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The Digital Radon Transform

The Radon transform for discretely given points of a g function at points
$ \left (t_{k},\omega _{l}  \right ) in sinogram space:
$ Rg \left (t_{k},\omega _{l}  \right )=\int g \left ( t_{k}\cos \vartheta _{l} -s\sin \vartheta _{l}\right, t_{k}\sin \vartheta _{l} +s\cos \vartheta _{l}  \right ))ds
$ Rg \left (t_{k},\omega _{l}  \right )\approx \sum_{i}^{N}  g \left ( t_{k}\cos \vartheta _{l} -s_{i}\sin \vartheta _{l}\right, t_{k}\sin \vartheta _{l} +s_{i}\cos \vartheta _{l} 
  \right ))\Delta s
$s_{i}=s_{0}+i\Delta s
With this notation for each i we need to interpolate the function g in 2D. If we choose the step size such that in dimension x we match grid points we can spare the interpolation step. Let us do the integration according to x:
$ x=t\cos \vartheta -s\sin \vartheta
$ ds=\frac{1}{\left | -sin\left ( \vartheta  \right ) \right |}dx
With that:
$ Rg \left (t_{k},\omega _{l}  \right )\approx \sum_{i}^{N}  g \left ( x_{0}+i\Delta x, t_{k}\sin \vartheta _{l} +\frac{t_{k}\cos \vartheta _{l}-\left ( x_{0}+i\Delta x \right )}{\sin \vartheta _{l}}\cos \vartheta _{l} 
  \right )\frac{1}{\left | sin\left ( \vartheta  \right ) \right |}\Delta x

From this from we see that if the angle falls around $ \vartheta0 or 1800,the expression renders unstable, and we should rather swith to mathcing grid lines in the y dimension.

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