A DCT is aFourier-related transform similar to thediscrete Fourier transform (DFT), but using onlyreal numbers. The DCTs are generally related toFourier 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 witheven 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 type-II DCT, which is often called simplythe DCT. This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT, is correspondingly often called simplythe inverse DCT orthe IDCT. Two related transforms are thediscrete sine transform (DST), which is equivalent to a DFT of real andodd functions, and themodified 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] aninteger approximation of the standard DCT,[2]: ix, xiii, 1, 141–304 used in severalISO/IEC andITU-T international standards.[1][2]
DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks.[3] DCT blocks sizes including 8x8pixels for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels.[1][4] The DCT has a strongenergy compaction property,[5][6] capable of achieving high quality at highdata compression ratios.[7][8] However, blockycompression artifacts can appear when heavy DCT compression is applied.
Since its introduction in 1974, there has been significant research on the DCT.[10] In 1977, Wen-Hsiung 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 theJoint Photographic Experts Group as the basis forJPEG's lossy image compression algorithm in 1992.[10][12]
Thediscrete sine transform (DST) was derived from the DCT, by replacing theNeumann condition atx=0 with aDirichlet condition.[2]: 35-36 The DST was described in the 1974 DCT paper by Ahmed, Natarajan and Rao.[5] A type-I DST (DST-I) was later described byAnil K. Jain in 1976, and a type-II DST (DST-II) 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 forinter-framemotion-compensatedvideo coding. They experimented with the DCT and thefast Fourier transform (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 perpixel for avideotelephone scene with image quality comparable to anintra-frame coder requiring 2-bit per pixel.[14][15] In 1979,Anil K. Jain and Jaswant R. Jain further developed motion-compensated DCT video compression,[16][17] also called block motion compensation.[17] This led to Chen developing a practical video compression algorithm, called motion-compensated DCT or adaptive scene coding, in 1981.[17] Motion-compensated DCT later became the standard coding technique for video compression from the late 1980s onwards.[18][19]
Nasir Ahmed also developed a lossless DCT algorithm with Giridhar Mandyam and Neeraj Magotra at theUniversity of New Mexico in 1995. This allows the DCT technique to be used forlossless compression of images. It is a modification of the original DCT algorithm, and incorporates elements of inverse DCT anddelta modulation. It is a more effective lossless compression algorithm thanentropy coding.[27] Lossless DCT is also known as LDCT.[28]
The DCT, and in particular the DCT-II, is often used in signal and image processing, especially for lossy compression, because it has a strongenergy compaction property.[5][6] In typical applications, most of the signal information tends to be concentrated in a few low-frequency components of the DCT. For strongly correlatedMarkov processes, the DCT can approach the compaction efficiency of theKarhunen-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 widely employed in solvingpartial differential equations byspectral methods, where the different variants of the DCT correspond to slightly different even and odd boundary conditions at the two ends of the array.
Color formatting — formattingluminance and color differences, color formats (such asYUV444 andYUV411),decoding operations such as the inverse operation between display color formats (YIQ,YUV,RGB)[1]
The DCT-II is an important image compression technique. It is used in image compression standards such asJPEG, andvideo compression standards such asH.26x,MJPEG,MPEG,DV,Theora andDaala. There, the two-dimensional DCT-II of blocks are computed and the results arequantized andentropy coded. In this case, 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 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.
