Complete-data filtered backprojection

Assume Countable Choice and let $\mu$ be the polar surface measure on $S^1$, with $\mu (E)=2{\lambda }_2(\{r\theta :\theta \in E,0<r\le 1\})$ for Borel $E\subseteq S^1$. The complete-data filtered-backprojection procedure takes as input the exact full sinogram $g=Rf$ of some $f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^2;\mathbb{C})$; all oriented normals $\theta \in S^1$ and all offsets $s\in \mathbb{R}$ are supplied. It returns the function $f_{\mathrm{F}\mathrm{B}\mathrm{P}}:{\mathbb{R}}^2\to \mathbb{C}$ defined by $f_{\mathrm{F}\mathrm{B}\mathrm{P}}(x)=\frac{1}{4\pi }{\int }_{S^1}q(\theta ,x\cdot \theta )d\mu (\theta ),$ where the filter output is $q(\theta ,s)={\int }_{\mathbb{R}}2\pi \mathrm{\mid }\sigma \mathrm{\mid }\widehat{g(\theta ,\cdot )}(\sigma )e^{2\pi is\sigma }d\sigma \mathrm{.}$ The exact instructions are: (1) for each $\theta$, take the one-dimensional Fourier transform of $g(\theta ,\cdot )$ with the negative-sign $2\pi$ convention; (2) multiply that transform by the declared ramp filter $2\pi \mathrm{\mid }\sigma \mathrm{\mid }$ and take its inverse Fourier integral to form $q(\theta ,\cdot )$; (3) evaluate $q(\theta ,\cdot )$ at $s=x\cdot \theta$ and integrate over the full polar surface measure in $\theta$; (4) divide by $4\pi$. For this input class the slice identity and Schwartz decay make the filter integral absolutely convergent and the backprojection integral well defined. This definition specifies the continuous full-data procedure; it gives no guarantee for finitely sampled, truncated, or noisy measurements. Exact recovery on the stated class is proved by the normal-operator theorem.

The Euclidean Radon transform on R²

The Fourier-slice identity for the Euclidean Radon transform

The Euclidean Radon normal operator and inversion formula

Mikko Salo, RICAM lecture notes v1, §2.1

Frank Natterer, “X-ray Tomography,” §3.1