Regression analysis-based approaches have been widely studied for face recognition (FR) in the past several years. More recently, to better deal with some difficult conditions such as occlusions and illumination, nuclear norm based matrix regression methods have been proposed to characterize the low-rank structure of the error image, which generalize the one-dimensional, pixel-based error model to the two-dimensional structure. These methods, however, are inherently devised for grayscale image based FR and without exploiting the color information which is proved beneficial for FR of color face images. Benefiting from quaternion representation, which is capable of encoding the cross-channel correlation of color images, we propose a novel color FR method by formulating the color FR problem as a nuclear norm based quaternion matrix regression (NQMR). We further develop a more robust model called R-NQMR by using the logarithm of the nuclear norm, instead of the original nuclear norm, which adaptively assigns weights on different singular values, and then extend it to deal with the mixed noise. The proposed models, then, are solved using the effective alternating direction multiplier method (ADMM). Experiments on several public face databases demonstrate the superior performance and efficacy of the proposed approaches for color FR, especially for some difficult conditions (occlusion, illumination and mixed noise) over some state-of-the-art regression analysis-based approaches.
Low-rank quaternion tensor completion method, a novel approach to recovery color videos and images is proposed in this paper. We respectively reconstruct a color image and a color video as a quaternion matrix (second-order tensor) and a third-order quaternion tensor by encoding the red, green, and blue channel pixel values on the three imaginary parts of a quaternion. Different from some traditional models which treat color pixel as a scalar and represent color channels separately, whereas, during the quaternion-based reconstruction, it is significant that the inherent color structures of colo…
As a new color image representation tool, quaternion has achieved excellent results in the color image processing, because it treats the color image as a whole rather than as a separate color space component, thus it can make full use of the high correlation among RGB channels. Recently, low-rank quaternion matrix completion (LRQMC) methods have proven very useful for color image inpainting. In this paper, we propose three novel LRQMC methods based on three quaternion-based bilinear factor (QBF) matrix norm minimization models. Specifically, we define quaternion double Frobenius norm (Q-DFN), …
As a new color image representation tool, quaternion has achieved excellent results in color image processing problems. In this paper, we propose a novel low-rank quaternion matrix completion algorithm to recover missing data of color image. Motivated by two kinds of low-rank approximation approaches (low-rank decomposition and nuclear norm minimization) in traditional matrix-based methods, we combine the two approaches in our quaternion matrix-based model. Furthermore, the nuclear norm of the quaternion matrix is replaced by the sum of Frobenius norm of its two low-rank factor quaternion matr…
In this letter, we propose a novel low-rank quaternion approximation (LRQA) model by directly constraining the quaternion rank prior for effectively removing the noise in color images. The LRQA model treats the color image holistically rather than independently for the color space components, thus it can fully utilize the high correlation among RGB channels. We design an iterative algorithm by using quaternion bilateral random projections (Q-BRP) to efficiently optimize the proposed model. The main advantage of Q-BRP is that the approximation of the low-rank quaternion matrix can be obtained q…
Quaternion matrices are employed successfully in many color image processing applications. In particular, a pure quaternion matrix can be used to represent red, green and blue channels of color images. A low-rank approximation for a pure quaternion matrix can be obtained by using the quaternion singular value decomposition. However, this approximation is not optimal in the sense that the resulting low-rank approximation matrix may not be pure quaternion, i.e., the low-rank matrix contains real component which is not useful for the representation of a color image. The main contribution of this …
Higher-order singular value decomposition (HOSVD) is one of the most efficient tensor decomposition techniques. It has the salient ability to represent high_dimensional data and extract features. In more recent years, the quaternion has proven to be a very suitable tool for color pixel representation as it can well preserve cross-channel correlation of color channels. Motivated by the advantages of the HOSVD and the quaternion tool, in this paper, we generalize the HOSVD to the quaternion domain and define quaternion-based HOSVD (QHOSVD). Due to the non-commutability of quaternion multiplicati…
Fujian Normal University（福建师范大学）；Center for Applied Mathematics of Fujian province (FJNU)（福建省应用数学中心（福建师范大学））；Jiangsu Normal University（江苏师范大学）；The Research Institute of Mathematical Science, Jiangsu Normal University（江苏师范大学数学科学研究院）；Fuzhou University of International Studies and Trade（福州外语外贸学院）
To address the non-negativity dropout problem of quaternion models, a novel quasi non-negative quaternion matrix factorization (QNQMF) model is presented for color image processing. To implement QNQMF, the quaternion projected gradient algorithm and the quaternion alternating direction method of multipliers are proposed via formulating QNQMF as the non-convex constraint quaternion optimization problems. Some properties of the proposed algorithms are studied. The numerical experiments on the color image reconstruction show that these algorithms encoded on the quaternion perform better than thes…
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 compression, and signal processing problems. With quaternion normal distribution based random sampling, the randomized QSVD algorithm projects a high-dimensional data to a low-dimensional subspace and then identifies an approximate range subspace of the quaternion matrix. The key statistical properties of quaternion Wishart distribution are proposed and used to perform the approximation error analysis of the…
Jiangsu Normal University（江苏师范大学）；Inner Mongolia University（内蒙古大学）；The University of Hong Kong（香港大学）；University of Electronic Science and Technology of China（电子科技大学）
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…
Structure preserving quaternion conjugate gradient-type methods for solving non-Hermitian quaternion linear systems
School of Mathematics and Statistics, Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University（福建师范大学数学与统计学院，分析数学及应用教育部重点实验室）；School of Mathematics and Statistics, Hainan University（海南大学数学与统计学院）；School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）
Shanghai University（上海大学）；Research Institute of USV Engineering（无人艇工程研究院）；Jiangsu Normal University（江苏师范大学）；Research Institute of Mathematical Science（数学科学研究院）
Quaternion singular value decomposition (QSVD) is a robust technique of digital watermarking which can extract high quality watermarks from watermarked images with low distortion. In this paper, QSVD technique is further investigated and an efficient robust watermarking scheme is proposed. The improved algebraic structure-preserving method is proposed to handle the problem of "explosion of complexity" occurred in the conventional QSVD design. Secret information is transmitted blindly by incorporating in QSVD two new strategies, namely, coefficient pair selection and adaptive embedding. Unlike …
