Error analysis for filtered back projection reconstructions in Besov spaces

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.