平面内从含噪离散数据反演Radon变换的局部重构分析
Local reconstruction analysis of inverting the Radon transform in the plane from noisy discrete data
AI总结：
本文分析二维含噪离散Radon变换数据经滤波反投影重构后的局部误差，证明其极限为高斯随机场并给出协方差，通过数值模拟验证理论。
摘要：
本文研究在二维空间中，当对采样步长为$\epsilon$的含噪离散Radon变换数据应用线性滤波反投影（FBP）算法时，重构误差$N_\epsilon^{\text{rec}}(x)$的性质。具体而言，我们分析平面内一个固定点$x_0$周围大小为$O(\epsilon)$的小邻域中的$x$所对应的$N_\epsilon^{\text{rec}}(x)$，其中测量噪声值$\eta_{k,j}$（即正弦图空间中的误差）是随机变量，这些变量相互独立但不一定同分布。在关于$\eta_{k,j}$前三阶矩的适当假设下，我们证明以下极限存在：$N^{\text{rec}}(\check{x};x_0)=\lim_{\epsilon\to0}N_\epsilon^{\text{rec}}(x_0+\epsilon\check{x})$，其中$\check{x}$属于有界域。这里，$N_\epsilon^{\text{rec}}$和$N^{\text{rec}}$被视为连续随机变量，极限在分布意义下理解。一旦极限确立，我们证明$N^{\text{rec}}$是一个零均值高斯随机场，并显式计算其协方差。此外，我们通过数值模拟和伪随机噪声验证了我们的理论。
英文摘要：
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.
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