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A discrete cosine transform (DCT) expresses a finite sequence of
data points In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...
in terms of a sum of
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

cosine
functions oscillating at different
frequencies Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparent ...

frequencies
. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

signal processing
and
data compression In signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electro ...
. It is used in most
digital media Digital media means any communication media that operate with the use of any of various encoded machine-readable data Machine-readable data, or computer-readable data, is data Data (; ) are individual facts, statistics, or items of informa ...
, including
digital images Digital usually refers to something using digits, particularly binary digits. Technology and computing Hardware *Digital electronics Digital electronics is a field of electronics Electronics comprises the physics, engineering, technology a ...
(such as
JPEG JPEG ( ) is a commonly used method of lossy compression In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

JPEG
and
HEIF High Efficiency Image File Format (HEIF) is a container format for storing individual images and image sequences. The standard covers multimedia files that can also include other media streams, such as timed text, audio and video. HEIF can st ...
, where small high-frequency components can be discarded),
digital video Digital video is an electronic representation of moving visual images (video) in the form of encoded digital data. This is in contrast to analog video, which represents moving visual images in the form of analog signals. Digital video comprises ...
(such as
MPEG The Moving Picture Experts Group (MPEG) is an alliance of working groups established jointly by International Organization for Standardization, ISO and International Electrotechnical Commission, IEC that sets standards for media coding, includ ...

MPEG
and
H.26x The Video Coding Experts Group or Visual Coding Experts Group (VCEG, also known as Question 6) is a working group of the (ITU-T) concerned with video coding standards. It is responsible for standardization of the "H.26x" line of video coding sta ...
),
digital audio Digital audio is a representation of sound recorded in, or converted into, Digital signal (signal processing), digital form. In digital audio, the sound wave of the audio signal is typically encoded as numerical Sampling (signal processing), s ...
(such as
Dolby Digital Dolby Digital, originally synonymous with Dolby AC-3, is the name for what has now become a family of audio compression technologies developed by Dolby Laboratories Dolby Laboratories, Inc. (often shortened to Dolby Labs and known simply a ...
,
MP3 MP3 (formally MPEG-1 Audio Layer III or MPEG-2 Audio Layer III) is a coding format for digital audio Digital audio is a representation of sound recorded in, or converted into, Digital signal (signal processing), digital form. In digital a ...

MP3
and
AAC AAC may refer to: Aviation * Advanced Aircraft Advanced Aircraft Corporation is an aircraft manufacturer based in Carlsbad, California. History AAC bought out Riley Aircraft in 1983 in aviation, 1983. Products The firm has specialised in conve ...
),
digital television Digital television (DTV) is the transmission of television audiovisual Audiovisual (AV) is electronic media 200px, Graphical representations of electrical audio data. Electronic media uses either analog (red) or digital (blue) signal pr ...
(such as
SDTV Standard-definition television (SDTV, SD, often shortened to standard definition) is a television system which uses a resolution that is not considered to be either high-definition television, high or enhanced-definition television, enhanced ...
,
HDTV High-definition television (HD or HDTV) describes a television system providing a substantially higher image resolution Image resolution is the detail an holds. The term applies to s, film images, and other types of images. Higher resolution m ...

HDTV
and VOD),
digital radio Digital radio is the use of digital technology to transmit or receive across the radio spectrum The radio spectrum is the part of the electromagnetic spectrum The electromagnetic spectrum is the range of frequencies (the spectrum A spec ...
(such as
AAC+ File:AAC profiles.svg, 250px, Evolution from MPEG-2 AAC-LC (Low Complexity) Profile and MPEG-4 AAC-LC MPEG-4 Part 3#MPEG-4 Audio Object Types, Object Type to AAC-HE v2 Profile. High-Efficiency Advanced Audio Coding (AAC-HE) is an audio coding f ...
and
DAB+ Digital Audio Broadcasting (DAB) is a digital radio Digital radio is the use of digital technology to transmit or receive across the radio spectrum. Digital transmission by radio waves includes digital broadcasting, and especially digital audi ...
), and
speech coding Speech coding is an application of data compression In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image ...
(such as
AAC-LD The MPEG-4 Low Delay Audio Coder (a.k.a. AAC Low Delay, or AAC-LD) is audio compression standard designed to combine the advantages of perceptual audio coding with the low delay necessary for two-way communication. It is closely derived from the ...
, Siren and Opus). DCTs are also important to numerous other applications in
science and engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, such as
digital signal processing Digital signal processing (DSP) is the use of digital processing Digital data, in information theory and information systems, is information represented as a string of discrete symbols each of which can take on one of only a finite number of ...
,
telecommunication Telecommunication is the transmission of information by various types of technologies over wire A wire is a single usually cylindrical A cylinder (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Gr ...
devices, reducing
network bandwidth In computing, bandwidth is the maximum rate of data transfer across a given path. Bandwidth may be characterized as network bandwidth, data bandwidth, or digital bandwidth. This definition of ''bandwidth'' is in contrast to the field of signal proc ...
usage, and
spectral method Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equation In mathematics, a differential equation is an equation that relates one or more function (mathema ...
s for the numerical solution of
partial differential equations In , a partial differential equation (PDE) is an equation which imposes relations between the various s of a . The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved ...
. The use of cosine rather than
sine In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

sine
functions is critical for compression, since it turns out (as described below) that fewer cosine functions are needed to approximate a typical
signal In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Sig ...
, whereas for differential equations the cosines express a particular choice of
boundary condition Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film Mathematics and physics * Boundary (top ...
s. In particular, a DCT is a
Fourier-related transform This is a list of linear transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chan ...
similar to the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
(DFT), but using only
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. The DCTs are generally related to Fourier Series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier Series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common. The most common variant of discrete cosine transform is the type-II DCT, which is often called simply "the DCT". This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT, is correspondingly often called simply "the inverse DCT" or "the IDCT". Two related transforms are the
discrete sine transformIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
(DST), which is equivalent to a DFT of real and ''odd'' functions, and the
modified discrete cosine transform The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine In mathematics, the trigonom ...
(MDCT), which is based on a DCT of ''overlapping'' data. Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT to MD signals. There are several algorithms to compute MD DCT. A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT. One of these is the integer DCT (IntDCT), an
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
approximation of the standard DCT, used in several
ISO/IEC ISO/IEC JTC 1 is a joint technical committee (JTC) of the International Organization for Standardization The International Organization for Standardization (ISO; ) is an international standard-setting body composed of representatives from var ...
and
ITU-T The ITU Telecommunication Standardization Sector (ITU-T) coordinates standards for telecommunications Telecommunication is the transmission of information by various types of technologies over , radio, , or other systems. It has its origin ...
international standards. DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks. DCT blocks can have a number of sizes, including 8x8
pixels In digital imaging, a pixel, pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in an all points addressable display device; so it is the smallest controllable elem ...

pixels
for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels. The DCT has a strong "energy compaction" property, capable of achieving high quality at high
data compression ratio Data compression ratio, also known as compression power, is a measurement of the relative reduction in size of data representation produced by a data compression algorithm. It is typically expressed as the division of uncompressed size by compressed ...
s. However, blocky
compression artifacts A compression artifact (or artefact) is a noticeable distortion of media (including images, audio, and video) caused by the application of lossy compression. Lossy data compression In signal processing, data compression, source coding, o ...

compression artifacts
can appear when heavy DCT compression is applied.


