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Digital geometry: discrete geometry and topology for computer imagery


Digital geometry is the study of geometric properties of discrete point sets, such as digitized objects in digital images. Since all the images treated in computers are digitized, we can not avoid the influence of such a digitization on their geometric properties. For digital objects, there are particular properties caused by the discontinuity and the finite size of the space of an image, which often provide combinatorial structures and can lead to discrete, combinatorial and/or arithmetic algorithms for solving geometric problems on these objects. Digital geometry may also implies the exact calculation in the integer domain, so that the results are guaranteed against calculation errors. Consequently, this framework allows us to achieve geometric computation in more reliable and efficient way for discrete point sets.

To analyze and synthesize digital images, conventional techniques often use geometric models defined in the continuous framework, while their calculations are performed digitally. Therefore, the obtained results can include errors of digitization and calculation, which may violate the geometric properties of the original object and thus lack reliability. In order to overcome such problems, we study combinatorial structures of information obtained from discrete point sets, which can be represented by graphs, for example. Using these combinatorial structures, we formulate geometric problems for discrete point sets in the discrete framework, and try to obtain more reliable results using discrete and combinatorial techniques.

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