arXiv subjects

Kit Ian Kou

Publications and source records attributed to Kit Ian Kou.

At least 37 records · Page 2Linked to original sources

arXiv2022

Quaternion Optimized Model with Sparse Regularization for Color Image Recovery

This paper addresses the color image completion problem in accordance with low-rank quatenrion matrix optimization that is characterized by sparse regularization in a transformed domain. This research was inspired by an appreciation of the fact that different signal types, including audio formats and images, possess structures that are inherently sparse in respect of their respective bases. Since color images can be processed as a whole in the quaternion domain, we depicted the sparsity of the color image in the quaternion discrete cosine transform (QDCT) domain. In addition, the representation of a low-rank structure that is intrinsic to the color image is a vital issue in the quaternion matrix completion problem. To achieve a more superior low-rank approximation, the quatenrion-based truncated nuclear norm (QTNN) is employed in the proposed model. Moreover, this model is facilitated by a competent alternating direction method of multipliers (ADMM) based on the algorithm. Extensive experimental results demonstrate that the proposed method can yield vastly superior completion performance in comparison with the state-of-the-art low-rank matrix/quaternion matrix approximation methods tested on color image recovery.

cs.CV↗

arXiv2022

Signal reconstruction from noisy multichannel samples

We consider the signal reconstruction problem under the case of the signals sampled in the multichannel way and with the presence of noise. Observing that if the samples are inexact, the rigorous enforcement of multichannel interpolation is inappropriate. Thus the reasonable smoothing and regularized corrections are indispensable. In this paper, we propose several alternative methods for the signal reconstruction from the noisy multichannel samples under different smoothing and regularization principles. We compare these signal reconstruction methods theoretically and experimentally in the various situations. To demonstrate the effectiveness of the proposed methods, the probability interpretation and the error analysis for these methods are provided. Additionally, the numerical simulations as well as some guidelines to use the methods are also presented.

cs.IT↗

arXiv2021

Quaternion-based dynamic mode decomposition for background modeling in color videos

Scene Background Initialization (SBI) is one of the challenging problems in computer vision. Dynamic mode decomposition (DMD) is a recently proposed method to robustly decompose a video sequence into the background model and the corresponding foreground part. However, this method needs to convert the color image into the grayscale image for processing, which leads to the neglect of the coupling information between the three channels of the color image. In this study, we propose a quaternion-based DMD (Q-DMD), which extends the DMD by quaternion matrix analysis, so as to completely preserve the inherent color structure of the color image and the color video. We exploit the standard eigenvalues of the quaternion matrix to compute its spectral decomposition and calculate the corresponding Q-DMD modes and eigenvalues. The results on the publicly available benchmark datasets prove that our Q-DMD outperforms the exact DMD method, and experiment results also demonstrate that the performance of our approach is comparable to that of the state-of-the-art ones.

cs.CV↗

arXiv2021

Quaternion tensor singular value decomposition using a flexible transform-based approach

A flexible transform-based tensor product named $\star_{\rm{QT}}$-product for $L$th-order ($L\geq 3$) quaternion tensors is proposed. Based on the $\star_{\rm{QT}}$-product, we define the corresponding singular value decomposition named TQt-SVD and the rank named TQt-rank of the $L$th-order ($L\geq 3$) quaternion tensor. Furthermore, with orthogonal quaternion transformations, the TQt-SVD can provide the best TQt-rank-$s$ approximation of any $L$th-order ($L\geq 3$) quaternion tensor. In the experiments, we have verified the effectiveness of the proposed TQt-SVD in the application of the best TQt-rank-$s$ approximation for color videos represented by third-order quaternion tensors.

math.NA↗

arXiv2021

Biquaternion Z Transform

In this work, the biquaternion Z transformation method is proposed to solve a class of biquaternion recurrence relations. Biqueternion Z transform is an natural extension of the complex Z transform. In the design process, special norm presentation is employed to analyze the region of convergence of the biquaternion geometry sequence. In addition, some useful properties have been given. It is shown that the proposed properties is helpful to understand the biquaternion Z transform. Finally, several examples have been given to illustrate the effectiveness of the proposed design method.