History

The discrete cosine transform (DCT) was first conceived by Nasir Ahmed, while working at
Kansas State University Kansas State University (KSU, Kansas State, or K-State) is a public university, public Land-grant university, land-grant research university with its main campus in Manhattan, Kansas. It was opened as the state's land-grant college in 1863 and ...
, and he proposed the concept to the
National Science Foundation The National Science Foundation (NSF) is an independent agency of the United States government Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group)The Independents were a group of ...

National Science Foundation
in 1972. He originally intended DCT for
image compression Image compression is a type of data compression In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image proc ...
. Ahmed developed a practical DCT algorithm with his PhD student T. Raj Natarajan and friend
K. R. Rao Kamisetty Ramamohan Rao was an Indian-American electrical engineer. He was a professor of Electrical Engineering at the University of Texas at Arlington (UT Arlington). Academically known as K. R. Rao, he is credited with the co-invention of ...
at the
University of Texas at Arlington The University of Texas at Arlington (UTA or UT Arlington) is a public research university in Arlington, Texas Arlington is a city in the U.S. state of Texas, located in Tarrant County. It forms part of the Mid-Cities region of the Dallas ...
in 1973, and they found that it was the most efficient algorithm for image compression. They presented their results in a January 1974 paper, titled ''Discrete Cosine Transform''. It described what is now called the type-II DCT (DCT-II), as well as the type-III inverse DCT (IDCT). It was a benchmark publication, and has been cited as a fundamental development in thousands of works since its publication. The basic research work and events that led to the development of the DCT were summarized in a later publication by Ahmed, "How I Came Up with the Discrete Cosine Transform". Since its introduction in 1974, there has been significant research on the DCT. In 1977, Wen-Hsiung Chen published a paper with C. Harrison Smith and Stanley C. Fralick presenting a fast DCT algorithm.Further developments include a 1978 paper by M.J. Narasimha and A.M. Peterson, and a 1984 paper by B.G. Lee. These research papers, along with the original 1974 Ahmed paper and the 1977 Chen paper, were cited by the
Joint Photographic Experts Group The Joint Photographic Experts Group (JPEG) is the joint committee between ISO The International Organization for Standardization (ISO ) is an international standard An international standard is a technical standard A technical standard is an ...
as the basis for
JPEG JPEG ( ) is a commonly used method of lossy compression In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

JPEG
's lossy image compression algorithm in 1992. In 1975, John A. Roese and Guner S. Robinson adapted the DCT for
inter-frame An inter frame is a frame in a video compression stream which is expressed in terms of one or more neighboring frames. The "inter" part of the term refers to the use of ''Inter frame prediction''. This kind of prediction tries to take advantage from ...
motion-compensated
video coding A video coding format (or sometimes video compression format) is a content representation format for storage or transmission of digital Digital usually refers to something using digits, particularly binary digits. Technology and computing Ha ...
. They experimented with the DCT and the
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in t ...
(FFT), developing inter-frame hybrid coders for both, and found that the DCT is the most efficient due to its reduced complexity, capable of compressing image data down to 0.25-
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
per
pixel In digital imaging Digital imaging or digital image acquisition is the creation of a representation of the visual characteristics of an object, such as a physical scene or the interior structure of an object. The term is often assumed to imp ...

pixel
for a videotelephone scene with image quality comparable to an intra-frame coder requiring 2-bit per pixel. In 1979, Anil K. Jain and Jaswant R. Jain further developed motion-compensated DCT video compression, also called block motion compensation. This led to Chen developing a practical video compression algorithm, called motion-compensated DCT or adaptive scene coding, in 1981. Motion-compensated DCT later became the standard coding technique for video compression from the late 1980s onwards. The integer DCT is used in
Advanced Video Coding Advanced Video Coding (AVC), also referred to as H.264 or MPEG-4 MPEG-4 is a method of defining compression of audio and visual (AV) digital data. It was introduced in late 1998 and designated a standard for a group of audio and video codin ...
(AVC), introduced in 2003, and
High Efficiency Video Coding High Efficiency Video Coding (HEVC), also known as H.265 and MPEG-H Part 2, is a video compression standard A video coding format (or sometimes video compression format) is a content representation format for storage or transmission of digita ...
(HEVC), introduced in 2013. The integer DCT is also used in the
High Efficiency Image Format High Efficiency Image File Format (HEIF) is a container format for storing individual images and image sequences. The standard covers multimedia files that can also include other media streams, such as timed text, audio and video. HEIF can sto ...
(HEIF), which uses a subset of the
HEVC High Efficiency Video Coding (HEVC), also known as H.265 and MPEG-H Part 2, is a video compression standard A video coding format (or sometimes video compression format) is a content representation format for storage or transmission of digita ...
video coding format for coding still images. A DCT variant, the
modified discrete cosine transform The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine In mathematics, the trigonom ...
(MDCT), was developed by John P. Princen, A.W. Johnson and Alan B. Bradley at the
University of Surrey The University of Surrey is a public research university A public university or public college is a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, higher (or Tertiary e ...
in 1987, following earlier work by Princen and Bradley in 1986. The MDCT is used in most modern audio compression formats, such as
Dolby Digital Dolby Digital, originally synonymous with Dolby AC-3, is the name for what has now become a family of audio compression technologies developed by Dolby Laboratories Dolby Laboratories, Inc. (often shortened to Dolby Labs and known simply a ...
(AC-3),
MP3 MP3 (formally MPEG-1 Audio Layer III or MPEG-2 Audio Layer III) is a coding format for digital audio Digital audio is a representation of sound recorded in, or converted into, Digital signal (signal processing), digital form. In digital a ...

