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), 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 AAC-LD, 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 Fourier-related 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 type-II DCT, which is often called simply the DCT. This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT, is correspondingly often called simply the inverse DCT or the IDCT. Two related transforms are the discrete sine 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), an integer approximation of the standard DCT, used in several ISO/IEC and ITU-T international standards.
DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks. DCT blocks sizes including 8x8 pixels for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels. The DCT has a strong energy compaction property, capable of achieving high quality at high data compression ratios.[1] However, blocky compression artifacts can appear when heavy DCT compression is applied.
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. 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. They presented their results in a January 1974 paper, titled Discrete Cosine Transform.[2] It described what is now called the type-II DCT (DCT-II), as well as the type-III inverse DCT (IDCT).
Since its introduction in 1974, there has been significant research on the DCT.[2] In 1977, Wen-Hsiung Chen published a paper with C. Harrison Smith and Stanley C. Fralick presenting a fast DCT algorithm.[3] [2] Further developments include a 1978 paper by M. J. Narasimha and A. M. Peterson, and a 1984 paper by B. G. Lee.[2] 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.[2] [4]
The discrete sine transform (DST) was derived from the DCT, by replacing the Neumann condition at x=0 with a Dirichlet condition. The DST was described in the 1974 DCT paper by Ahmed, Natarajan and Rao. A type-I DST (DST-I) was later described by Anil K. Jain in 1976, and a type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.[5]
In 1975, John A. Roese and Guner S. Robinson adapted the DCT for inter-frame motion-compensated video coding. They experimented with the DCT and the fast 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 per pixel for a videotelephone scene with image quality comparable to an intra-frame coder requiring 2-bit per pixel.[6] [7] In 1979, Anil K. Jain and Jaswant R. Jain further developed motion-compensated DCT video compression,[8] [9] also called block motion compensation.[9] This led to Chen developing a practical video compression algorithm, called motion-compensated DCT or adaptive scene coding, in 1981.[9] Motion-compensated DCT later became the standard coding technique for video compression from the late 1980s onwards.[10]
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,[11] following earlier work by Princen and Bradley in 1986.[12] The MDCT is used in most modern audio compression formats, such as Dolby Digital (AC-3), MP3 (which uses a hybrid DCT-FFT algorithm),[13] Advanced Audio Coding (AAC),[14] and Vorbis (Ogg).
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.[15] Lossless DCT is also known as LDCT.[16]
The DCT is the most widely used transformation technique in signal processing,[17] and by far the most widely used linear transform in data compression.[18] Uncompressed digital media as well as lossless compression have high memory and bandwidth requirements, which is significantly reduced by the DCT lossy compression technique,[19] [1] capable of achieving data compression ratios from 8:1 to 14:1 for near-studio-quality,[19] up to 100:1 for acceptable-quality content.[1] DCT compression standards are used in digital media technologies, such as digital images, digital photos,[20] [21] digital video,[22] [23] streaming media,[24] digital television, streaming television, video on demand (VOD),[1] digital cinema, high-definition video (HD video), and high-definition television (HDTV).[19] [25]
The DCT, and in particular the DCT-II, is often used in signal and image processing, especially for lossy compression, because it has a strong energy compaction property. In typical applications, most of the signal information tends to be concentrated in a few low-frequency components of the DCT. For strongly correlated Markov processes, the DCT can approach the compaction efficiency of the Karhunen-Loève transform (which is optimal in the decorrelation sense). As explained below, this stems from the boundary conditions implicit in the cosine functions.
DCTs are 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.
The DCT is widely used in many applications, which include the following.
