arXiv subjects

Kit Ian Kou

Publications and source records attributed to Kit Ian Kou.

At least 19 recordsLinked to original sources

arXiv2026

Tensor Orthogonal Subspace Split: Theory and Applications

Tensor representations have emerged as a fundamental paradigm for modeling multidimensional data by preserving intrinsic correlations across multiple modes. This paper proposes a novel theoretical framework, termed Tensor Orthogonal Subspace Split (TOSS), which explicitly splits a tensor, along a prescribed mode, into two orthogonal components: a dominant component lying in a prescribed subspace and a residual component lying in the corresponding orthogonal complement. We first present the general formulation of TOSS and systematically investigate its fundamental properties. As an important and practically meaningful special case, we further introduce the rank-one TOSS, which imposes a separable rank-one structure along the splitting mode and admits a clear geometric interpretation. This formulation naturally captures dominant consistent patterns while effectively isolating orthogonal residual component. The proposed framework establishes a unified theoretical foundation for tensor-domain orthogonal split and opens new avenues for structured tensor modeling across diverse applications. Building upon the developed TOSS theory, hyperspectral image restoration and color video background modeling are considered as two representative tasks, for which corresponding optimization models are formulated. Efficient algorithms are developed to solve the resulting problems. Extensive experimental results validate the effectiveness and superiority of the proposed approaches.

math.NA↗

arXiv2026

Quaternion Tensor Modeling for Joint Color-Polarization Demosaicking

Division-of-focal-plane (DoFP) color polarization cameras enable snapshot acquisition of color polarization mosaic images, but the inherently sparse sampling pattern makes color polarization demosaicking severely ill-posed. Existing methods often fail to jointly exploit the correlations among polarization channels and the physical constraints inherent in polarization imaging, resulting in noticeable demosaicking artifacts. To address this issue, a quaternion-tensor-based color polarization demosaicking (CPDM) method incorporating Stokes-domain total variation (TV) regularization is proposed. Correlation analysis shows that the correlations among polarization channels are stronger than those among color channels. Accordingly, the color polarization images acquired at $0^\circ$, $45^\circ$, $90^\circ$, and $135^\circ$ are encoded into the four components of a third-order quaternion tensor, with the color channels organized along its third mode. A low-rank prior is then imposed on the quaternion tensor to exploit the global structural redundancy in the color polarization data. Moreover, spatial gradients are mapped to the Stokes domain through an orthogonal transformation to separate intensity, polarization and residual variations, with adaptive quaternion weights enabling component-specific regularization and preserving the energy consistency of the reconstructed Stokes vectors. An efficient optimization algorithm is derived for the resulting model. Extensive experiments demonstrate the superior demosaicking performance of the proposed method.

cs.CV↗

arXiv2024

Advancements in Log-P-Analytic Functions: Landau-Type Theorems and Their Refinements

This work begins by introducing the groundbreaking concept of log-p-analytic functions. Following this introduction, we proceed to delineate four distinct formulations of Landau-type theorems, specifically crafted for the domain of poly-analytic functions. Among these, two theorems are distinguished by their exactitude, and a third theorem offers a refinement to the existing work of Abdulhadi and Hajj. Concluding the paper, we present four specialized versions of Landau-type theorems applicable to a subset of bounded log-p-analytic functions, resulting in the derivation of two precise outcomes.

math.CV↗

arXiv2024

FFT reconstruction of signals from MIMO sampled data

This paper introduces an innovative approach for signal reconstruction using data acquired through multi-input-multi-output (MIMO) sampling. First, we show that it is possible to perfectly reconstruct a set of periodic band-limited signals $\{x_r(t)\}_{r=1}^R$ from the samples of $\{y_m(t)\}_{m=1}^M$, which are the output signals of a MIMO system with inputs $\{x_r(t)\}_{r=1}^R$. Moreover, an FFT-based algorithm is designed to perform the reconstruction efficiently. It is demonstrated that this algorithm encompasses FFT interpolation and multi-channel interpolation as special cases. Then, we investigate the consistency property and the aliasing error of the proposed sampling and reconstruction framework to evaluate its effectiveness in reconstructing non-band-limited signals. The analytical expression for the averaged mean square error (MSE) caused by aliasing is presented. Finally, the theoretical results are validated by numerical simulations, and the performance of the proposed reconstruction method in the presence of noise is also examined.

