Analysis of reconstruction from noisy discrete generalized Radon data
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916255389188096 |
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| author | Katsevich, Alexander |
| author_facet | Katsevich, Alexander |
| contents | We consider a wide class of generalized Radon transforms $\mathcal R$, which act in $\mathbb{R}^n$ for any $n\ge 2$ and integrate over submanifolds of any codimension $N$, $1\le N\le n-1$. Also, we allow for a fairly general reconstruction operator $\mathcal A$. The main requirement is that $\mathcal A$ be a Fourier integral operator with a phase function, which is linear in the phase variable. We consider the task of image reconstruction from discrete data $g_{j,k} = (\mathcal R f)_{j,k} + η_{j,k}$. We show that the reconstruction error $N_ε^{\text{rec}}=\mathcal A η_{j,k}$ satisfies $N^{\text{rec}}(\check x;x_0)=\lim_{ε\to0}N_ε^{\text{rec}}(x_0+ε\check x)$, $\check x\in D$. Here $x_0$ is a fixed point, $D\subset\mathbb{R}^n$ is a bounded domain, and $η_{j,k}$ are independent, but not necessarily identically distributed, random variables. $N^{\text{rec}}$ and $N_ε^{\text{rec}}$ are viewed as continuous random functions of the argument $\check x$ (random fields), and the limit is understood in the sense of probability distributions. Under some conditions on the first three moments of $η_{j,k}$ (and some other not very restrictive conditions on $x_0$ and $\mathcal A$), we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and explicitly compute its covariance. We also present a numerical experiment with a cone beam transform in $\mathbb{R}^3$, which shows an excellent match between theoretical predictions and simulated reconstructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13269 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analysis of reconstruction from noisy discrete generalized Radon data Katsevich, Alexander Numerical Analysis Probability We consider a wide class of generalized Radon transforms $\mathcal R$, which act in $\mathbb{R}^n$ for any $n\ge 2$ and integrate over submanifolds of any codimension $N$, $1\le N\le n-1$. Also, we allow for a fairly general reconstruction operator $\mathcal A$. The main requirement is that $\mathcal A$ be a Fourier integral operator with a phase function, which is linear in the phase variable. We consider the task of image reconstruction from discrete data $g_{j,k} = (\mathcal R f)_{j,k} + η_{j,k}$. We show that the reconstruction error $N_ε^{\text{rec}}=\mathcal A η_{j,k}$ satisfies $N^{\text{rec}}(\check x;x_0)=\lim_{ε\to0}N_ε^{\text{rec}}(x_0+ε\check x)$, $\check x\in D$. Here $x_0$ is a fixed point, $D\subset\mathbb{R}^n$ is a bounded domain, and $η_{j,k}$ are independent, but not necessarily identically distributed, random variables. $N^{\text{rec}}$ and $N_ε^{\text{rec}}$ are viewed as continuous random functions of the argument $\check x$ (random fields), and the limit is understood in the sense of probability distributions. Under some conditions on the first three moments of $η_{j,k}$ (and some other not very restrictive conditions on $x_0$ and $\mathcal A$), we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and explicitly compute its covariance. We also present a numerical experiment with a cone beam transform in $\mathbb{R}^3$, which shows an excellent match between theoretical predictions and simulated reconstructions. |
| title | Analysis of reconstruction from noisy discrete generalized Radon data |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2405.13269 |