Discrete cosine transform: Difference between revisions

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{{Short description|Technique used in signal processing and data compression}}

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A '''discrete cosine transform''' ('''DCT''') expresses a finite sequence of [[data points]] in terms of a sum of [[cosine]] functions oscillating at different [[frequency|frequencies]]. The DCT, first proposed by [[Nasir Ahmed (engineer)|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 {{nowrap|[[H.26x]]}}), [[digital audio]] (such as [[Dolby Digital]], [[MP3]] and [[Advanced Audio Coding|AAC]]), [[digital television]] (such as [[SDTV]], [[HDTV]] and [[Video on demand|VOD]]), [[digital radio]] (such as [[AAC+]] and [[DAB+]]), and [[speech coding]] (such as [[AAC-LD]], [[Siren (codec)|Siren]] and [[Opus (audio format)|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 method]]s for the numerical solution of [[partial differential equations]].

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== History ==

The DCT was first conceived by [[Nasir Ahmed (engineer)|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]].<ref name="Ahmed" /><ref name="Stankovic"/> 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.<ref name="Ahmed"/> They presented their results in a January 1974 paper, titled ''Discrete Cosine Transform''.<ref name="pubDCT" /><ref name="pubRaoYip" /><ref name="t81">{{cite web|date=September 1992|title=T.81 – Digital compression and coding of continuous-tone still images – Requirements and guidelines|url=https://www.w3.org/Graphics/JPEG/itu-t81.pdf|access-date=12 July 2019|publisher=[[CCITT]]}}</ref> It described what is now called the type-II DCT (DCT-II),<ref name="Britanak2010" />{{rp|page = [https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA51 51]}} as well as the type-III inverse DCT (IDCT).<ref name="pubDCT"/>

Since its introduction in 1974, there has been significant research on the DCT.<ref name="t81"/> In 1977, Wen-Hsiung Chen published a paper with C. Harrison Smith and Stanley C. Fralick presenting a fast DCT algorithm.<ref name="A Fast Computational Algorithm for">{{cite journal |last1=Chen |first1=Wen-Hsiung |last2=Smith |first2=C. H. |last3=Fralick |first3=S. C. |title=A Fast Computational Algorithm for the Discrete Cosine Transform |journal=[[IEEE Transactions on Communications]] |date=September 1977 |volume=25 |issue=9 |pages=1004–1009 |doi=10.1109/TCOM.1977.1093941}}</ref><ref name="t81"/> Further developments include a 1978 paper by M. J. Narasimha and A. M. Peterson, and a 1984 paper by B. G. Lee.<ref name="t81"/> 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.<ref name="t81"/><ref name="chen">{{cite journal |last1=Smith |first1=C. |last2=Fralick |first2=S. |title=A Fast Computational Algorithm for the Discrete Cosine Transform |journal=IEEE Transactions on Communications |date=1977 |volume=25 |issue=9 |pages=1004–1009 |doi=10.1109/TCOM.1977.1093941 |issn=0090-6778}}</ref>

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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 equation]]s by [[spectral method]]s, 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 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.{{efn|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.{{efn|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.<!--[[User:Kvng/RTH]]-->

== Formal definition ==

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\qquad \text{ for } ~ k = 0,\ \ldots\ N-1 ~.</math>

Some authors further multiply the <math>x_0 </math> and <math> x_{N-1} </math> terms by <math> \sqrt{2\,}\, ,</math> and correspondingly multiply the <math> X_0 </math> and <math> X_{N-1}</math> terms by <math> 1/\sqrt{2\,} \,,</math> which, if one further multiplies by an overall scale factor of <math> display="inline">\sqrt{\tfrac{2}{N-1\,}\,} ,</math>, makes the DCT-I matrix [[orthogonal matrix|orthogonal]] 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 <math> 2(N-1) </math> real numbers with even symmetry. For example, a DCT-I of <math>N = 5 </math> real numbers <math> a\ b\ c\ d\ e </math> is exactly equivalent to a DFT of eight real numbers {{not a typo|<math> a\ b\ c\ d\ e\ d\ c\ b </math>}} (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 <math> N </math> less than 2, while all other DCT types are defined for any positive <math> N .</math>.

Thus, the DCT-I corresponds to the boundary conditions: <math> x_n </math> is even around <math> n = 0 </math> and even around <math> n = N - 1 </math>; similarly for <math> X_k .</math>.

=== DCT-II ===

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\qquad \text{ for } ~ k = 0,\ \dots\ N-1 ~.</math>

The DCT-II is probably the most commonly used form, and is often simply referred to as "the ''DCT"''.<ref name="pubDCT"/><ref name="pubRaoYip"/>

This transform is exactly equivalent (up to an overall scale factor of 2) to a [[Discrete Fourier transform|DFT]] of <math>4N</math> 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 <math>4N</math> inputs <math> y_n ,</math> where <math> y_{2n} = 0 ,</math>, <math> y_{2n+1} = x_n </math> for <math> 0 \leq n < N ,</math>, <math> y_{2N} = 0 ,</math>, and <math> y_{4N-n} = y_n </math> for <math> 0 < n < 2N .</math>. DCT-II transformation is also possible using 2{{mvar|N}}<math>2N</math> signal followed by a multiplication by half shift. This is demonstrated by [[John Makhoul|Makhoul]].{{cn|date=April 2025}}

