Quaternion Collaborative and Sparse Representation With Application to Color Face Recognition

Advancements in exponential synchronization and encryption techniques: Quaternion-Valued Artificial Neural Networks with two-sided coefficients

This paper presents cutting-edge advancements in exponential synchronization and encryption techniques, focusing on Quaternion-Valued Artificial Neural Networks (QVANNs) that incorporate two-sided coefficients. The study introduces a novel approach that harnesses the Cayley-Dickson representation method to simplify the complex equations inherent in QVANNs, thereby enhancing computational efficiency by exploiting complex number properties. The study employs the Lyapunov theorem to craft a resilient control system, showcasing its exponential synchronization by skillfully regulating the Lyapunov function and its derivatives. This management ensures the stability and synchronization of the network, which is crucial for reliable performance in various applications. Extensive numerical simulations are conducted to substantiate the theoretical framework, providing empirical evidence supporting the presented design and proofs. Furthermore, the paper explores the practical application of QVANNs in the encryption and decryption of color images, showcasing the network's capability to handle complex data processing tasks efficiently. The findings of this research not only contribute significantly to the development of complex artificial neural networks but pave the way for further exploration into systems with diverse delay types.

Tensor Nuclear Norm-Based Multi-Channel Atomic Representation for Robust Face Recognition

Numerous representation-based classification (RC) methods have been developed for face recognition due to their decent model interpretability and robustness against noise. Most existing RC methods primarily characterize the gray-scale reconstruction error image (single-channel data) in two ways: the one-dimensional (1D) pixel-based error model and the two-dimensional (2D) gray-scale image-matrix-based error model. The former measures the reconstruction error pixel by pixel, while the latter leverages 2D structural information of the gray-scale error image, such as the low-rank property. However, when applying these methods to different color channels of a test color face image (multi-channel data) separately and independently, they neglect the three-dimensional (3D) structural correlations among distinct color channels. In real-world scenarios, face images are often contaminated with complex noise, including contiguous occlusion and random pixel corruption, which pose significant challenges to these approaches and can lead to a decline in performance. In this paper, we propose a Tensor Nuclear Norm based Robust Multi-channel Atomic Representation (TNN-RMAR) framework with application to color face recognition. The proposed method has the following three critical ingredients: 1) We propose a 3D color image-tensor-based error model, which can take full advantage of the 3D structural information of the color error image. 2) To leverage the 3D structural information of the color error image, we model it as a 3-order tensor and exploit its low-rank property with the tensor nuclear norm. Given that multiple color channels in a color image are generally corrupted at the same positions, we design a tube-wise tailored loss function to further leverage its tube-wise structure. 3) We devise the multi-channel atomic norm (MAN) regularization for the representation coefficient matrix, which allows us to jointly harness the correlation information of coefficients in different color channels. In addition, we also devise an efficient algorithm to solve the TNN-RMAR framework based on the alternating direction method of multipliers (ADMM) framework. By leveraging TNN-RMAR as a general platform, we also develop several novel robust multi-channel RC methods. Experimental results on benchmark real-world databases validate the effectiveness and robustness of the proposed framework for robust color face recognition.

Aperiodically intermittent quantized control-based exponential synchronization of quaternion-valued inertial neural networks

Inertial neural networks are proposed via introducing an inertia term into the Hopfield models, which make their dynamic behavior more complex compared to the traditional first-order models. Besides, the aperiodically intermittent quantized control over conventional feedback control has its potential advantages on reducing communication blocking and saving control cost. Based on these facts, we are mainly devoted to exploring of exponential synchronization of quaternion-valued inertial neural networks under aperiodically intermittent quantized control. Firstly, a compact quaternion-valued aperiodically intermittent quantized control protocol is developed, which can mitigate significantly the complexity of theoretical derivation. Subsequently, several concise criteria involving matrix inequalities are formulated through constructing a type of Lyapunov functional and employing a direct analysis approach. The correctness of the obtained results eventually is verified by a typical example.