The DCT-II is an important image compression technique. It is used in image compression standards such as JPEG, and video compression standards such as, MJPEG, MPEG, DV, Theora and Daala. There, the two-dimensional DCT-II of
N x N
N
(0,0)
The integer DCT, an integer approximation of the DCT, is used in Advanced Video Coding (AVC),[26] introduced in 2003, and High Efficiency Video Coding (HEVC), 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. AVC uses 4 x 4 and 8 x 8 blocks. HEVC and HEIF use varied block sizes between 4 x 4 and 32 x 32 pixels., AVC is by far the most commonly used format for the recording, compression and distribution of video content, used by 91% of video developers, followed by HEVC which is used by 43% of developers.[27]
| Image compression standard | Year | Common applications | |
|---|---|---|---|
| 1992 | The most widely used image compression standard[28] [29] and digital image format.[30] | ||
| 2009 | Open XML Paper Specification | ||
| 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,[31] and supports animation with much more efficient compression than the animated GIF format.[32] | |
| 2014 | Based on HEVC compression | ||
| JPEG XL[33] | 2020 | A royalty-free raster-graphics file format that supports both lossy and lossless compression. |
| Year | Common applications | ||
|---|---|---|---|
| [34] [35] | 1988 | First of a family of video coding standards. Used primarily in older video conferencing and video telephone products. | |
| Motion JPEG (MJPEG)[36] | 1992 | QuickTime, video editing, non-linear editing, digital cameras | |
| MPEG-1 Video[37] | 1993 | Digital video distribution on CD or Internet video | |
| MPEG-2 Video | 1995 | Storage and handling of digital images in broadcast applications, digital television, HDTV, cable, satellite, high-speed Internet, DVD video distribution | |
| 1995 | Camcorders, digital cassettes | ||
| H.263 (MPEG-4 Part 2) | 1996 | Video telephony over public switched telephone network (PSTN),, Integrated Services Digital Network (ISDN)[38] [39] | |
| Advanced Video Coding (AVC,, MPEG-4) | 2003 | Popular HD video recording, compression and distribution format, Internet video, YouTube, Blu-ray Discs, HDTV broadcasts, web browsers, streaming television, mobile devices, consumer devices, Netflix,[40] video telephony, FaceTime[41] | |
| 2004 | Internet video, web browsers | ||
| 2006 | Windows media, Blu-ray Discs | ||
| 2007 | Professional video production.[42] | ||
| VP9 | 2010 | A video codec developed by Google used in the WebM container format with HTML5. | |
| High Efficiency Video Coding (HEVC,) | 2013 | Successor to the standard, having substantially improved compression capability | |
| 2013 | Research video format by Xiph.org | ||
| AV1[43] | 2018 | An open source format based on VP10 (VP9's internal successor), Daala and Thor; used by content providers such as YouTube[44] [45] and Netflix.[46] [47] |
| Speech coding standard | Year | Common applications | |
|---|---|---|---|
| AAC-LD (LD-MDCT) | 1999 | Mobile telephony, voice-over-IP (VoIP), iOS, FaceTime | |
| Siren | 1999 | VoIP, wideband audio, G.722.1 | |
| G.722.1 | 1999 | VoIP, wideband audio, G.722 | |
| G.729.1 | 2006 | G.729, VoIP, wideband audio, mobile telephony | |
| EVRC-WB | 2007 | Wideband audio | |
| G.718 | 2008 | VoIP, wideband audio, mobile telephony | |
| G.719 | 2008 | Teleconferencing, videoconferencing, voice mail | |
| CELT | 2011 | VoIP, mobile telephony | |
| Enhanced Voice Services (EVS) | 2014 | Mobile telephony, VoIP, wideband audio |
See also: ZPEG.
Multidimensional DCTs (MD DCTs) have several applications, mainly 3-D DCTs such as the 3-D DCT-II, which has several new applications like Hyperspectral Imaging coding systems, variable temporal length 3-D DCT coding, video coding algorithms, adaptive video coding and 3-D Compression. Due to enhancement in the hardware, software and introduction of several fast algorithms, the necessity of using MD DCTs is rapidly increasing. DCT-IV has gained popularity for its applications in fast implementation of real-valued polyphase filtering banks,[48] lapped orthogonal transform[49] and cosine-modulated wavelet bases.[50]
DCT plays an important role in digital signal processing specifically data compression. The DCT is widely implemented in digital signal processors (DSP), as well as digital signal processing software. Many companies have developed DSPs based on DCT technology. DCTs are widely used for applications such as encoding, decoding, video, audio, multiplexing, control signals, signaling, and analog-to-digital conversion. DCTs are also commonly used for high-definition television (HDTV) encoder/decoder chips.
