Discrete cosine transform: Difference between revisions

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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><!--[[User:Kvng/RTH]]-->

=== 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 DFT of <math>4N</math> real inputs of even symmetry, where the even-indexed elements are zero. That is, it is half of the 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}}<!--[[User:Kvng/RTH]]-->

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 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>