A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images (such as JPEG and HEIF), digital video (such as MPEG and H.26x), digital audio (such as Dolby Digital, MP3 and AAC), digital television (such as SDTV, HDTV and VOD), digital radio (such as AAC+ and DAB+), and speech coding (such as AACLD, Siren and Opus). DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.
A DCT is a Fourierrelated transform similar to the discrete Fourier transform (DFT), but using only real numbers. 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 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 typeII DCT, which is often called simply the DCT. This was the original DCT as first proposed by Ahmed. Its inverse, the typeIII DCT, is correspondingly often called simply the inverse DCT or the IDCT. Two related transforms are the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions, and the modified discrete cosine transform (MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT to multidimensional signals. A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT. One of these is the integer DCT (IntDCT),^{[1]} an integer approximation of the standard DCT,^{[2]}^{: ix, xiii, 1, 141–304 } used in several ISO/IEC and ITUT international standards.^{[1]}^{[2]}
DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks.^{[3]} DCT blocks sizes including 8x8 pixels for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels.^{[1]}^{[4]} The DCT has a strong energy compaction property,^{[5]}^{[6]} capable of achieving high quality at high data compression ratios.^{[7]}^{[8]} However, blocky compression artifacts can appear when heavy DCT compression is applied.
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Discrete Cosine Transform( DCT) Simple Explanation and Numerical DSP DCT

2D Discrete Cosine Transform(DCT)
Transcription
History
The DCT was first conceived by Nasir Ahmed, T. Natarajan and K. R. Rao while working at Kansas State University. The concept was proposed to the National Science Foundation in 1972. The DCT was originally intended for image compression.^{[9]}^{[1]} Ahmed developed a practical DCT algorithm with his PhD students T. Raj Natarajan, Wills Dietrich, and Jeremy Fries, and his friend Dr. K. R. Rao at the University of Texas at Arlington in 1973.^{[9]} They presented their results in a January 1974 paper, titled Discrete Cosine Transform.^{[5]}^{[6]}^{[10]} It described what is now called the typeII DCT (DCTII),^{[2]}^{: 51 } as well as the typeIII inverse DCT (IDCT).^{[5]}
Since its introduction in 1974, there has been significant research on the DCT.^{[10]} In 1977, WenHsiung Chen published a paper with C. Harrison Smith and Stanley C. Fralick presenting a fast DCT algorithm.^{[11]}^{[10]} Further developments include a 1978 paper by M. J. Narasimha and A. M. Peterson, and a 1984 paper by B. G. Lee.^{[10]} These research papers, along with the original 1974 Ahmed paper and the 1977 Chen paper, were cited by the Joint Photographic Experts Group as the basis for JPEG's lossy image compression algorithm in 1992.^{[10]}^{[12]}
The discrete sine transform (DST) was derived from the DCT, by replacing the Neumann condition at x=0 with a Dirichlet condition.^{[2]}^{: 3536 } The DST was described in the 1974 DCT paper by Ahmed, Natarajan and Rao.^{[5]} A typeI DST (DSTI) was later described by Anil K. Jain in 1976, and a typeII DST (DSTII) was then described by H.B. Kekra and J.K. Solanka in 1978.^{[13]}
In 1975, John A. Roese and Guner S. Robinson adapted the DCT for interframe motioncompensated video coding. They experimented with the DCT and the fast Fourier transform (FFT), developing interframe 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 0.25bit per pixel for a videotelephone scene with image quality comparable to an intraframe coder requiring 2bit per pixel.^{[14]}^{[15]} In 1979, Anil K. Jain and Jaswant R. Jain further developed motioncompensated DCT video compression,^{[16]}^{[17]} also called block motion compensation.^{[17]} This led to Chen developing a practical video compression algorithm, called motioncompensated DCT or adaptive scene coding, in 1981.^{[17]} Motioncompensated DCT later became the standard coding technique for video compression from the late 1980s onwards.^{[18]}^{[19]}
