图的离散线性标准变换
Discrete Linear Canonical Transform on Graphs
摘要：
随着谱理论和代数理论在离散信号处理技术中的广泛应用，在图信号处理领域，提出了越来越多的信号处理方法，如图傅里叶变换、图小波变换和加窗图傅里叶变换等。本文提出并设计了图上离散线性正则变换（GLCT）的定义，它是离散线性正则变换（DLCT）的扩展，就像图傅里叶变换（GFT）是离散傅里叶变换（DFT）的扩展一样。首先，基于DLCT特征分解方法的中心性和可扩展性，结合图chirp-傅里叶变换、图尺度变换和图分数傅里叶变换，提出了GLCT的定义。其次，我们推导并讨论了广义线性相关变换的性质和特殊情况。最后给出了图信号的GLCT变换实例，说明了变换的改进。
英文摘要：
With the wide application of spectral and algebraic theory in discrete signal processing techniques in the field of graph signal processing, an increasing number of signal processing methods have been proposed, such as the graph Fourier transform, graph wavelet transform and windowed graph Fourier transform. In this paper, we propose and design the definition of the discrete linear canonical transform on graphs (GLCT), which is an extension of the discrete linear canonical transform (DLCT), just as the graph Fourier transform (GFT) is an extension of the discrete Fourier transform (DFT). First, based on the centrality and scalability of the DLCT eigendecomposition approach, the definition of GLCT is proposed by combining graph chirp-Fourier transform, graph scale transform and graph fractional Fourier transform. Second, we derive and discuss the properties and special cases of GLCT. Finally, some GLCT examples of the graph signals are given to illustrate the improvement of the transformation.
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