Local reconstruction analysis of inverting the Radon transform in the plane from noisy discrete data
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Abhishek, Anuj Katsevich, Alexander Webber, James W. |
| author_facet | Abhishek, Anuj Katsevich, Alexander Webber, James W. |
| contents | In this paper, we investigate the reconstruction error, $N_\e^{\text{rec}}(x)$, when a linear, filtered back-projection (FBP) algorithm is applied to noisy, discrete Radon transform data with sampling step size $ε$ in two-dimensions. Specifically, we analyze $N_\e^{\text{rec}}(x)$ for $x$ in small, $O(\e)$-sized neighborhoods around a generic fixed point, $x_0$, in the plane, where the measurement noise values, $η_{k,j}$ (i.e., the errors in the sinogram space), are random variables. The latter are independent, but not necessarily identically distributed. We show, under suitable assumptions on the first three moments of the $η_{k,j}$, that the following limit exists: $N^{\text{rec}}(\chx;x_0) = \lim_{\e\to0}N_\e^{\text{rec}}(x_0+\e\chx)$, for $\check x$ in a bounded domain. Here, $N_\e^{\text{rec}}$ and $ N^{\text{rec}}$ are viewed as continuous random variables, and the limit is understood in the sense of distributions. Once the limit is established, we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and compute explicitly its covariance. In addition, we validate our theory using numerical simulations and pseudo random noise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local reconstruction analysis of inverting the Radon transform in the plane from noisy discrete data Abhishek, Anuj Katsevich, Alexander Webber, James W. Numerical Analysis In this paper, we investigate the reconstruction error, $N_\e^{\text{rec}}(x)$, when a linear, filtered back-projection (FBP) algorithm is applied to noisy, discrete Radon transform data with sampling step size $ε$ in two-dimensions. Specifically, we analyze $N_\e^{\text{rec}}(x)$ for $x$ in small, $O(\e)$-sized neighborhoods around a generic fixed point, $x_0$, in the plane, where the measurement noise values, $η_{k,j}$ (i.e., the errors in the sinogram space), are random variables. The latter are independent, but not necessarily identically distributed. We show, under suitable assumptions on the first three moments of the $η_{k,j}$, that the following limit exists: $N^{\text{rec}}(\chx;x_0) = \lim_{\e\to0}N_\e^{\text{rec}}(x_0+\e\chx)$, for $\check x$ in a bounded domain. Here, $N_\e^{\text{rec}}$ and $ N^{\text{rec}}$ are viewed as continuous random variables, and the limit is understood in the sense of distributions. Once the limit is established, we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and compute explicitly its covariance. In addition, we validate our theory using numerical simulations and pseudo random noise. |
| title | Local reconstruction analysis of inverting the Radon transform in the plane from noisy discrete data |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2403.12909 |