MP3
(which uses a hybrid DCT-
FFT A fast Fourier transform (FFT) is an algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm proceeds by successive subtract ...
algorithm), Advanced Audio Coding (AAC), and
Vorbis Vorbis is a free and open-source software Free and open-source software (FOSS) is software Software is a collection of instructions that tell a computer A computer is a machine that can be programmed to carry out sequences of ari ...
(
Ogg Ogg is a free, open Open or OPEN may refer to: citizen * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies ...
). The
discrete sine transformIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
(DST) was derived from the DCT, by replacing the Neumann condition at ''x=0'' with a
Dirichlet conditionIn mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary differential equation, ordinary or a partial differential equation ...
. The DST was described in the 1974 DCT paper by Ahmed, Natarajan and Rao. A type-I DST (DST-I) was later described by Anil K. Jain in 1976, and a type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978. Nasir Ahmed also developed a lossless DCT algorithm with Giridhar Mandyam and Neeraj Magotra at the
University of New Mexico The University of New Mexico (UNM; es, Universidad de Nuevo México) is a public In public relations Public relations (PR) is the practice of managing and disseminating information from an individual or an organization An or ...

University of New Mexico
in 1995. This allows the DCT technique to be used for
lossless compression Lossless compression is a class of data compression In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image ...
of images. It is a modification of the original DCT algorithm, and incorporates elements of inverse DCT and
delta modulation A delta modulation (DM or Δ-modulation) is an and conversion technique used for transmission of voice information where quality is not of primary importance. DM is the simplest form of (DPCM) where the difference between successive samples is e ...
. It is a more effective lossless compression algorithm than
entropy coding In information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamenta ...
. Lossless DCT is also known as LDCT.


Applications

The DCT is the most widely used transformation technique in
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

signal processing
, and by far the most widely used linear transform in
data compression In signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electro ...
. Uncompressed
digital media Digital media means any communication media that operate with the use of any of various encoded machine-readable data Machine-readable data, or computer-readable data, is data Data (; ) are individual facts, statistics, or items of informa ...
as well as
lossless compression Lossless compression is a class of data compression In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image ...
had impractically high
memory Memory is the faculty of the brain A brain is an organ Organ may refer to: Biology * Organ (anatomy) An organ is a group of Tissue (biology), tissues with similar functions. Plant life and animal life rely on many organs that co-exis ...
and
bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
requirements, which was significantly reduced by the highly efficient DCT
lossy compression In information technology, lossy compression or irreversible compression is the class of data compression, data encoding methods that uses inexact approximations and partial data discarding to represent the content. These techniques are used to r ...
technique, capable of achieving
data compression ratio Data compression ratio, also known as compression power, is a measurement of the relative reduction in size of data representation produced by a data compression algorithm. It is typically expressed as the division of uncompressed size by compressed ...
s from 8:1 to 14:1 for near-studio-quality, up to 100:1 for acceptable-quality content. DCT compression standards are used in digital media technologies, such as
digital images Digital usually refers to something using digits, particularly binary digits. Technology and computing Hardware *Digital electronics Digital electronics is a field of electronics Electronics comprises the physics, engineering, technology a ...
,
digital photo Digital photography uses cameras A camera is an optical Optics is the branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the ...
s,
digital video Digital video is an electronic representation of moving visual images (video) in the form of encoded digital data. This is in contrast to analog video, which represents moving visual images in the form of analog signals. Digital video comprises ...
,
streaming media Streaming media is multimedia Multimedia is a form of communication that combines different such as , , , , or into a single interactive presentation, in contrast to traditional mass media which ...
,
digital television Digital television (DTV) is the transmission of television audiovisual Audiovisual (AV) is electronic media 200px, Graphical representations of electrical audio data. Electronic media uses either analog (red) or digital (blue) signal pr ...
,
streaming television Streaming television is the digital distribution Digital distribution (also referred to as content delivery, online distribution, or electronic software distribution (ESD), among others) is the delivery or distribution of digital media conte ...
,
video-on-demand Video on demand (VOD) is a media distribution system that allows users to access videos without a traditional video playback device and the constraints of a typical static broadcasting schedule. In the 20th century, broadcasting in the form of o ...
(VOD),
digital cinema Digital cinema refers to adoption of digital Digital usually refers to something using digits, particularly binary digits. Technology and computing Hardware *Digital electronics Digital electronics is a field of electronics Electronics c ...
,
high-definition video High-definition video (HD video) is video Video is an electronic Electronic may refer to: *Electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of ele ...
(HD video), and
high-definition television High-definition television (HD or HDTV) describes a television system providing a substantially higher image resolution Image resolution is the detail an holds. The term applies to s, film images, and other types of images. Higher resolution m ...
(HDTV). The DCT, and in particular the DCT-II, is often used in signal and image processing, especially for lossy compression, because it has a strong "energy compaction" property: in typical applications, most of the signal information tends to be concentrated in a few low-frequency components of the DCT. For strongly correlated
Markov process A Markov chain or Markov process is a stochastic model In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory tre ...
es, the DCT can approach the compaction efficiency of the Karhunen-Loève transform (which is optimal in the decorrelation sense). As explained below, this stems from the boundary conditions implicit in the cosine functions. DCTs are also widely employed in solving
partial differential equations In , a partial differential equation (PDE) is an equation which imposes relations between the various s of a . The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved ...
by
spectral methods Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the fast Fourier transform. The idea is to write the solution of t ...
, where the different variants of the DCT correspond to slightly different even/odd boundary conditions at the two ends of the array. DCTs are also closely related to
Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined several equivalent ways; in this article the polynomials are defined ...
, and fast DCT algorithms (below) are used in
Chebyshev approximation In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler function (mathematics), functions, and with Quantitative property, quantitatively characterization (ma ...
of arbitrary functions by series of Chebyshev polynomials, for example in
Clenshaw–Curtis quadrature Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the Integrand#Terminology and notation, integrand in terms of Chebyshev polynomials. Equivalently, they emp ...
. The DCT is the coding standard for
multimedia Multimedia is a form of communication that combines different such as , , , , or into a single interactive presentation, in contrast to traditional mass media which featured little to no interaction fr ...

multimedia
telecommunication Telecommunication is the transmission of information by various types of technologies over wire A wire is a single usually cylindrical A cylinder (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Gr ...
devices. It is widely used for
bit rate In telecommunications and computing, bit rate (bitrate or as a variable ''R'') is the number of bits that are conveyed or processed per unit of time. The bit rate is expressed in the unit Data rate units, bit per second unit (symbol: ''bit/s' ...
reduction, and reducing
network bandwidth In computing, bandwidth is the maximum rate of data transfer across a given path. Bandwidth may be characterized as network bandwidth, data bandwidth, or digital bandwidth. This definition of ''bandwidth'' is in contrast to the field of signal proc ...
usage. DCT compression significantly reduces the amount of memory and bandwidth required for digital signals.


