## Geometrical Description of Pincushion Distortion

Depending on the value of the view angle (angle between the central X-ray beam and the object plane) two models are discussed. First, it is assumed that the central X-ray beam is perpendicular to the object plane. Second, an arbitrary view angle model is analyzed.

### Object at 0 View Angle

Let us first consider a case when the X-ray falls perpendicular to the object plane (i.e., the central X-ray beam is at the zenith position with respect to the patient). A point P x, y (Fig. 12) is projected onto the input phosphor surface at Pa x, y . The original distance d0 is magnified to dx and then lengthened to da (arc between S and Pa) by the input phosphor spherical surface. Since in this approach no other distortions are considered, a linear magnification is assumed.

Magnification along the central beam is dx £2 g d0 £1

where d0 is the distance from the object to the central point in the input phosphor, dx is the magnified distance of d0, and £1 and £2 are a distances from the focal spot FS to the object plane and input phosphor screen, respectively. From the geometry of Fig. 12, d, £ £2

From Eq. (10) and by the use of Eq. (11), the magnified distance dx is

With substitution of the relationship between the chord d,, arc 0, and radius R of the sphere, d, 2R sin 0/2

FIGURE 12 Geometrical distortions originated from projecting a flat object plate onto a curved input phosphor surface.

2fe2tg da/2R

14 and

If we substitute Eq. (14) in Eq. (9), the relationship between the original distance d0 and its projection onto the input phosphor surface da is given by d0

2R5itg da/2R

52 2R tg2 da/2R