# Variational Methods for Normal Integration

arXiv: https://arxiv.org/abs/1709.05965 · HTML: https://mlsanity.com/paper/1709.05965/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2017.

## Built on by (most cited first, 6 of 6)

- [Uncertainty-Aware Deep Multi-View Photometric Stereo](https://mlsanity.com/paper/2202.13071/) (2022)
- [Discontinuity-preserving Normal Integration with Auxiliary Edges](https://mlsanity.com/paper/2404.03138/) (2024)
- [Aerial Multi-View Stereo via Adaptive Depth Range Inference and Normal Cues](https://mlsanity.com/paper/2506.05655/) (2025)
- [Discontinuity-aware Normal Integration for Generic Central Camera Models](https://mlsanity.com/paper/2507.06075/) (2025)
- [An overview of some mathematical techniques and problems linking 3D vision to 3D printing](https://mlsanity.com/paper/2309.10549/) (2023)
- [Towards Fast and Scalable Normal Integration using Continuous Components](https://mlsanity.com/paper/2510.11508/) (2025)

## Abstract (arXiv)

The need for an efficient method of integration of a dense normal field is inspired by several computer vision tasks, such as shape-from-shading, photometric stereo, deflectometry, etc. Inspired by edge-preserving methods from image processing, we study in this paper several variational approaches for normal integration, with a focus on non-rectangular domains, free boundary and depth discontinuities. We first introduce a new discretization for quadratic integration, which is designed to ensure both fast recovery and the ability to handle non-rectangular domains with a free boundary. Yet, with this solver, discontinuous surfaces can be handled only if the scene is first segmented into pieces without discontinuity. Hence, we then discuss several discontinuity-preserving strategies. Those inspired, respectively, by the Mumford-Shah segmentation method and by anisotropic diffusion, are shown to be the most effective for recovering discontinuities.