# Image Labeling by Assignment

arXiv: https://arxiv.org/abs/1603.05285 · HTML: https://mlsanity.com/paper/1603.05285/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2016.

## Built on by (most cited first, 10 of 10)

- [Fisher Flow Matching for Generative Modeling over Discrete Data](https://mlsanity.com/paper/2405.14664/) (2024)
- [Generative Assignment Flows for Representing and Learning Joint Distributions of Discrete Data](https://mlsanity.com/paper/2406.04527/) (2024)
- [Nonparametric and Regularized Dynamical Wasserstein Barycenters for Sequential Observations](https://mlsanity.com/paper/2210.01918/) (2022)
- [Learning Linearized Assignment Flows for Image Labeling](https://mlsanity.com/paper/2108.02571/) (2021)
- [Self-Certifying Classification by Linearized Deep Assignment](https://mlsanity.com/paper/2201.11162/) (2022)
- [Sigma Flows for Image and Data Labeling and Learning Structured Prediction](https://mlsanity.com/paper/2408.15946/) (2024)
- [Two Models for Surface Segmentation using the Total Variation of the Normal Vector](https://mlsanity.com/paper/2412.00445/) (2024)
- [A Nonlocal Graph-PDE and Higher-Order Geometric Integration for Image Labeling](https://mlsanity.com/paper/2205.03991/) (2022)
- [Multilevel Geometric Optimization for Regularised Constrained Linear Inverse Problems](https://mlsanity.com/paper/2207.04934/) (2022)
- [Riemannian Patch Assignment Gradient Flows](https://mlsanity.com/paper/2504.13024/) (2025)

## Abstract (arXiv)

We introduce a novel geometric approach to the image labeling problem. Abstracting from specific labeling applications, a general objective function is defined on a manifold of stochastic matrices, whose elements assign prior data that are given in any metric space, to observed image measurements. The corresponding Riemannian gradient flow entails a set of replicator equations, one for each data point, that are spatially coupled by geometric averaging on the manifold. Starting from uniform assignments at the barycenter as natural initialization, the flow terminates at some global maximum, each of which corresponds to an image labeling that uniquely assigns the prior data. Our geometric variational approach constitutes a smooth non-convex inner approximation of the general image labeling problem, implemented with sparse interior-point numerics in terms of parallel multiplicative updates that converge efficiently.