A DCT variant, the modified discrete cosine transform (MDCT), was developed by John P. Princen, A.W. Johnson and Alan B. Bradley at the University of Surrey in 1987,^{[20]} following earlier work by Princen and Bradley in 1986.^{[21]} The MDCT is used in most modern audio compression formats, such as Dolby Digital (AC3),^{[22]}^{[23]} MP3 (which uses a hybrid DCTFFT algorithm),^{[24]} Advanced Audio Coding (AAC),^{[25]} and Vorbis (Ogg).^{[26]}
Nasir Ahmed also developed a lossless DCT algorithm with Giridhar Mandyam and Neeraj Magotra at the University of New Mexico in 1995. This allows the DCT technique to be used for lossless compression of images. It is a modification of the original DCT algorithm, and incorporates elements of inverse DCT and delta modulation. It is a more effective lossless compression algorithm than entropy coding.^{[27]} Lossless DCT is also known as LDCT.^{[28]}
Applications
The DCT is the most widely used transformation technique in signal processing,^{[29]} and by far the most widely used linear transform in data compression.^{[30]} Uncompressed digital media as well as lossless compression have high memory and bandwidth requirements, which is significantly reduced by the DCT lossy compression technique,^{[7]}^{[8]} capable of achieving data compression ratios from 8:1 to 14:1 for nearstudioquality,^{[7]} up to 100:1 for acceptablequality content.^{[8]} DCT compression standards are used in digital media technologies, such as digital images, digital photos,^{[31]}^{[32]} digital video,^{[18]}^{[33]} streaming media,^{[34]} digital television, streaming television, video on demand (VOD),^{[8]} digital cinema,^{[22]} highdefinition video (HD video), and highdefinition television (HDTV).^{[7]}^{[35]}
The DCT, and in particular the DCTII, is often used in signal and image processing, especially for lossy compression, because it has a strong energy compaction property.^{[5]}^{[6]} In typical applications, most of the signal information tends to be concentrated in a few lowfrequency components of the DCT. For strongly correlated Markov processes, the DCT can approach the compaction efficiency of the KarhunenLoève transform (which is optimal in the decorrelation sense). As explained below, this stems from the boundary conditions implicit in the cosine functions.
DCTs are widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even and odd boundary conditions at the two ends of the array.
DCTs are closely related to Chebyshev polynomials, and fast DCT algorithms (below) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials, for example in Clenshaw–Curtis quadrature.
General applications
The DCT is widely used in many applications, which include the following.
 Audio signal processing — audio coding, audio data compression (lossy and lossless),^{[36]} surround sound,^{[22]} acoustic echo and feedback cancellation, phoneme recognition, timedomain aliasing cancellation (TDAC)^{[37]}
 Digital audio^{[1]}
 Digital radio — Digital Audio Broadcasting (DAB+),^{[38]} HD Radio^{[39]}
 Speech processing — speech coding^{[40]}^{[41]} speech recognition, voice activity detection (VAD)^{[37]}
 Digital telephony — voiceoverIP (VoIP),^{[40]} mobile telephony, video telephony,^{[41]} teleconferencing, videoconferencing^{[1]}
 Biometrics — fingerprint orientation, facial recognition systems, biometric watermarking, fingerprintbased biometric watermarking, palm print identification/recognition^{[37]}
 Face detection — facial recognition^{[37]}
 Computers and the Internet — the World Wide Web, social media,^{[31]}^{[32]} Internet video^{[42]}
 Network bandwidth usage reducation^{[1]}
 Consumer electronics^{[37]} — multimedia systems,^{[1]} multimedia telecommunication devices,^{[1]} consumer devices^{[42]}
 Cryptography — encryption, steganography, copyright protection^{[37]}
 Data compression — transform coding, lossy compression, lossless compression^{[36]}
 Encoding operations — quantization, perceptual weighting, entropy encoding, variable encoding^{[1]}
 Digital media^{[34]} — digital distribution^{[43]}
 Streaming media^{[34]} — streaming audio, streaming video, streaming television, videoondemand (VOD)^{[8]}
 Forgery detection^{[37]}
 Geophysical transient electromagnetics (transient EM)^{[37]}
 Images — artist identification,^{[37]} focus and blurriness measure,^{[37]} feature extraction^{[37]}
 Color formatting — formatting luminance and color differences, color formats (such as YUV444 and YUV411), decoding operations such as the inverse operation between display color formats (YIQ, YUV, RGB)^{[1]}
