GEOMETRICAL IMAGE MODIFICATION One of the most common image processing operations is geometrical modification in which an image is spatially translated, scaled, rotated, nonlinearly warped, or viewed from a different perspective. | Digital Image Processing PIKS Inside Third Edition. William K. Pratt Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-37407-5 Hardback 0-471-22132-5 Electronic 13 GEOMETRICAL IMAGE MODIFICATION One of the most common image processing operations is geometrical modification in which an image is spatially translated scaled rotated nonlinearly warped or viewed from a different perspective. . TRANSLATION MINIFICATION MAGNIFICATION AND ROTATION Image translation scaling and rotation can be analyzed from a unified standpoint. Let G j k for 1 j J and 1 k K denote a discrete output image that is created by geometrical modification of a discrete input image F p q for 1 p P and 1 q Q. In this derivation the input and output images may be different in size. Geometrical image transformations are usually based on a Cartesian coordinate system representation in which the origin 0 0 is the lower left corner of an image while for a discrete image typically the upper left corner unit dimension pixel at indices 1 1 serves as the address origin. The relationships between the Cartesian coordinate representations and the discrete image arrays of the input and output images are illustrated in Figure . The output image array indices are related to their Cartesian coordinates by xk k 2 yk J 2- j 371 372 GEOMETRICAL IMAGE MODIFICATION FIGURE . Relationship between discrete image array and Cartesian coordinate representation. Similarly the input array relationship is given by Uq - 2 vp P 2 - p . Translation Translation of F p q with respect to its Cartesian origin to produce G j k involves the computation of the relative offset addresses of the two images. The translation address relationships are xk uq tx yj vp ty where tx and ty are translation offset constants. There are two approaches to this computation for discrete images forward and reverse address computation. In the forward approach uq and vp are computed for each