A common issue with DCT compression in digital media are blocky compression artifacts,[51] caused by DCT blocks.[52] The DCT algorithm can cause block-based artifacts when heavy compression is applied. Due to the DCT being used in the majority of digital image and video coding standards (such as the JPEG, and MPEG formats), DCT-based 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.[51] This can also cause the "mosquito noise" effect, commonly found in digital video (such as the MPEG formats).[53]
DCT blocks are often used in glitch art.[52] The artist Rosa Menkman makes use of DCT-based compression artifacts in her glitch art,[54] particularly the DCT blocks found in most digital media formats such as JPEG digital images and MP3 digital audio.[52] Another example is Jpegs by German photographer Thomas Ruff, which uses intentional JPEG artifacts as the basis of the picture's style.[55] [56]
Like any Fourier-related transform, discrete cosine transforms (DCTs) express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes. Like the discrete Fourier transform (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 Fourier-related 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
f(x)
x
x
f(x)
However, because DCTs operate on finite, discrete sequences, two issues arise that do not apply for the continuous cosine transform. First, one has to specify whether the function is even or odd at both the left and right boundaries of the domain (i.e. the min-n and max-n boundaries in the definitions below, respectively). Second, one has to specify around what point the function is even or odd. In particular, consider a sequence abcd of four equally spaced data points, and say that we specify an even left boundary. There are two sensible possibilities: either the data are even about the sample a, in which case the even extension is dcbabcd, or the data are even about the point halfway between a and the previous point, in which case the even extension is dcbaabcd (a is repeated).
These choices lead to all the standard variations of DCTs and also discrete sine 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 Fourier-related 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 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 discontinuities in a function reduce the rate of convergence of the Fourier series, so that more sinusoids are needed to represent the function with a given accuracy. The same principle governs the usefulness of the DFT and other transforms for signal compression; the smoother a function is, the fewer terms in its DFT or DCT are required to represent it accurately, and the more it can be compressed. (Here, we think of the DFT or DCT as approximations for the Fourier series or cosine series of a function, respectively, in order to talk about its "smoothness".) However, the implicit periodicity of the DFT means that discontinuities usually occur at the boundaries: any random segment of a signal is unlikely to have the same value at both the left and right boundaries. (A similar problem arises for the DST, in which the odd left boundary condition implies a discontinuity for any function that does not happen to be zero at that boundary.) In contrast, a DCT where both boundaries are even always yields a continuous extension at the boundaries (although the slope is generally discontinuous). This is why DCTs, and in particular DCTs of types I, II, V, and VI (the types that have two even boundaries) generally perform better for signal compression than DFTs and DSTs. In practice, a type-II DCT is usually preferred for such applications, in part for reasons of computational convenience.
f:\RN\to\RN
\R
~x0, \ldots xN~
X0,\ldots,XN
Xk =
| 1 | |
| 2 |
(x0+(-1)kxN-1)+
| N-2 | |
| \sum | |
| n=1 |
xn\cos\left[
| \pi | |
| N-1 |
nk\right] for~k=0, \ldots N-1~.
Some authors further multiply the
x0
xN-1
\sqrt{2},
X0
XN-1
1/\sqrt{2},
\sqrt{\tfrac{2}{N-1}},
The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a DFT of
2(N-1)
N=5
a b c d e
Note, however, that the DCT-I is not defined for
N
N.
Thus, the DCT-I corresponds to the boundary conditions:
xn
n=0
n=N-1
Xk.
Xk=
| N-1 | |
| \sum | |
| n=0 |
xn\cos\left[\tfrac{\pi}{N}\left(n+\tfrac{1}{2}\right)k\right] for~k=0, ... N-1~.
The DCT-II is probably the most commonly used form, and is often simply referred to as "the DCT".
This transform is exactly equivalent (up to an overall scale factor of 2) to a DFT of
4N
4N
yn,
y2n=0,
y2n+1=xn
0\leqn<N,
y2N=0,
y4N-n=yn
0<n<2N.
