Chapter 1 – Digital topology

Publisher Summary This chapter presents the construction of a consistent topology in the discrete space. It also provides robust basic definitions, such as connectivity and distance. Comparisons in relation to the continuous space are given and detailed throughout the study. Owing to the discrete nature of computers on which automated image processing is to be performed, images are typically given as sets of discrete points. To obtain a robust mathematical background for digital image processing, a formal study of such sets is to be developed. Then only, theoretical investigations can be carried out for presenting digital image processing operators. The acquisition step, whereby a continuous set is mapped onto a set of discrete points is introduced to characterize discrete sets, which represent binary digital images. Then, a topology is to be built in this context. This problem is addressed from the basis of neighborhood relationships to the definition of discrete sets. Based on the results derived, the construction of discrete distance functions is presented. Finally, the compatibility of such distance functions with Euclidean distance is studied. This last part also allows for the refinement of the definitions of discrete distance functions.