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Filter Design for the Filtered Backprojection

The formula for filtered backprojection can be generalized in a more practical way than was shown with the Riesz potentials, now we will look at the convolution form. Let $ g=\mathfrak{R}h the Radontransform in n dimensions. Let us prove that
$ \mathfrak{R}^{+}g*_{\mathbf{x}}f=\mathfrak{R}^{+}\left [ g*_{t}\mathfrak{R}f \right ]
where the indices to the convolution sign * indicate the variables of the convolution.
The LHS convolution with the our usual notations:
$\mathfrak{R}^{+}g*_{\mathbf{x}}f=\int\int  g\left ( \mathbf{\left (x-y  \right )\boldsymbol{\omega }}, \boldsymbol{\omega }\right )f\left ( \mathbf{y} \right )d\mathbf{y}d\boldsymbol{\omega }
Let us replace variable y with $ \mathbf{y}=t\boldsymbol{\omega }+\mathbf{z}, here z is perpendicular to $\boldsymbol{\omega }-ra. Then inserting:
$  \mathfrak{R}^{+}g*_{\mathbf{x}}f=\int \int\int  g\left ( \mathbf{x\boldsymbol{\omega }}-t, \boldsymbol{\omega }\right )f\left ( t\boldsymbol{\omega }+\mathbf{z} \right )dtd\mathbf{z}d\boldsymbol{\omega }=  \int\int  g\left ( \mathbf{x}\boldsymbol{\omega }-t, \boldsymbol{\omega }\right )\mathfrak{R}f\left ( t,\boldsymbol{\omega } \right )dtd\boldsymbol{\omega }= \int  g*_{t}\mathfrak{R}f\left ( \mathbf{x}\boldsymbol{\omega },\boldsymbol{\omega } \right )d\boldsymbol{\omega }=\mathfrak{R}^{+} g*_{t}\mathfrak{R}f\left ( \mathbf{x}\boldsymbol{\omega },\boldsymbol{\omega } \right )

Let us now choose the following V and v functions:
$ V= \mathfrak{R}^{+}v
$V*_{\mathbf{x}}f=\mathfrak{R}^{+}\left [ v*_{t}\mathfrak{R}f \right ]

Let us look for such V functions that in a given band limit approximate the Dirac delta function so the original f function would be restored. It can be proven that
$\mathfrak{F}V=\left |2\pi \xi   \right |^{n-1}\mathfrak{F}v
thus, if the Fourier transform of Vis a constant within the bandlimits, for v a family of filters can be designed. If we only allow for radial dependence of the filters we get our previous filtered backprojection formulas.


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