 Digital imaging — digital images, digital cameras, digital photography,^{[31]}^{[32]} highdynamicrange imaging (HDR imaging)^{[44]}
 Image compression^{[37]}^{[45]} — image file formats,^{[46]} multiview image compression, progressive image transmission^{[37]}
 Image processing — digital image processing,^{[1]} image analysis, contentbased image retrieval, corner detection, directional blockwise image representation, edge detection, image enhancement, image fusion, image segmentation, interpolation, image noise level estimation, mirroring, rotation, justnoticeable distortion (JND) profile, spatiotemporal masking effects, foveated imaging^{[37]}
 Image quality assessment — DCTbased quality degradation metric (DCT QM)^{[37]}
 Image reconstruction — directional textures auto inspection, image restoration, inpainting, visual recovery^{[37]}
 Medical technology
 Electrocardiography (ECG) — vectorcardiography (VCG)^{[37]}
 Medical imaging — medical image compression, image fusion, watermarking, brain tumor compression classification^{[37]}
 Pattern recognition^{[37]}
 Region of interest (ROI) extraction^{[37]}
 Signal processing — digital signal processing, digital signal processors (DSP), DSP software, multiplexing, signaling, control signals, analogtodigital conversion (ADC),^{[1]} compressive sampling, DCT pyramid error concealment, downsampling, upsampling, signaltonoise ratio (SNR) estimation, transmux, Wiener filter^{[37]}
 Complex cepstrum feature analysis^{[37]}
 DCT filtering^{[37]}
 Surveillance^{[37]}
 Vehicular black box camera^{[37]}
 Video
 Digital cinema^{[45]} — digital cinematography, digital movie cameras, video editing, film editing,^{[47]}^{[48]} Dolby Digital audio^{[1]}^{[22]}
 Digital television (DTV)^{[7]} — digital television broadcasting,^{[45]} standarddefinition television (SDTV), highdefinition TV (HDTV),^{[7]}^{[35]} HDTV encoder/decoder chips, ultra HDTV (UHDTV)^{[1]}
 Digital video^{[18]}^{[33]} — digital versatile disc (DVD),^{[45]} highdefinition (HD) video^{[7]}^{[35]}
 Video coding — video compression,^{[1]} video coding standards,^{[37]} motion estimation, motion compensation, interframe prediction, motion vectors,^{[1]} 3D video coding, local distortion detection probability (LDDP) model, moving object detection, Multiview Video Coding (MVC)^{[37]}
 Video processing — motion analysis, 3DDCT motion analysis, video content analysis, data extraction,^{[37]} video browsing,^{[49]} professional video production^{[50]}
 Watermarking — digital watermarking, image watermarking, video watermarking, 3D video watermarking, reversible data hiding, watermarking detection^{[37]}
 Wireless technology
 Mobile devices^{[42]} — mobile phones, smartphones,^{[41]} videophones^{[1]}
 Radio frequency (RF) technology — RF engineering, aperture arrays,^{[37]} beamforming, digital arithmetic circuits, directional sensing, space imaging^{[51]}
 Wireless sensor network (WSN) — wireless acoustic sensor networks^{[37]}
DCT visual media standards
The DCTII, also known as simply the DCT, is the most important image compression technique.^{[citation needed]} It is used in image compression standards such as JPEG, and video compression standards such as H.26x, MJPEG, MPEG, DV, Theora and Daala. There, the twodimensional DCTII of blocks are computed and the results are quantized and entropy coded. In this case, is typically 8 and the DCTII formula is applied to each row and column of the block. The result is an 8 × 8 transform coefficient array in which the element (topleft) is the DC (zerofrequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies.
The integer DCT, an integer approximation of the DCT,^{[2]}^{[1]} is used in Advanced Video Coding (AVC),^{[52]}^{[1]} introduced in 2003, and High Efficiency Video Coding (HEVC),^{[4]}^{[1]} introduced in 2013. The integer DCT is also used in the High Efficiency Image Format (HEIF), which uses a subset of the HEVC video coding format for coding still images.^{[4]} AVC uses 4x4 and 8x8 blocks. HEVC and HEIF use varied block sizes between 4x4 and 32x32 pixels.^{[4]}^{[1]} As of 2019^{[update]}, 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.^{[43]}
Image formats
Image compression standard  Year  Common applications 

JPEG^{[1]}  1992  The most widely used image compression standard^{[53]}^{[54]} and digital image format,^{[46]} 
JPEG XR  2009  Open XML Paper Specification 
WebP  2010  A graphic format that supports the lossy compression of digital images. Developed by Google. 
