1.
Discrete cosine transform
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A discrete cosine transform expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. DCTs are important to numerous applications in science and engineering, from lossy compression of audio and images, in particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform, but using only real numbers. DCTs are equivalent to DFTs of roughly twice the length, operating on data with even symmetry. There are eight standard DCT variants, of four are common. The most common variant of discrete cosine transform is the type-II DCT and its inverse, the type-III DCT, is correspondingly often called simply the inverse DCT or the IDCT. Multidimensional DCTs are developed to extend the concept of DCT on MD Signals, there are several algorithms to compute MD DCT. A new variety of fast algorithms are developed to reduce the computational complexity of implementing DCT. For strongly correlated Markov processes, the DCT can approach the efficiency of the Karhunen-Loève transform. As explained below, this stems from the boundary conditions implicit in the cosine functions, a related transform, the modified discrete cosine transform, or MDCT, is used in AAC, Vorbis, WMA, and MP3 audio compression. The DCT is used in JPEG image compression, MJPEG, MPEG, DV, Daala, there, the two-dimensional DCT-II of N × N blocks are computed and the results are quantized and entropy coded. In this case, N is typically 8 and the DCT-II formula is applied to each row, due to enhancement in the hardware, software and introduction of several fast algorithms, the necessity of using M-D DCTs is rapidly increasing. DCT-IV has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks, lapped orthogonal transform, like any Fourier-related transform, discrete cosine transforms express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes. Like the discrete Fourier transform, a DCT operates on a function at a number of discrete data points. The obvious distinction between a DCT and a DFT is that the former uses only cosine functions, while the latter uses both cosines and sines. However, this difference is merely a consequence of a deeper distinction. That is, once you write a function f as a sum of sinusoids, you can evaluate that sum at any x, the DFT, like the Fourier series, implies a periodic extension of the original function. A DCT, like a cosine transform, implies an even extension of the original function, however, because DCTs operate on finite, discrete sequences, two issues arise that do not apply for the continuous cosine transform. First, one has to specify whether the function is even or odd at both the left and right boundaries of the domain, second, one has to specify around what point the function is even or odd

2.
Signal processing
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According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. Oppenheim and Schafer further state that the digitalization or digital refinement of techniques can be found in the digital control systems of the 1940s and 1950s. Feature extraction, such as understanding and speech recognition. Quality improvement, such as reduction, image enhancement. Including audio compression, image compression, and video compression and this involves linear electronic circuits as well as non-linear ones. The former are, for instance, passive filters, active filters, additive mixers, integrators, non-linear circuits include compandors, multiplicators, voltage-controlled filters, voltage-controlled oscillators and phase-locked loops. Continuous-time signal processing is for signals that vary with the change of continuous domain, the methods of signal processing include time domain, frequency domain, and complex frequency domain. This technology was a predecessor of digital processing, and is still used in advanced processing of gigahertz signals. Digital signal processing is the processing of digitized discrete-time sampled signals, processing is done by general-purpose computers or by digital circuits such as ASICs, field-programmable gate arrays or specialized digital signal processors. Typical arithmetical operations include fixed-point and floating-point, real-valued and complex-valued, other typical operations supported by the hardware are circular buffers and look-up tables. Examples of algorithms are the Fast Fourier transform, finite impulse response filter, Infinite impulse response filter, nonlinear signal processing involves the analysis and processing of signals produced from nonlinear systems and can be in the time, frequency, or spatio-temporal domains. Nonlinear systems can produce complex behaviors including bifurcations, chaos, harmonics