Some authors further multiply the <math> X_0 </math> term by <math> 1/\sqrt{N\,} \, </math> and multiply the rest of the matrix by an overall scale factor of <math display="inline">\sqrt{{2}/{N}}</math> (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.<ref>{{cite web |url=https://www.mathworks.com/help/signal/ref/dct.html |title=Discrete cosine transform - MATLAB dct |website=www.mathworks.com |access-date=2019-07-11}}</ref> 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 (signal processing)|quantization]] step in JPEG<ref>{{cite book |isbn=9780442012724 |title=JPEG: Still Image Data Compression Standard |last1=Pennebaker |first1=William B. |last2=Mitchell |first2=Joan L. |date=31 December 1992|publisher=Springer }}</ref>), and a scaling can be chosen that allows the DCT to be computed with fewer multiplications.<ref>{{cite journal |url=https://search.ieice.org/bin/summary.php?id=e71-e_11_1095 |first1=Y. |last1=Arai |first2=T. |last2=Agui |first3=M. |last3=Nakajima |title=A fast DCT-SQ scheme for images |journal=IEICE Transactions |volume=71 |issue=11 |pages= 1095–1097 |year=1988}}</ref><ref>{{cite journal |doi=10.1016/j.sigpro.2008.01.004 |title=Type-II/III DCT/DST algorithms with reduced number of arithmetic operations |year=2008 |last1=Shao |first1=Xuancheng |last2=Johnson |first2=Steven G. |journal=Signal Processing |volume=88 |issue=6 |pages=1553–1564 |arxiv=cs/0703150 |bibcode=2008SigPr..88.1553S |s2cid=986733}}</ref>

The DCT-II implies the boundary conditions: <math> x_n </math> is even around <math> n = -1/2 </math> and even around <math> n = N - 1/2 \,;</math>; <math> X_k </math> is even around <math> k = 0 </math> and odd around <math> k = N .</math>.

=== DCT-III ===

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\qquad \text{ for } ~ k = 0,\ \ldots\ N-1 ~.</math>

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").<ref name="pubRaoYip"/>

Some authors divide the <math>x_0</math> term by <math>\sqrt{2}</math> instead of by 2 (resulting in an overall <math>x_0/\sqrt{2}</math> term) and multiply the resulting matrix by an overall scale factor of <math display="inline"> \sqrt{2/N}</math> (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: <math>x_n</math> is even around <math>n = 0</math> and odd around <math>n = N ;</math> <math>X_k</math> is even around <math>k = -1/2</math> and even around <math>k = N - 1/2.</math>

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A variant of the DCT-IV, where data from different transforms are ''overlapped'', is called the [[modified discrete cosine transform]] (MDCT).<ref>{{harvnb|Malvar|1992}}</ref>

The DCT-IV implies the boundary conditions: <math> x_n </math> is even around <math>n = -1/2</math> and odd around <math>n = N - 1/2;</math>; similarly for <math>X_k.</math>.<!--[[User:Kvng/RTH]]-->

=== 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|DFT]]sDFTs of even order (regardless of whether <math> N </math> is even or odd), since the corresponding DFT is of length <math> 2(N-1) </math> (for DCT-I) or <math> 4 N </math> (for DCT-II & III) or <math> 8 N </math> (for DCT-IV). The four additional types of discrete cosine transform<ref>{{harvnb|Martucci|1994}}</ref> correspond essentially to real-even DFTs of logically odd order, which have factors of <math> N \pm {1}/{2} </math> 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.

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Using the normalization conventions above, the inverse of DCT-I is DCT-I multiplied by 2/(''N''&nbsp;−&nbsp;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.<ref name="pubRaoYip"/>

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 <math display="inline">\sqrt{2/N}</math> so that the inverse does not require any additional multiplicative factor. Combined with appropriate factors of {{sqrt|2}} (see above), this can be used to make the transform matrix [[orthogonal matrix|orthogonal]].

== Multidimensional DCTs ==

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Specialized DCT algorithms, on the other hand, see widespread use for transforms of small, fixed sizes such as the {{nobr| 8 × 8 }} DCT-II 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 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 <math>~ 4N ~</math> 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 [[Fastest Fourier Transform in the West|FFTW]]) was described by {{harvtxt|Narasimha|Peterson|1978}} and {{harvtxt|Makhoul|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.{{efn|

The radix-4 step reduces the size <math>~ 4N ~</math> DFT to four size <math>~ N ~</math> 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 <math>~ N ~</math> FFT of real data plus <math>~ \mathcal{O}(N) ~</math> [[butterfly (FFT algorithm)|butterflies]], once the trivial and / or duplicate parts are eliminated and / or merged.

}}

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==Example of IDCT==

[[File:DCT filter comparison.png|thumb|right|An example showing eight different filters applied to a test image (top left) by multiplying its DCT spectrum (top right) with each filter.]]

Consider this {{resx|8x8}} [[grayscale image]] of capital letter A.

[[File:letter-a-8x8.png|frame|center|Original size, scaled 10x (nearest neighbor), scaled 10x (bilinear).]]

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* Takuya Ooura: General Purpose FFT Package, [http://www.kurims.kyoto-u.ac.jp/~ooura/fft.html 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, [http://drdobbs.com/parallel/184410889 Algorithm Alley].

* [httphttps://ltfat.sourceforge.net/ LTFAT] is a free Matlab/Octave toolbox with interfaces to the FFTW implementation of the DCTs and DSTs of type I-IV.

{{Compression Methods|state=expanded}}

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[[Category:Data compression]]

[[Category:Image compression]]

[[Category:Indian inventions]]

[[Category:H.26x]]

[[Category:JPEG]]