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Gaussian Smoothing
In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss). It is a widely used effect in graphics software, typically to reduce image noise and reduce detail. The visual effect of this blurring technique is a smooth blur resembling that of viewing the image through a translucent screen, distinctly different from the bokeh effect produced by an out-of-focus lens or the shadow of an object under usual illumination. Gaussian smoothing is also used as a pre-processing stage in computer vision algorithms in order to enhance image structures at different scales—see scale space representation and scale space implementation. Mathematics Mathematically, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function. This is also known as a two-dimensional Weierstrass transform. By contrast, convolving by a circle ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cappadocia Gaussian Blur
Cappadocia or Capadocia (; tr, Kapadokya), is a historical region in Central Anatolia, Turkey. It largely is in the provinces Nevşehir, Kayseri, Aksaray, Kırşehir, Sivas and Niğde. According to Herodotus, in the time of the Ionian Revolt (499 BC), the Cappadocians were reported as occupying a region from Mount Taurus to the vicinity of the Euxine (Black Sea). Cappadocia, in this sense, was bounded in the south by the chain of the Taurus Mountains that separate it from Cilicia, to the east by the upper Euphrates, to the north by Pontus, and to the west by Lycaonia and eastern Galatia. Van Dam, R. ''Kingdom of Snow: Roman rule and Greek culture in Cappadocia.'' Philadelphia: University of Pennsylvania Press, 2002, p.13 The name, traditionally used in Christian sources throughout history, continues in use as an international Tourism in Turkey, tourism concept to define a region of exceptional natural wonders, in particular characterized by fairy chimneys and a unique histo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transformation (mathematics)
In mathematics, a transformation is a function ''f'', usually with some geometrical underpinning, that maps a set ''X'' to itself, i.e. . Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transformations, such as rotations, reflections and translations. Partial transformations While it is common to use the term transformation for any function of a set into itself (especially in terms like "transformation semigroup" and similar), there exists an alternative form of terminological convention in which the term "transformation" is reserved only for bijections. When such a narrow notion of transformation is generalized to partial functions, then a partial transformation is a function ''f'': ''A'' → ''B'', where both ''A'' and ''B'' are subsets of some set ''X''. Algebraic structures The set of all transformations on a given base set, together with function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliasing
In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when a signal reconstructed from samples is different from the original continuous signal. Aliasing can occur in signals sampled in time, for instance digital audio, or the stroboscopic effect, and is referred to as temporal aliasing. It can also occur in spatially sampled signals (e.g. moiré patterns in digital images); this type of aliasing is called spatial aliasing. Aliasing is generally avoided by applying low-pass filters or anti-aliasing filters (AAF) to the input signal before sampling and when converting a signal from a higher to a lower sampling rate. Suitable reconstruction filtering should then be used when restoring the sampled signal to the continuous domain or converting a signal from a lower to a higher sampling rate. For spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Downsampling
In digital signal processing, downsampling, compression, and decimation are terms associated with the process of ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''compression'', or they can describe an entire process of bandwidth reduction (filtering) and sample-rate reduction. When the process is performed on a sequence of samples of a ''signal'' or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate (or density, as in the case of a photograph). ''Decimation'' is a term that historically means the '' removal of every tenth one''. But in signal processing, ''decimation by a factor of 10'' actually means ''keeping'' only every tenth sample. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, if compact disc audio at 44,100 samples/second is ''decimated'' by a factor of 5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Blur Before Downscaling
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymous adjective ''Gaussian'' is pronounced . Mathematics Algebra and linear algebra Geometry and differential geometry Number theory Cyclotomic fields *Gaussian period *Gaussian rational *Gauss sum, an exponential sum over Dirichlet characters ** Elliptic Gauss sum, an analog of a Gauss sum **Quadratic Gauss sum Analysis, numerical analysis, vector calculus and calculus of variations Complex analysis and convex analysis *Gauss–Lucas theorem *Gauss's continued fraction, an analytic continued fraction derived from the hypergeometric functions * Gauss's criterion – described oEncyclopedia of Mathematics* Gauss's hypergeometric theorem, an identity on hypergeometric series * Gauss plane Statistics *Gauss–K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Filter
Separability may refer to: Mathematics * Separable algebra, a generalization to associative algebras of the notion of a separable field extension * Separable differential equation, in which separation of variables is achieved by various means * Separable extension, in field theory, an algebraic field extension * Separable filter, a product of two or more simple filters in image processing * Separable ordinary differential equation, a class of equations that can be separated into a pair of integrals * Separable partial differential equation, a class of equations that can be broken down into differential equations in fewer independent variables * Separable permutation, a permutation that can be obtained by direct sums and skew sums of the trivial permutation * Separable polynomial, a polynomial whose number of distinct roots is equal to its degree * Separable sigma algebra, a separable space in measure theory * Separable space, a topological space that contains a countable, dense ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floor And Ceiling Functions
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or . For example, , , , and . Historically, the floor of has been–and still is–called the integral part or integer part of , often denoted (as well as a variety of other notations). Some authors may define the integral part as if is nonnegative, and otherwise: for example, and . The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part. For an integer, . Notation The ''integral part'' or ''integer part'' of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket notation in his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weighted Average
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox. Examples Basic example Given two school with 20 students, one with 30 test grades in each class as follows: :Morning class = :Afternoon class = The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concentric Circles
In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point), as may cylinders (sharing the same central axis). Geometric properties In the Euclidean plane, two circles that are concentric necessarily have different radii from each other.. However, circles in three-dimensional space may be concentric, and have the same radius as each other, but nevertheless be different circles. For example, two different meridians of a terrestrial globe are concentric with each other and with the globe of the earth (approximated as a sphere). More generally, every two great circles on a sphere are concentric with each other and with the sphere. By Euler's theorem in geometry on the distance between the circumcenter and incenter of a triangle, two concentric circles (with that distance being zero) are the cir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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