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Summed-area Table
A summed-area table is a data structure and algorithm for quickly and efficiently generating the sum of values in a rectangular subset of a grid. In the image processing domain, it is also known as an integral image. It was introduced to computer graphics in 1984 by Frank Crow for use with mipmaps. In computer vision it was popularized by Lewis and then given the name "integral image" and prominently used within the Viola–Jones object detection framework in 2001. Historically, this principle is very well known in the study of multi-dimensional probability distribution functions, namely in computing 2D (or ND) probabilities (area under the probability distribution) from the respective cumulative distribution functions. The algorithm As the name suggests, the value at any point (''x'', ''y'') in the summed-area table is the sum of all the pixels above and to the left of (''x'', ''y''), inclusive: I(x,y) = \sum_ i(x',y') where i(x,y) is the value of the pixel at (''x' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Image Application Example
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an ''antiderivative'', a function whose derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neuroimaging
Neuroimaging is the use of quantitative (computational) techniques to study the neuroanatomy, structure and function of the central nervous system, developed as an objective way of scientifically studying the healthy human brain in a non-invasive manner. Increasingly it is also being used for quantitative research studies of brain disease and psychiatric illness. Neuroimaging is highly multidisciplinary involving neuroscience, computer science, psychology and statistics, and is not a medical specialty. Neuroimaging is sometimes confused with neuroradiology. Neuroradiology is a medical specialty that uses non-statistical brain imaging in a clinical setting, practiced by radiologists who are medical practitioners. Neuroradiology primarily focuses on recognizing brain lesions, such as vascular diseases, strokes, tumors, and inflammatory diseases. In contrast to neuroimaging, neuroradiology is qualitative (based on subjective impressions and extensive clinical training) but sometime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefix Sum
In computer science, the prefix sum, cumulative sum, inclusive scan, or simply scan of a sequence of numbers is a second sequence of numbers , the summation, sums of Prefix (computer science), prefixes (running totals) of the input sequence: : : : :... For instance, the prefix sums of the natural numbers are the triangular numbers: : Prefix sums are trivial to compute in sequential models of computation, by using the formula to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort,. and they form the basis of the scan higher-order function in functional programming languages. Prefix sums have also been much studied in parallel algorithms, both as a test problem to be solved and as a useful primitive to be used as a subroutine in other parallel algorithms.. Abstractly, a prefix sum requires only a semigroup, binary associative operator ⊕, making it useful for many a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compensated Summation
In numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite-precision floating-point numbers, compared to the naive approach. This is done by keeping a separate ''running compensation'' (a variable to accumulate small errors), in effect extending the precision of the sum by the precision of the compensation variable. In particular, simply summing n numbers in sequence has a worst-case error that grows proportional to n, and a root mean square error that grows as \sqrt for random inputs (the roundoff errors form a random walk).. With compensated summation, using a compensation variable with sufficiently high precision the worst-case error bound is effectively independent of n, so a large number of values can be summed with an error that only depends on the floating-point precision of the result. The algorithm is attributed to William Kahan;. Ivo Babuška see ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Overflow
In computer programming, an integer overflow occurs when an arithmetic operation on integers attempts to create a numeric value that is outside of the range that can be represented with a given number of digits – either higher than the maximum or lower than the minimum representable value. The most common result of an overflow is that the least significant representable digits of the result are stored; the result is said to ''wrap'' around the maximum (i.e. modulo a power of the radix, usually two in modern computers, but sometimes ten or other number). On some processors like graphics processing units (GPUs) and digital signal processors (DSPs) which support saturation arithmetic, overflowed results would be ''clamped'', i.e. set to the minimum value in the representable range if the result is below the minimum and set to the maximum value in the representable range if the result is above the maximum, rather than wrapped around. An overflow condition may give results lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Image Example
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an ''antiderivative'', a function whose derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Deviation
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. The standard deviation is commonly used in the determination of what constitutes an outlier and what does not. Standard deviation may be abbreviated SD or std dev, and is most commonly represented in mathematical texts and equations by the lowercase Greek alphabet, Greek letter Sigma, σ (sigma), for the population standard deviation, or the Latin script, Latin letter ''s'', for the sample standard deviation. The standard deviation of a random variable, Sample (statistics), sample, statistical population, data set, or probability distribution is the square root of its variance. (For a finite population, v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voxel
In computing, a voxel is a representation of a value on a three-dimensional regular grid, akin to the two-dimensional pixel. Voxels are frequently used in the Data visualization, visualization and analysis of medical imaging, medical and scientific data (e.g. geographic information systems (GIS)). Voxels also have technical and artistic applications in video games, largely originating with surface rendering in ''Outcast (video game), Outcast'' (1999). ''Minecraft'' (2011) makes use of an entirely voxelated world to allow for a fully destructable and constructable environment. Voxel art, of the sort used in ''Minecraft'' and elsewhere, is a style and format of 3D art analogous to pixel art. As with pixels in a 2D bitmap, voxels themselves do not typically have their position (i.e. coordinates) explicitly encoded with their values. Instead, Rendering (computer graphics), rendering systems infer the position of a voxel based upon its position relative to other voxels (i.e., its pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summed Area Table
A summed-area table is a data structure and algorithm for quickly and efficiently generating the sum of values in a rectangular subset of a grid. In the image processing domain, it is also known as an integral image. It was introduced to computer graphics in 1984 by Frank Crow for use with mipmaps. In computer vision it was popularized by Lewis and then given the name "integral image" and prominently used within the Viola–Jones object detection framework in 2001. Historically, this principle is very well known in the study of multi-dimensional probability distribution functions, namely in computing 2D (or ND) probabilities (area under the probability distribution) from the respective cumulative distribution functions. The algorithm As the name suggests, the value at any point (''x'', ''y'') in the summed-area table is the sum of all the pixels above and to the left of (''x'', ''y''), inclusive: I(x,y) = \sum_ i(x',y') where i(x,y) is the value of the pixel at (''x'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Data Structure
In computer science, a data structure is a data organization and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the relationships among them, and the Function (computer programming), functions or Operator (computer programming), operations that can be applied to the data, i.e., it is an algebraic structure about data. Usage Data structures serve as the basis for abstract data types (ADT). The ADT defines the logical form of the data type. The data structure implements the physical form of the data type. Different types of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks. For example, Relational database, relational databases commonly use B-tree indexes for data retrieval, while compiler Implementation, implementations usually use hash tables to look up Identifier (computer languages), identifiers. Data s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