General applications

The DCT is widely used in many applications, which include the following.


DCT visual media standards

The DCT-II, also known as simply the DCT, is the most important
image compression Image compression is a type of data compression In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image proc ...
technique. It is used in image compression standards such as
JPEG JPEG ( ) is a commonly used method of lossy compression In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

JPEG
, and
video compression In signal processing, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either Lossy compression, lossy or Lossless comp ...
standards such as
H.26x The Video Coding Experts Group or Visual Coding Experts Group (VCEG, also known as Question 6) is a working group of the (ITU-T) concerned with video coding standards. It is responsible for standardization of the "H.26x" line of video coding sta ...
,
MJPEG Motion JPEG (M-JPEG or MJPEG) is a video compression format A video coding format (or sometimes video compression format) is a content representation format for storage or transmission of digital video content (such as in a data file or bits ...
,
MPEG The Moving Picture Experts Group (MPEG) is an alliance of working groups established jointly by International Organization for Standardization, ISO and International Electrotechnical Commission, IEC that sets standards for media coding, includ ...

MPEG
, ,
Theora Theora is a free Free may refer to: Concept * Freedom, having the ability to act or change without constraint * Emancipate, to procure political rights, as for a disenfranchised group * Free will, control exercised by rational agents over ...
and
Daala Daala is a video coding format under development by the Xiph.Org Foundation under the lead of Timothy B. Terriberry mainly sponsored by the Mozilla Corporation. Like Theora and Opus codec, Opus, Daala is available free of any royalties and its re ...
. There, the two-dimensional DCT-II of N \times N blocks are computed and the results are Quantization (signal processing), quantized and Entropy encoding, entropy coded. In this case, N is typically 8 and the DCT-II formula is applied to each row and column of the block. The result is an 8 × 8 transform coefficient array in which the (0,0) element (top-left) is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies.
Advanced Video Coding Advanced Video Coding (AVC), also referred to as H.264 or MPEG-4 MPEG-4 is a method of defining compression of audio and visual (AV) digital data. It was introduced in late 1998 and designated a standard for a group of audio and video codin ...
(AVC) uses the integer DCT (IntDCT), an integer approximation of the DCT. It uses 4x4 and 8x8 integer DCT blocks.
High Efficiency Video Coding High Efficiency Video Coding (HEVC), also known as H.265 and MPEG-H Part 2, is a video compression standard A video coding format (or sometimes video compression format) is a content representation format for storage or transmission of digita ...
(HEVC) and the
High Efficiency Image Format High Efficiency Image File Format (HEIF) is a container format for storing individual images and image sequences. The standard covers multimedia files that can also include other media streams, such as timed text, audio and video. HEIF can sto ...
(HEIF) use varied integer DCT block sizes between 4x4 and 32x32
pixels In digital imaging, a pixel, pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in an all points addressable display device; so it is the smallest controllable elem ...

pixels
. , AVC is by far the most commonly used format for the recording, compression and distribution of video content, used by 91% of video developers, followed by HEVC which is used by 43% of developers.


Image formats


Video formats


MDCT audio standards


General audio


Speech coding


MD DCT

Multidimensional DCTs (MD DCTs) have several applications, mainly 3-D DCTs such as the 3-D DCT-II, which has several new applications like Hyperspectral Imaging coding systems, variable temporal length 3-D DCT coding, Video coding (postal market), video coding algorithms, adaptive video coding and 3-D Compression. 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 and cosine-modulated wavelet bases.


Digital signal processing

DCT plays a very important role in
digital signal processing Digital signal processing (DSP) is the use of digital processing Digital data, in information theory and information systems, is information represented as a string of discrete symbols each of which can take on one of only a finite number of ...
. By using the DCT, the signals can be compressed. DCT can be used in electrocardiography for the compression of ECG signals. DCT2 provides a better compression ratio than DCT. The DCT is widely implemented in digital signal processors (DSP), as well as digital signal processing software. Many companies have developed DSPs based on DCT technology. DCTs are widely used for applications such as encoding, decoding, video, audio, multiplexing, control signals, signaling, and analog-to-digital conversion. DCTs are also commonly used for
high-definition television High-definition television (HD or HDTV) describes a television system providing a substantially higher image resolution Image resolution is the detail an holds. The term applies to s, film images, and other types of images. Higher resolution m ...
(HDTV) encoder/decoder integrated circuit, chips.


Compression artifacts

A common issue with DCT compression in
digital media Digital media means any communication media that operate with the use of any of various encoded machine-readable data Machine-readable data, or computer-readable data, is data Data (; ) are individual facts, statistics, or items of informa ...
are blocky
compression artifacts A compression artifact (or artefact) is a noticeable distortion of media (including images, audio, and video) caused by the application of lossy compression. Lossy data compression In signal processing, data compression, source coding, o ...

compression artifacts
, caused by DCT blocks. The DCT algorithm can cause block-based artifacts when heavy compression is applied. Due to the DCT being used in the majority of digital image and video coding standards (such as the
JPEG JPEG ( ) is a commonly used method of lossy compression In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

JPEG
,
H.26x The Video Coding Experts Group or Visual Coding Experts Group (VCEG, also known as Question 6) is a working group of the (ITU-T) concerned with video coding standards. It is responsible for standardization of the "H.26x" line of video coding sta ...
and
MPEG The Moving Picture Experts Group (MPEG) is an alliance of working groups established jointly by International Organization for Standardization, ISO and International Electrotechnical Commission, IEC that sets standards for media coding, includ ...

