# NeurTV: Total Variation on the Neural Domain

arXiv: https://arxiv.org/abs/2405.17241 · HTML: https://mlsanity.com/paper/2405.17241/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 65 references that are in the map.
- First posted on arXiv 27 May 2024.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Learning Continuous Image Representation with Local Implicit Image Function](https://mlsanity.com/paper/2012.09161/) (2020)
- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [Image Restoration using Total Variation Regularized Deep Image Prior](https://mlsanity.com/paper/1810.12864/) (2018)
- [Hyperspectral Image Restoration via Total Variation Regularized Low-rank Tensor Decomposition](https://mlsanity.com/paper/1707.02477/) (2017)
- [Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2212.00262/) (2022)
- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-shape Priori for Image Segmentation](https://mlsanity.com/paper/2002.03989/) (2020)
- [FastDeRain: A Novel Video Rain Streak Removal Method Using Directional Gradient Priors](https://mlsanity.com/paper/1803.07487/) (2018)
- [Unsupervised Image Deraining: Optimization Model Driven Deep CNN](https://mlsanity.com/paper/2203.13699/) (2022)
- [Elastica Models for Color Image Regularization](https://mlsanity.com/paper/2203.09995/) (2022)
- [Adaptive and Implicit Regularization for Matrix Completion](https://mlsanity.com/paper/2208.05640/) (2022)
- [S2S-WTV: Seismic Data Noise Attenuation Using Weighted Total Variation Regularized Self-Supervised Learning](https://mlsanity.com/paper/2212.13523/) (2022)

## Built on by (most cited first, 7 of 7)

- [Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives](https://mlsanity.com/paper/2505.15222/) (2025)
- [Cross-Frequency Implicit Neural Representation with Self-Evolving Parameters](https://mlsanity.com/paper/2504.10929/) (2025)
- [Low-Rank Tensor Recovery via Variational Schatten-p Quasi-Norm and Jacobian Regularization](https://mlsanity.com/paper/2506.22134/) (2025)
- [Rethinking Coupled Tensor Analysis for Hyperspectral Super-Resolution: Recoverable Modeling Under Endmember Variability](https://mlsanity.com/paper/2512.19489/) (2025)
- [Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2603.01034/) (2026)
- [OASIS: A Deep Learning Framework for Universal Spectroscopic Analysis Driven by Novel Loss Functions](https://mlsanity.com/paper/2509.11499/) (2025)
- [MG-SpaIR: Multi-grade Sparse-guided Implicit Representation for Training-Data-Free Image Restoration](https://mlsanity.com/paper/2607.00138/) (2026)

## Abstract (arXiv)

Recently, we have witnessed the success of total variation (TV) for many imaging applications. However, traditional TV is defined on the original pixel domain, which limits its potential. In this work, we suggest a new TV regularization defined on the neural domain. Concretely, the discrete data is implicitly and continuously represented by a deep neural network (DNN), and we use the derivatives of DNN outputs w.r.t. input coordinates to capture local correlations of data. As compared with classical TV on the original domain, the proposed TV on the neural domain (termed NeurTV) enjoys the following advantages. First, NeurTV is free of discretization error induced by the discrete difference operator. Second, NeurTV is not limited to meshgrid but is suitable for both meshgrid and non-meshgrid data. Third, NeurTV can more exactly capture local correlations across data for any direction and any order of derivatives attributed to the implicit and continuous nature of neural domain. We theoretically reinterpret NeurTV under the variational approximation framework, which allows us to build the connection between NeurTV and classical TV and inspires us to develop variants (e.g., space-variant NeurTV). Extensive numerical experiments with meshgrid data (e.g., color and hyperspectral images) and non-meshgrid data (e.g., point clouds and spatial transcriptomics) showcase the effectiveness of the proposed methods.