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## The Filtered Backprojection Method

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 l=0 case of this formula is:
 (2.25)

In other words:
Taking the one-dimensional Fourier Transform of the projection along the y axis 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 L is the length of the detector. This, in turn, implies that

 (2.30)

and therefore:

 = (2.31) = (2.32) = (2.33)

where

 (2.34)

is called a filtered projection, and plays the role of an inverse filter: multiplying by increases the influence of at high frequencies.
Integral (2.36) defines the map of density in the plane by accumulating all 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 is backprojected into 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 x' space that B(k') corresponds to is b(x'):
 b(x') = =

Now, remember that a Fourier transform of a convolution is equal to a product of Fourier transforms. Since and B(k') are Fourier transforms of and b(x'), their product is equal to Fourier transform of a convolution of the latter two functions:

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,
the filtered projection is the convolution of and b(x').

Next: A Discrete Formulation of Up: Computerised Tomography Previous: Computerised Tomography
Zdzislaw Meglicki
2001-02-26