arXiv subjects

Bing-Zhao Li

Publications and source records attributed to Bing-Zhao Li.

At least 19 recordsLinked to original sources

arXiv2026

Spectral Graph Uncertainty Principles via the Graph Fractional Fourier Transform

This paper develops a graph fractional uncertainty principle in the graph fractional Fourier transform (GFRFT) domain. We introduce localization operators in the vertex domain and the graph fractional spectral domain, and build an operator framework to characterize the joint localization of graph signals. A sandwiched joint localization operator is first constructed, whose largest eigenvalue quantifies the attainable simultaneous concentration in the two domains. Then, rotated localization operators and the numerical range are used to provide a geometric description of the admissible uncertainty region, together with a polygonal approximation method for its computation. Numerical examples show that the fractional order reshapes the vertex-graph fractional spectral localization trade-off, and enlarge or shrink the uncertainty region relative to the graph Fourier transform based case. These results generalize classical graph uncertainty principles to the GFRFT domain and provide a flexible tool for graph fractional signal analysis and graph-adapted localized representations.

math.GM↗

arXiv2026

The uncertainty principles of random signals related to the linear canonical transform

In this paper, we investigate uncertainty principles for random signals associated with the linear canonical transform (LCT). First, the LCT of random signals is formulated on the probability space. Based on this representation, the Heisenberg uncertainty principle is established to characterize the relationship between the expectations in the time and frequency domains. Furthermore, the Donoho-Stark uncertainty principle, developed from a measure theoretic perspective, reveals that a random signal cannot be simultaneously concentrated on arbitrarily small sets in both the time and frequency domains. The bounds obtained in these two uncertainty principles explicitly depend on the LCT parameters, indicating that the LCT offers greater flexibility than the Fourier transform (FT). The corresponding results in the fractional Fourier transform and FT domains are also given as special cases.

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arXiv2026

Graph Fractional Fourier Transform: A Unified and Efficient Sampling Theory

The graph Fourier transform (GFT) is a fundamental tool in graph signal processing and has recently been extended to the graph fractional Fourier transform (GFRFT). Existing sampling methods in the GFRFT domain are primarily designed to minimize error, whereas a wider range of alternative sampling strategies should be admitted. In this paper, a unified and efficient GFRFT sampling theory is proposed. First, a new definition of graph fractional bandlimited signals is introduced, with the corresponding graph fractional sampling and perfect reconstruction theorem, as well as the associated graph fractional localization operator. Next, several GFRFT sampling strategies are developed based on different criteria, including maximum cutoff frequency, minimum error, and maximum localized basis, along with the corresponding representations of their localization operators. Then, by exploiting a localization operator that jointly considers vertex and spectral localization, a fast sampling set selection method in the GFRFT domain is proposed. Finally, numerical experiments investigate the reconstruction errors and execution time of the proposed sampling methods and evaluate their performance in applications, demonstrating the effectiveness of the unified GFRFT sampling theory and its advantages over GFT methods.

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arXiv2026

Manifold Fractional Harmonic Transform for 3D Point Clouds

Point clouds can be regarded as discrete samples of smooth manifolds and are typically analyzed via the eigenfunctions of the Laplace-Beltrami operator. This paper extends manifold spectral analysis to the fractional domain, enabling continuous interpolation between the spatial and spectral domains for point cloud data. First, a point cloud manifold fractional harmonic transform (PMFHT) is proposed, with its fundamental properties rigorously derived, along with the associated convolution, correlation, and sampling theorems. These theoretical results establish a solid foundation for stable fractional-order spectral representation on manifolds. Second, within the PMFHT framework, two representative algorithms are developed. On the one hand, by integrating multi-order PMFHT with chaotic phase modulation, a point cloud encryption scheme is constructed, characterized by a large key space and high sensitivity to key perturbations. On the other hand, an optimal filter is designed in the fractional manifold spectral domain, leading to a maritime target detection method specifically tailored for point cloud data, which effectively suppresses sea clutter while preserving weak target energy under low signal-to-clutter ratio conditions. Finally, experiments on measured data validate the effectiveness of the proposed algorithms.

