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Multi-dimensional Graph Linear Canonical Transform

Na Li, Zhichao Zhang, Jie Han, Yunjie Chen, Chunzheng Cao

TL;DR

This paper proposes the two-dimensional graph linear canonical transform based on the central discrete dilated Hermite function and the two-dimensional graph linear canonical transform based on chirp multiplication-chirp convolution-chirp multiplication decomposition (2-D CM-CC-CM-GLCT).

Abstract

Many multi-dimensional (M-D) graph signals appear in the real world, such as digital images, sensor network measurements and temperature records from weather observation stations. It is a key challenge to design a transform method for processing these graph M-D signals in the linear canonical transform domain. This paper proposes the two-dimensional graph linear canonical transform based on the central discrete dilated Hermite function (2-D CDDHFs-GLCT) and the two-dimensional graph linear canonical transform based on chirp multiplication-chirp convolution-chirp multiplication decomposition (2-D CM-CC-CM-GLCT). Then, extending 2-D CDDHFs-GLCT and 2-D CM-CC-CM-GLCT to M-D CDDHFs-GLCT and M-D CM-CC-CM-GLCT. In terms of the computational complexity, additivity and reversibility, M-D CDDHFs-GLCT and M-D CM-CC-CM-GLCT are compared. Theoretical analysis shows that the computational complexity of M-D CM-CC-CM-GLCT algorithm is obviously reduced. Simulation results indicate that M-D CM-CC-CM-GLCT achieves comparable additivity to M-D CDDHFs-GLCT, while M-D CM-CC-CM-GLCT exhibits better reversibility. Finally, M-D GLCT is applied to data compression to show its application advantages. The experimental results reflect the superiority of M-D GLCT in the algorithm design and implementation of data compression.

Multi-dimensional Graph Linear Canonical Transform

TL;DR

This paper proposes the two-dimensional graph linear canonical transform based on the central discrete dilated Hermite function and the two-dimensional graph linear canonical transform based on chirp multiplication-chirp convolution-chirp multiplication decomposition (2-D CM-CC-CM-GLCT).

Abstract

Many multi-dimensional (M-D) graph signals appear in the real world, such as digital images, sensor network measurements and temperature records from weather observation stations. It is a key challenge to design a transform method for processing these graph M-D signals in the linear canonical transform domain. This paper proposes the two-dimensional graph linear canonical transform based on the central discrete dilated Hermite function (2-D CDDHFs-GLCT) and the two-dimensional graph linear canonical transform based on chirp multiplication-chirp convolution-chirp multiplication decomposition (2-D CM-CC-CM-GLCT). Then, extending 2-D CDDHFs-GLCT and 2-D CM-CC-CM-GLCT to M-D CDDHFs-GLCT and M-D CM-CC-CM-GLCT. In terms of the computational complexity, additivity and reversibility, M-D CDDHFs-GLCT and M-D CM-CC-CM-GLCT are compared. Theoretical analysis shows that the computational complexity of M-D CM-CC-CM-GLCT algorithm is obviously reduced. Simulation results indicate that M-D CM-CC-CM-GLCT achieves comparable additivity to M-D CDDHFs-GLCT, while M-D CM-CC-CM-GLCT exhibits better reversibility. Finally, M-D GLCT is applied to data compression to show its application advantages. The experimental results reflect the superiority of M-D GLCT in the algorithm design and implementation of data compression.
Paper Structure (23 sections, 47 equations, 5 figures, 7 tables)

This paper contains 23 sections, 47 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Basic Two-dimensional Graph Operations
  3. Cartesian product graph
  4. Two-dimensional graph Fourier transform
  5. Two-dimensional graph fractional Fourier transform
  6. Two-dimensional graph Chirp Multiplication
  7. Two-dimensional graph scaling transform
  8. Two-dimensional graph linear canonical transform
  9. 2-D CDDHFs-GLCT
  10. 2-D CM-CC-CM-GLCT
  11. when $\rm{det} \ \mathbf{B}\ne 0$
  12. when $\rm{det} \ \mathbf{B}= 0$
  13. Multi-dimensional graph linear canonical transform
  14. M-D CDDHFs-GLCT
  15. M-D CM-CC-CM-GLCT
  16. ...and 8 more sections

Figures (5)

  • Figure 1: 2-D graph signals $\mathbf{x}_{1}$ to $\mathbf{x}_{4}$.
  • Figure 2: Normalized mean-square errors (NMSEs) of the additivity property for 50 different sets of $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$.
  • Figure 3: Normalized mean-square errors (NMSEs) of the reversibility property for 50 different sets of $\mathbf{M}$.
  • Figure 4: RE, NRMS and CC of 2-D GFRFT with different rotation angles $\alpha$ for different compression parameters $\gamma$.
  • Figure 5: Comparison between the 2-D GFRFT and the 2-D CM-CC-CM-GLCT with different compression parameters $\gamma$.