High Efficiency Image Format (HEIF)  2013  Image file format based on HEVC compression. It improves compression over JPEG,^{[4]} and supports animation with much more efficient compression than the animated GIF format.^{[55]} 
BPG  2014  Based on HEVC compression 
JPEG XL^{[56]}  2020  A royaltyfree rastergraphics file format that supports both lossy and lossless compression. 
Video formats
Video coding standard  Year  Common applications 

H.261^{[57]}^{[58]}  1988  First of a family of video coding standards. Used primarily in older video conferencing and video telephone products. 
Motion JPEG (MJPEG)^{[59]}  1992  QuickTime, video editing, nonlinear editing, digital cameras 
MPEG1 Video^{[60]}  1993  Digital video distribution on CD or Internet video 
MPEG2 Video (H.262)^{[60]}  1995  Storage and handling of digital images in broadcast applications, digital television, HDTV, cable, satellite, highspeed Internet, DVD video distribution 
DV  1995  Camcorders, digital cassettes 
H.263 (MPEG4 Part 2)^{[57]}  1996  Video telephony over public switched telephone network (PSTN), H.320, Integrated Services Digital Network (ISDN)^{[61]}^{[62]} 
Advanced Video Coding (AVC / H.264 / MPEG4)^{[1]}^{[52]}  2003  Most common HD video recording/compression/distribution format, Internet video, YouTube, Bluray Discs, HDTV broadcasts, web browsers, streaming television, mobile devices, consumer devices, Netflix,^{[42]} video telephony, FaceTime^{[41]} 
Theora  2004  Internet video, web browsers 
VC1  2006  Windows media, Bluray Discs 
Apple ProRes  2007  Professional video production.^{[50]} 
VP9  2010  A video codec developed by Google used in the WebM container format with HTML5. 
High Efficiency Video Coding (HEVC / H.265)^{[1]}^{[4]}  2013  The emerging successor to the H.264/MPEG4 AVC standard, having substantially improved compression capability. 
Daala  2013  Research video format by Xiph.org. 
AV1^{[63]}  2018  An open source format based on VP10 (VP9's internal successor), Daala and Thor; used by content providers such as YouTube^{[64]}^{[65]} and Netflix.^{[66]}^{[67]} 
MDCT audio standards
General audio
Speech coding
Speech coding standard  Year  Common applications 

AACLD (LDMDCT)^{[77]}  1999  Mobile telephony, voiceoverIP (VoIP), iOS, FaceTime^{[41]} 
Siren^{[40]}  1999  VoIP, wideband audio, G.722.1 
G.722.1^{[78]}  1999  VoIP, wideband audio, G.722 
G.729.1^{[79]}  2006  G.729, VoIP, wideband audio,^{[79]} mobile telephony 
EVRCWB^{[38]}^{: 31, 478] }  2007  Wideband audio 
G.718^{[80]}  2008  VoIP, wideband audio, mobile telephony 
G.719^{[38]}  2008  Teleconferencing, videoconferencing, voice mail 
CELT^{[81]}  2011  VoIP,^{[82]}^{[83]} mobile telephony 
Enhanced Voice Services (EVS)^{[84]}  2014  Mobile telephony, VoIP, wideband audio 
MD DCT
Multidimensional DCTs (MD DCTs) have several applications, mainly 3D DCTs such as the 3D DCTII, which has several new applications like Hyperspectral Imaging coding systems,^{[85]} variable temporal length 3D DCT coding,^{[86]} video coding algorithms,^{[87]} adaptive video coding^{[88]} and 3D Compression.^{[89]} Due to enhancement in the hardware, software and introduction of several fast algorithms, the necessity of using MD DCTs is rapidly increasing. DCTIV has gained popularity for its applications in fast implementation of realvalued polyphase filtering banks,^{[90]} lapped orthogonal transform^{[91]}^{[92]} and cosinemodulated wavelet bases.^{[93]}
Digital signal processing
DCT plays a very important role in digital signal processing. 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 analogtodigital conversion. DCTs are also commonly used for highdefinition television (HDTV) encoder/decoder chips.^{[1]}
Compression artifacts
A common issue with DCT compression in digital media are blocky compression artifacts,^{[94]} caused by DCT blocks.^{[3]} The DCT algorithm can cause blockbased 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, H.26x and MPEG formats), DCTbased blocky compression artifacts are widespread in digital media. 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 quantized. This process can cause blocking artifacts, primarily at high data compression ratios.^{[94]} This can also cause the "mosquito noise" effect, commonly found in digital video (such as the MPEG formats).^{[95]}
DCT blocks are often used in glitch art.^{[3]} The artist Rosa Menkman makes use of DCTbased compression artifacts in her glitch art,^{[96]} particularly the DCT blocks found in most digital media formats such as JPEG digital images and MP3 digital audio.^{[3]} Another example is Jpegs by German photographer Thomas Ruff, which uses intentional JPEG artifacts as the basis of the picture's style.^{[97]}^{[98]}
Informal overview
Like any Fourierrelated 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 (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 conditions from the DFT or other related transforms.