cs.IT↗

arXiv2024

Randomized Quaternion UTV Decomposition and Randomized Quaternion Tensor UTV Decomposition

In this paper, the quaternion matrix UTV (QUTV) decomposition and quaternion tensor UTV (QTUTV) decomposition are proposed. To begin, the terms QUTV and QTUTV are defined, followed by the algorithms. Subsequently, by employing random sampling from the quaternion normal distribution, randomized QUTV and randomized QTUTV are generated to provide enhanced algorithmic efficiency. These techniques produce decompositions that are straightforward 9 to understand and require minimal cost. Furthermore, theoretical analysis is discussed. Specifically, the upper bounds for approximating QUTV on the rank-K and QTUTV on the TQt-rank K errors are provided, followed by deterministic error bounds and average-case error bounds for the randomized situations, which demonstrate the correlation between the accuracy of the low-rank approximation and the singular values. Finally, numerous numerical experiments are presented to verify that the proposed algorithms work more efficiently and with similar relative errors compared to other comparable decomposition methods. For the novel decompositions, the theory analysis offers a solid theoretical basis and the experiments show significant potential for the associated processing tasks of color images and color videos.

math.NA↗

arXiv2024

The Extended Paley-Wiener Theorem over the Hardy-Sobolev Spaces

We examine how the square-integrable function subspaces are transformed using the holomorphic Fourier transform. On account of this, the extended Paley-Wiener theorem over the Hardy-Sobolev spaces is produced. The theorem also asserts that the reproducing kernel of the Hardy-Sobolev spaces can be found. We discuss the relationship between the disc and the upper half-plane.

math.FA↗

arXiv2024

Efficient quaternion CUR method for low-rank approximation to quaternion matrix

The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix approximation is sometimes considerable due to the computation of the quaternion singular value decomposition (QSVD), which limits their application to real large-scale data. To address this deficiency, an efficient quaternion matrix CUR (QMCUR) method for low-rank approximation is suggested, which provides significant acceleration in color image processing. We first explore the QMCUR approximation method, which uses actual columns and rows of the given quaternion matrix, instead of the costly QSVD. Additionally, two different sampling strategies are used to sample the above-selected columns and rows. Then, the perturbation analysis is performed on the QMCUR approximation of noisy versions of low-rank quaternion matrices. Extensive experiments on both synthetic and real data further reveal the superiority of the proposed algorithm compared with other algorithms for getting low-rank approximation, in terms of both efficiency and accuracy.

math.NA↗

arXiv2023

Quaternion tensor left ring decomposition and application for color image inpainting

In recent years, tensor networks have emerged as powerful tools for solving large-scale optimization problems. One of the most promising tensor networks is the tensor ring (TR) decomposition, which achieves circular dimensional permutation invariance in the model through the utilization of the trace operation and equitable treatment of the latent cores. On the other hand, more recently, quaternions have gained significant attention and have been widely utilized in color image processing tasks due to their effectiveness in encoding color pixels by considering the three color channels as a unified entity. Therefore, in this paper, based on the left quaternion matrix multiplication, we propose the quaternion tensor left ring (QTLR) decomposition, which inherits the powerful and generalized representation abilities of the TR decomposition while leveraging the advantages of quaternions for color pixel representation. In addition to providing the definition of QTLR decomposition and an algorithm for learning the QTLR format, the paper further proposes a low-rank quaternion tensor completion (LRQTC) model and its algorithm for color image inpainting based on the defined QTLR decomposition. Finally, extensive experiments on color image inpainting demonstrate that the proposed LRQTC method is highly competitive.

cs.CV↗

arXiv2023

$L_{2,1}$-Norm Regularized Quaternion Matrix Completion Using Sparse Representation and Quaternion QR Decomposition

Color image completion is a challenging problem in computer vision, but recent research has shown that quaternion representations of color images perform well in many areas. These representations consider the entire color image and effectively utilize coupling information between the three color channels. Consequently, low-rank quaternion matrix completion (LRQMC) algorithms have gained significant attention. We propose a method based on quaternion Qatar Riyal decomposition (QQR) and quaternion $L_{2,1}$-norm called QLNM-QQR. This new approach reduces computational complexity by avoiding the need to calculate the QSVD of large quaternion matrices. We also present two improvements to the QLNM-QQR method: an enhanced version called IRQLNM-QQR that uses iteratively reweighted quaternion $L_{2,1}$-norm minimization and a method called QLNM-QQR-SR that integrates sparse regularization. Our experiments on natural color images and color medical images show that IRQLNM-QQR outperforms QLNM-QQR and that the proposed QLNM-QQR-SR method is superior to several state-of-the-art methods.

