Matrix equation representation of the convolution equation and its unique solvability
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929391420833792 |
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| author | Satake, Yuki Sogabe, Tomohiro Kemmochi, Tomoya Zhang, Shao-Liang |
| author_facet | Satake, Yuki Sogabe, Tomohiro Kemmochi, Tomoya Zhang, Shao-Liang |
| contents | We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15359 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Matrix equation representation of the convolution equation and its unique solvability Satake, Yuki Sogabe, Tomohiro Kemmochi, Tomoya Zhang, Shao-Liang Numerical Analysis We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation. |
| title | Matrix equation representation of the convolution equation and its unique solvability |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2306.15359 |