Quaternion Singular Value Decomposition Research Articles

A novel structure-preserving algorithm for the singular value decomposition of biquaternion matrices

In this paper, we study the singular value decomposition of biquaternion matrices. We prove that the singular value decomposition of biquaternion matrices can be equivalently converted to the singular value decomposition of quaternion adjoint matrices. By means of the Cheng product, we propose a novel representation called Q -representation and design a Q structure-preserving algorithm to deal with the singular value decomposition of quaternion matrices, which makes use of high-level operations. The algorithm is more efficient than that in quaternion toolbox for matlab using quaternion arithmetics and real structure-preserving algorithm. The Q structure-preserving algorithm for singular value decomposition of biquaternion matrices is presented through the application of the Q structure-preserving algorithm for singular value decomposition of quaternion matrices. Numerical experiments are provided to demonstrate the efficiency of the Q structure-preserving algorithm. We apply the Q structure-preserving algorithm of quaternion singular value decomposition to improve the image denoising process.

Fixed-precision randomized quaternion singular value decomposition algorithm for low-rank quaternion matrix approximations

The fixed-precision randomized quaternion singular value decomposition algorithm (FPRQSVD) is presented to compute the low-rank quaternion matrix approximation. The FPRQSVD algorithm estimates the appropriate rank according to the given tolerance without a predetermined rank parameter, and computes the corresponding low-rank quaternion matrix approximation automatically. Error analysis indicates that the FPRQSVD algorithm is mathematically equivalent to the randomized quaternion singular value decomposition algorithm. Numerical experiments are given to demonstrate that the FPRQSVD algorithm is feasible and effective. Moreover, we use it to deal with the problem of color image inpainting.

Neural network models for the quaternion singular value decompositions

We develop two neural network models for computing the quaternion singular value decomposition (QSVD), which can be shown to arise as gradient flows or Riemannian-gradient flows on the product of Stiefel manifold over the quaternion skew-field and their geometric and dynamical properties are investigated. Numerical experiments including color image compression indicate the feasibility and effectiveness of the proposed neural network models.

Dual quaternion singular value decomposition based on bidiagonalization to a dual number matrix using dual quaternion householder transformations

We propose a practical method for computing the singular value decomposition of dual quaternion matrices. The dual quaternion Householder matrix is first proposed, and by combining the properties of dual quaternions, we can implement the transformation of a dual quaternion matrix to a bidiagonalized dual number matrix. We have proven that the singular values of a dual quaternion matrix are same to the singular values of its bidiagonalized dual number matrix. Numerical experiment demonstrates the effectiveness of our proposed method for computing the singular value decomposition of dual quaternion matrices.

A novel joint denoising method for gear fault diagnosis with improved quaternion singular value decomposition

Singular value decomposition (SVD) has drawn increasing attention in recent years as an effective signal noise reduction method. However, it is inapplicable to multivariate signals with abundant fault information. Existing quaternion singular value decomposition (QSVD) can decompose multivariate signals simultaneously, but the methods based on QSVD dependent on the embedding dimension and reconstructed components order seriously, and they are not suitable to noisy signals. To solve the problem, a novel joint denoising method improved quaternion singular value decomposition (IQSVD) is proposed, which can determine two parameters adaptively. Firstly, to select the embedding dimension, the improved power spectral density (IPSD) is proposed with considering frequency features and time-domine traits together. And then the periodic intensity (PI) is used to determine the reconstructed components by searching the component composing the most abundant periodic part. Finally, combined with envelope spectrum analysis, the gear fault is diagnosed. The results of the gear simulated and experimental multivariate signals verify the effectiveness of proposed method.

Color image watermarking based on a fast structure-preserving algorithm of quaternion singular value decomposition

This paper presents a new color image watermarking algorithm based on a fast structure-preserving algorithm of quaternion singular value decomposition (QSVD). We first propose F structure-preserving algorithm of QSVD by using quaternion Householder transformation, then apply it to color image watermarking, finally compare it with other algorithms and get better invisibility, security and almost robustness.

