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Gaussian Blur
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 circ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cappadocia Gaussian Blur
Cappadocia (; , from ) is a historical region in Central Anatolia region, Turkey. It is largely in the provinces of Nevşehir Province, Nevşehir, Kayseri Province, Kayseri, Aksaray Province, Aksaray, Kırşehir Province, Kırşehir, Sivas Province, Sivas and Niğde Province, Niğde. Today, the touristic Cappadocia Region is located in Nevşehir Province, Nevşehir province. According to Herodotus, in the time of the Ionian Revolt (499 BC), the Cappadocians were reported as occupying a region from the Taurus Mountains to the vicinity of the Euxine (Black Sea). Cappadocia, in this sense, was bounded in the south by the chain of mountains that separate it from Cilicia, to the east by the upper Euphrates, to the north by the Pontus (region), 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 Christianity, Chr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transformation (mathematics)
In mathematics, a transformation, transform, or self-map 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliasing
In signal processing and related disciplines, aliasing is a phenomenon that a reconstructed signal from samples of the original signal contains low frequency components that are not present in the original one. This is caused when, in the original signal, there are components at frequency exceeding a certain frequency called Nyquist frequency, f_s / 2, where f_s is the sampling frequency ( undersampling). This is because typical reconstruction methods use low frequency components while there are a number of frequency components, called aliases, which sampling result in the identical sample. 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 in digital audio or the stroboscopic effect, and is referred to as temporal aliasing. Aliasing in spatially sampled signals (e.g., moiré patterns in digital images) is referre ... [...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 o ... [...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–Kuzmin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member of a #Related asymptotic notations, family of notations invented by German mathematicians Paul Gustav Heinrich Bachmann, Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '':wikt:Ordnung#German, Ordnung'', meaning the order of approximation. In computer science, big O notation is used to Computational complexity theory, 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 arithmetic function, arithmetical function and a better understood approximation; one well-known exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Filter
A separable filter in image processing can be written as product of two more simple filters. Typically a 2-dimensional convolution operation is separated into two 1-dimensional filters. This reduces the computational costs on an N\times M image with a m\times n filter from \mathcal(M\cdot N\cdot m\cdot n) down to \mathcal(M\cdot N\cdot (m + n)). Examples 1. A two-dimensional smoothing filter: : \frac \begin 1 \\ 1 \\ 1 \end * \frac \begin 1 & 1 & 1 \end = \frac \begin 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end 2. Another two-dimensional smoothing filter with stronger weight in the middle: : \frac \begin 1 \\ 2 \\ 1 \end * \frac \begin 1 & 2 & 1 \end = \frac \begin 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end 3. The Sobel operator, used commonly for edge detection: : \begin 1 \\ 2 \\ 1 \end * \begin 1 & 0 & -1 \end = \begin 1 & 0 & -1 \\ 2 & 0 & -2 \\ 1 & 0 & -1 \end This works also for the Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floor And Ceiling Functions
In mathematics, 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, for floor: , , and for ceiling: , and . The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations). However, the same term, ''integer part'', is also used for truncation towards zero, which differs from the floor function for negative numbers. For an integer , . Although and produce graphs that appear exactly alike, they are not the same when the value of is an exact integer. For example, when , . However, if , then , while . 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev's Inequality
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) provides an upper bound on the probability of deviation of a random variable (with finite variance) from its mean. More specifically, the probability that a random variable deviates from its mean by more than k\sigma is at most 1/k^2, where k is any positive constant and \sigma is the standard deviation (the square root of the variance). The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers. Its practical usage is similar to the 68–95–99.7 rule, which applies only to normal distributions. Chebyshev's inequality is more general, stating that a minimum of just 75% of values must lie within two standard deviations of the ... [...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'' when they share the same center. Any pair of (possibly unalike) objects with well-defined centers can be concentric, including circles, spheres, regular polygons, regular polyhedra, parallelograms, cones, conic sections, and quadrics. Geometric objects are ''coaxial'' if they share the same axis (line of symmetry). Geometric objects with a well-defined axis include circles (any line through the center), spheres, cylinders, conic sections, and surfaces of revolution. Concentric objects are often part of the broad category of ''whorled patterns'', which also includes ''spirals'' (a curve which emanates from a point, moving farther away as it revolves around the point). 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 nevertheles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
