Besov空间中滤波反投影重建的误差分析
Error analysis for filtered back projection reconstructions in Besov spaces
AI总结：
针对Sobolev空间刻画CT重建结果非最优的问题，该文在Besov空间下分析FBP重建的近似误差，推导了固有近似误差的Lp范数上界，并将结果推广至总重建误差估计。
摘要：
滤波反投影（FBP）方法是计算机断层扫描（CT）中应用最广泛的重建算法。由于该逆问题的不适定性，对于给定的含噪数据只能实现近似重建。对由此产生的重建误差的研究曾是20世纪90年代极为活跃的研究领域，近年来又在最优滤波器设计及一般Sobolev空间中FBP近似误差估计方面重新受到关注。\n 然而，Sobolev空间的选择在刻画典型CT重建结果方面并非最优。一种广泛使用的模型是特征函数之和，这类函数用Besov空间$\u200b\mathrm{B}^{α,p}_q(\mathbb{R}^2)$建模更为合适。尤其是$α\approx 1$的$\u200b\mathrm{B}^{α,1}_1(\mathbb{R}^2)$是图像分析中描述自然图像的首选模型。\n 在含噪Radon数据的情况下，FBP总重建误差$$\\|f-f_L^δ\\| \le \\|f-f_L\\|+ \\|f_L - f_L^δ\\|$$可分解为近似误差和数据误差，其中$L$作为正则化参数。本文研究了目标函数$f \in \mathrm{L}^1(\mathbb{R}^2) \cap \mathrm{B}^{α,p}_q(\mathbb{R}^2)$（其中$α$为正的非整数，$1 \leq p,q \leq \infty$）的FBP重建近似误差。我们证明，在对所用低通滤波器的窗函数$W$做出合理假设的条件下，FBP固有近似误差$f-f_L$的$\u200b\mathrm{L}^p$范数可由下式给出上界：\begin{equation*} \\|f - f_L\\|_{\mathrm{L}^p(\mathbb{R}^2)} \leq c_{α,q,W} \\, L^{-α} \\, |f|_{\mathrm{B}^{α,p}_q(\mathbb{R}^2)} \end{equation*} 随后通过经典方法将该结果推广到总重建误差的估计。
英文摘要：
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 $\mathrm{B}^{α,p}_q(\mathbb{R}^2)$. In particular $\mathrm{B}^{α,1}_1(\mathbb{R}^2)$ with $α\approx 1$ is a preferred model in image analysis for describing natural images. In case of noisy Radon data the total FBP reconstruction error $$\|f-f_L^δ\| \le \|f-f_L\|+ \|f_L - f_L^δ\|$$ 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 $f \in \mathrm{L}^1(\mathbb{R}^2) \cap \mathrm{B}^{α,p}_q(\mathbb{R}^2)$ with positive $α\not\in \mathbb{N}$ 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_{α,q,W} \, L^{-α} \, |f|_{\mathrm{B}^{α,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.
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