The integer DCT, an integer approximation of the DCT,[2][1] is used inAdvanced Video Coding (AVC),[52][1] introduced in 2003, andHigh Efficiency Video Coding (HEVC),[4][1] introduced in 2013. The integer DCT is also used in theHigh Efficiency Image Format (HEIF), which uses a subset of theHEVC video coding format for coding still images.[4] AVC uses 4 x 4 and 8 x 8 blocks. HEVC and HEIF use varied block sizes between 4 x 4 and 32 x 32pixels.[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]
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,[85] variable temporal length 3-D DCT coding,[86]video coding algorithms,[87] adaptive video coding[88] and 3-D Compression.[89] Due to enhancement in the hardware, software and introduction of several fast algorithms, the necessity of using MD DCTs is rapidly increasing.DCT-IV has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks,[90] lapped orthogonal transform[91][92] and cosine-modulated wavelet bases.[93]
A common issue with DCT compression indigital media are blockycompression artifacts,[94] caused by DCT blocks.[3] 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 blocks is taken within each block and the resulting DCT coefficients arequantized. This process can cause blocking artifacts, primarily at highdata compression ratios.[94] This can also cause themosquito noise effect, commonly found indigital video.[95]
DCT blocks are often used inglitch art.[3] The artistRosa Menkman makes use of DCT-based compression artifacts in her glitch art,[96] particularly the DCT blocks found in mostdigital media formats such asJPEG digital images andMP3 audio.[3] Another example isJpegs by German photographerThomas Ruff, which uses intentionalJPEG artifacts as the basis of the picture's style.[97][98]
Like any Fourier-related transform, DCTs express a function or a signal in terms of a sum ofsinusoids with differentfrequencies andamplitudes. Like the DFT, a DCT operates on a function at a finite number ofdiscrete 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 ofcomplex exponentials). However, this visible difference is merely a consequence of a deeper distinction: a DCT implies differentboundary conditions from the DFT or other related transforms.
The Fourier-related transforms that operate on a function over a finitedomain, such as the DFT or DCT or aFourier series, can be thought of as implicitly defining anextension 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 aperiodic extension of the original function. A DCT, like acosine transform, implies aneven extension of the original function.
Illustration of the implicit even/odd extensions of DCT input data, forN=11 data points (red dots), for the four most common types of DCT (types I-IV). Note the subtle differences at the interfaces between the data and the extensions: in DCT-II and DCT-IV both the end points are replicated in the extensions but not in DCT-I or DCT-III (and a zero point is inserted at the sign reversal extension in DCT-III).
However, because DCTs operate onfinite,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 atboth 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 aroundwhat point the function is even or odd. In particular, consider a sequenceabcd of four equally spaced data points, and say that we specify an evenleft boundary. There are two sensible possibilities: either the data are even about the samplea, in which case the even extension isdcbabcd, or the data are even about the pointhalfway betweena and the previous point, in which case the even extension isdcbaabcd (a is repeated).
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. These choices lead to all the standard variations of DCTs and alsodiscrete sine transforms (DSTs). Half of these possibilities, those where theleft 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 solvepartial differential equations byspectral methods, the boundary conditions are directly specified as a part of the problem being solved. Or, for the 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 theenergy 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 anydiscontinuities in a function reduce therate 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.[a] 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.[b] In contrast, a DCT whereboth boundaries are evenalways yields a continuous extension at the boundaries (although theslope 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.
Formally, the discrete cosine transform is alinear, invertiblefunction (where denotes the set ofreal numbers), or equivalently an invertibleN ×Nsquare matrix. There are several variants of the DCT with slightly modified definitions. TheN real numbers are transformed into theN real numbers according to one of the formulas:
Some authors further multiply the and terms by and correspondingly multiply the and terms by which, if one further multiplies by an overall scale factor of, makes the DCT-I matrixorthogonal but breaks the direct correspondence with a real-evenDFT.
The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a DFT of real numbers with even symmetry. For example, a DCT-I of real numbers 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 less than 2, while all other DCT types are defined for any positive.
Thus, the DCT-I corresponds to the boundary conditions: is even around and even around; similarly for.
The DCT-II is probably the most commonly used form, and is often simply referred to as theDCT.[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 even-indexed elements are zero. That is, it is half of the DFT of the inputs where, for,, and for. DCT-II transformation is also possible using signal followed by a multiplication by half shift. This is demonstrated byMakhoul.[citation needed]
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 DCT-III). This makes the DCT-II matrixorthogonal, but breaks the direct correspondence with a real-even DFT of half-shifted input. This is the normalization used byMatlab.[99] In many applications, such asJPEG, the scaling is arbitrary because scale factors can be combined with a subsequent computational step (e.g. thequantization step in JPEG[100]), and a scaling can be chosen that allows the DCT to be computed with fewer multiplications.[101][102]
The DCT-II implies the boundary conditions: is even around and even around; is even around and odd around.
