Radiation Imaging - Introduction to Biomedical Engineering - page 1035

calculated and the measured data is below a specified error limit. A number of different
versions of the ART were developed and used with first- and second-generation CAT scan-
ners. Later-generation scanners used analytic reconstruction techniques, since the iterative
methods were computationally slow and had convergence problems in the presence of noise.
Analytic techniques include the Fourier transform, back-projection, filtered back-projection,
and convolution back-projection approaches. All of the analytic methods differ from the iter-
ative methods in that the image is reconstructed directly from the attenuation projection data.
Analytic techniques use the central section theorem and the two-dimensional Fourier trans-
form, which is shown in Figure 15.15. Given an image f ( x, y ), a single projection is taken along

the x direction, forming a projection g ( y ) described by

1

g ð y Þ¼ f ð x , y Þ dx

1

This projection represents an array of line integrals as shown in Figure 15.15. The two-
dimensional Fourier transform of f ( x , y ) is given by

ð 1

F ð u , v Þ¼ f ð x , y Þ exp ½ j 2 pð ux þ vy Þ dxdy

1

In the Fourier domain, along the line u ¼ 0, this transform becomes

ð 1

F ð 0, v Þ¼ f ð x , y Þ exp ð j 2 p vy Þ dxdy

1

which can be rewritten as

ð 1

F ð 0, v Þ¼ f ð x , y Þ dx = exp ð j 2p vy Þ dy

1

or

½ g ð y Þ 1

F ð 0, v Þ¼ F

1

FIGURE 15.15 A projection via the central-sectioning theorem.