cs.CV↗

arXiv2023

The Color Clifford Hardy Signal: Application to Color Edge Detection and Optical Flow

This paper introduces the idea of the color Clifford Hardy signal, which can be used to process color images. As a complex analytic function's high-dimensional analogue, the color Clifford Hardy signal inherits many desirable qualities of analyticity. A crucial tool for getting the color and structural data is the local feature representation of a color image in the color Clifford Hardy signal. By looking at the extended Cauchy-Riemann equations in the high-dimensional space, it is possible to see the connection between the different parts of the color Clifford Hardy signal. Based on the distinctive and important local amplitude and local phase generated by the color Clifford Hardy signal, we propose five methods to identify the edges of color images with relation to a certain color. To prove the superiority of the offered methodologies, numerous comparative studies employing image quality assessment criteria are used. Specifically by using the multi-scale structure of the color Clifford Hardy signal, the proposed approaches are resistant to a variety of noises. In addition, a color optical flow detection method with anti-noise ability is provided as an example of application.

eess.IV↗

arXiv2023

Quaternion Matrix Completion Using Untrained Quaternion Convolutional Neural Network for Color Image Inpainting

The use of quaternions as a novel tool for color image representation has yielded impressive results in color image processing. By considering the color image as a unified entity rather than separate color space components, quaternions can effectively exploit the strong correlation among the RGB channels, leading to enhanced performance. Especially, color image inpainting tasks are highly beneficial from the application of quaternion matrix completion techniques, in recent years. However, existing quaternion matrix completion methods suffer from two major drawbacks. First, it can be difficult to choose a regularizer that captures the common characteristics of natural images, and sometimes the regularizer that is chosen based on empirical evidence may not be the optimal or efficient option. Second, the optimization process of quaternion matrix completion models is quite challenging because of the non-commutativity of quaternion multiplication. To address the two drawbacks of the existing quaternion matrix completion approaches mentioned above, this paper tends to use an untrained quaternion convolutional neural network (QCNN) to directly generate the completed quaternion matrix. This approach replaces the explicit regularization term in the quaternion matrix completion model with an implicit prior that is learned by the QCNN. Extensive quantitative and qualitative evaluations demonstrate the superiority of the proposed method for color image inpainting compared with some existing quaternion-based and tensor-based methods.

eess.IV↗

arXiv2023

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. Instead of decomposition, a direct analytical method is proposed to achieve FXTSYN of UCQVMNNs using one-norm smoothly. Then apply the set-valued map and the differential inclusion theorem to handle discontinuity problems of drive-response systems. The novel nonlinear controllers together with the Lyapunov function are designed to achieve the control goal. Using the FXTSYN theory and inequality techniques, some criteria of FXTSYN for UCQVMNNs are given. Furthermore, the estimated settling time is obtained explicitly. Finally, numerical simulations are presented to demonstrate the correctness, effectiveness and practicability of the obtained theoretical results.

eess.SY↗

arXiv2022

Quaternion Tensor Completion with Sparseness for Color Video Recovery

A novel low-rank completion algorithm based on the quaternion tensor is proposed in this paper. This approach uses the TQt-rank of quaternion tensor to maintain the structure of RGB channels throughout the entire process. In more detail, the pixels in each frame are encoded on three imaginary parts of a quaternion as an element in a quaternion matrix. Each quaternion matrix is then stacked into a quaternion tensor. A logarithmic function and truncated nuclear norm are employed to characterize the rank of the quaternion tensor in order to promote the low rankness of the tensor. Moreover, by introducing a newly defined quaternion tensor discrete cosine transform-based (QTDCT) regularization to the low-rank approximation framework, the optimized recovery results can be obtained in the local details of color videos. In particular, the sparsity of the quaternion tensor is reasonably characterized by l1 norm in the QDCT domain. This strategy is optimized via the two-step alternating direction method of multipliers (ADMM) framework. Numerical experimental results for recovering color videos show the obvious advantage of the proposed method over other potential competing approaches.

eess.IV↗

arXiv2022

Convolution theorems associated with quaternion linear canonical transform and applications

