Page 129 - Phase Space Optics Fundamentals and Applications

110 Chapter Four

x θ
R (x , θ)
g θ

y θ y g(x, y)
x θ

θ x

FIGURE 4.1 Projection scheme for the definition of the Radon transform.

say, g(x, y), its Radon transform is defined as a generalized marginal

+∞

R {g (x, y) ,x , } = R g (x , ) = g (x, y) dy (4.8)
−∞
where, as presented in Fig. 4.1, x and y are the coordinates rotated
by an angle . It is easy to see from this figure that

R g (x , + ) = R g (−x , ) (4.9)

Thus, the reduced domain ∈ (0, ) is used for R g (x , ). Note that the
integration in the above definition is performed along straight lines
characterized, for a given pair (x , ), by

x x
y = − for = 0,
sin tan 2
x = x for = 0 (4.10)

y = x for =
2