(a) Show that the Radon transform [Eq. (5.11-3)] of the unit impulse δ(x, y)is a straight...

(a) Show that the Radon transform [Eq. (5.11-3)] of the unit impulse δ(x, y)is a straight vertical line in the ρθ-plane passing through the origin.

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(b) Show that the radon transform of the impulse δ(x − x0, y − y0)is a sinusoidal curve in the ru-plane.

EXAMPLE 5.17:

Using the Radon transform to obtain the projection of a circular region.

Before proceeding, we illustrate how to use the Radon transform to obtain an analytical expression for the projection of the circular object in Fig. 5.38(a):

where A is a constant and r is the radius of the object. We assume that the circle is centered on the origin of the xy-plane. Because the object is circularly symmetric, its projections are the same for all angles, so all we have to do is obtain the projection for θ= 0°. Equation (5.11-3) then becomes

FIGURE 5.38 (a) A

disk and (b) a plot of its Radon transform, derived analytically. Here we were able to plot the transform because it depends only on one variable. When g depends on both p and 9, the Radon transform becomes an image whose axes are p and 9, and the intensity of a pixel is proportional to the value of g at the location of that pixel.

where the second line follows from Eq. (4.2-10). As noted earlier, this is a line integral (along the line L(p, 0) in this case). Also, note that g(p, 8) = 0 when

|ρ| > r. When |ρ| ≤ r the integral is evaluated from to Therefore,

Carrying out the integration yields

where we used the fact mentioned above that g(p,θ) = 0 when |ρ| > r. Figure 5.38(b) shows the result, which agrees with the projections illustrated in Figs. 5.32 and 5.33. Note that g(p,θ) = g(ρ); that is, g is independent of θ because the object is symmetric about the origin.

(5.11-3)