Page 298 - Introduction to Information Optics

5.9. Radon Transform L 8 3

where b is the Dirac delta function. This delta function incorporates the
parallel projection constraint such that the position r would kept on the path
f • R/R — R, where the direction of/? is normal to the path, as shown in Fig. 5,3,
One property of the Radon transform is symmetry as,

g(-R,<j)) = g(R, $ + K).

5.9.2. IMAGE RECONSTRUCTION

It is important now to examine the reconstruction of an object from its
Radon transform. This is the inverse Radon transform. One method of
reconstruction is based on the central slice theorem. The ID Fourier transform
of the projection g(R, (f>) with respect to R produces the Fourier transform of
the object /(r, 0), because using Eq. (5.37) this ID Fourier transform becomes

I 4>) exp(- i2npR) dR= \ /(r, 0) exp(-ilnupr cos(</> - f?))r dr d0
'o Jo

i2n(ux + vyj) dx dy — F(n, t>),

(5.38)

where x = r cos 0, y = sin 6 and w = p cos </>, y = p sin 0. The object can then
be recovered from F(x, y) by the inverse Fourier transform.
In the projection plane in image space, if for a given radiation projection
direction we rotated the Cartesian coordinate system (x, y) by the angle (f) to
(x', v'), the projection R = x', and integral (5.37) become ID, as

0(K,0)= |/(x',yW. (5.39)

Equivalently, if we rotate the Fourier transform in the frequency space by an
angle (f>, the frequency components (u, v) become (u', v'), and the integral in Eq.
(5.38) becomes

g(R, 0) e\p(-i2npR)dR - /(.x, /) e\p(~inu'x') dx'dy' = F(u', 0). (5.40)

This result shows that at each given projection angle, the ID Fourier transform
of the projection of a 2D object slice is directly one line through the 2D Fourier