math.CA↗

arXiv2021

A Robust Lane Detection Associated with Quaternion Hardy Filter

In this article, a robust color-edge feature extraction method based on the Quaternion Hardy filter is proposed. The Quaternion Hardy filter is an emerging edge detection theory. It is along with the Poisson and conjugate Poisson smoothing kernels to handle various types of noise. Combining with the Quaternion Hardy filter, Jin's color gradient operator and Hough transform, the color-edge feature detection algorithm is proposed and applied to the lane marking detection. Experiments are presented to demonstrate the validity of the proposed algorithm. The results are accurate and robust with respect to the complex environment lane markings.

cs.CV↗

arXiv2021

Low Rank Quaternion Matrix Recovery via Logarithmic Approximation

In color image processing, image completion aims to restore missing entries from the incomplete observation image. Recently, great progress has been made in achieving completion by approximately solving the rank minimization problem. In this paper, we utilize a novel quaternion matrix logarithmic norm to approximate rank under the quaternion matrix framework. From one side, unlike the traditional matrix completion method that handles RGB channels separately, the quaternion-based method is able to avoid destroying the structure of images via putting the color image in a pure quaternion matrix. From the other side, the logarithmic norm induces a more accurate rank surrogate. Based on the logarithmic norm, we take advantage of not only truncated technique but also factorization strategy to achieve image restoration. Both strategies are optimized based on the alternating minimization framework. The experimental results demonstrate that the use of logarithmic surrogates in the quaternion domain is more superior in solving the problem of color images completion.

eess.IV↗

arXiv2021

Weighted Truncated Nuclear Norm Regularization for Low-Rank Quaternion Matrix Completion

In recent years, quaternion matrix completion (QMC) based on low-rank regularization has been gradually used in image de-noising and de-blurring.Unlike low-rank matrix completion (LRMC) which handles RGB images by recovering each color channel separately, the QMC models utilize the connection of three channels by processing them as a whole. Most of the existing quaternion-based methods formulate low-rank QMC (LRQMC) as a quaternion nuclear norm (a convex relaxation of the rank) minimization problem.The main limitation of these approaches is that the singular values being minimized simultaneously so that the low-rank property could not be approximated well and efficiently. To achieve a more accurate low-rank approximation, the matrix-based truncated nuclear norm has been proposed and also been proved to have the superiority. In this paper, we introduce a quaternion truncated nuclear norm (QTNN) for LRQMC and utilize the alternating direction method of multipliers (ADMM) to get the optimization.We further propose weights to the residual error quaternion matrix during the update process for accelerating the convergence of the QTNN method with admissible performance. The weighted method utilizes a concise gradient descent strategy which has a theoretical guarantee in optimization. The effectiveness of our method is illustrated by experiments on real visual datasets.

eess.IV↗

arXiv2021

A Robust Color Edge Detection Algorithm Based on Quaternion Hardy Filter

This paper presents a robust filter called quaternion Hardy filter (QHF) for color image edge detection. The QHF can be capable of color edge feature enhancement and noise resistance. It is flexible to use QHF by selecting suitable parameters to handle different levels of noise. In particular, the quaternion analytic signal, which is an effective tool in color image processing, can also be produced by quaternion Hardy filtering with specific parameters. Based on the QHF and the improved Di Zenzo gradient operator, a novel color edge detection algorithm is proposed. Importantly, it can be efficiently implemented by using the fast discrete quaternion Fourier transform technique. From the experimental results, we conclude that the minimum PSNR improvement rate is 2.3% and minimum SSIM improvement rate is 30.2% on the Dataset 3. The experiments demonstrate that the proposed algorithm outperforms several widely used algorithms.

cs.CV↗

arXiv2021

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. 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. The defined QHOSVD can be widely used in various visual data processing with color pixels. 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.

eess.IV↗

arXiv2020

Sampling expansions associated with quaternion difference equations

Starting with a quaternion difference equation with boundary conditions, a parameterized sequence which is complete in finite dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of discrete transform, we form a quaternion function space whose elements have sampling expansions. Moreover, through formulating boundary-value problems, we make a connection between a class of tridiagonal quaternion matrices and polynomials with quaternion coefficients. We show that for a tridiagonal symmetric quaternion matrix, one can always associate a quaternion characteristic polynomial whose roots are eigenvalues of the matrix. Several examples are given to illustrate the results.