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arXiv2026

Two-Channel Filter Banks on Joint Time-Vertex Graphs with Oversampled Graph Laplacian Matrix

To address the limitations of conventional critically sampled graph filter banks in joint time-vertex signal processing, which require decomposing the joint graph into bipartite subgraphs and thus cannot fully exploit all temporal and spatial edges in a single-stage transform, we introduce the joint time-vertex oversampled graph Laplacian matrix. This operator enables the construction of bipartite extensions that preserve all edges of the original joint graph and supports redundant multiresolution representations. Based on this operator, we design two-channel joint time-vertex oversampled graph filter banks and develop efficient oversampling extensions using a $K$-coloring strategy. The proposed framework is applied to both graph signal and image/video denoising, modeling images as graph signals to leverage structural relationships. Extensive experiments demonstrate its effectiveness in decomposition, reconstruction, and denoising, achieving notable performance improvements over critically sampled and existing methods.

math.GM↗

arXiv2025

Fractional harmonic transform on point cloud manifolds

Three-dimensional point clouds can be viewed as discrete samples of smooth manifolds, allowing spectral analysis using the Laplace-Beltrami operator (LBO). However, the traditional point cloud manifold harmonic transform (PMHT) is limited by its fixed basis functions and single spectral representation, which restricts its ability to capture complex geometric features. This paper proposes a point cloud manifold fractional harmonic transform (PMFHT), which generalizes PMHT by introducing fractional-order parameters and constructs a continuously adjustable intermediate fractional-order spectral domain between the spatial domain and the frequency domain. This fractional-order framework supports more flexible transformation and filtering operations. Experiments show that choosing different transformation orders can enrich the spectral representation of point clouds and achieve excellent results in tasks such as filtering and feature enhancement. Therefore, PMFHT not only expands the theoretical framework of point cloud spectral analysis, but also provides a powerful new tool for manifold geometry processing.

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arXiv2025

The 2p order Heisenberg-Pauli-Weyl uncertainty principles related to the offset linear canonical transform

The uncertainty principle is one of the fundamental tools for time-frequency analysis in signal processing, revealing the intrinsic trade-off between time and frequency resolutions. With the continuous development of various advanced time-frequency analysis methods based on the Fourier transform, investigating uncertainty principles associated with these methods has become one of the most interesting topics. This paper studies the uncertainty principles related to the offset linear canonical transform, including the Plancherel-Parseval-Rayleigh identity, the $2p$ order Heisenberg-Pauli-Weyl uncertainty principle and the sharpened Heisenberg-Weyl uncertainty principle. Numerical simulations are also proposed to validate the derived results.

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arXiv2025

Sampling of Graph Signals Based on Joint Time-Vertex Fractional Fourier Transform

With the growing demand for non-Euclidean data analysis, graph signal processing (GSP) has gained significant attention for its capability to handle complex time-varying data. This paper introduces a novel sampling method based on the joint time-vertex fractional Fourier transform (JFRFT), enhancing signal representation in time-frequency analysis and GSP. The JFRFT sampling theory is established by deriving conditions for the perfect recovery of jointly bandlimited signals, along with an optimal sampling set selection strategy. To further enhance the efficiency of large-scale time-vertex signal processing, the design of localized sampling operators is investigated. Numerical simulations and real data experiments validate the superior performance of the proposed methods in terms of recovery accuracy and computational efficiency, offering new insights into efficient time-varying signal processing.