Robust dual color images watermarking scheme with hyperchaotic encryption based on quaternion DFrAT and genetic algorithm

A robust dual color images watermarking algorithm is designed based on quaternion discrete fractional angular transform (QDFrAT) and genetic algorithm. To guarantee the watermark security, the original color watermark image is encrypted with a 4D hyperchaotic system. A pure quaternion matrix is acquired by performing the discrete wavelet transform (DWT), the block division and the discrete cosine transform on the original color cover image. The quaternion matrix is operated by the QDFrAT to improve the robustness and the security of the watermarking scheme with the optimal transform angle and the fractional order. Then the singular value matrix is obtained by the quaternion singular value decomposition (QSVD) to further enhance the scheme’s stability. The encryption watermark is also processed by DWT and QSVD. Afterward, the singular value matrix of the encryption watermark is embedded into the singular value matrix of the host image by the optimal scaling factor. Moreover, the values to balance imperceptibility and robustness are optimized with a genetic algorithm. It is shown that the proposed color image watermarking scheme performs well in imperceptibility, security, robustness and embedding capacity.

Quaternion higher-order singular value decomposition and its applications in color image processing

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. On the other hand, 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 multiplication, QHOSVD is not a trivial extension of the HOSVD. They have similar but different calculation procedures. Theoretically, QHOSVD is a proper tensor generalization of the quaternion singular value decomposition (QSVD), and a proper quaternion generalization of the standard HOSVD. From the application point of view, the defined QHOSVD can be widely used in various visual data processing with color pixels. As examples, in this paper, we present two applications of the defined QHOSVD in color image processing—multi-focus color image fusion and color image denoising. The experimental results on the two applications respectively demonstrate the competitive performance of the proposed methods over some existing ones.

Adaptive quaternion multivariate local characteristic-scale decomposition and its application to gear fault diagnosis

Local characteristic-scale decomposition (LCD) has been widely applied in the field of gear fault diagnosis. However, LCD is only suitable for processing single channel signals and has poor noise robustness. Gear fault information collected by single channel of the sensor is not sufficient when the background noise is severe, based on which, a novel adaptive multivariate signal noise reduction method, quaternion singular value decomposition fault information spectrum (QSVDFIS), is defined in this paper. Based on QSVDFIS, a new multivariate signal decomposition method, adaptive quaternion multivariate local characteristic-scale decomposition (AQMLCD), is proposed in this paper. AQMLCD can adaptively reduce the noise of multivariate signal components, effectively fuse the fault information of each channel. Compared with multivariate empirical mode decomposition (MEMD), the noise robustness of AQMLCD is significantly improved, while the computation efficiency of AQMLCD is similar to that of MEMD. AQMLCD is applied to gear fault simulation and experimental signals, and the results illustrate the superiority of AQMLCD method.

Randomized Quaternion Singular Value Decomposition for Low-Rank Matrix Approximation

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 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 algorithm. Theoretical results show that the randomized QSVD algorithm can trace dominant singular value decomposition triplets of a quaternion matrix with acceptable accuracy. Numerical experiments also indicate the rationality of proposed theories. Applied to color face recognition problems, the randomized QSVD algorithm obtains higher recognition accuracies and behaves more efficient than the known Lanczos-based partial QSVD and a quaternion version of the fast frequent directions algorithm.

A blind dual color images watermarking based on quaternion singular value decomposition

A blind dual color images watermarking based on quaternion singular value decomposition

Robust image watermarking algorithm based on QWT and QSVD using 2D Chebyshev-Logistic map

In this paper, a novel watermarking algorithm based on quaternion wavelet transform(QWT) and quaternion singular value decomposition(QSVD) is proposed for copyright protection. Firstly, color images are converted from RGB space to YCbCr space. Then one-level QWT transform is applied to the luminance component Y. The LL component is split to 4×4 non-overlapping blocks. Every block is decomposed by QSVD, and the bits of watermark are embedded in singular vectors or singular values. Moreover, 2D Chebyshev-Logistic map is proposed to encrypt the watermarks and scramble the embedding positions. In the watermarking extraction process, both blind and semi-blind watermarking extraction methods are introduced. Experiments show that proposed watermarking algorithm has good robust against common attacks and has good invisibility.