math.CA↗

arXiv2020

A Robust Color Edge Detection Algorithm Based on Quaternion Hardy Filter

This paper presents a robust filter called quaternion Hardy filter (QHF) for color image edge detection. The QHF can be capable of color edge feature enhancement and noise resistance. It is flexible to use QHF by selecting suitable parameters to handle different levels of noise. In particular, the quaternion analytic signal, which is an effective tool in color image processing, can also be produced by quaternion Hardy filtering with specific parameters. Based on the QHF and the improved Di Zenzo gradient operator, a novel color edge detection algorithm is proposed. Importantly, it can be efficiently implemented by using the fast discrete quaternion Fourier transform technique. The experiments demonstrate that the proposed algorithm outperforms several widely used algorithms.

eess.SP↗

arXiv2020

Constrained low-rank quaternion approximation for color image denoising by bilateral random projections

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 quite accurately in an inexpensive way. Furthermore, color image denoising is further based on nonlocal self-similarity (NSS) prior. The experimental results on color image denoising illustrate the effectiveness and superiority of the proposed method.

eess.IV↗

arXiv2020

Quaternion-based bilinear factor matrix norm minimization for color image inpainting

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), quaternion double nuclear norm (Q-DNN) and quaternion Frobenius/nuclear norm (Q-FNN), and then show their relationship with quaternion-based matrix Schatten-p (Q- Schatten-p ) norm for certain p values. The proposed methods can avoid computing quaternion singular value decompositions (QSVD) for large quaternion matrices, and thus can effectively reduce the calculation time compared with existing (LRQMC) methods. The experimental results demonstrate the superior performance of the proposed methods over some state-of-the-art low-rank (quaternion) matrix completion methods.

eess.IV↗

arXiv2020

A Unified Analysis of Linear Quaternion Dynamic Equations on Time Scales

Over the last years, considerable attention has been paid to the role of the quaternion differential equations (QDEs) which extend the ordinary differential equations. The theory of QDEs was recently well established and it has wide applications in physics and life science. This paper establishes a systematic frame work for the theory of linear quaternion dynamic equations on time scales (QDETS), which can be applied to wave phenomena modeling, fluid dynamics and filter design. The algebraic structure of the solutions to the QDETS is actually a left- or right- module, not a linear vector space. On the non-commutativity of the quaternion algebra, many concepts and properties of the classical dynamic equations on time scales (DETS) can not be applied. They should be redefined accordingly. Using $q$-determinant, a novel definition of Wronskian is introduced under the framework of quaternions which is different from the standard one in DETS. Liouville's formula for QDETS is also analyzed. Upon these, the solutions to the linear QDETS are established. The Putzer's algorithms to evaluate the fundamental solution matrix for homogeneous QDETS are presented. Furthermore, the variation of constants formula to solve the nonhomogeneous QDETs is given. Some concrete examples are provided to illustrate the feasibility of the proposed algorithms.

math.CA↗

arXiv2020

Quaternion Hardy Functions for Local Images Feature Representation

This paper is concerned with the applications of local features of the quaternion Hardy function. The feature information can be provided by the polar form of the quaternion Hardy function, such as the local attenuation and local phase vector. By using the generalized Cauchy-Riemann equations of local features, four various kinds of edge detectors are newly developed. The experiment results show that the amplitude-based edge detectors perform the envelope of test images while the phase-based edge detectors give the detail of the test images. Our proposed algorithms greatly outperform existing schemes.

eess.IV↗

arXiv2020

Low-rank quaternion tensor completion for recovering color videos and images

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 color images and color videos can be completely preserved. Under the definition of Tucker rank, the global low-rank prior to quaternion tensor is encoded as the nuclear norm of unfolding quaternion matrices. Then, by applying the ADMM framework, we provide the tensor completion algorithm for any order quaternion tensors, which theoretically can be well used to recover missing entries of any multidimensional data with color structures. Simulation results for color videos and color images recovery show the superior performance and efficiency of the proposed method over some state-of-the-art existing ones.

math.NA↗

arXiv2020

Quaternion matrix regression for color face recognition

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.

math.NA↗