MPEG
formats), DCT-based blocky compression artifacts are widespread in
digital media Digital media means any communication media that operate with the use of any of various encoded machine-readable data Machine-readable data, or computer-readable data, is data Data (; ) are individual facts, statistics, or items of informa ...
. In a DCT algorithm, an image (or frame in an image sequence) is divided into square blocks which are processed independently from each other, then the DCT of these blocks is taken, and the resulting DCT coefficients are Quantization (signal processing), quantized. This process can cause blocking artifacts, primarily at high
data compression ratio Data compression ratio, also known as compression power, is a measurement of the relative reduction in size of data representation produced by a data compression algorithm. It is typically expressed as the division of uncompressed size by compressed ...
s. This can also cause the "mosquito noise" effect, commonly found in
digital video Digital video is an electronic representation of moving visual images (video) in the form of encoded digital data. This is in contrast to analog video, which represents moving visual images in the form of analog signals. Digital video comprises ...
(such as the MPEG formats). DCT blocks are often used in glitch art. The artist Rosa Menkman makes use of DCT-based compression artifacts in her glitch art, particularly the DCT blocks found in most
digital media Digital media means any communication media that operate with the use of any of various encoded machine-readable data Machine-readable data, or computer-readable data, is data Data (; ) are individual facts, statistics, or items of informa ...
formats such as
JPEG JPEG ( ) is a commonly used method of lossy compression In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

JPEG
digital images Digital usually refers to something using digits, particularly binary digits. Technology and computing Hardware *Digital electronics Digital electronics is a field of electronics Electronics comprises the physics, engineering, technology a ...
and
MP3 MP3 (formally MPEG-1 Audio Layer III or MPEG-2 Audio Layer III) is a coding format for digital audio Digital audio is a representation of sound recorded in, or converted into, Digital signal (signal processing), digital form. In digital a ...

MP3
digital audio Digital audio is a representation of sound recorded in, or converted into, Digital signal (signal processing), digital form. In digital audio, the sound wave of the audio signal is typically encoded as numerical Sampling (signal processing), s ...
. Another example is ''Jpegs'' by German photographer Thomas Ruff, which uses intentional
JPEG JPEG ( ) is a commonly used method of lossy compression In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

JPEG
artifacts as the basis of the picture's style.


Informal overview

Like any Fourier-related transform, discrete cosine transforms (DCTs) express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes. Like the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
(DFT), a DCT operates on a function at a finite 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 (in the form of complex exponentials). However, this visible difference is merely a consequence of a deeper distinction: a DCT implies different
boundary condition Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film Mathematics and physics * Boundary (top ...
s from the DFT or other related transforms. The Fourier-related transforms that operate on a function over a finite domain of a function, domain, such as the DFT or DCT or a Fourier series, can be thought of as implicitly defining an ''extension'' of that function outside the domain. That is, once you write a function f(x) as a sum of sinusoids, you can evaluate that sum at any x, even for x where the original f(x) was not specified. The DFT, like the Fourier series, implies a periodic function, periodic extension of the original function. A DCT, like a Sine and cosine transforms, 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 (i.e. the min-''n'' and max-''n'' boundaries in the definitions below, respectively). Second, one has to specify around ''what point'' the function is even or odd. In particular, consider a sequence ''abcd'' of four equally spaced data points, and say that we specify an even ''left'' boundary. There are two sensible possibilities: either the data are even about the sample ''a'', in which case the even extension is ''dcbabcd'', or the data are even about the point ''halfway'' between ''a'' and the previous point, in which case the even extension is ''dcbaabcd'' (''a'' is repeated). These choices lead to all the standard variations of DCTs and also
discrete sine transformIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s (DSTs). Each boundary can be either even or odd (2 choices per boundary) and can be symmetric about a data point or the point halfway between two data points (2 choices per boundary), for a total of 2 × 2 × 2 × 2 = 16 possibilities. Half of these possibilities, those where the ''left'' boundary is even, correspond to the 8 types of DCT; the other half are the 8 types of DST. These different boundary conditions strongly affect the applications of the transform and lead to uniquely useful properties for the various DCT types. Most directly, when using Fourier-related transforms to solve partial differential equations by
spectral method Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equation In mathematics, a differential equation is an equation that relates one or more function (mathema ...
s, the boundary conditions are directly specified as a part of the problem being solved. Or, for the Modified discrete cosine transform, MDCT (based on the type-IV DCT), the boundary conditions are intimately involved in the MDCT's critical property of time-domain aliasing cancellation. In a more subtle fashion, the boundary conditions are responsible for the "energy compactification" properties that make DCTs useful for image and audio compression, because the boundaries affect the rate of convergence of any Fourier-like series. In particular, it is well known that any Classification of discontinuities, discontinuities in a function reduce the rate of convergence of the Fourier series, so that more sinusoids are needed to represent the function with a given accuracy. The same principle governs the usefulness of the DFT and other transforms for signal compression; the smoother a function is, the fewer terms in its DFT or DCT are required to represent it accurately, and the more it can be compressed. (Here, we think of the DFT or DCT as approximations for the Fourier series or cosine series of a function, respectively, in order to talk about its "smoothness".) However, the implicit periodicity of the DFT means that discontinuities usually occur at the boundaries: any random segment of a signal is unlikely to have the same value at both the left and right boundaries. (A similar problem arises for the DST, in which the odd left boundary condition implies a discontinuity for any function that does not happen to be zero at that boundary.) In contrast, a DCT where ''both'' boundaries are even ''always'' yields a continuous extension at the boundaries (although the slope is generally discontinuous). This is why DCTs, and in particular DCTs of types I, II, V, and VI (the types that have two even boundaries) generally perform better for signal compression than DFTs and DSTs. In practice, a type-II DCT is usually preferred for such applications, in part for reasons of computational convenience.


Formal definition

Formally, the discrete cosine transform is a linear, invertible function (mathematics), function f : \R^ \to \R^ (where \R denotes the set of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s), or equivalently an invertible × square matrix. There are several variants of the DCT with slightly modified definitions. The real numbers ~ x_0,\ \ldots\ x_ ~ are transformed into the real numbers X_0,\, \ldots,\, X_ according to one of the formulas:


DCT-I

:X_k = \frac (x_0 + (-1)^k x_) + \sum_^ x_n \cos \left[\, \frac \, n \, k \,\right] \qquad \text ~ k = 0,\ \ldots\ N-1 ~. Some authors further multiply the x_0 and x_ terms by \sqrt\, , and correspondingly multiply the X_0 and X_ terms by 1/\sqrt \,, which makes the DCT-I matrix orthogonal matrix, orthogonal. If one further multiplies by an overall scale factor of \sqrt , but breaks the direct correspondence with a real-even Discrete Fourier transform, DFT. The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a Discrete Fourier transform, DFT of 2(N-1) real numbers with even symmetry. For example, a DCT-I of N = 5 real numbers a\ b\ c\ d\ e is exactly equivalent to a DFT of eight real numbers (even symmetry), divided by two. (In contrast, DCT types II-IV involve a half-sample shift in the equivalent DFT.) Note, however, that the DCT-I is not defined for N less than 2. (All other DCT types are defined for any positive N . Thus, the DCT-I corresponds to the boundary conditions: x_n is even around n = 0 and even around n = N - 1 ; similarly for X_k .


