Download Computational Proximity: Excursions in the Topology of by James F. Peters PDF

By James F. Peters

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 ways aside. more often than not in computational proximity, the booklet begins with a few kind of proximity house (topological area built with a proximity relation) that has an inherent geometry. In CP, kinds 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 concept are important subject matters within the research of nearness and separation of actual aswell as summary units. CP has a hefty visible content material. purposes of CP in desktop imaginative and prescient, multimedia, mind job, biology, social networks, and cosmology are incorporated. The ebook has been derived from the lectures of the writer in a graduate direction at the topology of electronic photos taught during the last a number of years. a number of the scholars have supplied vital insights and beneficial feedback. the subjects during this monograph introduce many kinds of proximities with a computational flavour (especially, what has develop into often called the robust touch relation), many nuances of topological areas, and point-free geometry.

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

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L (p) = (x1 , y1 ) (Coordinates of point p). This means that each planar point or single-set region with n-features has a description defined by a feature vector in an n + 2-dimensional feature space. Let Φ(A), Φ(p) denote feature vectors for a single-set region A and point p, respectively, in a space X. Then Φ(A) = (φL (A), φ1 (A), . . , Φ(A) = ((xM , yM ), φ1 (A), . . , φn (A)) (Region A feature vector), with the centroid of A at location ((xM , yM )). Φ(p) = (φL (p), φ1 (p), . . , Φ(p) = ((x, y), φ1 (p), .

A probe φ maps a member of a set to a value in R (reals). There are two basic types of probes, namely, point-based probes and set-based probes. A point-based probe maps a point (singleton in a nonempty set) to a feature value of the point that is a real number. A set-based probe maps a nonempty set (non-singleton set of points) to a feature value of the set that is a real number. For more about this, see Sect. 1. , [15]). This notion of descriptive nearness is analogous to the similarity between art gallery paintings such as the closeness of descriptions of mountain winterscapes.

Descriptive Intersection [7, Sect. 3, p. 84]. (Φ) Φ(A) = {Φ(x) ∈ Rn : x ∈ A}, set of feature vectors. (∩) A ∩ B = {x ∈ A ∪ B : Φ(x) ∈ Φ(A)&Φ(x) ∈ Φ(B)}. Φ Φ The descriptive proximity δΦ was introduced in [48, 58, 59]. Let Φ(x) be a feature vector for x ∈ X, a nonempty set of non-abstract points such as picture points. A δΦ B reads A is descriptively near B, provided Φ(x) = Φ(y) for at least one pair of points, ˘ x ∈ A, y ∈ B. The proximity δ in the Cech, Efremovi˘c, and Wallman proximities is replaced by δΦ .

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