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arXiv2025

Anti-interrupted sampling repeater jamming via linear canonical Wigner distribution lightweight LFM detection

Interrupted sampling repeater jamming (ISRJ) poses a serious threat to radar target detection. Traditional time-frequency (TF) domain anti-jamming methods are prone to TF aliasing in multi-component signal scenarios, and cannot effectively suppress ISRJ with energy close to the real target under low signal-to-noise ratio (SNR) conditions. To address these challenges, this paper proposes an anti-jamming method based on generalized linear canonical Wigner distribution (GLWD) line detection. By setting the parameters reasonably, the TF image of GLWD can have excellent TF resolution and energy concentration, greatly improving the signal separation and SNR. Furthermore, in order to enhance the detection capability of the target LFM signal, the existing mobile line segment detection (M-LSD) is improved and the mobile long line segment detection (M-LLSD) is proposed. M-LLSD can detect the target signal more easily and reduce the sensitivity to the jamming signal, so as to efficiently and accurately extract the TF position information of the target signal. Finally, a TF filter is constructed based on the mapping between GLWD and short-time Fourier transform (STFT), performing filtering in the STFT domain to suppress jamming. Simulations and experiments show that the method can effectively suppress such difficult-to-distinguish jamming and is suitable for real-time radar anti-jamming with good robustness.

math.GM↗

arXiv2025

Reconstruction of Graph Signals on Complex Manifolds with Kernel Methods

Graph signals are widely used to describe vertex attributes or features in graph-structured data, with applications spanning the internet, social media, transportation, sensor networks, and biomedicine. Graph signal processing (GSP) has emerged to facilitate the analysis, processing, and sampling of such signals. While kernel methods have been extensively studied for estimating graph signals from samples provided on a subset of vertices, their application to complex-valued graph signals remains largely unexplored. This paper introduces a novel framework for reconstructing graph signals using kernel methods on complex manifolds. By embedding graph vertices into a higher-dimensional complex ambient space that approximates a lower-dimensional manifold, the framework extends the reproducing kernel Hilbert space to complex manifolds. It leverages Hermitian metrics and geometric measures to characterize kernels and graph signals. Additionally, several traditional kernels and graph topology-driven kernels are proposed for reconstructing complex graph signals. Finally, experimental results on synthetic and real-world datasets demonstrate the effectiveness of this framework in accurately reconstructing complex graph signals, outperforming conventional kernel-based approaches. This work lays a foundational basis for integrating complex geometry and kernel methods in GSP.

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arXiv2025

The Graph Fractional Fourier Transform in Hilbert Space

Graph signal processing (GSP) leverages the inherent signal structure within graphs to extract high-dimensional data without relying on translation invariance. It has emerged as a crucial tool across multiple fields, including learning and processing of various networks, data analysis, and image processing. In this paper, we introduce the graph fractional Fourier transform in Hilbert space (HGFRFT), which provides additional fractional analysis tools for generalized GSP by extending Hilbert space and vertex domain Fourier analysis to fractional order. First, we establish that the proposed HGFRFT extends traditional GSP, accommodates graphs on continuous domains, and facilitates joint time-vertex domain transform while adhering to critical properties such as additivity, commutativity, and invertibility. Second, to process generalized graph signals in the fractional domain, we explore the theory behind filtering and sampling of signals in the fractional domain. Finally, our simulations and numerical experiments substantiate the advantages and enhancements yielded by the HGFRFT.

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arXiv2024

A novel STAP algorithm via volume cross-correlation function on the Grassmann manifold

The performance of space-time adaptive processing (STAP) is often degraded by factors such as limited sample size and moving targets. Traditional clutter covariance matrix (CCM) estimation relies on Euclidean metrics, which fail to capture the intrinsic geometric and structural properties of the covariance matrix, thus limiting the utilization of structural information in the data. To address these issues, the proposed algorithm begins by constructing Toeplitz Hermitian positive definite (THPD) matrices from the training samples. The Brauer disc (BD) theorem is then employed to filter out THPD matrices containing target signals, retaining only clutter-related matrices. These clutter matrices undergo eigendecomposition to construct the Grassmann manifold, enabling CCM estimation through the volume cross-correlation function (VCF) and gradient descent method. Finally, the filter weight vector is computed for filtering. By fully leveraging the structural information in radar data, this approach significantly enhances both accuracy and robustness of clutter suppression. Experimental results on simulated and measured data demonstrate superior performance of the proposed algorithm in heterogeneous environments.

math.GM↗

arXiv2024

Clifford-valued linear canonical Stockwell transform

We present a new Clifford-valued linear canonical Stockwell transform aimed at providing efficient and focused representation of Clifford-valued functions in high-dimensional time-frequency analysis. This transform improves upon the windowed Fourier and wavelet transforms by incorporating angular, scalable, and localized windows, allowing for greater directional flexibility in multi-scale signal analysis within the Clifford domain. Using operator theory, we explore the core properties of the proposed transform, such as the inner product relation, reconstruction formula, and interval theorem. Practical examples are included to confirm the validity of the derived results.