3.
Opus (audio format)
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Opus is a lossy audio coding format developed by the Xiph. Opus combines the speech-oriented linear predictive coding SILK algorithm and the lower-latency, MDCT-based CELT algorithm, bitrate, audio bandwidth, complexity, and algorithm can all be adjusted seamlessly in each frame. Its delay is exceptionally low compared to competing codecs, which well over 100 ms. As an open format standardized through RFC6716, an implementation called libopus is available under the New BSD License. The reference has both fixed-point and floating-point optimizations for low- and high-end devices, with SIMD optimizations on platforms that support them, all known software patents that cover Opus are licensed under royalty-free terms. Opus supports constant and variable bitrate encoding from 6 kbit/s to 510 kbit/s, frame sizes from 2.5 ms to 60 ms, an Opus stream can support up to 255 audio channels, and it allows channel coupling between channels in groups of two using mid-side coding. Unlike Vorbis, Opus does not require large codebooks for each file, making it more efficient for short clips of audio. As an open standard, the algorithms are documented. Qualcomm, Huawei, France Telecom, and Ericsson have claimed that their patents may apply, which Xiphs legal counsel denies, the Opus license automatically and retroactively terminates for any entity that attempts to file a patent suit. In Opus, both were modified to support more frame sizes, as well as further improvements and integration. Better tone detection is a project to improve quality. The format has three different modes, speech, hybrid, and CELT, the third mode is pure-CELT, designed for general audio. SILK is inherently VBR and cannot hit a target, while CELT can always be encoded to any specific number of bytes, enabling hybrid. Opus was originally specified for encapsulation in Ogg containers, specified as audio/ogg, codecs=opus, matroska, WebM, MPEG-TS, and MP4 all officially support Opus streams. The Opus spec allows the sample rates, Opus was proposed for the standardization of a new audio format at the IETF. It is based on two separate standard proposals from the Xiph. Org Foundation and Skype Technologies S. A. Its main developers are Jean-Marc Valin, Koen Vos, and Timothy B, among others, Juin-Hwey Chen, Gregory Maxwell, and Christopher Montgomery were also involved. The development of the CELT part of the format back to thoughts on a successor for Vorbis under the working name Ghost

4.
JPEG XR
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JPEG XR is a still-image compression standard and file format for continuous tone photographic images, based on technology originally developed and patented by Microsoft under the name HD Photo. It supports both lossy and lossless compression, and is the image format for Ecma-388 Open XML Paper Specification documents. Microsoft first announced Windows Media Photo at WinHEC2006, and then renamed it to HD Photo in November of that year, in July 2007, the Joint Photographic Experts Group and Microsoft announced HD Photo to be under consideration to become a JPEG standard known as JPEG XR. On 16 March 2009, JPEG XR was given final approval as ITU-T Recommendation T.832 and starting in April 2009, on 19 June 2009, it passed an ISO/IEC Final Draft International Standard ballot, resulting in final approval as International Standard ISO/IEC 29199-2. The ITU-T updated its publication with a corrigendum approved in December 2009, in 2010, after completion of the image coding specification, the ITU-T and ISO/IEC also published a motion format specification, a conformance test set, and reference software for JPEG XR. In 2011, they published a report describing the workflow architecture for the use of JPEG XR images in applications. Lossless compression JPEG XR also supports lossless compression, the signal processing steps in JPEG XR are the same for both lossless and lossy coding. This makes the lossless mode simple to support and enables the trimming of some bits from a compressed image to produce a lossy compressed image. Tile structure support A JPEG XR coded image can be segmented into tile regions, the data for each region can be decoded separately. This enables rapid access to parts of an image without needing to decode the entire image, when a type of tiling referred to as soft tiling is used, the tile region structuring can be changed without fully decoding the image and without introducing additional distortion. For support of using an RGB color space, JPEG XR includes an internal conversion to the YCgCo color space. 16-bit and 32-bit fixed point color component codings are supported in JPEG XR. Moreover, 16-bit and 32-bit floating point color component codings are supported in JPEG XR, in these cases the image is interpreted as floating point data, although the JPEG XR encoding and decoding steps are all performed using only integer operations. The shared-exponent floating point color format known as RGBE is also supported, in addition to RGB and CMYK formats, JPEG XR also supports grayscale and multi-channel color encodings with an arbitrary number of channels. The color representations, in most cases, are transformed to a color representation. The transformation is reversible, so that this color transformation step does not introduce distortion. Transparency map support An alpha channel may be present to represent transparency, full decoding is also unnecessary for certain editing operations such as cropping, horizontal or vertical flips, or cardinal rotations. The tile structure for access to image regions can also be changed without full decoding, metadata support A JPEG XR image file may optionally contain an embedded ICC color profile, to achieve consistent color representation across multiple devices

