# Semi-Global Weighted Least Squares in Image Filtering

## TL;DR

This paper introduces Semi-Global Weighted Least Squares (SG-WLS), an efficient approximation method for image filtering that significantly reduces computational costs while maintaining performance close to the original WLS model.

## Contribution

We propose SG-WLS, a novel iterative approach using one-dimensional WLS subsystems with a special neighborhood construction, enabling faster and more memory-efficient image filtering.

## Key findings

- SG-WLS is approximately 20 times faster than the original WLS model.
- Memory usage of SG-WLS is significantly lower, at most a small fraction of WLS.
- SG-WLS achieves comparable filtering performance to the full 2D WLS model.

## Abstract

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 $\sim20$ times faster than the WLS model. For an image of $M\times N$, the memory cost of SG-WLS is at most at the magnitude of $max\{\frac{1}{M}, \frac{1}{N}\}$ 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.