2D Fan-Beam Tomography

Fan-parallel rebinning methods

to be finished..

The filter-backproject (FBP) approach for tomographic scan

The fan-beam FBP method uses the following three steps.

1. Compute weighted projections for each $\beta$. \begin{equation} \tilde{p} (s, \beta) \triangleq p(s, \beta) \frac{w_{2\pi} (s, \beta) J(s)}{W_1 ^2 (s)}. \end{equation}
2. Filter those weighted projections (along $s$) for each $\beta$ using the modified ramp filter \begin{equation} \check{p} (s, \beta) \triangleq \tilde{p} (s, \beta) * g_{*} (s),\ \forall \beta. \end{equation}
3. Perform a weighted backprojection of those filtered projections: \begin{equation} f(x, y) = \int_0 ^{2\pi} \frac{1}{W_2^2 (x, y, \beta) L_{\beta} ^2 (x, y)} \check{p} (s_{\beta} (x, y), \beta) \mathrm{d} \beta. \end{equation}