DCT-II

:X_k = \sum_^ x_n \cos \left[\, \tfrac \left( n + \frac \right) k \, \right] \qquad \text ~ k = 0,\ \dots\ N-1 ~. The DCT-II is probably the most commonly used form, and is often simply referred to as "the DCT". This transform is exactly equivalent (up to an overall scale factor of 2) to a Discrete Fourier transform, DFT of 4N real inputs of even symmetry where the even-indexed elements are zero. That is, it is half of the Discrete Fourier transform, DFT of the 4N inputs y_n , where y_ = 0 , y_ = x_n for 0 \leq n < N , y_ = 0 , and y_ = y_n for 0 < n < 2N . DCT-II transformation is also possible using 2 signal followed by a multiplication by half shift. This is demonstrated by John Makhoul, Makhoul. Some authors further multiply the X_0 term by 1/\sqrt \, . and multiply the resulting matrix by an overall scale factor of \sqrt (see below for the corresponding change in DCT-III). This makes the DCT-II matrix orthogonal matrix, orthogonal, but breaks the direct correspondence with a real-even Discrete Fourier transform, DFT of half-shifted input. This is the normalization used by Matlab, for example, see. In many applications, such as
JPEG JPEG ( ) is a commonly used method of lossy compression In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

JPEG
, the scaling is arbitrary because scale factors can be combined with a subsequent computational step (e.g. the Quantization (signal processing), quantization step in JPEG), and a scaling can be chosen that allows the DCT to be computed with fewer multiplications. The DCT-II implies the boundary conditions: x_n is even around n = -1/2 and even around n = N - 1/2 \,; X_k is even around k = 0 and odd around k = N .


DCT-III

: X_k = / x_0 + \sum_^ x_n \cos \left[\, \tfrac \, n \left( k + / \right) \,\right] \qquad \text ~ k = 0,\ \dots\ N-1 ~. Because it is the inverse of DCT-II (up to a scale factor, see below), this form is sometimes simply referred to as "the inverse DCT" ("IDCT"). Some authors divide the ~ x_0 ~ term by \sqrt instead of by 2 (resulting in an overall x_0/\sqrt term) and multiply the resulting matrix by an overall scale factor of \sqrt (see above for the corresponding change in DCT-II), so that the DCT-II and DCT-III are transposes of one another. This makes the DCT-III matrix orthogonal matrix, orthogonal, but breaks the direct correspondence with a real-even Discrete Fourier transform, DFT of half-shifted output. The DCT-III implies the boundary conditions: x_n is even around n = 0 and odd around n = N ; X_k is even around k = -/ and even around k = N - /.


DCT-IV

: X_k = \sum_^ x_n \cos \left[\, \tfrac \, \left( n + / \right) \left( k + / \right) \,\right] \qquad \text ~ k = 0,\ \ldots\ N-1 ~. The DCT-IV matrix becomes orthogonal matrix, orthogonal (and thus, being clearly symmetric, its own inverse) if one further multiplies by an overall scale factor of \sqrt . A variant of the DCT-IV, where data from different transforms are ''overlapped'', is called the
modified discrete cosine transform The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine In mathematics, the trigonom ...
(MDCT). The DCT-IV implies the boundary conditions: x_n is even around n = -/ and odd around n = N - / ; similarly for X_k .


DCT V-VIII

DCTs of types I–IV treat both boundaries consistently regarding the point of symmetry: they are even/odd around either a data point for both boundaries or halfway between two data points for both boundaries. By contrast, DCTs of types V-VIII imply boundaries that are even/odd around a data point for one boundary and halfway between two data points for the other boundary. In other words, DCT types I–IV are equivalent to real-even Discrete Fourier transform, DFTs of even order (regardless of whether N is even or odd), since the corresponding DFT is of length 2(N-1) (for DCT-I) or 4 N (for DCT-II & III) or 8 N (for DCT-IV). The four additional types of discrete cosine transform correspond essentially to real-even DFTs of logically odd order, which have factors of N \pm / in the denominators of the cosine arguments. However, these variants seem to be rarely used in practice. One reason, perhaps, is that Fast Fourier transform, FFT algorithms for odd-length DFTs are generally more complicated than Fast Fourier transform, FFT algorithms for even-length DFTs (e.g. the simplest radix-2 algorithms are only for even lengths), and this increased intricacy carries over to the DCTs as described below. (The trivial real-even array, a length-one DFT (odd length) of a single number  , corresponds to a DCT-V of length N = 1 .)


Inverse transforms

Using the normalization conventions above, the inverse of DCT-I is DCT-I multiplied by 2/(''N'' − 1). The inverse of DCT-IV is DCT-IV multiplied by 2/''N''. The inverse of DCT-II is DCT-III multiplied by 2/''N'' and vice versa. Like for the discrete Fourier transform, DFT, the normalization factor in front of these transform definitions is merely a convention and differs between treatments. For example, some authors multiply the transforms by \sqrt so that the inverse does not require any additional multiplicative factor. Combined with appropriate factors of (see above), this can be used to make the transform matrix orthogonal matrix, orthogonal.


Multidimensional DCTs

Multidimensional variants of the various DCT types follow straightforwardly from the one-dimensional definitions: they are simply a separable product (equivalently, a composition) of DCTs along each dimension.