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).[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 DCT-II), so that the DCT-II and DCT-III are transposes of one another. This makes the DCT-III matrixorthogonal, but breaks the direct correspondence with a real-even DFT of half-shifted output.
The DCT-III implies the boundary conditions: is even around and odd around is even around and even around
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 DFTs of even order (regardless of whether is even or odd), since the corresponding DFT is of length (for DCT-I) or (for DCT-II & III) or (for DCT-IV). The four additional types of discrete cosine transform[104] correspond essentially to real-even 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 thatFFT algorithms for odd-length DFTs are generally more complicated thanFFT 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 numbera , corresponds to a DCT-V of length)
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.[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 matrixorthogonal.
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.
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):
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.
The3-D DCT-II is only the extension of2-D DCT-II in three dimensional space and mathematically can be calculated by the formula
The inverse of3-D DCT-II is3-D DCT-III and can be computed from the formula given by
Technically, computing a two-, three- (or -multi) dimensional DCT by sequences of one-dimensional DCTs along each dimension is known as arow-column algorithm. As withmultidimensional 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.
In order to apply the VR DIF algorithm the input data is to be formulated and rearranged as follows.[105][106] The transform sizeN × N × N is assumed to be 2.
The four basic stages of computing 3-D DCT-II using VR DIF Algorithm.
where
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.
The original 3-D DCT-II 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 3-D DCT-II can be expressed as
The whole 3-D DCT calculation needs stages, and each stage involves butterflies. The whole 3-D 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 post-additions (recursive additions) which can be calculated directly after the butterfly stage or after the bit-reverse stage are given by[106]
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 and respectively. From Table 1, it can be seen that the total number
TABLE 1Comparison of VR DIF & RCF Algorithms for computing 3D-DCT-II
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 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.[107] Therefore, although the above proposed 3-D 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 3-D DCT algorithms. It can be implemented in place using a single butterfly and possesses the properties of theCooley–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-styleCooley–Tukey FFT algorithms.
The image to the right shows a combination of horizontal and vertical frequencies for an 8 × 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( 8×8 ) is transformed to alinear combination of these 64 frequency squares.
The M-D DCT-IV is just an extension of 1-D DCT-IV on toM dimensional domain. The 2-D DCT-IV of a matrix or an image is given by
for and
We can compute the MD DCT-IV using the regular row-column 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 1-D DCTs directly. MD DCT-IV also has several applications in various fields.
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 thefast Fourier transform (FFT). One can also compute DCTs via FFTs combined with pre- and post-processing 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 forpower-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 (Frigo & Johnson 2005). Algorithms based on theCooley–Tukey FFT algorithm are most common, but any other FFT algorithm is also applicable. For example, theWinograd 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 (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 lengthsN with FFT-based algorithms.[c]Specialized DCT algorithms, on the other hand, see widespread use for transforms of small, fixed sizes such as the 8 × 8 DCT-II used inJPEG 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 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 inFFTPACK andFFTW) was described byNarasimha & Peterson (1978) andMakhoul (1980), 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.[d]Because the even-indexed elements are zero, this radix-4 step is exactly the same as a split-radix step. If the subsequent size real-data FFT is also performed by a real-datasplit-radix algorithm (as inSorensen et al. (1987)), then the resulting algorithm actually matches what was long the lowest published arithmetic count for the power-of-two DCT-II ( real-arithmetic operations[e]).
A recent reduction in the operation count to also uses a real-data 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 function-call 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.)
Original size, scaled 10x (nearest neighbor), scaled 10x (bilinear).Basis functions of the discrete cosine transformation with corresponding coefficients (specific for our image). DCT of the image =.
Each basis function is multiplied by its coefficient and then this product is added to the final image.
On the left is the final image. In the middle is the weighted function (multiplied by a coefficient) which is added to the final image. On the right is the current function and corresponding coefficient. Images are scaled (using bilinear interpolation) by factor 10×.