The Fourierrelated transforms that operate on a function over a finite 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 as a sum of sinusoids, you can evaluate that sum at any , even for where the original was not specified. 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 (i.e. the minn and maxn 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 transforms (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 Fourierrelated transforms to solve partial differential equations by spectral methods, the boundary conditions are directly specified as a part of the problem being solved. Or, for the MDCT (based on the typeIV DCT), the boundary conditions are intimately involved in the MDCT's critical property of timedomain 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 Fourierlike series.
In particular, it is well known that any 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 typeII 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 (where denotes the set of real numbers), or equivalently an invertible N × N square matrix. There are several variants of the DCT with slightly modified definitions. The N real numbers are transformed into the N real numbers according to one of the formulas:
DCTI
Some authors further multiply the and terms by and correspondingly multiply the and terms by which makes the DCTI matrix orthogonal, if one further multiplies by an overall scale factor of but breaks the direct correspondence with a realeven DFT.
The DCTI is exactly equivalent (up to an overall scale factor of 2), to a DFT of real numbers with even symmetry. For example, a DCTI of real numbers is exactly equivalent to a DFT of eight real numbers (even symmetry), divided by two. (In contrast, DCT types IIIV involve a halfsample shift in the equivalent DFT.)
Note, however, that the DCTI is not defined for less than 2, while all other DCT types are defined for any positive
Thus, the DCTI corresponds to the boundary conditions: is even around and even around ; similarly for
DCTII
The DCTII is probably the most commonly used form, and is often simply referred to as "the DCT".^{[5]}^{[6]}
This transform is exactly equivalent (up to an overall scale factor of 2) to a DFT of real inputs of even symmetry where the evenindexed elements are zero. That is, it is half of the DFT of the inputs where for and for DCTII transformation is also possible using 2N signal followed by a multiplication by half shift. This is demonstrated by Makhoul.
Some authors further multiply the term by and multiply the rest of the matrix by an overall scale factor of (see below for the corresponding change in DCTIII). This makes the DCTII matrix orthogonal, but breaks the direct correspondence with a realeven DFT of halfshifted input. This is the normalization used by Matlab, for example, see.^{[99]} In many applications, such as JPEG, the scaling is arbitrary because scale factors can be combined with a subsequent computational step (e.g. the quantization step in JPEG^{[100]}), and a scaling can be chosen that allows the DCT to be computed with fewer multiplications.^{[101]}^{[102]}
The DCTII implies the boundary conditions: is even around and even around is even around and odd around
DCTIII
Because it is the inverse of DCTII up to a scale factor (see below), this form is sometimes simply referred to as "the inverse DCT" ("IDCT").^{[6]}
Some authors divide the term by instead of by 2 (resulting in an overall term) and multiply the resulting matrix by an overall scale factor of (see above for the corresponding change in DCTII), so that the DCTII and DCTIII are transposes of one another. This makes the DCTIII matrix orthogonal, but breaks the direct correspondence with a realeven DFT of halfshifted output.
The DCTIII implies the boundary conditions: is even around and odd around is even around and even around
DCTIV
The DCTIV matrix becomes orthogonal (and thus, being clearly symmetric, its own inverse) if one further multiplies by an overall scale factor of
A variant of the DCTIV, where data from different transforms are overlapped, is called the modified discrete cosine transform (MDCT).^{[103]}
The DCTIV implies the boundary conditions: is even around and odd around similarly for
DCT VVIII
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 VVIII 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 realeven DFTs of even order (regardless of whether is even or odd), since the corresponding DFT is of length (for DCTI) or (for DCTII & III) or (for DCTIV). The four additional types of discrete cosine transform^{[104]} correspond essentially to realeven DFTs of logically odd order, which have factors of in the denominators of the cosine arguments.
