Nonlinear Hierarchical Matrix Factorization-Based Tensor Ring Approximation for Multi-dimensional Image Recovery

IF 2.8 2区 数学 Q1 MATHEMATICS, APPLIED
Wei-Hao Wu, Ting-Zhu Huang, Xi-Le Zhao, Hao Zhang, Zhi-Long Han
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引用次数: 0

Abstract

Recently, tensor ring (TR) approximation has received increasing attention in multi-dimensional image processing. In TR approximation, the key backbone is the shallow matrix factorizations, which approximate the circular unfolding of the multi-dimensional image. However, the shallow matrix factorization limits the standard TR approximation’s ability to represent images with complex details and textures. To address this limitation, we propose a nonlinear hierarchical matrix factorization-based tensor ring (NHTR) approximation. Specifically, instead of the shallow matrix factorization, we introduce the nonlinear hierarchical matrix factorization in NHTR approximation to approximate circularly \(\lceil \frac{N}{2}\rceil \)-modes unfoldings of an N-th order tensor. Benefiting from the powerful expressive capability of the nonlinear hierarchical matrix factorization, the proposed NHTR approximation can faithfully capture fine details of the clean image compared to classical tensor ring approximation. Empowered with the proposed NHTR, we build a multi-dimensional image recovery model and establish a theoretical error bound between the recovered image and the clean image based on the proposed model. To solve the highly nonlinear and hierarchical optimization problem, we develop an efficient alternating minimization-based algorithm. Experiments on multispectral images and color videos conclusively demonstrate the superior performance of our method over the compared state-of-the-art methods in multi-dimensional image recovery.

Abstract Image

基于非线性层次矩阵因数分解的张量环逼近法实现多维图像复原
近来,张量环(TR)近似在多维图像处理中受到越来越多的关注。在 TR 近似中,关键的支柱是浅矩阵因式分解,它近似于多维图像的环形展开。然而,浅矩阵因式分解限制了标准 TR 近似表示具有复杂细节和纹理的图像的能力。为了解决这个问题,我们提出了一种基于分层矩阵因式分解的非线性张量环(NHTR)近似方法。具体来说,我们在 NHTR 近似中引入了非线性分层矩阵因式分解,而不是浅层矩阵因式分解,以近似 N 阶张量的环形(\lceil \frac{N}{2}\rceil \)模式展开。得益于非线性分层矩阵因式分解强大的表达能力,与经典的张量环近似相比,所提出的 NHTR 近似能忠实地捕捉干净图像的精细细节。借助所提出的 NHTR,我们建立了一个多维图像复原模型,并基于所提出的模型建立了恢复图像与干净图像之间的理论误差约束。为了解决高度非线性和分层优化问题,我们开发了一种基于交替最小化的高效算法。在多光谱图像和彩色视频上的实验证明,我们的方法在多维图像复原方面的性能优于同类先进方法。
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来源期刊
Journal of Scientific Computing
Journal of Scientific Computing 数学-应用数学
CiteScore
4.00
自引率
12.00%
发文量
302
审稿时长
4-8 weeks
期刊介绍: Journal of Scientific Computing is an international interdisciplinary forum for the publication of papers on state-of-the-art developments in scientific computing and its applications in science and engineering. The journal publishes high-quality, peer-reviewed original papers, review papers and short communications on scientific computing.
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