Some authors further multiply the
X0
1/\sqrt{N}
The DCT-II implies the boundary conditions:
xn
n=-1/2
n=N-1/2;
Xk
k=0
k=N.
Xk= \tfrac{1}{2}x0+
| N-1 | |
| \sum | |
| n=1 |
xn\cos\left[\tfrac{\pi}{N}\left(k+\tfrac{1}{2}\right)n\right] for~k=0, \ldots N-1~.
Because it is the inverse of DCT-II up to a scale factor (see below), this form is sometimes simply referred to as "the inverse DCT" ("IDCT").
Some authors divide the
x0
\sqrt{2}
x0/\sqrt{2}
The DCT-III implies the boundary conditions:
xn
n=0
n=N;
Xk
k=-1/2
k=N-1/2.
Xk=
| N-1 | |
| \sum | |
| n=0 |
xn\cos\left[\tfrac{\pi}{N}\left(n+\tfrac{1}{2}\right)\left(k+\tfrac{1}{2}\right)\right] fork=0, \ldots N-1~.
The DCT-IV 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 DCT-IV, where data from different transforms are overlapped, is called the modified discrete cosine transform (MDCT).
The DCT-IV implies the boundary conditions:
xn
n=-1/2
n=N-1/2;
Xk.
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
N
2(N-1)
4N
8N
N\pm{1}/{2}
However, these variants seem to be rarely used in practice. One reason, perhaps, is that FFT algorithms for odd-length DFTs are generally more complicated than FFT algorithms for even-length DFTs (e.g. the simplest radix-2 algorithms are only for even lengths), and this increased intricacy carries over to the DCTs as described below.
(The trivial real-even array, a length-one DFT (odd length) of a single number , corresponds to a DCT-V of length
N=1.
Using the normalization conventions above, the inverse of DCT-I is DCT-I multiplied by 2/(N − 1). The inverse of DCT-IV is DCT-IV multiplied by 2/N. The inverse of DCT-II is DCT-III multiplied by 2/N and vice versa.
Like for the 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 (see above), this can be used to make the transform matrix orthogonal.
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):
| \begin{align} X | |
| k1,k2 |
&=
| N1-1 | |
| \sum | |
| n1=0 |
\left(
| N2-1 | |
| \sum | |
| n2=0 |
| x | |
| n1,n2 |
\cos\left[
| \pi | |
| N2 |
| \left(n | ||||
|
\right)k2\right]\right) \cos\left[
| \pi | |
| N1 |
| \left(n | ||||
|
\right)k1\right]\\ &=
| N1-1 | |
| \sum | |
| n1=0 |
| N2-1 | |
| \sum | |
| n2=0 |
| x | |
| n1,n2 |
\cos\left[
| \pi | |
| N1 |
| \left(n | ||||
|
\right)k1\right] \cos\left[
| \pi | |
| N2 |
| \left(n | ||||
|
\right)k2\right]. \end{align}
The inverse of a multi-dimensional DCT is just a separable product of the inverses of the corresponding one-dimensional DCTs (see above), e.g. the one-dimensional inverses applied along one dimension at a time in a row-column algorithm.
The 3-D DCT-II is only the extension of 2-D DCT-II in three dimensional space and mathematically can be calculated by the formula
| X | |
| k1,k2,k3 |
=
| N1-1 | |
| \sum | |
| n1=0 |
| N2-1 | |
| \sum | |
| n2=0 |
| N3-1 | |
| \sum | |
| n3=0 |
| x | |
| n1,n2,n3 |
\cos\left[
| \pi | |
| N1 |
| \left(n | ||||
|
\right)k1\right] \cos\left[
| \pi | |
| N2 |
| \left(n | ||||
|
\right)k2\right] \cos\left[
| \pi | |
| N3 |
| \left(n | ||||
|
\right)k3\right], forki=0,1,2,...,Ni-1.