Robust face recognition using quaternion interval type II fuzzy logic-based feature extraction on colour images

In this paper, we propose a new robust and fast learning technique by investigating the effect of integration of quaternion and interval type II fuzzy logic along with non-iterative, parameter free deterministic learning machine (DLM) pertaining to face recognition problem. The traditional learning techniques did not account colour information and degree of pixel wise association of individual pixel of a colour face image in their network. Therefore, this paper presents a new technique named quaternion interval type II based deterministic learning machine (QIntTyII-DLM), which considers the interrelationship between three colour channels viz. red, green, and blue (RGB) by representing each colour pixel of a colour image in quaternion number sequence. Here, quaternion vector representation of a colour face image is fuzzified using interval type II fuzzy logic. This reduces the redundancy between pixels of different colour channels and also transforms colour channels of the image to orthogonal colour space. Thereafter, classification is performed using DLM. Experiments performed (on four standard datasets AR, Georgia Tech, Indian, face (female) and faces 94 (male) face datasets) and comparison done with other existing techniques proves that the proposed technique gives better results in terms of percentage error rate (reduces approximately 10-12%) and computational speed.

Fixed-time synchronization for quaternion-valued memristor-based neural networks with mixed delays

In this paper, the fixed-time synchronization (FXTSYN) of unilateral coefficients quaternion-valued memristor-based neural networks (UCQVMNNs) with mixed delays is investigated. A direct analytical approach is suggested to obtain FXTSYN of UCQVMNNs utilizing one-norm smoothness in place of decomposition. When dealing with drive-response system discontinuity issues, use the set-valued map and the differential inclusion theorem. To accomplish the control objective, innovative nonlinear controllers and the Lyapunov functions are designed. Furthermore, some criteria of FXTSYN for UCQVMNNs are given using inequality techniques and the novel FXTSYN theory. And the accurate settling time is obtained explicitly. Finally, in order to show that the obtained theoretical results are accurate, useful, and applicable, numerical simulations are presented at the conclusion.

Modal Regression-Based Graph Representation for Noise Robust Face Hallucination

Manifold learning-based face hallucination technologies have been widely developed during the past decades. However, the conventional learning methods always become ineffective in noise environment due to the least-square regression, which usually generates distorted representations for noisy inputs they employed for error modeling. To solve this problem, in this article, we propose a modal regression-based graph representation (MRGR) model for noisy face hallucination. In MRGR, the modal regression-based function is incorporated into graph learning framework to improve the resolution of noisy face images. Specifically, the modal regression-induced metric is used instead of the least-square metric to regularize the encoding errors, which admits the MRGR to robust against noise with uncertain distribution. Moreover, a graph representation is learned from feature space to exploit the inherent typological structure of patch manifold for data representation, resulting in more accurate reconstruction coefficients. Besides, for noisy color face hallucination, the MRGR is extended into quaternion (MRGR-Q) space, where the abundant correlations among different color channels can be well preserved. Experimental results on both the grayscale and color face images demonstrate the superiority of MRGR and MRGR-Q compared with several state-of-the-art methods.

Mittag-Leffler stability of fractional-order quaternion-valued memristive neural networks with generalized piecewise constant argument

This paper studies the global Mittag-Leffler (M-L) stability problem for fractional-order quaternion-valued memristive neural networks (FQVMNNs) with generalized piecewise constant argument (GPCA). First, a novel lemma is established, which is used to investigate the dynamic behaviors of quaternion-valued memristive neural networks (QVMNNs). Second, by using the theories of differential inclusion, set-valued mapping, and Banach fixed point, several sufficient criteria are derived to ensure the existence and uniqueness (EU) of the solution and equilibrium point for the associated systems. Then, by constructing Lyapunov functions and employing some inequality techniques, a set of criteria are proposed to ensure the global M-L stability of the considered systems. The obtained results in this paper not only extends previous works, but also provides new algebraic criteria with a larger feasible range. Finally, two numerical examples are introduced to illustrate the effectiveness of the obtained results.

Quaternion Scalar and Vector Norm Decomposition: Quaternion PCA for Color Face Recognition

This paper proposes a decomposition called quaternion scalar and vector norm decomposition (QSVND) for approximation problems in color image processing. Different from traditional quaternion norm approximations that are always the single objective models (SOM), QSVND is adopted to transform the SOM into the bi-objective model (BOM). Furthermore, regularization is used to solve the BOM problem as a common scalarization method, which converts the BOM into a more reasonable SOM. This can handle over-fitting or under-fitting problems neglected in this kind of research for quaternion representation (QR) in color image processing. That is how to treat redundancy caused by the extra scalar part when the vector part of a quaternion is used to represent a color pixel. We apply QSVND to quaternion principal component analysis (QPCA) for color face recognition (FR), which can deal with the phenomenon of under-fitting of vector part norm approximation. Comparisons with the competing approaches on AR, FERET, FEI, and KDEF&AKDEF databases consistently show the superiority of the proposed approach for color FR.

