arXiv subjects

Zhigang Jia

Publications and source records attributed to Zhigang Jia.

At least 19 recordsLinked to original sources

arXiv2025

A Novel Wasserstein Quaternion Generative Adversarial Network for Color Image Generation

Color image generation has a wide range of applications, but the existing generation models ignore the correlation among color channels, which may lead to chromatic aberration problems. In addition, the data distribution problem of color images has not been systematically elaborated and explained, so that there is still the lack of the theory about measuring different color images datasets. In this paper, we define a new quaternion Wasserstein distance and develop its dual theory. To deal with the quaternion linear programming problem, we derive the strong duality form with helps of quaternion convex set separation theorem and quaternion Farkas lemma. With using quaternion Wasserstein distance, we propose a novel Wasserstein quaternion generative adversarial network. Experiments demonstrate that this novel model surpasses both the (quaternion) generative adversarial networks and the Wasserstein generative adversarial network in terms of generation efficiency and image quality.

cs.CV↗

arXiv2025

Blind Deconvolution for Color Images Using Normalized Quaternion Kernels

In this work, we address the challenging problem of blind deconvolution for color images. Existing methods often convert color images to grayscale or process each color channel separately, which overlooking the relationships between color channels. To handle this issue, we formulate a novel quaternion fidelity term designed specifically for color image blind deconvolution. This fidelity term leverages the properties of quaternion convolution kernel, which consists of four kernels: one that functions similarly to a non-negative convolution kernel to capture the overall blur, and three additional convolution kernels without constraints corresponding to red, green and blue channels respectively model their unknown interdependencies. In order to preserve image intensity, we propose to use the normalized quaternion kernel in the blind deconvolution process. Extensive experiments on real datasets of blurred color images show that the proposed method effectively removes artifacts and significantly improves deblurring effect, demonstrating its potential as a powerful tool for color image deconvolution.

cs.CV↗

arXiv2025

On Eigenvector Computation and Eigenvalue Reordering for the Non-Hermitian Quaternion Eigenvalue Problem

In this paper we present several additions to the quaternion QR algorithm, including algorithms for eigenvector computation and eigenvalue reordering. A key outcome of the eigenvalue reordering algorithm is that the aggressive early deflation (AED) technique, which significantly enhances the convergence of the QR algorithm, is successfully applied to the quaternion eigenvalue problem. We conduct numerical experiments to demonstrate the efficiency and effectiveness of the proposed algorithms.

math.NA↗

arXiv2025

A fragile zero-watermarking method based on dual quaternion matrix decomposition

Medical images play a crucial role in assisting diagnosis, remote consultation, and academic research. However, during the transmission and sharing process, they face serious risks of copyright ownership and content tampering. Therefore, protecting medical images is of great importance. As an effective means of image copyright protection, zero-watermarking technology focuses on constructing watermarks without modifying the original carrier by extracting its stable features, which provides an ideal approach for protecting medical images. This paper aims to propose a fragile zero-watermarking model based on dual quaternion matrix decomposition, which utilizes the operational relationship between the standard part and the dual part of dual quaternions to correlate the original carrier image with the watermark image, and generates zero-watermarking information based on the characteristics of dual quaternion matrix decomposition, ultimately achieving copyright protection and content tampering detection for medical images.

eess.IV↗

arXiv2025

A Novel Adaptive Low-Rank Matrix Approximation Method for Image Compression and Reconstruction

Low-rank matrix approximation plays an important role in various applications such as image processing, signal processing and data analysis. The existing methods require a guess of the ranks of matrices that represent images or involve additional costs to determine the ranks. A novel efficient orthogonal decomposition with automatic basis extraction (EOD-ABE) is proposed to compute the optimal low-rank matrix approximation with adaptive identification of the optimal rank. By introducing a randomized basis extraction mechanism, EOD-ABE eliminates the need for additional rank determination steps and can compute a rank-revealing approximation to a low-rank matrix. With a computational complexity of $O(mnr)$, where $m$ and $n$ are the dimensions of the matrix and $r$ is its rank, EOD-ABE achieves significant speedups compared to the state-of-the-art methods. Experimental results demonstrate the superior speed, accuracy and robustness of EOD-ABE and indicate that EOD-ABE is a powerful tool for fast image compression and reconstruction and hyperspectral image dimensionality reduction in large-scale applications.

math.NA↗

arXiv2025

A New Cross-Space Total Variation Regularization Model for Color Image Restoration with Quaternion Blur Operator