^Here, we think of the DFT or DCT as approximations for theFourier series orcosine series of a function, respectively, in order to talk about its smoothness.
^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.
^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 built-in hardware optimization.
^The radix-4 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 plusbutterflies, 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 DCT-II 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 byArai, Agui & Nakajima (1988) for the size-8 case used in JPEG.
^Smith, C.; Fralick, S. (1977). "A Fast Computational Algorithm for the Discrete Cosine Transform".IEEE Transactions on Communications.25 (9):1004–1009.doi:10.1109/TCOM.1977.1093941.ISSN0090-6778.
^Roese, John A.; Robinson, Guner S. (30 October 1975). Tescher, Andrew G. (ed.). "Combined Spatial And Temporal Coding Of Digital Image Sequences".Efficient Transmission of Pictorial Information.0066. International Society for Optics and Photonics:172–181.Bibcode:1975SPIE...66..172R.doi:10.1117/12.965361.S2CID62725808.
^abc"History of Video Compression".ITU-T. Joint Video Team (JVT) of ISO/IEC MPEG & ITU-T VCEG (ISO/IEC JTC1/SC29/WG11 and ITU-T SG16 Q.6). July 2002. pp. 11,24–9, 33,40–1,53–6. Retrieved3 November 2019.
^Princen, John P.; Johnson, A.W.; Bradley, Alan B. (1987). "Subband/Transform coding using filter bank designs based on time domain aliasing cancellation".ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing. Vol. 12. pp. 2161–2164.doi:10.1109/ICASSP.1987.1169405.S2CID58446992.
^Princen, J.; Bradley, A. (1986). "Analysis/Synthesis filter bank design based on time domain aliasing cancellation".IEEE Transactions on Acoustics, Speech, and Signal Processing.34 (5):1153–1161.doi:10.1109/TASSP.1986.1164954.
^abBritanak, V. (2011). "On Properties, Relations, and Simplified Implementation of Filter Banks in the Dolby Digital (Plus) AC-3 Audio Coding Standards".IEEE Transactions on Audio, Speech, and Language Processing.19 (5):1231–1241.doi:10.1109/TASL.2010.2087755.S2CID897622.
^Mandyam, Giridhar D.; Ahmed, Nasir; Magotra, Neeraj (17 April 1995). Rodriguez, Arturo A.; Safranek, Robert J.; Delp, Edward J. (eds.). "DCT-based scheme for lossless image compression".Digital Video Compression: Algorithms and Technologies 1995.2419. International Society for Optics and Photonics:474–478.Bibcode:1995SPIE.2419..474M.doi:10.1117/12.206386.S2CID13894279.
^Muchahary, D.; Mondal, A. J.; Parmar, R. S.; Borah, A. D.; Majumder, A. (2015). "A Simplified Design Approach for Efficient Computation of DCT".2015 Fifth International Conference on Communication Systems and Network Technologies. pp. 483–487.doi:10.1109/CSNT.2015.134.ISBN978-1-4799-1797-6.S2CID16411333.
^abcdMcKernan, Brian (2005).Digital cinema: the revolution in cinematography, postproduction, distribution.McGraw-Hill. p. 58.ISBN978-0-07-142963-4.DCT is used in most of the compression systems standardized by the Moving Picture Experts Group (MPEG), is the dominant technology for image compression. In particular, it is the core technology of MPEG-2, the system used for DVDs, digital television broadcasting, that has been used for many of the trials of digital cinema.
^Potluri, U. S.; Madanayake, A.; Cintra, R. J.; Bayer, F. M.; Rajapaksha, N. (17 October 2012). "Multiplier-free DCT approximations for RF multi-beam digital aperture-array space imaging and directional sensing".Measurement Science and Technology.23 (11): 114003.doi:10.1088/0957-0233/23/11/114003.ISSN0957-0233.S2CID119888170.
^abWang, Hanli; Kwong, S.; Kok, C. (2006). "Efficient prediction algorithm of integer DCT coefficients forH.264/AVC optimization".IEEE Transactions on Circuits and Systems for Video Technology.16 (4):547–552.doi:10.1109/TCSVT.2006.871390.S2CID2060937.
