A data-dependent regularization method based on the graph Laplacian
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arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913555127730176 |
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| author | Bianchi, Davide Evangelista, Davide Aleotti, Stefano Donatelli, Marco Piccolomini, Elena Loli Li, Wenbin |
| author_facet | Bianchi, Davide Evangelista, Davide Aleotti, Stefano Donatelli, Marco Piccolomini, Elena Loli Li, Wenbin |
| contents | We investigate a variational method for ill-posed problems, named $\texttt{graphLa+}Ψ$, which embeds a graph Laplacian operator in the regularization term. The novelty of this method lies in constructing the graph Laplacian based on a preliminary approximation of the solution, which is obtained using any existing reconstruction method $Ψ$ from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data and noise. We demonstrate that $\texttt{graphLa+}Ψ$ is a regularization method and rigorously establish both its convergence and stability properties.
We present selected numerical experiments in 2D computerized tomography, wherein we integrate the $\texttt{graphLa+}Ψ$ method with various reconstruction techniques $Ψ$, including Filter Back Projection ($\texttt{graphLa+FBP}$), standard Tikhonov ($\texttt{graphLa+Tik}$), Total Variation ($\texttt{graphLa+TV}$), and a trained deep neural network ($\texttt{graphLa+Net}$). The $\texttt{graphLa+}Ψ$ approach significantly enhances the quality of the approximated solutions for each method $Ψ$. Notably, $\texttt{graphLa+Net}$ is outperforming, offering a robust and stable application of deep neural networks in solving inverse problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16936 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A data-dependent regularization method based on the graph Laplacian Bianchi, Davide Evangelista, Davide Aleotti, Stefano Donatelli, Marco Piccolomini, Elena Loli Li, Wenbin Numerical Analysis 47A52, 05C90, 68T07, 92C55 We investigate a variational method for ill-posed problems, named $\texttt{graphLa+}Ψ$, which embeds a graph Laplacian operator in the regularization term. The novelty of this method lies in constructing the graph Laplacian based on a preliminary approximation of the solution, which is obtained using any existing reconstruction method $Ψ$ from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data and noise. We demonstrate that $\texttt{graphLa+}Ψ$ is a regularization method and rigorously establish both its convergence and stability properties. We present selected numerical experiments in 2D computerized tomography, wherein we integrate the $\texttt{graphLa+}Ψ$ method with various reconstruction techniques $Ψ$, including Filter Back Projection ($\texttt{graphLa+FBP}$), standard Tikhonov ($\texttt{graphLa+Tik}$), Total Variation ($\texttt{graphLa+TV}$), and a trained deep neural network ($\texttt{graphLa+Net}$). The $\texttt{graphLa+}Ψ$ approach significantly enhances the quality of the approximated solutions for each method $Ψ$. Notably, $\texttt{graphLa+Net}$ is outperforming, offering a robust and stable application of deep neural networks in solving inverse problems. |
| title | A data-dependent regularization method based on the graph Laplacian |
| topic | Numerical Analysis 47A52, 05C90, 68T07, 92C55 |
| url | https://arxiv.org/abs/2312.16936 |