formulation is based on a piecewise convex approximation and therefore needs far fewer labels. In comparison to recent MRF-based approaches, our method is formulated in a spatially continuous setting and shows less grid bias. Moreover, in a local sense, our formulation is the tightest possible convex relaxation. It is easy to implement and allows an efficient primal-dual optimization on GPUs. We show the effectiveness of our approach on several computer vision problems.