The inverse of 3-D DCT-II is 3-D DCT-III and can be computed from the formula given by
| x | |
| n1,n2,n3 |
=
| N1-1 | |
| \sum | |
| k1=0 |
| N2-1 | |
| \sum | |
| k2=0 |
| N3-1 | |
| \sum | |
| k3=0 |
| X | |
| k1,k2,k3 |
\cos\left[
| \pi | |
| N1 |
| \left(n | ||||
|
\right)k1\right] \cos\left[
| \pi | |
| N2 |
| \left(n | ||||
|
\right)k2\right] \cos\left[
| \pi | |
| N3 |
| \left(n | ||||
|
\right)k3\right], forni=0,1,2,...,Ni-1.
Technically, computing a two-, three- (or -multi) dimensional DCT by sequences of one-dimensional DCTs along each dimension is known as a row-column algorithm. As with multidimensional FFT algorithms, however, there exist other methods to compute the same thing while performing the computations in a different order (i.e. interleaving/combining the algorithms for the different dimensions). Owing to the rapid growth in the applications based on the 3-D DCT, several fast algorithms are developed for the computation of 3-D DCT-II. Vector-Radix algorithms are applied for computing M-D DCT to reduce the computational complexity and to increase the computational speed. To compute 3-D DCT-II efficiently, a fast algorithm, Vector-Radix Decimation in Frequency (VR DIF) algorithm was developed.
In order to apply the VR DIF algorithm the input data is to be formulated and rearranged as follows.[61] [62] The transform size N × N × N is assumed to be 2.
\begin{array}{lcl}\tilde{x}(n1,n2,n3)=x(2n1,2n2,2n3)\ \tilde{x}(n1,n2,N-n3-1)=x(2n1,2n2,2n3+1)\\ \tilde{x}(n1,N-n2-1,n3)=x(2n1,2n2+1,2n3)\\ \tilde{x}(n1,N-n2-1,N-n3-1)=x(2n1,2n2+1,2n3+1)\\ \tilde{x}(N-n1-1,n2,n3)=x(2n1+1,2n2,2n3)\\ \tilde{x}(N-n1-1,n2,N-n3-1)=x(2n1+1,2n2,2n3+1)\\ \tilde{x}(N-n1-1,N-n2-1,n3)=x(2n1+1,2n2+1,2n3)\\ \tilde{x}(N-n1-1,N-n2-1,N-n3-1)=x(2n1+1,2n2+1,2n3+1)\\ \end{array}
where
0\leqn1,n2,n3\leq
| N | |
| 2 |
-1
The figure to the adjacent shows the four stages that are involved in calculating 3-D DCT-II using VR DIF algorithm. The first stage is the 3-D reordering using the index mapping illustrated by the above equations. The second stage is the butterfly calculation. Each butterfly calculates eight points together as shown in the figure just below, where
c(\varphii)=\cos(\varphii)
The original 3-D DCT-II now can be written as
X(k1,k2,k3)=\sum
| N-1 | |
| n1=1 |
| N-1 | |
| \sum | |
| n2=1 |
| N-1 | |
| \sum | |
| n3=1 |
\tilde{x}(n1,n2,n3)\cos(\varphik1)\cos(\varphik2)\cos(\varphik3)
where
\varphii=
| \pi | |
| 2N |
(4Ni+1),andi=1,2,3.
If the even and the odd parts of
k1,k2
k3
X(k1,k2,k3)=\sum
| \tfracN2-1 | |
| n1=1 |
| \tfracN2-1 | |
| \sum | |
| n2=1 |
| \tfracN2-1 | |
| \sum | |
| n1=1 |
\tilde{x}ijl(n1,n2,n3)\cos(\varphi(2k1+i)\cos(\varphi(2k2+j) \cos(\varphi(2k3+l))
where
\tilde{x}ijl(n1,n2,n3)=\tilde{x}(n1,n2,n
| l\tilde{x}\left(n | |
| 1,n |
2,n
|
\right)
| j\tilde{x}\left(n | |
| +(-1) | |
| 1,n |
|
| j+l | |
| ,n | |
| 3\right)+(-1) |
\tilde{x}\left(n1,n
|
| ,n | ||||
|
\right)
| i\tilde{x}\left(n | ||||
| +(-1) | ||||
|
,n2,n
| i+j | |
| 3\right)+(-1) |
| \tilde{x}\left(n | + | ||||
|
| n | |
| 2 |
,n2,n3\right)
+(-1)i+l
| \tilde{x}\left(n | ||||
|
,n2,n
|
\right)
+(-1)i+j+l
| \tilde{x}\left(n | ||||
|
| ,n | ||||
|
| ,n | ||||
|
\right)wherei,j,l=0or1.