However, these variants seem to be rarely used in practice. One reason, perhaps, is that FFT algorithms for oddlength DFTs are generally more complicated than FFT algorithms for evenlength DFTs (e.g. the simplest radix2 algorithms are only for even lengths), and this increased intricacy carries over to the DCTs as described below.
(The trivial realeven array, a lengthone DFT (odd length) of a single number a , corresponds to a DCTV of length )
Inverse transforms
Using the normalization conventions above, the inverse of DCTI is DCTI multiplied by 2/(N − 1). The inverse of DCTIV is DCTIV multiplied by 2/N. The inverse of DCTII is DCTIII multiplied by 2/N and vice versa.^{[6]}
Like for the 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 so that the inverse does not require any additional multiplicative factor. Combined with appropriate factors of √2 (see above), this can be used to make the transform matrix orthogonal.
Multidimensional DCTs
Multidimensional variants of the various DCT types follow straightforwardly from the onedimensional definitions: they are simply a separable product (equivalently, a composition) of DCTs along each dimension.
MD DCTII
For example, a twodimensional DCTII of an image or a matrix is simply the onedimensional DCTII, from above, performed along the rows and then along the columns (or vice versa). That is, the 2D DCTII is given by the formula (omitting normalization and other scale factors, as above):
 The inverse of a multidimensional DCT is just a separable product of the inverses of the corresponding onedimensional DCTs (see above), e.g. the onedimensional inverses applied along one dimension at a time in a rowcolumn algorithm.
The 3D DCTII is only the extension of 2D DCTII in three dimensional space and mathematically can be calculated by the formula
The inverse of 3D DCTII is 3D DCTIII and can be computed from the formula given by
Technically, computing a two, three (or multi) dimensional DCT by sequences of onedimensional DCTs along each dimension is known as a rowcolumn algorithm. As with 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 3D DCT, several fast algorithms are developed for the computation of 3D DCTII. VectorRadix algorithms are applied for computing MD DCT to reduce the computational complexity and to increase the computational speed. To compute 3D DCTII efficiently, a fast algorithm, VectorRadix Decimation in Frequency (VR DIF) algorithm was developed.
3D DCTII VR DIF
In order to apply the VR DIF algorithm the input data is to be formulated and rearranged as follows.^{[105]}^{[106]} The transform size N × N × N is assumed to be 2.
 where
The figure to the adjacent shows the four stages that are involved in calculating 3D DCTII using VR DIF algorithm. The first stage is the 3D 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 .
The original 3D DCTII now can be written as
where
If the even and the odd parts of and and are considered, the general formula for the calculation of the 3D DCTII can be expressed as
where
Arithmetic complexity
The whole 3D DCT calculation needs stages, and each stage involves butterflies. The whole 3D DCT requires 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 and the total number of real additions i.e. including the postadditions (recursive additions) which can be calculated directly after the butterfly stage or after the bitreverse stage are given by^{[106]}
The conventional method to calculate MDDCTII is using a RowColumnFrame (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 and respectively. From Table 1, it can be seen that the total number
Transform Size  3D VR Mults  RCF Mults  3D VR Adds  RCF Adds 

8 × 8 × 8  2.625  4.5  10.875  10.875 
16 × 16 × 16  3.5  6  15.188  15.188 
32 × 32 × 32  4.375  7.5  19.594  19.594 
64 × 64 × 64  5.25  9  24.047  24.047 
of multiplications associated with the 3D 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 3D DCT VR algorithm more efficient and better suited for 3D applications that involve the 3D DCTII such as video compression and other 3D 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.^{[107]} Therefore, although the above proposed 3D VR algorithm does not achieve the theoretical lower bound on the number of multiplications,^{[108]} it has a simpler computational structure as compared to other 3D DCT algorithms. It can be implemented in place using a single butterfly and possesses the properties of the Cooley–Tukey FFT algorithm in 3D. Hence, the 3D VR presents a good choice for reducing arithmetic operations in the calculation of the 3D DCTII, while keeping the simple structure that characterize butterflystyle Cooley–Tukey FFT algorithms.
The image to the right shows a combination of horizontal and vertical frequencies for an 8 × 8 twodimensional 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 topleft square yields a halfcycle increase in the horizontal frequency. Another move to the right yields two halfcycles. A move down yields two halfcycles horizontally and a halfcycle vertically. The source data ( 8×8 ) is transformed to a linear combination of these 64 frequency squares.