A Novel Zero Watermarking Based on DT-CWT and Quaternion for HDR Image

This paper presents a high dynamic range (HDR) image zero watermarking method based on dual tree complex wavelet transform (DT-CWT) and quaternion. In order to be against tone mapping (TM), DT-CWT is used to transform the three RGB color channels of the HDR image for obtaining the low-pass sub-bands, respectively, since DT-CWT can extract the contour of the HDR image and the contour change of the HDR image is small after TM. The HDR image provides a wide dynamic range, and thus, three-color channel correlations are higher than inner-relationships and the quaternion is used to consider three color channels as a whole to be transformed. Quaternion fast Fourier transform (QFFT) and quaternion singular value decomposition (QSVD) are utilized to decompose the HDR image for obtaining robust features, which is fused with a binary watermark to generate a zero watermark for copyright protection. Furthermore, the binary watermark is scrambled for the security by using the Arnold transform. Experimental results denote that the proposed zero-watermarking method is robust to TM and other image processing attacks, and can protect the HDR image efficiently.

Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices

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 a real component which is not useful for the representation of a color image. The main contribution of this paper is to find an optimal rank-$r$ pure quaternion matrix approximation for a pure quaternion matrix (a color image). Our idea is to use a projection on a low-rank quaternion matrix manifold and a projection on a quaternion matrix with zero real component, and develop an alternating projections algorithm to find such optimal low-rank pure quaternion matrix approximation. The convergence of the projection algorithm can be established by showing that the low-rank quaternion matrix manifold and the zero real component quaternion matrix manifold has a nontrivial intersection point. Numerical examples on synthetic pure quaternion matrices and color images are presented to illustrate the projection algorithm can find optimal low-rank pure quaternion approximation for pure quaternion matrices or color images.

A 3D Image Quality Assessment Method Based on Vector Information and SVD of Quaternion Matrix under Cloud Computing Environment

With the increasing demands of end-users to the visual perception in three-dimension (3D) image, quality assessment for 3D imageis dominantly required as the feedback information for multimedia transmission systems. In this paper, a novel full-reference quality assessment method by considering the depth and integral color information of 3D image under cloud computing environment is proposed. Based on the property of the depth information in 3D image, the depth map is firstly separated into different planes according to the perception of human visual system (HVS). Then, after express the image pixels of every separated plane through quaternions, the structural and energy information are separated by quaternion singular value decomposition (QSVD). The distortion of structural and energy in every plane are calculated in various formulas respectively. The final result is calculated in terms of the global score, which synthesizes the structural and energy distortion scores in every individual depth plane. It should be pointed out that the chrominance information is employed in our mechanism to evaluate the color image quality because of its useful characteristic for 3D color image quality assessment, and its spatial correlation is used for calculating structural distortion through vector cross-product. Our experimental results confirm that the proposed method has achieves better performance under cloud computing environments compared with other existing 3D image quality assessment methods.

Quaternion-based sparse tight frame for multicomponent signal recovery

Multicomponent noise attenuation often presents more severe processing challenges than scalar data owing to the uncorrelated random noise in each component. Meanwhile, weak signals merged in the noise are easier to degrade using the scalar processing workflows while ignoring their possible supplement from other components. For seismic data preprocessing, transform-based approaches have achieved improved performance on mitigating noise while preserving the signal of interest, especially when using an adaptive basis trained by dictionary-learning methods. We have developed a quaternion-based sparse tight frame (QSTF) with the help of quaternion matrix and tight frame analyses, which can be used to process the vector-valued multicomponent data by following a vectorial processing workflow. The QSTF is conveniently trained through iterative sparsity-based regularization and quaternion singular-value decomposition. In the quaternion-based sparse domain, multicomponent signals are orthogonally represented, which preserve the nonlinear relationships among multicomponent data to a greater extent as compared with the scalar approaches. We test the performance of our method on synthetic and field multicomponent data, in which component-wise, concatenated, and long-vector models of multicomponent data are used as comparisons. Our results indicate that more features, specifically the weak signals merged in the noise, are better recovered using our method than others.