Quantized intermittent control tactics for exponential synchronization of quaternion-valued memristive delayed neural networks

This article studies the global exponential synchronization (GES) of quaternion-valued memristive delayed neural networks (QVMDNNs) by quantized intermittent control (QIC). Without decomposing the original systems into usual real-valued or complex-valued ones, the discussed system is directly processed in both cases of the differential inclusion theory. Based on two QIC strategies and applying Lyapunov functional method and inequality techniques, several new delay-dependent criteria in the form of real-valued algebraic inequalities are directly derived to assure the GES of the concerned system. Compared to the traditional feedback control, the QIC strategy can reduce the control expenses and shorten time to achieve the GES. Ultimately, two examples are given to validate the effectiveness and advantages of the proposed outcomes.

Generalized and Discriminative Collaborative Representation for Multiclass Classification

This article presents a generalized collaborative representation-based classification (GCRC) framework, which includes many existing representation-based classification (RC) methods, such as collaborative RC (CRC) and sparse RC (SRC) as special cases. This article also advances the GCRC theory by exploring theoretical conditions on the general regularization matrix. A key drawback of CRC and SRC is that they fail to use the label information of training data and are essentially unsupervised in computing the representation vector. This largely compromises the discriminative ability of the learned representation vector and impedes the classification performance. Guided by the GCRC theory, we propose a novel RC method referred to as discriminative RC (DRC). The proposed DRC method has the following three desirable properties: 1) discriminability: DRC can leverage the label information of training data and is supervised in both representation and classification, thus improving the discriminative ability of the representation vector; 2) efficiency: it has a closed-form solution and is efficient in computing the representation vector and performing classification; and 3) theory: it also has theoretical guarantees for classification. Experimental results on benchmark databases demonstrate both the efficacy and efficiency of DRC for multiclass classification.

Hallucinating Color Face Image by Learning Graph Representation in Quaternion Space

Recently, learning-based representation techniques have been well exploited for grayscale face image hallucination. For color images, the previous methods only handle the luminance component or each color channel individually, without considering the abundant correlations among different channels as well as the inherent geometrical structure of data manifold. In this article, we propose a learning-based model in quaternion space with graph representation for color face hallucination. Instead of the spatial domain, the color image is represented in the quaternion domain to preserve correlations among different color channels. Moreover, a quaternion graph is learned to smooth the quaternion feature space, which helps to not only stabilize the linear system but also enclose the inherent topology structure of quaternion patch manifold. Besides, considering that single low-resolution (LR) image patch can just provide limited informative information in representation, we propose to simultaneously encode the query smaller LR patch as well as a larger patch containing the surrounding pixels seated at the same position in the objective. The larger patch with rich patterns is used to compensate the lost information in the query LR patch, which further enhances the manifold consistency assumption between the LR and HR patch spaces. The experimental results demonstrated the efficiency of the proposed method in hallucinating color face images.

Fixed/Preassigned-time synchronization of quaternion-valued neural networks via pure power-law control

The fixed-time synchronization and preassigned-time synchronization of quaternion-valued neural networks are concerned in this article. By developing fixed-time stability and proposing a pure power-law control scheme, some simple conditions are obtained to realize fixed-time synchronization of quaternion-valued neural networks and the upper bound of the synchronized time is provided. Furthermore, the preassigned-time synchronization of quaternion-valued neural networks is investigated based on pure power-law control design, where the synchronization time is preassigned in advance and the control gains are finite. Note that the designed controllers in this paper are the pure power-law forms, which are simpler and more effective compared with the traditional design composed of the linear part and power-law part. Eventually, an example is given to illustrate the feasibility and validity of the results obtained.

Color Image Recovery Using Low-Rank Quaternion Matrix Completion Algorithm

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 a 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 the Frobenius norm of its two low-rank factor quaternion matrices. Based on the relationship between the quaternion matrix and its equivalent complex matrix, the problem eventually is converted from the quaternion number domain to the complex number domain. An alternating minimization method is applied to solve the model. Simulation results on color image recovery show the superior performance and efficiency of the proposed algorithm over some tensor-based and quaternion-based ones.