5.
MP3
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Compared to CD quality digital audio, MP3 compression commonly achieves 75 to 95% reduction in size. MP3 files are thus 1/4 to 1/20 the size of the digital audio stream. This is important for both transmission and storage concerns, the basis for such comparison is the CD digital audio format which requires 1411200 bit/s. A commonly used MP3 encoding setting is CBR128 kbit/s resulting in file size 1/11 of the original CD-quality file, the MP3 lossy compression works by reducing the accuracy of certain parts of a continuous sound that are considered to be beyond the auditory resolution ability of most people. This method is referred to as perceptual coding or psychoacoustics. It uses psychoacoustic models to discard or reduce the precision of less audible to human hearing. MP3 was designed by the Moving Picture Experts Group as part of its MPEG-1 standard, the first subgroup for audio was formed by several teams of engineers at Fraunhofer IIS, University of Hanover, AT&T-Bell Labs, Thomson-Brandt, CCETT, and others. MPEG-1 Audio, which included MPEG-1 Audio Layer I, II and III was approved as a draft of ISO/IEC standard in 1991, finalised in 1992. A backwards compatible MPEG-2 Audio extension with lower sample and bit rates was published in 1995, MP3 is a streaming or broadcast format meaning that individual frames can be lost without affecting the ability to decode successfully delivered frames. Storing an MP3 stream in a file enables time-shifted playback, the MP3 lossy audio data compression algorithm takes advantage of a perceptual limitation of human hearing called auditory masking. In 1894, the American physicist Alfred M. Mayer reported that a tone could be rendered inaudible by another tone of lower frequency, in 1959, Richard Ehmer described a complete set of auditory curves regarding this phenomenon. Ernst Terhardt et al. created an algorithm describing auditory masking with high accuracy and this work added to a variety of reports from authors dating back to Fletcher, and to the work that initially determined critical ratios and critical bandwidths. A wide variety of compression algorithms were reported in IEEEs refereed Journal on Selected Areas in Communications. The genesis of the MP3 technology is described in a paper from Professor Hans Musmann who chaired the ISO MPEG Audio group for several years. The immediate predecessors of MP3 were Optimum Coding in the Frequency Domain, the first practical implementation of an audio perceptual coder in hardware, was an implementation of a psychoacoustic transform coder based on Motorola 56000 DSP chips. Another predecessor of the MP3 format and technology is to be found in the perceptual codec MUSICAM based on an integer arithmetics 32 sub-bands filterbank, driven by a psychoacoustic model. It was primarily designed for Digital Audio Broadcasting and digital TV and this codec incorporated into a broadcasting system using COFDM modulation was demonstrated on air and on the field together with Radio Canada and CRC Canada during the NAB show in 1991. F. As a doctoral student at Germanys University of Erlangen-Nuremberg, Karlheinz Brandenburg began working on music compression in the early 1980s

6.
Linear map
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In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE

7.
Vorbis
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Vorbis is a free and open-source software project headed by the Xiph. Org Foundation. The project produces an audio coding format and software reference encoder/decoder for lossy audio compression, Vorbis is most commonly used in conjunction with the Ogg container format and it is therefore often referred to as Ogg Vorbis. Vorbis is a continuation of audio compression development started in 1993 by Chris Montgomery, intensive development began following a September 1998 letter from the Fraunhofer Society announcing plans to charge licensing fees for the MP3 audio format. The Vorbis project started as part of the Xiphophorus companys Ogg project, Chris Montgomery began work on the project and was assisted by a growing number of other developers. They continued refining the source code until the Vorbis file format was frozen for 1.0 in May 2000, originally licensed as LGPL, in 2001 the Vorbis license was changed to the BSD license to encourage adoption with endorsement of Richard Stallman. A stable version of the software was released on July 19,2002. The Xiph. Org Foundation maintains an implementation, libvorbis. There are also some fine-tuned forks, most notably aoTuV, that offer better audio quality and these improvements are periodically merged back into the reference codebase. Vorbis is named after a Discworld character, Exquisitor Vorbis in Small Gods by Terry Pratchett, the Ogg format, however, is not named after Nanny Ogg, another Discworld character, the name is in fact derived from ogging, jargon that arose in the computer game Netrek. The Vorbis format has proven popular among supporters of free software and they argue that its higher fidelity and completely free nature, unencumbered by patents, make it a well-suited replacement for patented and restricted formats like MP3. Vorbis has different uses for consumer products, many video game titles store in-game audio as Vorbis, including Amnesia, The Dark Descent, Grand Theft Auto, San Andreas, Halo, Combat Evolved, Minecraft, and World of Warcraft, among others. Popular software players support Vorbis playback either natively or through an external plugin, a number of websites, including Wikipedia, use it. Others include Jamendo and Mindawn, as well as national radio stations like JazzRadio, Absolute Radio, NPR, Radio New Zealand. The Spotify audio streaming service uses Vorbis for its audio streams, also, the French music site Qobuz offers its customers the possibility to download their purchased songs in Vorbis format, as does the American music site Bandcamp. However, by 2014, not many further significant tests had been made, listening tests have attempted to find the best quality lossy audio codecs at certain bitrates. Mid to low bitrates, private tests in 2005 at 80 kbit/s and 96 kbit/s showed that aoTuV Vorbis had a better quality than other lossy audio formats, high bitrates, most people do not hear significant differences. However, trained listeners can often hear significant differences between codecs at identical bitrates, and aoTuV Vorbis performed better than LC-AAC, MP3, due to the ever-evolving nature of audio codecs, the results of many of these tests have become outdated. Listening tests are carried out as ABX tests, i. e. the listener has to identify an unknown sample X as being A or B

