Radon Transformation - Digital Signal Processing for Medical Imaging Using Matlab - page 11

Fig. 1.9 Illustration of reconstruction of the image obtained by cumulative summation of filtered
parallel beam backprojected images

1.3.1 Relationship Between Parallel Beam
and Fanbeam Projection

Conceptually (refer Sect. 1.2 and Fig. 1.13 ), the used in parallel beam projection is

the angle (measured in the anticlockwise direction) between the Y-axis and lineseg-
ment joining the source and the detector ( SD ). Also l is the perpendicular distance
with sign measured from the origin to the line segment SD . The line integration
obtained from the line segment SD 1inFig. 1.14 corresponds to

θ

δ =− δ 1
while using Fan beam projection. The same line integration, if it is computed using
the parallel beam projection, corresponds to

β = 0 and

− θ = δ 1 and l = C sin δ 1 . Note that the

relationship holds for β = 0. If the reference line segment is rotated by an non-zero

angle β in the anticlockwise direction (refer Fig. 1.14 ), we get l = C sin ( − δ 1 ) and

θ = − δ 1 . In general, line integration obtained from the fanbeam projection with

the particular

B

δ and β equals to the line integration obtained from the parallel beam

projection with . Thus the reconstruction formula used

for parallel beam projection technique can be used for formulating the reconstruction
formula for Fan beam projection technique as described below.

θ = β + δ and l = C sin (δ)