M-D DCT-II

For example, a two-dimensional DCT-II of an image or a matrix is simply the one-dimensional DCT-II, from above, performed along the rows and then along the columns (or vice versa). That is, the 2D DCT-II is given by the formula (omitting normalization and other scale factors, as above): : \begin X_ &= \sum_^ \left( \sum_^ x_ \cos \left[\frac \left(n_2+\frac\right) k_2 \right]\right) \cos \left[\frac \left(n_1+\frac\right) k_1 \right]\\ &= \sum_^ \sum_^ x_ \cos \left[\frac \left(n_1+\frac\right) k_1 \right] \cos \left[\frac \left(n_2+\frac\right) k_2 \right] . \end :The inverse of a multi-dimensional DCT is just a separable product of the inverses of the corresponding one-dimensional DCTs (see above), e.g. the one-dimensional inverses applied along one dimension at a time in a row-column algorithm. The ''3-D DCT-II'' is only the extension of ''2-D DCT-II'' in three dimensional space and mathematically can be calculated by the formula : X_ = \sum_^ \sum_^ \sum_^ x_ \cos \left[\frac \left(n_1+\frac\right) k_1 \right] \cos \left[\frac \left(n_2+\frac\right) k_2 \right] \cos \left[\frac \left(n_3+\frac\right) k_3 \right],\quad \text k_i = 0,1,2,\dots,N_i-1. The inverse of 3-D DCT-II is 3-D DCT-III and can be computed from the formula given by : x_ = \sum_^ \sum_^ \sum_^ X_ \cos \left[\frac \left(n_1+\frac\right) k_1 \right] \cos \left[\frac \left(n_2+\frac\right) k_2 \right] \cos \left[\frac \left(n_3+\frac\right) k_3 \right],\quad \text n_i=0,1,2,\dots,N_i-1. Technically, computing a two-, three- (or -multi) dimensional DCT by sequences of one-dimensional DCTs along each dimension is known as a ''row-column'' algorithm. As with fast Fourier transform#Multidimensional FFTs, multidimensional FFT algorithms, however, there exist other methods to compute the same thing while performing the computations in a different order (i.e. interleaving/combining the algorithms for the different dimensions). Owing to the rapid growth in the applications based on the 3-D DCT, several fast algorithms are developed for the computation of 3-D DCT-II. Vector-Radix algorithms are applied for computing M-D DCT to reduce the computational complexity and to increase the computational speed. To compute 3-D DCT-II efficiently, a fast algorithm, Vector-Radix Decimation in Frequency (VR DIF) algorithm was developed.


3-D DCT-II VR DIF

In order to apply the VR DIF algorithm the input data is to be formulated and rearranged as follows. The transform size ''N × N × N'' is assumed to be 2. : \begin\tilde(n_1,n_2,n_3) =x(2n_1,2n_2,2n_3)\\ \tilde(n_1,n_2,N-n_3-1)=x(2n_1,2n_2,2n_3+1)\\ \tilde(n_1,N-n_2-1,n_3)=x(2n_1,2n_2+1,2n_3)\\ \tilde(n_1,N-n_2-1,N-n_3-1)=x(2n_1,2n_2+1,2n_3+1)\\ \tilde(N-n_1-1,n_2,n_3)=x(2n_1+1,2n_2,2n_3)\\ \tilde(N-n_1-1,n_2,N-n_3-1)=x(2n_1+1,2n_2,2n_3+1)\\ \tilde(N-n_1-1,N-n_2-1,n_3)=x(2n_1+1,2n_2+1,2n_3)\\ \tilde(N-n_1-1,N-n_2-1,N-n_3-1)=x(2n_1+1,2n_2+1,2n_3+1)\\ \end :where 0\leq n_1,n_2,n_3 \leq \frac -1 The figure to the adjacent shows the four stages that are involved in calculating 3-D DCT-II using VR DIF algorithm. The first stage is the 3-D reordering using the index mapping illustrated by the above equations. The second stage is the butterfly calculation. Each butterfly calculates eight points together as shown in the figure just below, where c(\varphi_i)=\cos(\varphi_i). The original 3-D DCT-II now can be written as :X(k_1,k_2,k_3)=\sum_^\sum_^\sum_^\tilde(n_1,n_2,n_3) \cos(\varphi k_1)\cos(\varphi k_2)\cos(\varphi k_3) where \varphi_i= \frac(4N_i+1),\text i= 1,2,3. If the even and the odd parts of k_1,k_2 and k_3 and are considered, the general formula for the calculation of the 3-D DCT-II can be expressed as :X(k_1,k_2,k_3)=\sum_^\sum_^\sum_^\tilde_(n_1,n_2,n_3) \cos(\varphi (2k_1+i)\cos(\varphi (2k_2+j) \cos(\varphi (2k_3+l)) where : \tilde_(n_1,n_2,n_3)=\tilde(n_1,n_2,n_3)+(-1)^l\tilde\left(n_1,n_2,n_3+\frac\right) : +(-1)^j\tilde\left(n_1,n_2+\frac,n_3\right)+(-1)^\tilde\left(n_1,n_2+\frac,n_3+\frac\right) : +(-1)^i\tilde\left(n_1+\frac,n_2,n_3\right)+(-1)^\tilde\left(n_1+\frac+\frac,n_2,n_3\right) : +(-1)^\tilde\left(n_1+\frac,n_2,n_3+\frac\right) : +(-1)^\tilde\left(n_1+\frac,n_2+\frac,n_3+\frac\right) \text i,j,l= 0 \text 1.


= Arithmetic complexity

= The whole 3-D DCT calculation needs ~ [\log_2 N] ~ stages, and each stage involves ~ \tfrac\ N^3 ~ butterflies. The whole 3-D DCT requires ~ \left[ \tfrac\ N^3 \log_2 N \right] ~ butterflies to be computed. Each butterfly requires seven real multiplications (including trivial multiplications) and 24 real additions (including trivial additions). Therefore, the total number of real multiplications needed for this stage is ~ \left[ \tfrac\ N^3\ \log_2 N \right] ~, and the total number of real additions i.e. including the post-additions (recursive additions) which can be calculated directly after the butterfly stage or after the bit-reverse stage are given by ~ \underbrace_\text+\underbrace_\text = \left[\fracN^3 \log_2N-3N^3+3N^2\right] ~. The conventional method to calculate MD-DCT-II is using a Row-Column-Frame (RCF) approach which is computationally complex and less productive on most advanced recent hardware platforms. The number of multiplications required to compute VR DIF Algorithm when compared to RCF algorithm are quite a few in number. The number of Multiplications and additions involved in RCF approach are given by ~\left[\fracN^3 \log_2 N \right]~ and ~ \left[\fracN^3 \log_2 N - 3N^3 + 3N^2 \right] ~, respectively. From Table 1, it can be seen that the total number of multiplications associated with the 3-D DCT VR algorithm is less than that associated with the RCF approach by more than 40%. In addition, the RCF approach involves matrix transpose and more indexing and data swapping than the new VR algorithm. This makes the 3-D DCT VR algorithm more efficient and better suited for 3-D applications that involve the 3-D DCT-II such as video compression and other 3-D image processing applications. The main consideration in choosing a fast algorithm is to avoid computational and structural complexities. As the technology of computers and DSPs advances, the execution time of arithmetic operations (multiplications and additions) is becoming very fast, and regular computational structure becomes the most important factor. Therefore, although the above proposed 3-D VR algorithm does not achieve the theoretical lower bound on the number of multiplications, it has a simpler computational structure as compared to other 3-D DCT algorithms. It can be implemented in place using a single butterfly and possesses the properties of the Cooley–Tukey FFT algorithm in 3-D. Hence, the 3-D VR presents a good choice for reducing arithmetic operations in the calculation of the 3-D DCT-II, while keeping the simple structure that characterize butterfly-style Cooley–Tukey FFT algorithms. The image to the right shows a combination of horizontal and vertical frequencies for an (~ N_1 = N_2 = 8 ~) two-dimensional DCT. Each step from left to right and top to bottom is an increase in frequency by 1/2 cycle. For example, moving right one from the top-left square yields a half-cycle increase in the horizontal frequency. Another move to the right yields two half-cycles. A move down yields two half-cycles horizontally and a half-cycle vertically. The source data is transformed to a linear combination of these 64 frequency squares.