The cross-channel deblurring problem in color image processing is difficult to solve due to the complex coupling and structural blurring of color pixels. Until now, there are few efficient algorithms that can reduce color artifacts in deblurring process. To solve this challenging problem, we present a novel cross-space total variation (CSTV) regularization model for color image deblurring by introducing a quaternion blur operator and a cross-color space regularization functional. The existence and uniqueness of the solution are proved and a new L-curve method is proposed to find a balance of regularization terms on different color spaces. The Euler-Lagrange equation is derived to show that CSTV has taken into account the coupling of all color channels and the local smoothing within each color channel. A quaternion operator splitting method is firstly proposed to enhance the ability of color artifacts reduction of the CSTV regularization model. This strategy also applies to the well-known color deblurring models. Numerical experiments on color image databases illustrate the efficiency and effectiveness of the new model and algorithms. The color images restored by them successfully maintain the color and spatial information and are of higher quality in terms of PSNR, SSIM, MSE and CIEde2000 than the restorations of the-state-of-the-art methods.

cs.CV↗

arXiv2024

Flexible Quaternion Generalized Minimal Residual Method for Ill-Posed Quaternion Inverse Problems

The main goal of this paper is to propose a new quaternion total variation regularization model for solving linear ill-posed quaternion inverse problems, which arise from three-dimensional signal filtering or color image processing. The quaternion total variation term in the model is represented by collaborative total variation regularization and approximated by a quaternion iteratively reweighted norm. A novel flexible quaternion generalized minimal residual method is presented to quickly solve this model. An improved convergence theory is established to obtain a sharp upper bound of the residual norm of quaternion minimal residual method (QGMRES). The convergence theory is also presented for preconditioned QGMRES. Numerical experiments indicate the superiority of the proposed model and algorithms over the state-of-the-art methods in terms of iteration steps, CPU time, and the quality criteria of restored color images.

math.NA↗

arXiv2024

Quaternion Generative Adversarial Neural Networks and Applications to Color Image Inpainting

Color image inpainting is a challenging task in imaging science. The existing method is based on real operation, and the red, green and blue channels of the color image are processed separately, ignoring the correlation between each channel. In order to make full use of the correlation between each channel, this paper proposes a Quaternion Generative Adversarial Neural Network (QGAN) model and related theory, and applies it to solve the problem of color image inpainting with large area missing. Firstly, the definition of quaternion deconvolution is given and the quaternion batch normalization is proposed. Secondly, the above two innovative modules are applied to generate adversarial networks to improve stability. Finally, QGAN is applied to color image inpainting and compared with other state-of-the-art algorithms. The experimental results show that QGAN has superiority in color image inpainting with large area missing.

cs.CV↗

arXiv2023

A New Low-Rank Learning Robust Quaternion Tensor Completion Method for Color Video Inpainting Problem and Fast Algorithms

The color video inpainting problem is one of the most challenging problem in the modern imaging science. It aims to recover a color video from a small part of pixels that may contain noise. However, there are less of robust models that can simultaneously preserve the coupling of color channels and the evolution of color video frames. In this paper, we present a new robust quaternion tensor completion (RQTC) model to solve this challenging problem and derive the exact recovery theory. The main idea is to build a quaternion tensor optimization model to recover a low-rank quaternion tensor that represents the targeted color video and a sparse quaternion tensor that represents noise. This new model is very efficient to recover high dimensional data that satisfies the prior low-rank assumption. To solve the case without low-rank property, we introduce a new low-rank learning RQTC model, which rearranges similar patches classified by a quaternion learning method into smaller tensors satisfying the prior low-rank assumption. We also propose fast algorithms with global convergence guarantees. In numerical experiments, the proposed methods successfully recover color videos with eliminating color contamination and keeping the continuity of video scenery, and their solutions are of higher quality in terms of PSNR and SSIM values than the state-of-the-art algorithms.

cs.CV↗

arXiv2022

Quasi Non-Negative Quaternion Matrix Factorization with Application to Color Face Recognition

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 these algorithms encoded on the red, green and blue channels. Furthermore, we apply the proposed algorithms to the color face recognition. Numerical results indicate that the accuracy rate of face recognition on the quaternion model is better than on the red, green and blue channels of color image as well as single channel of gray level images for the same data, when large facial expressions and shooting angle variations are presented.

cs.CV↗

arXiv2022

Non-Local Robust Quaternion Matrix Completion for Color Images and Videos Inpainting

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.

cs.CV↗

arXiv2021

Randomized Quaternion Singular Value Decomposition for Low-Rank 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 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 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 fast frequent directions algorithm.

math.NA↗

arXiv2021

On condition numbers of the total least squares problem with linear equality constraint

