By James F. Peters

ISBN-10: 3319302604

ISBN-13: 9783319302607

ISBN-10: 3319302620

ISBN-13: 9783319302621

This e-book introduces computational proximity (CP) as an algorithmic method of discovering nonempty units of issues which are both just about one another or a long way aside. regularly in computational proximity, the ebook starts off with a few type of proximity house (topological area built with a proximity relation) that has an inherent geometry. In CP, forms of close to units are thought of, particularly, spatially close to units and descriptivelynear units. it truly is proven that connectedness, boundedness, mesh nerves, convexity, shapes and form idea are valuable themes within the research of nearness and separation of actual aswell as summary units. CP has a hefty visible content material. functions of CP in machine imaginative and prescient, multimedia, mind task, biology, social networks, and cosmology are integrated. The booklet has been derived from the lectures of the writer in a graduate direction at the topology of electronic photos taught during the last numerous years. a number of the scholars have supplied vital insights and beneficial feedback. the subjects during this monograph introduce many types of proximities with a computational flavour (especially, what has turn into referred to as the robust touch relation), many nuances of topological areas, and point-free geometry.

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**Extra info for Computational Proximity: Excursions in the Topology of Digital Images**

**Example text**

Surface 3D Voronoï meshes are introduced P. Alliez and others in [78, starting on p. 54]. J. Hilhorst [79]. For more about Voronoï diagrams, see, for example, [54, Sect. 1]. An important recent development in the tessellation of plane surfaces is the introduction of Möbius diagrams containing regions with curved edges This leads to curved Voronoï diagrams [54]. Let p ∈ R2 , the Euclidean plane and let λi , μi be two real numbers, i = 1, . . , n. For a point x in the Euclidean plane, the distance δi (x) from x to a Möbius site ωi is defined by δi (x) = λi (x − pi )2 − μi .

11. An important fringe benefit of the CP approach in digital image processing is the discovery of image patterns. There are many forms of image patterns revealed by computational proximity methods that provide a basis for classifying images. Examples of geometric image patterns are connected sets of points that form what are known as mesh nerves. The notion of nerve, mesh nerve, proximal nerves, and nervous systems in digital images are introduced in this chapter. 1 Computational Proximity Framework 5 Fig.

In simple terms, this means sets that share elements, have strong proximity. For example, the shaded parts , , in contain points in the Euclidean plane that are shared by the tightly twisted (overlapping) shapes A and B. For an example of proximal but not strongly proximal physical objects, see the adjacent Feudi di San Gregorio wine casks in Fig. 7. The adjacent wine casks touch each other but do not have have common points. This illustrates one of the great subtleties of traditional spatial proximities.

### Computational Proximity: Excursions in the Topology of Digital Images by James F. Peters

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