ABSTRACT
One of the most common image processing operations is geometrical modification in which an image is spatially translated, scaled in size, rotated or nonlinearly warped (1).
12.1. BASIC GEOMETRICAL METHODS
Image translation, size scaling and rotation can be analyzed from a unified standpoint. Let for and denote a discrete destination image that is created by geometrical modification of a discrete source image for and . In this derivation, the source and destination images may be different in size. Geometrical image transformations are usually based on a Cartesian coordinate system representation in which pixels are of unit dimension, and the origin is at the center of the upper left corner pixel of an image array. The relationships between the Cartesian coordinate representations and the discrete image array of the destination image are illustrated in Figure 12.1-1. The destination image array indices are related to their Cartesian coordinates by
(12.1-1a)
. (12.1-1b)
D j k,( ) 0 j J 1-≤ ≤ 0 k K 1-≤ ≤ S p q,( )
0 p P 1-≤ ≤ 0 q Q 1-≤ ≤
0 0,( )
D j k,( )
Similarly, the source array relationship is given by
(12.1-2a)
. (12.1-2b)
12.1.1. Translation
Translation of with respect to its Cartesian origin to produce involves the computation of the relative offset addresses of the two images. The translation address relationships are
(12.1-3a)
(12.1-3b)
where and are translation offset constants. There are two approaches to this computation for discrete images: forward and reverse address computation. In the forward approach, and are computed for each source pixel and substituted into Eq. 12.1-3 to obtain and . Next, the destination array addresses are computed by inverting Eq. 12.1-1. The composite computation
FIGURE 12.1-1. Relationship between discrete image array and Cartesian coordinate representation of a destination image D(j, k).
