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Relaization of the filtered backprojection

The Radon inversion formula in 2D can be stated as follows:

$ f\left ( x,y \right )= \frac{1}{\left (2\pi   \right )^{2}} \int_{0}^{\pi}\int_{-\infty }^{\infty} \left |r  \right |e^{ir\boldsymbol{x}\boldsymbol{\omega} }\mathfrak{F}\left (Rf  \right )_{t}\left [ r \right\boldsymbol{\omega} ]drd\theta in polar coordinates.

Before the backprojection -as we have seen-, the Radon transform obtained as a scanner acquisition should be Fourier transformed, multiplied by$\left |r \right |, then inverse Fourier transformed inthe r variable. This one is a filtering in the frequency domain with filter $\left |r \right |. We can use the notation for this process of $\mathfrak{G} filter operator, and instead of the |r| filter function we could introduce a more general notation: $ v\left ( r \right ).

Wigh the previous notation thus:
$ \mathfrak{G}=\mathfrak{F}_{r}^{-1}\left [\mathfrak{F}_{t}\left [ \mathfrak{R}f\left ( t \right ) \right ]\left ( r )v( r) \right ]

According to the Fourier convolution theorem a multiplication in the frequency domain can be equivalently calculated as a convolution in the spatial domain.
The filter operator can be written as follows:
$\mathfrak{G}=\int_{-\infty }^{\infty} \left (\mathfrak{R}f \right )\left ( t, \right\vartheta )\phi \left ( t-\boldsymbol{x}\boldsymbol{\omega} \right ) dt =\mathfrak{R}f*_{t}\phi\left ( \boldsymbol{x}\boldsymbol{\omega} \right )
Summarized, the reconstruction goes as follows:
$ f\left ( x,y \right )= \frac{1}{ 2\pi } \int_{0}^{\pi} \left \mathfrak{G} (Rf \right )\left ( \boldsymbol{x}\boldsymbol{\omega}, \right\boldsymbol{\omega} ) d\theta.

Filter types

As the \left |r  \right | as a hight pass filter aplifies noises, the filter \left |r  \right | can be replaced by a numerically more stable counterpart. The simplest filter is the Ram-Lak (Ramachandran-Lakshminarayan) filter, that zeros out the filter above a certain band limit:
$ v_{Ram-Lak}\left ( r \right )= \left\{\begin{matrix}
\left |r  \right | & :ha \left |r  \right |< L \\ 
 0& : \textsl{otherwise}
If the filtering is done in the spatial domain:

\phi_{Ram-Lak} \left ( Lt \right )= \frac{L^{^{2}}}{4\pi ^{2}}\left \{ sinc\left ( Lt \right )  \right -\frac{1}{2}sinc^{2}\left ( \frac{Lt}{2} \right )\}=\frac{L^{^{2}}}{4\pi ^{2}}  \{ \frac{\sin \left ( Lt \right )}{Lt}  \right +\frac{\cos\left ( Lt \right )-1}{(Lt)^{2}} \}
on the RHS an equivalent form is given.

  • Ram-Lak filter in frequency and in the spatial domain:
Ram-Lak filter in the frequency domain
Ram-Lak filter in the spatial domain

Some other filter types:

  • Shepp_Logan

Frequency domain:
$ v_{Shepp-Logan}=\left |r  \right |\textup{sinc}\left ( r\frac{\pi}{2} \right ) |  0\leq r\leq 1
Spatial domain:
$\phi_{Shepp-Logan} =\frac{L^{2}}{2\pi^{3}}\frac{\frac{\pi}{2}-Lt\sin\left ( Lt \right )}{\left (\frac{\pi}{2}  \right )^{2}-(Lt)^{2}}

Shepp-Logan filter in frequency domain
Shepp-Logan filter in the spatial domain

  • cosine

Frequency domain:
$ v_{cos}=\left |r  \right |\cos\left ( r\frac{\pi}{2} \right ) |  0\leq r\leq 1
Spatial domain:
$ \phi_{Cos} \left ( Lt \right )=\phi_{Ram-Lak} \left ( Lt-\frac{\pi}{2} \right )+\phi_{Ram-Lak} \left ( Lt+\frac{\pi}{2} \right )

The number of possible filters are infinite, you could use e.g. Hanning, and Hamming filters. The target function necessary for designing filters cannot be derived directly from the Fourier Inversion Formula, we need to tread a step back and find a formula that also shows the quality aspects of the approximate inverse Radon transforms. Before we would do that we have to clarify how the CST and the Fourier Inversion Formula looks like in multiple dimensions and then how the inversion formulae should be interpreted.

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