Semi-Global Weighted Least Squares in Image Filtering

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/8connected 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 max{ 1 M , 1 N } of that of the WLS model. We show the effectiveness and efficiency of our SG-WLS in a range of applications.

Image smoothing via unsupervised learning

“…For example, [Farbman et al 2008] proposed an edge-preserving operator in a weighted least square (WLS) optimization framework, which prevents the local image regions from being over-sharpened with an L 2 norm. Similar schemes have been achieved more efficiently by [Liu et al 2017;Min et al 2014]. These works are devoted to extracting and manipulating the image details using image smoothing for various applications such as detail enhancement, HDR tone mapping, etc.…”

“…6, Article 259. Publication date: November 2018. arXiv:1811.02804v1 [cs.CV] 7 Nov 2018 259:2 • Qingnan, F. et al Liu et al 2017;Min et al 2014;Xu et al 2011. Despite the great improvements, however, their smoothing results are still not perfect, and no existing algorithm can serve as an image smoothing panacea for various applications.…”

Find Outliers of Image Edge Consistency by Weighted Local Linear Regression with Equality Constraints

“…Smooth weight is inversely correlated with the distance between and . WLS are continuously improved, such as efficient semi-global WLS [ 15 ] and constrained WLS [ 16 ], to keep pace with its applications. However, the data weight correlated with the fidelity of is usually undefined.…”

Directional relative total variation for structure–texture decomposition

“…However, the texture layer may receive similar penalties so that TV regulariser can't remove textures completely. To overcome the drawback of the TV model, a number of state-of-the-art methods have been proposed, such as weighted least squares (WLS) [18], semi-global WLS [19], l 0 gradient minimisation [20], and relative TV (RTV) [21]. The success of WLS optimisation is leading to a smoothness weight for TV regularisation term based on the gradient to penalise structure and texture differently.…”