Graph Fourier transform based on ℓ1 norm variation minimization

Yang, Lihua; Qi, Anna; Huang, Chao; Huang, Jianfeng

doi:10.1016/j.acha.2020.04.001

Cited by 21 publications

(6 citation statements)

References 11 publications

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“…In [16], [17], authors use the eigenvectors of the adjacency matrix W to form the Fourier basis. In our previous work [25], we propose a method to compute the graph Fourier basis by minimizing the…”

Section: B Graph Fourier Transform and Filtersmentioning

confidence: 99%

Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs Based on an Optimization Model

Junxia¹,

Yang²

2022

Preprint

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Section: B Graph Fourier Transform and Filtersmentioning

confidence: 99%

Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs Based on an Optimization Model

Junxia¹,

Yang²

2022

Preprint

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“…• When the vertices are assigned real numbers, it is the traditional GSP [1]- [15]; • When the vertices are assigned to a specified finite closed interval [a, b] that satisfies the Lebesgue measure, it is a joint time-vertex GSP [16], [17]; • When the vertices are assigned a new graph structure that is in a Hilbert space and has a discrete metric in L 2 or ℓ 2 space [24]- [26].…”

Section: Introductionmentioning

confidence: 99%

Quantum entangled fractional Fourier transform based on the IWOP technique

Zhang¹,

Li²,

Zhao³

et al. 2023

Chinese Phys. B

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In our previous papers, the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics, and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform. The core function of the coordinate–momentum exchange operators in the addition law of fractional Fourier transform was analyzed too. In this paper, the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators (IWOP) are used to establish the entanglement fractional Fourier transform theory to the extent of quantum. A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

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“…To process such signals, one needs to extend the well-developed theory of classical signal processing to graph signals. There have been a lot of researches on graph signal processing, including graph shift operators [4,5,6,7,8,9,10,11,12,13], graph filters [14,15,16,17,18], graph Fourier transforms [19,20,21], windowed graph Fourier transforms [4,22], graph wavelets [23,24,25,26,27], graph signal sampling [28,29,30,31,32,33,34,35,36,37], multiscale analysis [38,39,40], and approximation theory for graph signals [41,42,43].…”

Section: Introductionmentioning

confidence: 99%

Atomic Filter: a Weak Form of Shift Operator for Graph Signals

Yang¹,

Zhang²,

Zhang³

et al. 2022

Preprint

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The shift operation plays a crucial role in the classical signal processing. It is the generator of all the filters and the basic operation for time-frequency analysis, such as windowed Fourier transform and wavelet transform. With the rapid development of internet technology and big data science, a large amount of data are expressed as signals defined on graphs. In order to establish the theory of filtering, windowed Fourier transform and wavelet transform in the setting of graph signals, we need to extend the shift operation of classical signals to graph signals.It is a fundamental problem since the vertex set of a graph is usually not a vector space and the addition operation cannot be defined on the vertex set of the graph. In this paper, based on our understanding on the core role of shift operation in classical signal processing we propose the concept of atomic filters, which can be viewed as a weak form of the shift operator for graph signals. Then, we study the conditions such that an atomic filter is norm-preserving, periodic, or real-preserving. The property of real-preserving holds naturally in the classical signal processing, but no the research has been reported on this topic in the graph signal setting. With these conditions we propose the concept of normal atomic filters for graph signals, which degenerates into the classical shift operator under mild conditions if the graph is circulant. Typical examples of graphs that have or have not normal atomic filters are given. Finally, as an application, atomic filters are utilized to construct time-frequency atoms which constitute a frame of the graph signal space.

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Graph Fourier transform based on ℓ1 norm variation minimization

Cited by 21 publications

References 11 publications

Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs Based on an Optimization Model

Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs Based on an Optimization Model

Quantum entangled fractional Fourier transform based on the IWOP technique

Atomic Filter: a Weak Form of Shift Operator for Graph Signals

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