Assume a computerised tomography set up with rays parallel to the *y*-axis
directed upwards, see Figure 2.1.
Assume absorbing density in the plane to be ,
where
function
has a compact support. Signal
intensity registered on the *x* axis,
will then
be:

(2.23) |

Now consider a two-dimensional Fourier transform of
function :

(2.24) |

The

(2.25) |

In other words:

Taking the one-dimensional Fourier Transform of the projection along theyaxis gives us the (k,0) line in the Fourier space of .

We can now rotate our set-up and obtain lines in the Fourier space
of
under any angle by calculating one-dimensional
Fourier transforms of measured parallel projections, and in this
way effectively reconstructing the whole *M*(*k*, *l*). Once we have
*M*(*k*, *l*) we can then obtain
by taking
the *inverse* Fourier Transform of *M*(*k*, *l*).

It is convenient in this case to write it all down in terms of the rotation angle .

Consider rotating the system of coordinates as follows:

(2.26) |

Then:

(2.27) |

and

= | |||

= | |||

= |

The inverse transform of yields :

(2.28) |

A closer inspection shows that it is not necessary to rotate the apparatus through the whole , because swapping the source and the detector does not change the resulting attenuation, i.e.,

(2.29) |

where

(2.30) |

and therefore:

where

(2.34) |

is called a

Integral (2.36) defines the map of density in the plane byaccumulatingall of the filtered projections for all angles from 0 to . Each filtered projection contributes to the density along the line of constant for a particular value of . And so each filtered projection isbackprojectedinto the plane.

In order to avoid aliasing problems associated with the Nyquist critical
frequency
,
we shall introduce a new filter
function defined as follows:

(2.35) |

and convolve it with in the expression that defines the filtered projection:

(2.36) |

The function in the

b(x') |
= | ||

= |

Now, remember that a Fourier transform of a convolution is equal to a product of Fourier transforms. Since and

Since is the inverse Fourier Transform of , the Fourier Transform in the equation (2.42) gets undone and we're simply left with:

(2.37) |

In other words,

thefiltered projectionis theconvolutionofandb(x').