# Sublabel-Accurate Discretization of Nonconvex Free-Discontinuity Problems

@article{Mllenhoff2017SublabelAccurateDO, title={Sublabel-Accurate Discretization of Nonconvex Free-Discontinuity Problems}, author={Thomas M{\"o}llenhoff and Daniel Cremers}, journal={2017 IEEE International Conference on Computer Vision (ICCV)}, year={2017}, pages={1192-1200} }

In this work we show how sublabel-accurate multilabeling approaches [15, 18] can be derived by approximating a classical label-continuous convex relaxation of nonconvex free-discontinuity problems. This insight allows to extend these sublabel-accurate approaches from total variation to general convex and nonconvex regularizations. Furthermore, it leads to a systematic approach to the discretization of continuous convex relaxations. We study the relationship to existing discretizations and to…

## 12 Citations

Sublabel-Accurate Multilabeling Meets Product Label Spaces

- Computer ScienceGCPR
- 2021

This paper presents a combination of two approaches designed to make liftings more scalable, namely product-space relaxations and sublabel-accurate discretizations, which significantly outperforms baseline methods and finds solutions with lower energies given the same amount of memory.

Sublabel-Accurate Convex Relaxation with Total Generalized Variation Regularization

- Computer ScienceGCPR
- 2018

The proposed formulation extends a recent sublabel-accurate relaxation for multi-label problems and thus allows for accurate solutions using only a small number of labels, significantly improving over previous approaches towards lifting the total generalized variation.

Functional Liftings of Vectorial Variational Problems with Laplacian Regularization

- MathematicsSSVM
- 2019

A functional lifting-based convex relaxation of variational problems with Laplacian-based second-order regularization that encompasses the discretization-first sublabel-accurate continuous multilabeling approach as a special case is proposed.

Lifting methods for manifold-valued variational problems

- Mathematics, Computer ScienceHandbook of Variational Methods for Nonlinear Geometric Data
- 2020

This work provides a review of lifting methods in a refined framework based on a finite element discretization of the range, which extends the concept of sublabel-accurate lifting to manifolds and generalizes existing methods for total variation regularization to support general convex regularization.

Inverse Scale Space Iterations for Non-convex Variational Problems Using Functional Lifting

- MathematicsSSVM
- 2021

This work applies the classical Bregman iteration to a lifted version of the functional with sublabel-accurate discretization, and provides a condition for the subgradients of the regularizer under which this lifted iteration reduces to the standard BRegman iteration.

Efficient and Flexible Sublabel-Accurate Energy Minimization

- Computer ScienceArXiv
- 2022

This work proposes an efficient sublabel-accurate method that utilizes the best properties of both continuous and discrete models and shows the flexibility of the proposed approach to general pairwise smoothness terms, so that it is applicable to a wide range of regularizations.

Inverse Scale Space Iterations for Non-Convex Variational Problems: The Continuous and Discrete Case

- MathematicsArXiv
- 2022

Non-linear filtering approaches allow to obtain decompositions of images with respect to a nonclassical notion of scale, induced by the choice of a convex, absolutely one-homogeneous regularizer. The…

Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus

- Mathematics2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
- 2019

This work recalls that functionals with polyconvex Lagrangians can be reparametrized as convex one-homogeneous functionals on the graph of the function and proposes a discretization of the resulting infinite-dimensional optimization problem using Whitney forms, which generalizes recent "sublabel-accurate" multilabeling approaches.

Fast Convex Relaxations using Graph Discretizations

- Computer ScienceBMVC
- 2020

This methodology is discussed in detail and examples in multi-label segmentation by minimal partitions and stereo estimation are shown, where it is demonstrated that the proposed graph discretization technique can reduce the runtime as well as the memory consumption by up to a factor of 10 in comparison to classical pixelwise discretizations.

Composite Optimization by Nonconvex Majorization-Minimization

- Computer Science, MathematicsSIAM J. Imaging Sci.
- 2018

This work considers a natural class of nonconvex majorizers for these functions, and shows that these majorizers are still sufficient for a globally convergent optimization scheme, and illustrates the behavior of the algorithm for depth super-resolution from raw time-of-flight data.

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