One approach for reconstructing the image is simply to take the inverse Radon transform of the projections. This involves two steps; the image is back projected and then filtered with a two dimensional ramp filter.

The back projection operator, is defined as

This operator represents the accumulation of the projections that pass through the point (x,y). As in the example above, the back projection is a blurred version of the desired image.

The image can be now unblurred with a two-dimensional ramp filter. Because two-dimensional filtering is computationally intensive, a better a better approach is to reverse the order of the filtering and the back projection. Figure 6 illustrates both approaches. The second approach is possible due to the projection slice theorem, which states that the one-dimensional Fourier transform with respect to s of is equal to the central slice at angle of the two-dimensional Fourier transform of f(x,y),

The projection slice theorem implies that the Radon transform of the
two-dimensional convolution of two functions is equal to the
one-dimensional convolution of their Radon transforms. Convolution
has the nice property of being symmetric, i.e *f***g*=*g***f*. This can be
used to find a closed formula for the inverse Radon transform. It also
implies a simpler inverse Radon scheme than before. It can be
summarized with the following two points:

- Do a one-dimensional high-pass filtering of each projection using a ramp filter.
- Back project the outputs of the filter.

**Figure 6:** Illustration of the two possible cases for filtered
back projection. Most algorithms today does a one-dimensional filtering of the
projections before the back projection.

Thu May 8 12:28:25 CDT 1997