PRESERVING TOPOLOGY BY A DIGITIZATION PROCESS

The main task of digital image processing is to recognize properties o f real objects based on their digital images. These images are obtaine d by some sampling device, like a CCD camera, and represented as finit e sets of points that are assigned some value in a gray-level or color scale. Based on technical properties of sampling devices, these point s are usually assumed to form a square grid and are modeled as finite subsets of Z(2). Therefore, a fundamental question in digital image pr ocessing is which features in the digital image correspond, under cert ain conditions, to properties of the underlying objects. In practical applications this question is mostly answered by visually judging the obtained digital images. In this paper we present a comprehensive answ er to this question with respect to topological properties. In particu lar, we derive conditions relating properties of real objects to the g rid size of the sampling device which guarantee that a real object and its digital image are topologically equivalent. These conditions also imply that two digital images of a given object are topologically equ ivalent. This means, for example, that shifting or rotating an object or the camera cannot lead to topologically different images, i.e., top ological properties of obtained digital images are invariant under shi fting and rotation.