Robust Sparse Representation in Quaternion Space

Sparse representation has achieved great success across various fields including signal processing, machine learning and computer vision. However, most existing sparse representation methods are confined to the real valued data. This largely limit their applicability to the quaternion valued data, which has been widely used in numerous applications such as color image processing. Another critical issue is that their performance may be severely hampered due to the data noise or outliers in practice. To tackle the problems above, in this work we propose a robust quaternion valued sparse representation (RQVSR) method in a fully quaternion valued setting. To handle the quaternion noises, we first define a new robust estimator referred as quaternion Welsch estimator to measure the quaternion residual error. Compared to the conventional quaternion mean square error, it can largely suppress the impact of large data corruption and outliers. To implement RQVSR, we have overcome the difficulties raised by the noncommutativity of quaternion multiplication and developed an effective algorithm by leveraging the half-quadratic theory and the alternating direction method of multipliers framework. The experimental results show the effectiveness and robustness of the proposed method for quaternion sparse signal recovery and color image reconstruction.

Prior Knowledge-Based Probabilistic Collaborative Representation for Visual Recognition

Collaborative representation is an effective way to design classifiers for many practical applications. In this paper, we propose a novel classifier, called the prior knowledge-based probabilistic collaborative representation-based classifier (PKPCRC), for visual recognition. Compared with existing classifiers which use the collaborative representation strategy, the proposed PKPCRC further includes characteristics of training samples of each class as prior knowledge. Four types of prior knowledge are developed from the perspectives of image distance and representation capacity. They adaptively accommodate the contribution of each class and result in an accurate representation to classify a query sample. Experiments and comparisons on four challenging databases demonstrate that PKPCRC outperforms several state-of-the-art classifiers.

Two-Dimensional Quaternion Sparse Discriminant Analysis

Linear discriminant analysis has been incorporated with various representations and measurements for dimension reduction and feature extraction. In this paper, we propose two-dimensional quaternion sparse discriminant analysis (2D-QSDA) that meets the requirements of representing RGB and RGB-D images. 2D-QSDA advances in three aspects: 1) including sparse regularization, 2D-QSDA relies only on the important variables, and thus shows good generalization ability to the out-of-sample data which are unseen during the training phase; 2) benefited from quaternion representation, 2D-QSDA well preserves the high order correlation among different image channels and provides a unified approach to extract features from RGB and RGB-D images; 3) the spatial structure of the input images is retained via the matrix-based processing. We tackle the constrained trace ratio problem of 2D-QSDA by solving a corresponding constrained trace difference problem, which is then transformed into a quaternion sparse regression (QSR) model. Afterward, we reformulate the QSR model to an equivalent complex form to avoid the processing of the complicated structure of quaternions. A nested iterative algorithm is designed to learn the solution of 2D-QSDA in the complex space and then we convert this solution back to the quaternion domain. To improve the separability of 2D-QSDA, we further propose 2D-QSDAw using the weighted pairwise between-class distances. Extensive experiments on RGB and RGB-D databases demonstrate the effectiveness of 2D-QSDA and 2D-QSDAw compared with peer competitors.

Synchronization control of quaternion-valued memristive neural networks with and without event-triggered scheme

In this paper, the real-valued memristive neural networks (MNNs) are extended to quaternion field, a new class of neural networks named quaternion-valued memristive neural networks (QVMNNs) is then established. The problem of master-slave synchronization of this type of networks is investigated in this paper. Two types of controllers are designed: the traditional feedback controller and the event-triggered controller. Corresponding synchronization criteria are then derived based on Lyapunov method. Moreover, it is demonstrated that Zeno behavior can be avoided in case of the event-triggered strategy proposed in this work. Finally, corresponding simulation examples are proposed to demonstrate the correctness of the proposed results derived in this work.

Low-rank quaternion approximation for color image processing

Low-rank matrix approximation (LRMA)-based methods have made a great success for grayscale image processing. When handling color images, LRMA either restores each color channel independently using the monochromatic model or processes the concatenation of three color channels using the concatenation model. However, these two schemes may not make full use of the high correlation among RGB channels. To address this issue, we propose a novel low-rank quaternion approximation (LRQA) model. It contains two major components: first, instead of modeling a color image pixel as a scalar in conventional sparse representation and LRMA-based methods, the color image is encoded as a pure quaternion matrix, such that the cross-channel correlation of color channels can be well exploited; second, LRQA imposes the low-rank constraint on the constructed quaternion matrix. To better estimate the singular values of the underlying low-rank quaternion matrix from its noisy observation, a general model for LRQA is proposed based on several nonconvex functions. Extensive evaluations for color image denoising and inpainting tasks verify that LRQA achieves better performance over several state-of-the-art sparse representation and LRMA-based methods in terms of both quantitative metrics and visual quality.