^Alakuijala, Jyrki; Sneyers, Jon; Versari, Luca; Wassenberg, Jan (22 January 2021)."JPEG XL White Paper"(PDF).JPEG Org.Archived(PDF) from the original on 2 May 2021. Retrieved14 Jan 2022.Variable-sized DCT (square or rectangular from 2x2 to 256x256) serves as a fast approximation of the optimal decorrelating transform.
^YouTube Developers (15 September 2018)."AV1 Beta Launch Playlist".YouTube. Retrieved14 January 2022.The first videos to receive YouTube's AV1 transcodes.
^Hazra, Sudip; Mateti, Prabhaker (September 13–16, 2017)."Challenges in Android Forensics". In Thampi, Sabu M.; Pérez, Gregorio Martínez; Westphall, Carlos Becker; Hu, Jiankun; Fan, Chun I.; Mármol, Félix Gómez (eds.).Security in Computing and Communications: 5th International Symposium, SSCC 2017. Springer. pp. 286–299 (290).doi:10.1007/978-981-10-6898-0_24.ISBN9789811068980.
^Abousleman, G. P.; Marcellin, M. W.; Hunt, B. R. (January 1995), "Compression of hyperspectral imagery using 3-D DCT and hybrid DPCM/DCT",IEEE Trans. Geosci. Remote Sens.,33 (1):26–34,Bibcode:1995ITGRS..33...26A,doi:10.1109/36.368225
^Yeo, B.; Liu, B. (May 1995), "Volume rendering of DCT-based compressed 3D scalar data",IEEE Transactions on Visualization and Computer Graphics,1:29–43,doi:10.1109/2945.468390
^Chan, S.C.; Liu, W.; Ho, K.I. (2000). "Perfect reconstruction modulated filter banks with sum of powers-of-two coefficients".2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353). Vol. 2. pp. 73–76.doi:10.1109/ISCAS.2000.856261.hdl:10722/46174.ISBN0-7803-5482-6.S2CID1757438.
^Queiroz, R. L.; Nguyen, T. Q. (1996). "Lapped transforms for efficient transform/subband coding".IEEE Trans. Signal Process.44 (5):497–507.
^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/ip-f-2.1990.0063.
^abAlshibami, O.; Boussakta, S. (July 2001). "Three-dimensional algorithm for the 3-D DCT-III".Proc. Sixth Int. Symp. Commun., Theory Applications:104–107.
^Guoan Bi; Gang Li; Kai-Kuang Ma; Tan, T.C. (2000). "On the computation of two-dimensional DCT".IEEE Transactions on Signal Processing.48 (4):1171–1183.Bibcode:2000ITSP...48.1171B.doi:10.1109/78.827550.
^Feig, E.; Winograd, S. (July 1992a). "On the multiplicative complexity of discrete cosine transforms".IEEE Transactions on Information Theory.38 (4):1387–1391.doi:10.1109/18.144722.
^Nussbaumer, H.J. (1981).Fast Fourier transform and convolution algorithms (1st ed.). New York: Springer-Verlag.
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). "Real-valued fast Fourier transform algorithms".IEEE Transactions on Acoustics, Speech, and Signal Processing.35 (6):849–863.CiteSeerX10.1.1.205.4523.doi:10.1109/TASSP.1987.1165220.
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,ISBN978-0-89006-467-2
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.
Cheng, L. Z.; Zeng, Y. H. (2003). "New fast algorithm for multidimensional type-IV DCT".IEEE Transactions on Signal Processing.51 (1):213–220.doi:10.1109/TSP.2002.806558.
Wen-Hsiung 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,ISBN978-0-521-88068-8, archived fromthe original on 2011-08-11, retrieved2011-08-13
Matteo Frigo andSteven G. Johnson:FFTW,FFTW Home Page. A free (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,FFT Package 1-dim / 2-dim. 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,Algorithm Alley.
LTFAT is a free Matlab/Octave toolbox with interfaces to the FFTW implementation of the DCTs and DSTs of type I-IV.