The whole 3-D DCT calculation needs
~[log2N]~
~\tfrac{1}{8} N3~
~\left[\tfrac{1}{8} N3log2N\right]~
~\left[
| 3 log | |
| \tfrac{7}{8} N | |
| 2 |
N\right]~,
~\underbrace{\left[
| 3 | |
| 2 |
N3log2N\right]}
|
N3
| 3+3N | |
| log | |
| 2N-3N |
| 2\right]} | |
| Recursive |
=\left[
| 9 | |
| 2 |
N3
| 3+3N | |
| log | |
| 2N-3N |
2\right]~.
The conventional method to calculate MD-DCT-II is using a Row-Column-Frame (RCF) approach which is computationally complex and less productive on most advanced recent hardware platforms. The number of multiplications required to compute VR DIF Algorithm when compared to RCF algorithm are quite a few in number. The number of Multiplications and additions involved in RCF approach are given by
| ~\left[ | 3 |
| 2 |
N3log2N\right]~
~\left[
| 9 | |
| 2 |
N3log2N-3N3+3N2\right]~,
| 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 |
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.[63] Therefore, although the above proposed 3-D VR algorithm does not achieve the theoretical lower bound on the number of multiplications,[64] it has a simpler computational structure as compared to other 3-D DCT algorithms. It can be implemented in place using a single butterfly and possesses the properties of the Cooley–Tukey FFT algorithm in 3-D. Hence, the 3-D VR presents a good choice for reducing arithmetic operations in the calculation of the 3-D DCT-II, while keeping the simple structure that characterize butterfly-style Cooley–Tukey FFT algorithms.
The image to the right shows a combination of horizontal and vertical frequencies for an
(~N1=N2=8~)
The M-D DCT-IV is just an extension of 1-D DCT-IV on to dimensional domain. The 2-D DCT-IV of a matrix or an image is given by
Xk,\ell=
| N-1 | |
| \sum | |
| n=0 |
| M-1 | |
| \sum | |
| m=0 |
xn,m\cos\left(
| (2m+1)(2k+1) \pi | |
| 4N |
\right)\cos\left(
| (2n+1)(2\ell+1) \pi | |
| 4M |
\right)~,
for
~~k=0, 1, 2 \ldots N-1~~
~~\ell=0, 1, 2, \ldots M-1~.
We can compute the MD DCT-IV using the regular row-column method or we can use the polynomial transform method[65] 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
~l{O}(N2)~
~l{O}(NlogN)~
~l{O}(N)~
~l{O}(NlogN)~
The most efficient algorithms, in principle, are usually those that are specialized directly for the DCT, as opposed to using an ordinary FFT plus
~l{O}(N)~
While DCT algorithms that employ an unmodified FFT often have some theoretical overhead compared to the best specialized DCT algorithms, the former also have a distinct advantage: Highly optimized FFT programs are widely available. Thus, in practice, it is often easier to obtain high performance for general lengths with FFT-based algorithms.Specialized DCT algorithms, on the other hand, see widespread use for transforms of small, fixed sizes such as the DCT-II used in JPEG compression, or the small DCTs (or MDCTs) typically used in audio compression. (Reduced code size may also be a reason to use a specialized DCT for embedded-device applications.)
In fact, even the DCT algorithms using an ordinary FFT are sometimes equivalent to pruning the redundant operations from a larger FFT of real-symmetric data, and they can even be optimal from the perspective of arithmetic counts. For example, a type-II DCT is equivalent to a DFT of size
~4N~
~N~
~2Nlog2N-N+2~
A recent reduction in the operation count to
~\tfrac{17}{9}Nlog2N+l{O}(N)
~N~,
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.