8.
Video compression picture types
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In the field of video compression a video frame is compressed using different algorithms with different advantages and disadvantages, centered mainly around amount of data compression. These different algorithms for video frames are called picture types or frame types, the three major picture types used in the different video algorithms are I, P and B. They are different in the characteristics, I‑frames are the least compressible. P‑frames can use data from previous frames to decompress and are more compressible than I‑frames, B‑frames can use both previous and forward frames for data reference to get the highest amount of data compression. There are three types of pictures used in compression, I‑frames, P‑frames and B‑frames. An I‑frame is an Intra-coded picture, in effect a fully specified picture, P‑frames and B‑frames hold only part of the image information, so they need less space to store than an I‑frame and thus improve video compression rates. A P‑frame holds only the changes in the image from the previous frame, for example, in a scene where a car moves across a stationary background, only the cars movements need to be encoded. The encoder does not need to store the unchanging background pixels in the P‑frame, P‑frames are also known as delta‑frames. A B‑frame saves even more space by using differences between the current frame and both the preceding and following frames to specify its content. While the terms frame and picture are often used interchangeably, strictly speaking, a frame is a complete image captured during a known time interval, and a field is the set of odd-numbered or even-numbered scanning lines composing a partial image. For example, in 1080 full HD mode, there are 1080 lines of pixels, an odd field consists of pixel information for lines 1,3. And even field has pixel information of lines 2,4, frames that are used as a reference for predicting other frames are referred to as reference frames. In the latest international standard, known as H. 264/MPEG-4 AVC, a slice is a spatially distinct region of a frame that is encoded separately from any other region in the same frame. In that standard, instead of I-frames, P-frames, and B-frames, there are I-slices, P-slices, also in H.264 are found several additional types of frames/slices, SI‑frames/slices, Facilitates switching between coded streams, contains SI-macroblocks. SI- SP‑frames will allow for increases in the error resistance, when such frames are used along with a smart decoder, it is possible to recover the broadcast streams of damaged DVDs. I-frames are coded without reference to any frame except themselves, may be generated by an encoder to create a random access point. May also be generated when differentiating image details prohibit generation of effective P or B-frames, typically require more bits to encode than other frame types. Often, I‑frames are used for access and are used as references for the decoding of other pictures

9.
Linear algebra
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Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, the set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns, such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics, for instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces, combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Because linear algebra is such a theory, nonlinear mathematical models are sometimes approximated by linear models. The study of linear algebra first emerged from the study of determinants, determinants were used by Leibniz in 1693, and subsequently, Gabriel Cramer devised Cramers Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, the study of matrix algebra first emerged in England in the mid-1800s. In 1844 Hermann Grassmann published his Theory of Extension which included foundational new topics of what is called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for womb, while studying compositions of linear transformations, Arthur Cayley was led to define matrix multiplication and inverses. Crucially, Cayley used a letter to denote a matrix. In 1882, Hüseyin Tevfik Pasha wrote the book titled Linear Algebra, the first modern and more precise definition of a vector space was introduced by Peano in 1888, by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its form in the first half of the twentieth century. The use of matrices in quantum mechanics, special relativity, the origin of many of these ideas is discussed in the articles on determinants and Gaussian elimination. Linear algebra first appeared in American graduate textbooks in the 1940s, following work by the School Mathematics Study Group, U. S. high schools asked 12th grade students to do matrix algebra, formerly reserved for college in the 1960s. In France during the 1960s, educators attempted to teach linear algebra through finite-dimensional vector spaces in the first year of secondary school and this was met with a backlash in the 1980s that removed linear algebra from the curriculum. To better suit 21st century applications, such as mining and uncertainty analysis