Novel types of convolution operators for quaternion linear canonical transform (QLCT) are proposed. Type one and two are defined in the spatial and QLCT spectral domains, respectively. They are distinct in the quaternion space and are consistent once in complex or real space. Various types of convolution formulas are discussed. Consequently, the QLCT of the convolution of two quaternionic functions can be implemented by the product of their QLCTs, or the summation of the products of their QLCTs. As applications, correlation operators and theorems of the QLCT are derived. The proposed convolution formulas are used to solve Fredholm integral equations with special kernels. Some systems of second-order partial differential equations, which can be transformed into the second-order quaternion partial differential equations, can be solved by the convolution formulas as well. As a final point, we demonstrate that the convolution theorem facilitates the design of multiplicative filters.

math.CA↗

arXiv2022

Low Rank Quaternion Matrix Completion Based on Quaternion QR Decomposition and Sparse Regularizer

Matrix completion is one of the most challenging problems in computer vision. Recently, quaternion representations of color images have achieved competitive performance in many fields. Because it treats the color image as a whole, the coupling information between the three channels of the color image is better utilized. Due to this, low-rank quaternion matrix completion (LRQMC) algorithms have gained considerable attention from researchers. In contrast to the traditional quaternion matrix completion algorithms based on quaternion singular value decomposition (QSVD), we propose a novel method based on quaternion Qatar Riyal decomposition (QQR). In the first part of the paper, a novel method for calculating an approximate QSVD based on iterative QQR is proposed (CQSVD-QQR), whose computational complexity is lower than that of QSVD. The largest $r \ (r>0)$ singular values of a given quaternion matrix can be computed by using CQSVD-QQR. Then, we propose a new quaternion matrix completion method based on CQSVD-QQR which combines low-rank and sparse priors of color images. Experimental results on color images and color medical images demonstrate that our model outperforms those state-of-the-art methods.

eess.IV↗

arXiv2022

Monogenic Signal Associated with Linear Canonical Transform and Application to Edge Detection Problems

Monogenic signal is regarded as a generalization of analytic signal from the one dimensional space to the high dimensional space. It is defined by an original signal with the combination of Riesz transform. Then it provides the signal features representation, such as the local attenuation and the local phase vector. The main objective of this study is to analyze the local phase vector and the local attenuation in the high dimensional spaces. The differential phase congruency is applied for the edge detection problems.

math.CA↗

arXiv2022

Quaternion Tensor Train Rank Minimization with Sparse Regularization in a Transformed Domain for Quaternion Tensor Completion

The tensor train rank (TT-rank) has achieved promising results in tensor completion due to its ability to capture the global low-rankness of higher-order (>3) tensors. On the other hand, recently, quaternions have proven to be a very suitable framework for encoding color pixels, and have obtained outstanding performance in various color image processing tasks. In this paper, the quaternion tensor train (QTT) decomposition is presented, and based on that the quaternion TT-rank (QTT-rank) is naturally defined, which are the generalizations of their counterparts in the real number field. In addition, to utilize the local sparse prior of the quaternion tensor, a general and flexible transform framework is defined. Combining both the global low-rank and local sparse priors of the quaternion tensor, we propose a novel quaternion tensor completion model, i.e., QTT-rank minimization with sparse regularization in a transformed domain. Specifically, we use the quaternion weighted nuclear norm (QWNN) of mode-n canonical unfolding quaternion matrices to characterize the global low-QTT-rankness, and the l1-norm of the quaternion tensor in a transformed domain to characterize the local sparse property. Moreover, to enable the QTT-rank minimization to handle color images and better handle color videos, we generalize KA, a tensor augmentation method, to quaternion tensors and define quaternion KA (QKA), which is a helpful pretreatment step for QTT-rank based optimization problems. The numerical experiments on color images and color videos inpainting tasks indicate the advantages of the proposed method over the state-of-the-art ones.

eess.IV↗

arXiv2022

3-D generalized analytic signal associated with linear canonical transform in Clifford biquaternion domain

The analytic signal is a useful mathematical tool. It separates qualitative and quantitative information of a signal in form of the local phase and local amplitude. The Clifford Fourier transform (CFT) plays a vital role in the representation of multidimensional signals. By generalizing the CFT to the Clifford linear canonical transform (CLCT), we present a new type of Clifford biquaternionic analytic signal. Due to the advantages of more freedom, the envelop detection problems of 3D images, with the help of this new analytic signal, can get a better visual appearance. Synthesis examples are presented to demonstrate these advantages.

math.CV↗