Geometric Transformations

Abstract

In this chapter, techniques for remapping pixel locations are described. These techniques are known as geometric transformations of the images, necessary in different applications of the artificial vision, both to correct any geometric distortions or to introduce desired visual effects. Geometric correction is usually introduced during the acquisition, for example, images acquired while the objects or the sensors are moving, as in the case of satellite and/or aerial acquisitions. In all cases, the geometrical operator must be able to reproduce as accurately as possible the image with the same initial information content through the image resampling process. The class of affine transformations that covers different transformations is presented: translation, rotation, resizing, and shearing. Elementary transformations can be combined in different ways by multiplying the matrices in different orders, producing a different number of transformations. Furthermore, projective transformations are described, which are known as a subclass of warping techniques. The perspective transformation (already considered in the chapter of the image formation process) which projects 3D points on a plane is also described from an algebraic-geometric point of view. Perspective transformations play a central role in image processing because they provide an approximation to the way an image is formed by looking at a 3D world. The projective and perspective transformation essentially changes the geometric and radiometric characteristics with which the image was acquired. This leads to the need for image resampling which is the process of regaining a discrete image from a set of pixels geometrically transformed into another discrete set of pixel coordinates. Finally, the method of resampling images with the various methods of interpolation and image sampling is described.