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A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection f from the space onto itself that multiplies all distances by the same positive real number r, so that for any two points x and y we have ((), ()) = (,), where d(x,y) is the Euclidean distance from x to y. [16]
From a spatial point of view, nearness (a.k.a. proximity) is considered a generalization of set intersection.For disjoint sets, a form of nearness set intersection is defined in terms of a set of objects (extracted from disjoint sets) that have similar features within some tolerance (see, e.g., §3 in).
The principles of similarity and proximity often work together to form a Visual Hierarchy. Either principle can dominate the other, depending on the application and combination of the two. For example, in the grid to the left, the similarity principle dominates the proximity principle; the rows are probably seen before the columns.
In statistics and related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity exists, usually such measures are in some sense the inverse of distance metrics : they take on large values for similar ...
Standard (trivial) self-similarity [1] In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many ...
Proximity without the first axiom is called quasi-proximity (but then Axioms 2 and 4 must be stated in a two-sided fashion). If A δ B {\displaystyle A\;\delta \;B} we say A {\displaystyle A} is near B {\displaystyle B} or A {\displaystyle A} and B {\displaystyle B} are proximal ; otherwise we say A {\displaystyle A} and B {\displaystyle B} are ...
The Tweedie convergence theorem can be used to explain the origin of the variance to mean power law, 1/f noise and multifractality, features associated with self-similar processes. [9] Ethernet traffic data is often self-similar. [5] Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network ...
The rule of similarity states that images that are similar to each other can be grouped together as being the same type of object or part of the same object. Therefore, the more similar two images or objects are, the more likely it will be that they can be grouped together. For example, two squares among many circles will be grouped together.