Filtered back projection (FBP) methods are the most widely used reconstruction algorithms in computerized tomography (CT). The ill-posedness of this inverse problem allows only an approximate reconstruction for given noisy data. Studying the resulting reconstruction error has been a most active field of research in the 1990s and has recently been revived in terms of optimal filter design and estimating the FBP approximation errors in general Sobolev spaces. However, the choice of Sobolev spaces is suboptimal for characterizing typical CT reconstructions. A widely used model are sums of characteristic functions, which are better modelled in terms of Besov spaces $\mathrm{B}^{\alpha,p}_q(\mathbb{R}^2)$. In particular $\mathrm{B}^{\alpha,1}_1(\mathbb{R}^2)$ with $\alpha \approx 1$ is a preferred model in image analysis for describing natural images. In case of noisy Radon data the total FBP reconstruction error $$\|f-f_L^\delta\| \le \|f-f_L\|+ \|f_L - f_L^\delta\|$$ splits into an approximation error and a data error, where $L$ serves as regularization parameter. In this paper, we study the approximation error of FBP reconstructions for target functions $f \in \mathrm{L}^1(\mathbb{R}^2) \cap \mathrm{B}^{\alpha,p}_q(\mathbb{R}^2)$ with positive $\alpha \not\in \mathbb{N}$ and $1 \leq p,q \leq \infty$. We prove that the $\mathrm{L}^p$-norm of the inherent FBP approximation error $f-f_L$ can be bounded above by \begin{equation*} \|f - f_L\|_{\mathrm{L}^p(\mathbb{R}^2)} \leq c_{\alpha,q,W} \, L^{-\alpha} \, |f|_{\mathrm{B}^{\alpha,p}_q(\mathbb{R}^2)} \end{equation*} under suitable assumptions on the utilized low-pass filter's window function $W$. This then extends by classical methods to estimates for the total reconstruction error.

中文翻译：

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Besov空间中滤波反投影重建的误差分析

滤波反投影 (FBP) 方法是计算机断层扫描 (CT) 中使用最广泛的重建算法。这个逆问题的不适定性只允许对给定的噪声数据进行近似重建。研究由此产生的重建误差一直是 1990 年代最活跃的研究领域，最近又在优化滤波器设计和估计一般 Sobolev 空间中的 FBP 近似误差方面重新兴起。然而，Sobolev 空间的选择对于表征典型的 CT 重建是次优的。一个广泛使用的模型是特征函数的总和，最好用 Besov 空间 $\mathrm{B}^{\alpha,p}_q(\mathbb{R}^2)$ 建模。特别是 $\mathrm{B}^{\alpha,1}_1(\mathbb{R}^2)$ 和 $\alpha \approx 1$ 是图像分析中用于描述自然图像的首选模型。在有噪声的氡数据的情况下，总 FBP 重建误差 $$\|f-f_L^\delta\| \le \|f-f_L\|+ \|f_L - f_L^\delta\|$$ 分为近似误差和数据误差，其中 $L$ 作为正则化参数。在本文中，我们研究了目标函数的 FBP 重建的近似误差 $f \in \mathrm{L}^1(\mathbb{R}^2) \cap \mathrm{B}^{\alpha,p}_q (\mathbb{R}^2)$ 与正 $\alpha \not\in \mathbb{N}$ 和 $1 \leq p,q \leq \infty$。我们证明了固有 FBP 近似误差 $f-f_L$ 的 $\mathrm{L}^p$-范数可以通过 \begin{equation*} \|f - f_L\|_{\mathrm{L }^p(\mathbb{R}^2)} \leq c_{\alpha,q,W} \, L^{-\alpha} \, |f|_{\mathrm{B}^{\alpha, p}_q(\mathbb{R}^2)} \end{equation*} 在使用的​​低通滤波器的窗函数 $W$ 的适当假设下。