scholarly journals Discrete Laplacian Operator and Its Applications in Signal Processing

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 89692-89707
Author(s):  
Waseem Waheed ◽  
Guang Deng ◽  
Bo Liu
Keyword(s):  
Signal Processing ◽  
Laplacian Operator ◽  
Discrete Laplacian

scholarly journals Dataset Denoising Based on Manifold Assumption

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Zhonghua Hao ◽  
Shiwei Ma ◽  
Hui Chen ◽  
Jingjing Liu
Keyword(s):  
Point Cloud ◽  
Geometric Distribution ◽  
Lagrange Equation ◽  
Optimal Solution ◽  
Classification Problem ◽  
Machine Learning Algorithms ◽  
High Dimensional ◽  
Laplacian Operator ◽  
Discrete Laplacian ◽  
Denoising Method

Learning the knowledge hidden in the manifold-geometric distribution of the dataset is essential for many machine learning algorithms. However, geometric distribution is usually corrupted by noise, especially in the high-dimensional dataset. In this paper, we propose a denoising method to capture the “true” geometric structure of a high-dimensional nonrigid point cloud dataset by a variational approach. Firstly, we improve the Tikhonov model by adding a local structure term to make variational diffusion on the tangent space of the manifold. Then, we define the discrete Laplacian operator by graph theory and get an optimal solution by the Euler–Lagrange equation. Experiments show that our method could remove noise effectively on both synthetic scatter point cloud dataset and real image dataset. Furthermore, as a preprocessing step, our method could improve the robustness of manifold learning and increase the accuracy rate in the classification problem.

scholarly journals Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaoying Han ◽  
Peter E. Kloeden
Keyword(s):  
Weighted Space ◽  
Lattice System ◽  
Laplacian Operator ◽  
Pullback Attractor ◽  
Forward Dynamics ◽  
Discrete Laplacian ◽  
Lattice Systems ◽  
Nonempty Compact ◽  
Omega Limit Set ◽  
Dense Inclusion

<p style='text-indent:20px;'>A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences <inline-formula><tex-math id="M1">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. First the existence of a pullback attractor in <inline-formula><tex-math id="M2">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula> is established by utilizing the dense inclusion of <inline-formula><tex-math id="M3">\begin{document}$ \ell^2 \subset {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.</p>

Sparse Image and Signal Processing

2009 ◽  
Author(s):  
Jean-Luc Starck ◽  
Fionn Murtagh ◽  
Jalal Fadili
Keyword(s):  
Signal Processing

Signal Processing for In-Line Inspection of Gas Transmission Pipelines

1996 ◽  
Vol 8 (1) ◽  
pp. 233-247
Author(s):  
S. Mandayam ◽  
L. Udpa ◽  
S. S. Udpa ◽  
W. Lord
Keyword(s):  
Signal Processing ◽  
Transmission Pipelines
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