ABSTRACT
Figure 14.1b is an image of the set of projections [pθ(r)] for θ = 0°–180° for the CT image in Figure 14.1a. e top row in Figure 14.1b is at 0° and the bottom row at 180°. Columns index positions within projections (r). e sinusoidal appearance of paths formed by points 1 and 2 led us to call this image a sinogram. e 0° projection is a posterior view, while the 90° projection is a view from the right. Values at each x′ = r in the projection
pθ(r) are integrations across the object along a line perpendicular to the projection (i.e., line integrals along y′). e equation for mapping a point in the x-y image to a point in the r-θ sinogram (Figure 14.1) is as follows:
r rxy( ) cos( )q j q= × - , (14.1)
where
r x y
y
x xy = + =
æ è ç
ö ø ÷
-2 2 1f tan ,
with rxy as the distance from the origin to the point x, y, φ as the counterclockwise (CCW) angle from the positive x-axis to the point, and θ as the angle of the projection. e three parameters of Equation 14.1 (r, θ, and ϕ) determine the key features of the sinusoidal path followed by a point in the object:
• e amplitude of the sinusoid is determined by distance from the axis of rotation (rxy).
