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## A Discrete Formulation of Filtered Backprojection

Our first step is to replace

with

 (2.38)

where a single pair yields such (x,y) that

 (2.39)

so that going through all measured values of will eventually cover the whole plane .

1.
Points for small values of k' are much denser than points for large values of k', hence the importance of the filtering function. Filtering with weights the data to compensate for the sparser fill of the frequency domain at larger .
2.
Filter B(k') does the same job, whereas its relative b(x') is equivalent, but operates in the spatial domain.

The next step is to discretise

with

 (2.40)

But the index k has only meaning for x'0 through x'K-1, and the positions for which we have any measurements of are .

Convolution is a symmetric operation, i.e., . This can be proven simply by subsitution. This means that we can move x'n - x'k to p getting:

 (2.41)

where we have restricted k to run between -(K - 1) and (K - 1), because, for example, for x'n = x0 we can subtract a negative x'-(K-1) and still stay within the measured region, and for x'n = x'K-1 we can subtract x'K-1 and go back to x'0. But we are still going to hit some combinations of x'n and x'k that are going to push us beyond the physically meaningful region [x'0, x'K-1].

What then? We are simply going to pad our vector p with zeros.

The discrete values of bk are going to be:

 (2.42)

In summary:

1.
to compute for a given nwe have to evaluate a scalar product of p and b, both of length 2K-1
2.
to compute for all n will have to perform floating point multiplications.

This is going to be expensive.

Because of the low cost of FFT it is cheaper to

1.
Take FFT of a padded data function to form
2.
Discretise

This algorithm is going to be cheaper, because its major cost is associated with FFTs and these will run like .

What next?

1.
For a given angle the vector gives values of C at points (x,y) such that

 (2.43)

2.
The filtered projection values are mapped onto their closest grid box (xg, yg), and then summed within that box to produce density .

Next: FFTPACK Up: Computerised Tomography Previous: The Filtered Backprojection Method
Zdzislaw Meglicki
2001-02-26