Digital Geometry

Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean space.

Simply put, digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, the raster display of a computer, or in newspapers are in fact digital images.

Its main application areas are computer graphics and image analysis.

Main aspects of study are:

• Constructing digitized representations of objects, with the emphasis on precision and efficiency (either by means of synthesis, see, for example, Bresenham's line algorithm or digital disks, or by means of digitization and subsequent processing of digital images).
• Study of properties of digital sets; see, for example, Pick's theorem, digital convexity, digital straightness, or digital planarity.
• Transforming digitized representations of objects, for example (A) into simplified shapes such as (i) skeletons, by repeated removal of simple points such that the digital topology of an image does not change, or (ii) medial axis, by calculating local maxima in a distance transform of the given digitized object representation, or (B) into modified shapes using mathematical morphology.
• Reconstructing "real" objects or their properties (area, length, curvature, volume, surface area, and so forth) from digital images.
• Study of digital curves, digital surfaces, and digital manifolds.
• Designing tracking algorithms for digital objects.
• Functions on digital space.

Digital geometry heavily overlaps with discrete geometry and may be considered as a part thereof.