A Bintree Representation of Generalized Binary Digital Images

A d-dimensional binary digital image can most easily be modelled by a d-dimensional binary array. Each element of the array represents a (d-dimensional) pixel which is normally called "black" ("white") if the element's value is 1 (0). For most purposes it is necessary to introduce topological notions (adjacency, connectedness, etc.) for digital images and to study their properties. This is the main topic of digital topology KR89]. Depending on the speciic point of view, pixels are then normally understood as elements of Z d , as points with integer coordinates in R d , or as d-dimensional closed unit cubes. The resulting diiculties are well-known and rather easily explained (cf. Pav82], BN84], and especially Kov89]). Bie90] avoids these diiculties by giving up the requirement that all pixels of a digital image have to be of the same type. As this approach does not conform to the conventional deenitions of digital images Fiu89], the resulting "images" are called generalized binary digital images or hyperimages. Bie90] starts from the pixel understood as a d-dimensional closed unit cube and replaces it by pixels which are relatively open unit cubes of dimensions 0; 1; :::; d. For the case d = 2, Figure 1 shows that an "old" pixel is the disjoint union of 9 "new" pixels of which four are 0-dimensional, four 1-dimensional, and one 2-dimensional. It is practical to distinguish between the "horizontal" and "vertical" 1-dimensional "new" pixels. Consequently we get for d = 2 four types of pixels which we denote by the symbols , |, j,. Figure 2 shows a conventional 2-dimensional binary digital image, with its 36 pixels understood as closed unit squares. Figure 3 shows the corresponding hyperimage where the numbers of pixels belonging to the four types are 49, 42, 42, and 36. The two gures clearly show the most important advantage of hyperimages compared to conventional digital images: The "old" pixels in Figure 2 form only a subdivision of the whole image (diierent "old" pixels are not necessarily disjoint sets), whereas the "new" pixels in Figure 3 form a partition of the hyperimage. Like a conventional digital image, also a hyperimage can most easily be modelled by a d-dimensional binary array. Again, every element of the array represents a pixel which is understood as a unit cube R d. But now, all these unit cubes are relatively open and their dimension is no longer necessarily = …