MDDCTIV
The MD DCTIV is just an extension of 1D DCTIV on to M dimensional domain. The 2D DCTIV of a matrix or an image is given by
 for and
We can compute the MD DCTIV using the regular rowcolumn method or we can use the polynomial transform method^{[109]} 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 1D DCTs directly. MD DCTIV also has several applications in various fields.
Computation
Although the direct application of these formulas would require operations, it is possible to compute the same thing with only complexity by factorizing the computation similarly to the fast Fourier transform (FFT). One can also compute DCTs via FFTs combined with pre and postprocessing steps. In general, 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 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 poweroftwo sizes) are typically closely related to FFT algorithms – since DCTs are essentially DFTs of realeven 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 (Frigo & Johnson 2005). 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 minimalmultiplication algorithms for the DFT, albeit generally at the cost of more additions, and a similar algorithm was proposed by (Feig & Winograd 1992a) 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 (Duhamel & Vetterli 1990).
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 N with FFTbased algorithms.^{[a]} Specialized DCT algorithms, on the other hand, see widespread use for transforms of small, fixed sizes such as the 8 × 8 DCTII used in 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 embeddeddevice applications.)
In fact, even the DCT algorithms using an ordinary FFT are sometimes equivalent to pruning the redundant operations from a larger FFT of realsymmetric data, and they can even be optimal from the perspective of arithmetic counts. For example, a typeII DCT is equivalent to a DFT of size with realeven symmetry whose evenindexed 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 Narasimha & Peterson (1978) and Makhoul (1980), and this method in hindsight can be seen as one step of a radix4 decimationintime Cooley–Tukey algorithm applied to the "logical" realeven DFT corresponding to the DCTII.^{[b]} Because the evenindexed elements are zero, this radix4 step is exactly the same as a splitradix step. If the subsequent size realdata FFT is also performed by a realdata splitradix algorithm (as in Sorensen et al. (1987)), then the resulting algorithm actually matches what was long the lowest published arithmetic count for the poweroftwo DCTII ( realarithmetic operations^{[c]}).
A recent reduction in the operation count to also uses a realdata FFT.^{[110]} 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 functioncall overhead in invoking a separate FFT routine might be significant for small 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  Contains a potentially easier to understand example of DCT transformation
 List of Fourierrelated transforms
 Modified discrete cosine transform
Notes
 ^ Algorithmic performance on modern hardware is typically not principally determined by simple arithmetic counts, and optimization requires substantial engineering effort to make best use, within its intrinsic limits, of available builtin hardware optimization.
 ^ The radix4 step reduces the size DFT to four size DFTs of real data, two of which are zero, and two of which are equal to one another by the even symmetry. Hence giving a single size FFT of real data plus butterflies, once the trivial and / or duplicate parts are eliminated and / or merged.
 ^ The precise count of real arithmetic operations, and in particular the count of real multiplications, depends somewhat on the scaling of the transform definition. The count is for the DCTII definition shown here; two multiplications can be saved if the transform is scaled by an overall factor. Additional multiplications can be saved if one permits the outputs of the transform to be rescaled individually, as was shown by Arai, Agui & Nakajima (1988) for the size8 case used in JPEG.
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 ^ Pennebaker, William B.; Mitchell, Joan L. (31 December 1992). JPEG: Still Image Data Compression Standard. Springer. ISBN 9780442012724.
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 ^ Shao, Xuancheng; Johnson, Steven G. (2008). "TypeII/III DCT/DST algorithms with reduced number of arithmetic operations". Signal Processing. 88 (6): 1553–1564. arXiv:cs/0703150. doi:10.1016/j.sigpro.2008.01.004. S2CID 986733.
 ^ Malvar 1992
 ^ Martucci 1994
 ^ Chan, S.C.; Ho, K.L. (1990). "Direct methods for computing discrete sinusoidal transforms". IEE Proceedings F  Radar and Signal Processing. 137 (6): 433. doi:10.1049/ipf2.1990.0063.
 ^ ^{a} ^{b} Alshibami, O.; Boussakta, S. (July 2001). "Threedimensional algorithm for the 3D DCTIII". Proc. Sixth Int. Symp. Commun., Theory Applications: 104–107.
