论文标题
BESOV空间中过滤后的投影重建的错误分析
Error analysis for filtered back projection reconstructions in Besov spaces
论文作者
论文摘要
过滤后的投影（FBP）方法是计算机断层扫描（CT）中使用最广泛的重建算法。这个反问题的不良性仅允许给定嘈杂数据的近似重建。研究所得的重建误差一直是1990年代最活跃的研究领域，最近在最佳滤波器设计和估算一般Sobolev空间的FBP近似误差方面已恢复。 但是，Sobolev空间的选择是表征典型CT重建的次优。一个广泛使用的模型是特征函数的总和，它可以更好地建模，以besov空间$ \ mathrm {b}^{α，p} _q（\ Mathbb {r}^2）$。尤其是$ \ mathrm {b}^{α，1} _1（\ mathbb {r}^2）$，$α\约1 $是描述自然图像的图像分析中的首选模型。 如果有嘈杂的ra数据，总FBP重建错误$$ \ | f-f_l^δ\ | \ le \ | f -f_l \ | |+ \ | f_l -f_l^δ\ | $$将近似错误和数据误差分解为数据误差，其中$ l $用作正则化参数。在本文中，我们研究了目标函数的FBP重建的近似误差$ f \ in \ Mathrm {l}^1（\ Mathbb {r}^2）\ cap \ cap \ mathrm {b}^{b}^{α，p} _q（\ alsbb} _q（\ mathbb {\ mathbb {r}^2） $ 1 \ leq p，q \ leq \ infty $。我们证明，$ \ mathrm {l}^p $ - 固有的fbp近似错误$ f-f_l $可以通过\ begin {equation*} \ | f_l \ | ____________ { l^{ - α} \，| f | _ {\ mathrm {b}^{α，p} _q（\ mathbb {r}^2）} \ end {equin {qore {qore {qore {equination*}在适当的假设上，在使用的低通滤波器窗口窗口函数上。然后，通过经典方法扩展到总重建误差的估计。
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.