Two-Dimensional Quaternion PCA and Sparse PCA

Benefited from quaternion representation that is able to encode the cross-channel correlation of color images, quaternion principle component analysis (QPCA) was proposed to extract features from color images while reducing the feature dimension. A quaternion covariance matrix (QCM) of input samples was constructed, and its eigenvectors were derived to find the solution of QPCA. However, eigen-decomposition leads to the fixed solution for the same input. This solution is susceptible to outliers and cannot be further optimized. To solve this problem, this paper proposes a novel quaternion ridge regression (QRR) model for two-dimensional QPCA (2D-QPCA). We mathematically prove that this QRR model is equivalent to the QCM model of 2D-QPCA. The QRR model is a general framework and is flexible to combine 2D-QPCA with other technologies or constraints to adapt different requirements of real-world applications. Including sparsity constraints, we then propose a quaternion sparse regression model for 2D-QSPCA to improve its robustness for classification. An alternating minimization algorithm is developed to iteratively learn the solution of 2D-QSPCA in the equivalent complex domain. In addition, 2D-QPCA and 2D-QSPCA can preserve the spatial structure of color images and have a low computation cost. Experiments on several challenging databases demonstrate that 2D-QPCA and 2D-QSPCA are effective in color face recognition, and 2D-QSPCA outperforms the state of the arts.

Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches

In this paper, we investigate the global stability of quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of quaternion multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: $ij=-ji=k,~jk=-kj=i$ , $ki=-ik=j$ , $i^{2}=j^{2}=k^{2}=ijk=-1$ . With the Lyapunov function method, some criteria are, respectively, presented to ensure the global $\mu $ -stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of quaternion multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where quaternion self-conjugate matrices and quaternion positive definite matrices are used. Some new sufficient conditions in the form of quaternion-valued LMI are, respectively, established for the global $\mu $ -stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.

Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks with mixed delays via state-feedback control

This paper studies the drive-response synchronization for quaternion-valued shunting inhibitory cellular neural networks (QVSICNNs) with mixed delays. First, QVSICNN is decomposed into an equivalent real-valued system in order to avoid the non-commutativity of the multiplicity. Then, the existence of almost periodic solutions is obtained based on the Banach fixed point theorem. An novel state-feedback controller is designed to ensure the global exponential almost periodic synchronization. At the end of the paper, an example is given to illustrate the effectiveness of the obtained results.

Quaternion Locality-Constrained Coding for Color Face Hallucination

Recently, the locality linear coding (LLC) has attracted more and more attentions in the areas of image processing and computer vision. However, the conventional LLC with real setting is just designed for the grayscale image. For the color image, it usually treats each color channel individually or encodes the monochrome image by concatenating all the color channels, which ignores the correlations among different channels. In this paper, we propose a quaternion-based locality-constrained coding (QLC) model for color face hallucination in the quaternion space. In QLC, the face images are represented as quaternion matrices. By transforming the channel images into an orthogonal feature space and encoding the coefficients in the quaternion domain, the proposed QLC is expected to learn the advantages of both quaternion algebra and locality coding scheme. Hence, the QLC cannot only expose the true topology of image patch manifold but also preserve the inherent correlations among different color channels. Experimental results demonstrated that our proposed QLC method achieved superior performance in color face hallucination compared with other state-of-the-art methods.

Quaternion-Michelson Descriptor for Color Image Classification

In this paper, we develop a simple yet powerful framework called quaternion-Michelson descriptor (QMD) to extract local features for color image classification. Unlike traditional local descriptors extracted directly from the original (raw) image space, QMD is derived from the Michelson contrast law and the quaternionic representation (QR) of color images. The Michelson contrast is a stable measurement of image contents from the viewpoint of human perception, while QR is able to handle all the color information of the image holisticly and to preserve the interactions among different color channels. In this way, QMD integrates both the merits of Michelson contrast and QR. Based on the QMD framework, we further propose two novel quaternionic Michelson contrast binary pattern descriptors from different perspectives. Experiments and comparisons on different color image classification databases demonstrate that the proposed framework and descriptors outperform several state-of-the-art methods.