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Dilation (morphology)
Dilation (usually represented by ⊕) is one of the basic operations in mathematical morphology. Originally developed for binary images, it has been expanded first to grayscale images, and then to complete lattices. The dilation operation usually uses a structuring element for probing and expanding the shapes contained in the input image. Binary dilation In binary morphology, dilation is a shift-invariant (translation invariant) operator, equivalent to Minkowski addition. A binary image is viewed in mathematical morphology as a subset of a Euclidean space R''d'' or the integer grid Z''d'', for some dimension ''d''. Let ''E'' be a Euclidean space or an integer grid, ''A'' a binary image in ''E'', and ''B'' a structuring element regarded as a subset of R''d''. The dilation of ''A'' by ''B'' is defined by ::A \oplus B = \bigcup_ A_b, where ''A''''b'' is the translation of ''A'' by ''b''. Dilation is commutative, also given by A \oplus B = B\oplus A = \bigcup_ B_a. If ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 employed as well on graphs, surface meshes, solids, and many other spatial structures. Topological and geometrical continuous-space concepts such as size, shape, convexity, connectivity, and geodesic distance, were introduced by MM on both continuous and discrete spaces. MM is also the foundation of morphological image processing, which consists of a set of operators that transform images according to the above characterizations. The basic morphological operators are erosion, dilation, opening and closing. MM was originally developed for binary images, and was later extended to grayscale functions and images. The subsequent generalization to complete lattices is widely accepted today as MM's theoretical foundation. History Mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closing (morphology)
In mathematical morphology, the closing of a set (binary image) ''A'' by a structuring element ''B'' is the erosion (morphology), erosion of the dilation (morphology), dilation of that set, :A\bullet B = (A\oplus B)\ominus B, \, where \oplus and \ominus denote the dilation and erosion, respectively. In image processing, closing is, together with opening (morphology), opening, the basic workhorse of morphological signal noise, noise removal. Opening removes small objects, while closing removes small holes. Properties * It is idempotent, that is, (A\bullet B)\bullet B=A\bullet B. * It is increasing, that is, if A\subseteq C, then A\bullet B \subseteq C\bullet B. * It is ''extensive'', i.e., A\subseteq A\bullet B. * It is Translational invariance, translation invariant. See also *Mathematical morphology *Dilation (morphology), Dilation *Erosion (morphology), Erosion *Opening (morphology), Opening *Top-hat_transform, Top-hat transformation Bibliography * ''Image Analysis and Mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buffer (GIS)
In geographic information systems (GIS) and spatial analysis, buffer analysis is the determination of a zone around a geographic feature containing locations that are within a specified distance of that feature, the buffer zone (or just buffer). A buffer is likely the most commonly used tool within the proximity analysis methods.Wade, T. and Smmer, S. eds. A to Z GIS' History The buffer operation has been a core part of GIS functionality since the original integrated GIS software packages of the late 1970s and early 1980s, such as ARC/INFO, Odyssey, and MOSS. Although it has been one of the most widely used GIS operations in subsequent years, in a wide variety of applications, there has been little published research on the tool itself, except for the occasional development of a more efficient algorithm. Basic algorithm The fundamental method to create a buffer around a geographic feature stored in a vector data model, with a given radius ''r'' is as follows: * Single point: C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Definitions Let (P, \leq) be a preordered set and let S \subseteq P. An element g \in P is said to be if g \in S and if it also satisfies: :s \leq g for all s \in S. By using \,\geq\, instead of \,\leq\, in the above definition, the definition of a least element of S is obtained. Explicitly, an element l \in P is said to be if l \in S and if it also satisfies: :l \leq s for all s \in S. If (P, \leq) is even a partially ordered set then S can have at most one greatest element and it can have at most one least element. Whenever a greatest element of S exists and is unique then this element is called greatest element of S. The terminology least element of S is defined simila ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Definitions Let (P, \leq) be a preordered set and let S \subseteq P. An element g \in P is said to be if g \in S and if it also satisfies: :s \leq g for all s \in S. By using \,\geq\, instead of \,\leq\, in the above definition, the definition of a least element of S is obtained. Explicitly, an element l \in P is said to be if l \in S and if it also satisfies: :l \leq s for all s \in S. If (P, \leq) is even a partially ordered set then S can have at most one greatest element and it can have at most one least element. Whenever a greatest element of S exists and is unique then this element is called greatest element of S. The terminology least element of S is defined simila ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and maxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of Comparability, comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Lattice
In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' Specifically, every non-empty finite lattice is complete. Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra. Complete lattices must not be confused with complete partial orders (''cpo''s), which constitute a strictly more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (''locales''). Formal definition A partially ordered set (''L'', ≤) is a ''complete lattice'' if every subset ''A'' of ''L'' has both a greatest lower bound (the infimum, also called the ''meet'') and a least upper bound (the supremum, also called the ''j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Statistics
In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. Notation and examples For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. If the sample values are :6, 9, 3, 8, the order statistics would be denoted :x_=3,\ \ x_=6,\ \ x_=8,\ \ x_=9,\, where the subscript enclosed in parentheses indicates the th order statistic of the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the ''local'' or ''relative'' extrema), or on the entire domain (the ''global'' or ''absolute'' extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗, if for all ''x'' in ''X''. The value of the function at a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grayscale Morphological Dilation
In digital photography, computer-generated imagery, and colorimetry, a grayscale image is one in which the value of each pixel is a single sample representing only an ''amount'' of light; that is, it carries only intensity information. Grayscale images, a kind of black-and-white or gray monochrome, are composed exclusively of shades of gray. The contrast ranges from black at the weakest intensity to white at the strongest. Grayscale images are distinct from one-bit bi-tonal black-and-white images, which, in the context of computer imaging, are images with only two colors: black and white (also called ''bilevel'' or '' binary images''). Grayscale images have many shades of gray in between. Grayscale images can be the result of measuring the intensity of light at each pixel according to a particular weighted combination of frequencies (or wavelengths), and in such cases they are monochromatic proper when only a single frequency (in practice, a narrow band of frequencies) is cap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
