The image nonlocal self-similarity (NSS) prior refers to the fact that a local patch often has many nonlocal similar patches to it across the image and has been widely applied in many recently proposed machining learning algorithms for image processing. However, there is no theoretical analysis on its working principle in the literature. In this paper, we discover a potential causality between NSS and low-rank property of color images, which is also available to grey images. A new patch group based NSS prior scheme is proposed to learn explicit NSS models of natural color images. The numerical low-rank property of patched matrices is also rigorously proved. The NSS-based QMC algorithm computes an optimal low-rank approximation to the high-rank color image, resulting in high PSNR and SSIM measures and particularly the better visual quality. A new tensor NSS-based QMC method is also presented to solve the color video inpainting problem based on quaternion tensor representation. The numerical experiments on color images and videos indicate the advantages of NSS-based QMC over the state-of-the-art methods.

The main aim of this paper is to develop the quaternion generalized minimal residual method (QGMRES) for solving quaternion linear systems. Quaternion linear systems arise from three-dimensional or...

This paper presents a randomized quaternion singular value decomposition (QSVD) algorithm for low-rank matrix approximation problems, which are widely used in color face recognition, video compress...

Quaternion singular value decomposition (QSVD) is a robust technique of digital watermarking that extracts high quality watermarks from watermarked images with low distortion. However, the existing QSVD-based watermarking schemes face the obstacle of “explosion of complexity” and have much room for improvement in terms of real-time, invisibility, and robustness. In this paper, we overcome such obstacle by introducing a new real structure-preserving QSVD algorithm and propose a novel QSVD-based watermarking scheme with high efficiency. Secret information is transmitted blindly by incorporating two new strategies: coefficient pair selection and adaptive embedding. The highly correlated coefficient pairs determined by the normalized cross-correlation method reduce the impact of embedding by reducing the maximum modification of the coefficient values, resulting in high fidelity of the watermarked image. Large-size 8-color binary watermark and QR code effectively verify that the proposed watermarking scheme can resist various image attacks in numerical experiments. Two keys designed by Logistic chaotic map ensure the security of the watermarking system. Under the premise of considering the correlation of color channels, the proposed watermarking scheme not only performs well in real-time and invisibility, but also has satisfactory advantages in robustness compared with the state-of-the-art methods.