Image Quality Assessment Based on Quaternion Singular Value Decomposition

We propose an image quality assessment metric based on quaternion singular value decomposition that represents a color image as a quaternion matrix, separates image noise information using singular value decomposition and extracts features from both the whole image and its noise information. In the proposed method, the color image and its local variance are represented by using quaternion and then performing singular value decomposition. Later, 75% of singular values are taken as image noise information. We extract a luminance comparison, contrast comparison, structure comparison, phase congruency and gradient magnitude from whole color images and extract the peak signal-to-noise ratio from image noise information as features. Finally, these features are used as the input to a kernel extreme learning machine to predict the quality of the tested images. Extensive experiments performed on four benchmark image quality assessment databases demonstrate that the proposed metric achieves high consistency with the subjective evaluations and outperforms state-of-the-art image quality assessment metrics.

Robust Dual-Color Watermarking Based on Quaternion Singular Value Decomposition

This paper proposes a robust dual-color watermarking based on quaternion singular value decomposition (QSVD), which can embed large payloads into color images with low distortion, and can obtain strong robustness to process color image in a holistic manner. First, two notes are proposed for designing the proposed watermarking scheme, one of which is about three strong correlations found in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$U$ </tex-math></inline-formula> that can be used for watermark embedding, and the other is analyzing the feasibility of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$ </tex-math></inline-formula> compensation for QSVD-based watermarking scheme. In addition, a fast structure-preserving algorithm is used to calculate the singular value decomposition (SVD) of a quaternion matrix, which makes the procedure computationally more flexible and efficient. Then a new watermarking scheme is proposed to protect the copyright of color images. This scheme uses quaternion to make the color image channels correlated so that the proposed watermarking scheme has strong anti-attack performance. Experimental results show that the proposed dual-color watermarking is not only imperceptible but also robust to some common attacks, and the performance of the proposed method outperforms other methods considered in this work.

Robust Image Hashing With Singular Values Of Quaternion SVD

Abstract Image hashing is an efficient technique of many multimedia systems, such as image retrieval, image authentication and image copy detection. Classification between robustness and discrimination is one of the most important performances of image hashing. In this paper, we propose a robust image hashing with singular values of quaternion singular value decomposition (QSVD). The key contribution is the innovative use of QSVD, which can extract stable and discriminative image features from CIE L*a*b* color space. In addition, image features of a block are viewed as a point in the Cartesian coordinates and compressed by calculating the Euclidean distance between its point and a reference point. As the Euclidean distance requires smaller storage than the original block features, this technique helps to make a discriminative and compact hash. Experiments with three open image databases are conducted to validate efficiency of our image hashing. The results demonstrate that our image hashing can resist many digital operations and reaches a good discrimination. Receiver operating characteristic curve comparisons illustrate that our image hashing outperforms some state-of-the-art algorithms in classification performance.

Quaternion-based weighted nuclear norm minimization for color image denoising

The quaternion method plays an important role in color image processing, because it represents the color image as a whole rather than as a separate color space component, thus naturally handling the coupling among color channels. The weighted nuclear norm minimization (WNNM) scheme assigns different weights to different singular values, leading to more reasonable image representation method. In this paper, we propose a novel quaternion weighted nuclear norm minimization (QWNNM) model and algorithm under the low rank sparse framework. The proposed model represents the color image as a low rank quaternion matrix, where quaternion singular value decomposition can be calculated by its equivalent complex matrix. We solve the QWNNM by adaptively assigning different singular values with different weights. Color image denoising is implemented by QWNNM based on non-local similarity priors. In this new color space, the inherent color structure can be well preserved during image reconstruction. For high noise levels, we apply a Gaussian low pass filter (LPF) to the noisy image as a preprocessing before QWNNM, which reduces the iteration numbers and improves the denoised results. The experimental results clearly show that the proposed method outperforms K-SVD, QKSVD and WNNM in terms of both quantitative criteria and visual perceptual.