# Semi-Global Weighted Least Squares in Image Filtering

arXiv: https://arxiv.org/abs/1705.01674 · HTML: https://mlsanity.com/paper/1705.01674/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 May 2017.

## Built on by (most cited first, 5 of 5)

- [Facial Demorphing via Identity Preserving Image Decomposition](https://mlsanity.com/paper/2408.10993/) (2024)
- [Hierarchical Image Peeling: A Flexible Scale-space Filtering Framework](https://mlsanity.com/paper/2104.01534/) (2021)
- [Pyramid Texture Filtering](https://mlsanity.com/paper/2305.06525/) (2023)
- [A Generalized Framework for Edge-preserving and Structure-preserving Image Smoothing](https://mlsanity.com/paper/2107.07058/) (2021)
- [Deep Image Prior with L0 Gradient Regularizer for Image Smoothing](https://mlsanity.com/paper/2601.13400/) (2026)

## Abstract (arXiv)

Solving the global method of Weighted Least Squares (WLS) model in image filtering is both time- and memory-consuming. In this paper, we present an alternative approximation in a time- and memory- efficient manner which is denoted as Semi-Global Weighed Least Squares (SG-WLS). Instead of solving a large linear system, we propose to iteratively solve a sequence of subsystems which are one-dimensional WLS models. Although each subsystem is one-dimensional, it can take two-dimensional neighborhood information into account due to the proposed special neighborhood construction. We show such a desirable property makes our SG-WLS achieve close performance to the original two-dimensional WLS model but with much less time and memory cost. While previous related methods mainly focus on the 4-connected/8-connected neighborhood system, our SG-WLS can handle a more general and larger neighborhood system thanks to the proposed fast solution. We show such a generalization can achieve better performance than the 4-connected/8-connected neighborhood system in some applications. Our SG-WLS is ~20 times faster than the WLS model. For an image of M× N, the memory cost of SG-WLS is at most at the magnitude of maxfrac1M, frac1N of that of the WLS model. We show the effectiveness and efficiency of our SG-WLS in a range of applications. The code is publicly available at: https://github.com/wliusjtu/Semi-Global-Weighted-Least-Squares-in-Image-Filtering.