Digital geometry deals with
discrete sets (usually discrete
point sets) considered to be
digitized models
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
or
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
s 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
Digital usually refers to something using discrete digits, often binary digits.
Technology and computing Hardware
*Digital electronics, electronic circuits which operate using digital signals
**Digital camera, which captures and stores digital i ...
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
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick i ...
, 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
Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it can be empl ...
.
* 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.
* Curve sketching, a method of drawing a curve pixel by pixel.
Digital geometry heavily overlaps with
discrete geometry and may be considered as a part thereof.
Digital space
A 2D digital space usually means a 2D grid space that only contains integer points in 2D Euclidean space. A 2D image is a function on a 2D digital space (See
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
).
In Rosenfeld and Kak's book, digital connectivity are defined as the relationship among elements in digital space. For example, 4-connectivity and 8-connectivity in 2D. Also see
pixel connectivity. A digital space and its (digital-)connectivity determine a
digital topology.
In digital space, the digitally continuous function (A. Rosenfeld, 1986) and the
gradually varied function (L. Chen, 1989) were proposed, independently.
A digitally continuous function means a function in which the value (an integer) at a digital point is the same or off by at most 1 from its neighbors. In other words, if ''x'' and ''y'' are two adjacent points in a digital space, , ''f''(''x'') − ''f''(''y''), ≤ 1.
A gradually varied function is a function from a digital space
to
where