MD-DCT-IV

The M-D DCT-IV is just an extension of 1-D DCT-IV on to  dimensional domain. The 2-D DCT-IV of a matrix or an image is given by : X_ = \sum_^ \; \sum_^ \ x_ \cos\left(\ \frac \ \right) \cos\left(\ \frac \ \right) ~, : for ~~ k = 0,\ 1,\ 2\ \ldots\ N-1 ~~ and ~~ \ell= 0,\ 1,\ 2,\ \ldots\ M-1 ~. We can compute the MD DCT-IV using the regular row-column method or we can use the polynomial transform method for the fast and efficient computation. The main idea of this algorithm is to use the Polynomial Transform to convert the multidimensional DCT into a series of 1-D DCTs directly. MD DCT-IV also has several applications in various fields.


Computation

Although the direct application of these formulas would require ~ \mathcal(N^2) ~ operations, it is possible to compute the same thing with only ~ \mathcal(N \log N ) ~ complexity by factorizing the computation similarly to the
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in t ...
(FFT). One can also compute DCTs via FFTs combined with ~\mathcal(N)~ pre- and post-processing steps. In general, ~\mathcal(N \log N )~ methods to compute DCTs are known as fast cosine transform (FCT) algorithms. The most efficient algorithms, in principle, are usually those that are specialized directly for the DCT, as opposed to using an ordinary FFT plus ~ \mathcal(N) ~ extra operations (see below for an exception). However, even "specialized" DCT algorithms (including all of those that achieve the lowest known arithmetic counts, at least for power of two, power-of-two sizes) are typically closely related to FFT algorithms – since DCTs are essentially DFTs of real-even data, one can design a fast DCT algorithm by taking an FFT and eliminating the redundant operations due to this symmetry. This can even be done automatically . Algorithms based on the Cooley–Tukey FFT algorithm are most common, but any other FFT algorithm is also applicable. For example, the Winograd FFT algorithm leads to minimal-multiplication algorithms for the DFT, albeit generally at the cost of more additions, and a similar algorithm was proposed by for the DCT. Because the algorithms for DFTs, DCTs, and similar transforms are all so closely related, any improvement in algorithms for one transform will theoretically lead to immediate gains for the other transforms as well . While DCT algorithms that employ an unmodified FFT often have some theoretical overhead compared to the best specialized DCT algorithms, the former also have a distinct advantage: Highly optimized FFT programs are widely available. Thus, in practice, it is often easier to obtain high performance for general lengths with FFT-based algorithms. Specialized DCT algorithms, on the other hand, see widespread use for transforms of small, fixed sizes such as the DCT-II used in
JPEG JPEG ( ) is a commonly used method of lossy compression In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

JPEG
compression, or the small DCTs (or MDCTs) typically used in audio compression. (Reduced code size may also be a reason to use a specialized DCT for embedded-device applications.) In fact, even the DCT algorithms using an ordinary FFT are sometimes equivalent to pruning the redundant operations from a larger FFT of real-symmetric data, and they can even be optimal from the perspective of arithmetic counts. For example, a type-II DCT is equivalent to a DFT of size ~ 4N ~ with real-even symmetry whose even-indexed elements are zero. One of the most common methods for computing this via an FFT (e.g. the method used in FFTPACK and FFTW) was described by and , and this method in hindsight can be seen as one step of a radix-4 decimation-in-time Cooley–Tukey algorithm applied to the "logical" real-even DFT corresponding to the DCT-II. Because the even-indexed elements are zero, this radix-4 step is exactly the same as a split-radix step. If the subsequent size ~ N ~ real-data FFT is also performed by a real-data split-radix FFT algorithm, split-radix algorithm (as in ), then the resulting algorithm actually matches what was long the lowest published arithmetic count for the power-of-two DCT-II (~ 2 N \log_2 N - N + 2 ~ real-arithmetic operations). A recent reduction in the operation count to ~ \tfrac N \log_2 N + \mathcal(N) also uses a real-data FFT. So, there is nothing intrinsically bad about computing the DCT via an FFT from an arithmetic perspective – it is sometimes merely a question of whether the corresponding FFT algorithm is optimal. (As a practical matter, the function-call overhead in invoking a separate FFT routine might be significant for small ~ N ~, but this is an implementation rather than an algorithmic question since it can be solved by unrolling or inlining.)


Example of IDCT

Consider this 8x8 grayscale image of capital letter A. Each basis function is multiplied by its coefficient and then this product is added to the final image.


See also

* Discrete wavelet transform * JPEG#Discrete cosine transform, JPEGDiscretecosinetransformContains a potentially easier to understand example of DCT transformation * List of Fourier-related transforms * Modified discrete cosine transform


Notes


References


Further reading

* * * * * * * * * * * * * * *


External links

* Syed Ali Khayam
The Discrete Cosine Transform (DCT): Theory and Application


* Matteo Frigo and Steven G. Johnson: ''FFTW'', http://www.fftw.org/. A free (GNU General Public License, GPL) C library that can compute fast DCTs (types I-IV) in one or more dimensions, of arbitrary size. * Takuya Ooura: General Purpose FFT Package, http://www.kurims.kyoto-u.ac.jp/~ooura/fft.html. Free C & FORTRAN libraries for computing fast DCTs (types II–III) in one, two or three dimensions, power of 2 sizes. * Tim Kientzle: Fast algorithms for computing the 8-point DCT and IDCT, http://drdobbs.com/parallel/184410889.
LTFAT
is a free Matlab/Octave toolbox with interfaces to the FFTW implementation of the DCTs and DSTs of type I-IV. {{DEFAULTSORT:Discrete Cosine Transform Digital signal processing Fourier analysis Discrete transforms Data compression Image compression Indian inventions H.26x JPEG Lossy compression algorithms Video compression