# Variational reaction-diffusion systems for semantic segmentation

arXiv: https://arxiv.org/abs/1604.00092 · HTML: https://mlsanity.com/paper/1604.00092/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Apr 2016.

## Built on by (most cited first, 1 of 1)

- [PDE-Constrained Optimization for Neural Image Segmentation with Physics Priors](https://mlsanity.com/paper/2602.01069/) (2026)

## Abstract (arXiv)

A novel global energy model for multi-class semantic image segmentation is proposed that admits very efficient exact inference and derivative calculations for learning. Inference in this model is equivalent to MAP inference in a particular kind of vector-valued Gaussian Markov random field, and ultimately reduces to solving a linear system of linear PDEs known as a reaction-diffusion system. Solving this system can be achieved in time scaling near-linearly in the number of image pixels by reducing it to sequential FFTs, after a linear change of basis. The efficiency and differentiability of the model make it especially well-suited for integration with convolutional neural networks, even allowing it to be used in interior, feature-generating layers and stacked multiple times. Experimental results are shown demonstrating that the model can be employed profitably in conjunction with different convolutional net architectures, and that doing so compares favorably to joint training of a fully-connected CRF with a convolutional net.