This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and componentwise condition numbers of the TLSE problem are derived. Computable expressions and upper bounds for these condition numbers are also given to avoid the costly Kronecker product-based operations. The results unify the ones for the TLS problem. For TLSE problems with equilibratory input data, numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition numbers-based estimates are much tighter.

math.NA↗

arXiv2020

A contribution to condition numbers of the multidimensional total least squares problem with linear equality constraint

This paper is devoted to condition numbers of the multidimensional total least squares problem with linear equality constraint (TLSE). Based on the perturbation theory of invariant subspace, the TLSE problem is proved to be equivalent to a multidimensional unconstrained weighed total least squares problem in the limit sense. With a limit technique, Kronecker-product-based formulae for normwise, mixed and componentwise condition numbers of the minimum Frobenius norm TLSE solution are given. Compact upper bounds of these condition numbers are provided to reduce the storage and computation cost. All expressions and upper bounds of these condition numbers unify the ones for the single-dimensional TLSE problem and multidimensional total least squares problem. Some numerical experiments are performed to illustrate our results.

math.NA↗

arXiv2020

The Multi-Symplectic Lanczos Algorithm and Its Applications to Color Image Processing

Low-rank approximations of original samples are playing more and more an important role in many recently proposed mathematical models from data science. A natural and initial requirement is that these representations inherit original structures or properties. With this aim, we propose a new multi-symplectic method based on the Lanzcos bidiagonalization to compute the partial singular triplets of JRS-symmetric matrices. These singular triplets can be used to reconstruct optimal low-rank approximations while preserving the intrinsic multi-symmetry. The augmented Ritz and harmonic Ritz vectors are used to perform implicit restarting to obtain a satisfactory bidiagonal matrix for calculating the $k$ largest or smallest singular triplets, respectively. We also apply the new multi-symplectic Lanczos algorithms to color face recognition and color video compressing and reconstruction. Numerical experiments indicate their superiority over the state-of-the-art algorithms.

math.NA↗

arXiv2019

Advanced Variations of Two-Dimensional Principal Component Analysis for Face Recognition

The two-dimensional principal component analysis (2DPCA) has become one of the most powerful tools of artificial intelligent algorithms. In this paper, we review 2DPCA and its variations, and propose a general ridge regression model to extract features from both row and column directions. To enhance the generalization ability of extracted features, a novel relaxed 2DPCA (R2DPCA) is proposed with a new ridge regression model. R2DPCA generates a weighting vector with utilizing the label information, and maximizes a relaxed criterion with applying an optimal algorithm to get the essential features. The R2DPCA-based approaches for face recognition and image reconstruction are also proposed and the selected principle components are weighted to enhance the role of main components. Numerical experiments on well-known standard databases indicate that R2DPCA has high generalization ability and can achieve a higher recognition rate than the state-of-the-art methods, including in the deep learning methods such as CNNs, DBNs, and DNNs.

cs.CV↗

arXiv2018

Sample-Relaxed Two-Dimensional Color Principal Component Analysis for Face Recognition and Image Reconstruction

A sample-relaxed two-dimensional color principal component analysis (SR-2DCPCA) approach is presented for face recognition and image reconstruction based on quaternion models. A relaxation vector is automatically generated according to the variances of training color face images with the same label. A sample-relaxed, low-dimensional covariance matrix is constructed based on all the training samples relaxed by a relaxation vector, and its eigenvectors corresponding to the $r$ largest eigenvalues are defined as the optimal projection. The SR-2DCPCA aims to enlarge the global variance rather than to maximize the variance of the projected training samples. The numerical results based on real face data sets validate that SR-2DCPCA has a higher recognition rate than state-of-the-art methods and is efficient in image reconstruction.

cs.CV↗

arXiv2018

Perturbation Analysis of An Eigenvector-Dependent Nonlinear Eigenvalue Problem With Applications?

The eigenvector-dependent nonlinear eigenvalue problem (NEPv) $A(P)V=VΛ$, where the columns of $V\in\mathbb{C}^{n\times k}$ are orthonormal, $P=VV^{\mathrm{H}}$, $A(P)$ is Hermitian, and $Λ=V^{\mathrm{H}}A(P)V$, arises in many important applications, such as the discretized Kohn-Sham equation in electronic structure calculations and the trace ratio problem in linear discriminant analysis. In this paper, we perform a perturbation analysis for the NEPv, which gives upper bounds for the distance between the solution to the original NEPv and the solution to the perturbed NEPv. A condition number for the NEPv is introduced, which reveals the factors that affect the sensitivity of the solution. Furthermore, two computable error bounds are given for the NEPv, which can be used to measure the quality of an approximate solution. The theoretical results are validated by numerical experiments for the Kohn-Sham equation and the trace ratio optimization.

math.NA↗