# Unsupervised Learning of the Total Variation Flow

arXiv: https://arxiv.org/abs/2206.04406 · HTML: https://mlsanity.com/paper/2206.04406/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 64 references that are in the map.
- First posted on arXiv 9 Jun 2022.

## Builds on (most cited first, 19 of 19)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [A large annotated medical image dataset for the development and evaluation of segmentation algorithms](https://mlsanity.com/paper/1902.09063/) (2019)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [DeepView: View Synthesis with Learned Gradient Descent](https://mlsanity.com/paper/1906.07316/) (2019)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problems](https://mlsanity.com/paper/1904.05417/) (2019)
- [Spectral Decompositions using One-Homogeneous Functionals](https://mlsanity.com/paper/1601.02912/) (2016)
- [Multiscale Segmentation via Bregman Distances and Nonlinear Spectral Analysis](https://mlsanity.com/paper/1604.06665/) (2016)
- [Nonlinear Spectral Image Fusion](https://mlsanity.com/paper/1703.08001/) (2017)
- [Deeply Learned Spectral Total Variation Decomposition](https://mlsanity.com/paper/2006.10004/) (2020)
- [Separation Surfaces in the Spectral TV Domain for Texture Decomposition](https://mlsanity.com/paper/1511.04687/) (2015)
- [Solving Image PDEs with a Shallow Network](https://mlsanity.com/paper/2110.08327/) (2021)

## Built on by (most cited first, 3 of 3)

- [DeepTV: A neural network approach for total variation minimization](https://mlsanity.com/paper/2409.05569/) (2024)
- [Hidden Knowledge: Mathematical Methods for the Extraction of the Fingerprint of Medieval Paper from Digital Images](https://mlsanity.com/paper/2303.03794/) (2023)
- [Ensemble of Weak Spectral Total Variation Learners: a PET-CT Case Study](https://mlsanity.com/paper/2507.08735/) (2025)

## Abstract (arXiv)

The total variation (TV) flow generates a scale-space representation of an image based on the TV functional. This gradient flow observes desirable features for images, such as sharp edges and enables spectral, scale, and texture analysis. Solving the TV flow is challenging; one reason is the the non-uniqueness of the subgradients. The standard numerical approach for TV flow requires solving multiple non-smooth optimisation problems. Even with state-of-the-art convex optimisation techniques, this is often prohibitively expensive and strongly motivates the use of alternative, faster approaches. Inspired by and extending the framework of physics-informed neural networks (PINNs), we propose the TVflowNET, an unsupervised neural network approach, to approximate the solution of the TV flow given an initial image and a time instance. The TVflowNET requires no ground truth data but rather makes use of the PDE for optimisation of the network parameters. We circumvent the challenges related to the non-uniqueness of the subgradients by additionally learning the related diffusivity term. Our approach significantly speeds up the computation time and we show that the TVflowNET approximates the TV flow solution with high fidelity for different image sizes and image types. Additionally, we give a full comparison of different network architecture designs as well as training regimes to underscore the effectiveness of our approach.