 ^ Guoan Bi; Gang Li; KaiKuang Ma; Tan, T.C. (2000). "On the computation of twodimensional DCT". IEEE Transactions on Signal Processing. 48 (4): 1171–1183. Bibcode:2000ITSP...48.1171B. doi:10.1109/78.827550.
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Further reading
 Narasimha, M.; Peterson, A. (June 1978). "On the Computation of the Discrete Cosine Transform". IEEE Transactions on Communications. 26 (6): 934–936. doi:10.1109/TCOM.1978.1094144.
 Makhoul, J. (February 1980). "A fast cosine transform in one and two dimensions". IEEE Transactions on Acoustics, Speech, and Signal Processing. 28 (1): 27–34. doi:10.1109/TASSP.1980.1163351.
 Sorensen, H.; Jones, D.; Heideman, M.; Burrus, C. (June 1987). "Realvalued fast Fourier transform algorithms". IEEE Transactions on Acoustics, Speech, and Signal Processing. 35 (6): 849–863. CiteSeerX 10.1.1.205.4523. doi:10.1109/TASSP.1987.1165220.
 Plonka, G.; Tasche, M. (January 2005). "Fast and numerically stable algorithms for discrete cosine transforms". Linear Algebra and Its Applications. 394 (1): 309–345. doi:10.1016/j.laa.2004.07.015.
 Duhamel, P.; Vetterli, M. (April 1990). "Fast fourier transforms: A tutorial review and a state of the art". Signal Processing (Submitted manuscript). 19 (4): 259–299. doi:10.1016/01651684(90)90158U.
 Ahmed, N. (January 1991). "How I came up with the discrete cosine transform". Digital Signal Processing. 1 (1): 4–9. doi:10.1016/10512004(91)90086Z.
 Feig, E.; Winograd, S. (September 1992b). "Fast algorithms for the discrete cosine transform". IEEE Transactions on Signal Processing. 40 (9): 2174–2193. Bibcode:1992ITSP...40.2174F. doi:10.1109/78.157218.
 Malvar, Henrique (1992), Signal Processing with Lapped Transforms, Boston: Artech House, ISBN 9780890064672
 Martucci, S. A. (May 1994). "Symmetric convolution and the discrete sine and cosine transforms". IEEE Transactions on Signal Processing. 42 (5): 1038–1051. Bibcode:1994ITSP...42.1038M. doi:10.1109/78.295213.
 Oppenheim, Alan; Schafer, Ronald; Buck, John (1999), DiscreteTime Signal Processing (2nd ed.), Upper Saddle River, N.J: Prentice Hall, ISBN 9780137549207
 Frigo, M.; Johnson, S. G. (February 2005). "The Design and Implementation of FFTW3" (PDF). Proceedings of the IEEE. 93 (2): 216–231. CiteSeerX 10.1.1.66.3097. doi:10.1109/JPROC.2004.840301. S2CID 6644892.
 Boussakta, Said.; Alshibami, Hamoud O. (April 2004). "Fast Algorithm for the 3D DCTII" (PDF). IEEE Transactions on Signal Processing. 52 (4): 992–1000. Bibcode:2004ITSP...52..992B. doi:10.1109/TSP.2004.823472. S2CID 3385296.
 Cheng, L. Z.; Zeng, Y. H. (2003). "New fast algorithm for multidimensional typeIV DCT". IEEE Transactions on Signal Processing. 51 (1): 213–220. doi:10.1109/TSP.2002.806558.
 WenHsiung Chen; Smith, C.; Fralick, S. (September 1977). "A Fast Computational Algorithm for the Discrete Cosine Transform". IEEE Transactions on Communications. 25 (9): 1004–1009. doi:10.1109/TCOM.1977.1093941.
 Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 12.4.2. Cosine Transform", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 9780521880688, archived from the original on 20110811, retrieved 20110813
External links
 Syed Ali Khayam: The Discrete Cosine Transform (DCT): Theory and Application
 Implementation of MPEG integer approximation of 8x8 IDCT (ISO/IEC 230022)
 Matteo Frigo and Steven G. Johnson: FFTW, FFTW Home Page. A free (GPL) C library that can compute fast DCTs (types IIV) in one or more dimensions, of arbitrary size.
 Takuya Ooura: General Purpose FFT Package, FFT Package 1dim / 2dim. 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 8point DCT and IDCT, Algorithm Alley.
 LTFAT is a free Matlab/Octave toolbox with interfaces to the FFTW implementation of the DCTs and DSTs of type IIV.