  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.