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arXiv2024

Linear Canonical Jacobi-Dunkl Transform: Theory and Applications

This paper aims to develop an innovative method for harmonic analysis by introducing the linear canonical Jacobi-Dunkl transform (LCJDT), which integrates both the Jacobi-Dunkl transform (JDT) and the linear canonical transform (LCT). Firstly, the kernel function of the LCJDT is derived, and its fundamental properties are examined. Subsequently, the LCJDT is established, along with an investigation of its essential properties, including the inversion formula, Parseval's theorem, differentiation, the convolution theorem, and the uncertainty principle. Finally, the potential application of the LCJDT in solving the heat equation is explored.

math.GM↗

arXiv2024

Discrete Linear Canonical Transform on Graphs: Uncertainty Principle and Sampling

With an increasing influx of classical signal processing methodologies into the field of graph signal processing, approaches grounded in discrete linear canonical transform have found application in graph signals. In this paper, we initially propose the uncertainty principle of the graph linear canonical transform (GLCT), which is based on a class of graph signals maximally concentrated in both vertex and graph spectral domains. Subsequently, leveraging the uncertainty principle, we establish conditions for recovering bandlimited signals of the GLCT from a subset of samples, thereby formulating the sampling theory for the GLCT. We elucidate interesting connections between the uncertainty principle and sampling. Further, by employing sampling set selection and experimental design sampling strategies, we introduce optimal sampling operators in the GLCT domain. Finally, we evaluate the performance of our methods through simulations and numerical experiments across applications.

math.GM↗

arXiv2024

Linear canonical space-time transform and convolution theorems

Following the idea of the fractional space-time Fourier transform, a linear canonical space-time transform for 16-dimensional space-time $C\ell_{3,1}$-valued signals is investigated in this paper. First, the definition of the proposed linear canonical space-time transform is given, and some related properties of this transform are obtained. Second, the convolution operator and the corresponding convolution theorem are proposed. Third, the convolution theorem associated with the two-sided linear canonical space-time transform is derived.

math.GM↗

arXiv2023

Polar Linear Canonical Wavelet Transform: Theory and Its Application

The polar wavelet transform (PWT) has been proven to be a powerful mathematical tool for signal and image processing in recent years. Due to the increasing demand for directional representations of signals in engineering, it is impossible to fully exploit the intrinsic directional features of signals to describe high-dimensional signals like images. Focusing on this problem, the polar linear canonical wavelet transform (PLCWT) is proposed in this paper. Firstly, the theory of the PLCWT is investigated in detail, including its definition, basic properties and inversion formula. Secondly, the convolution and correlation theorems of the PLCWT are derived. Thirdly, uncertainty principles related to the PLCWT are obtained. Finally, the potential application of the PLCWT in image edge detection is discussed. Simulation results verify the correctness and effectiveness of the proposed method.

math.GM↗

arXiv2023

Sampling theorems associated with offset linear canonical transform by polar coordinates

The sampling theorem for the offset linear canonical transform (OLCT) of bandlimited functions in polar coordinates is an important signal analysis tool in many fields of signal processing and medical imaging. This study investigates two sampling theorems for interpolating bandlimited and highest frequency bandlimited functions in the OLCT and offset linear canonical Hankel transform (OLCHT) domains by polar coordinates. Based on the classical Stark's interpolation formulas, we derive the sampling theorems for bandlimited functions in the OLCT and OLCHT domains, respectively. The first interpolation formula is concise and applicable. Due to the consistency of the OLCHT order, the second interpolation formula is superior to the first interpolation formula in computational complexity.

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