Discrete cosine transform

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.

From the perspective of algebraic signal processing, the relationship between the DCT and the DFT reflects their different underlying algebraic structures: the DFT basis functions are the irreducible representations of the cyclic group, making the DFT the natural spectral transform for signals with periodic (circular) boundary conditions, while the DCT basis functions are the irreducible representations of the dihedral group, making the DCT natural for signals with symmetric (even) boundary conditions.

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), However, blocky compression artifacts can appear when heavy DCT compression is applied.

History

The DCT was first conceived by Nasir Ahmed 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. It described what is now called the type-II DCT (DCT-II),

The discrete sine transform (DST) was derived from the DCT, by replacing the Neumann condition at x=0 with a Dirichlet condition.

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. In 1979, Anil K. Jain and Jaswant R. Jain further developed motion-compensated DCT video compression, also called block motion compensation.

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, following earlier work by Princen and Bradley in 1986. The MDCT is used in most modern audio compression formats, such as Dolby Digital (AC-3), Advanced Audio Coding (AAC), and Vorbis (Ogg). Lossless DCT is also known as LDCT.

Applications

The DCT is the most widely used transformation technique in signal processing, and by far the most widely used linear transform in data compression. Uncompressed digital media as well as lossless compression have high memory and bandwidth requirements, which is significantly reduced by the DCT lossy compression technique, digital video, streaming media, digital television, streaming television, video on demand (VOD),

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. surround sound,

Speech processing — speech coding focus and blurriness measure,
Image compression — digital cinematography, digital movie cameras, video editing, film editing, Dolby Digital audio professional video production
Wireless sensor network (WSN) — wireless acoustic sensor networks

Image formats

{| class="wikitable"

! Image compression standard !! Year !! Common applications

| JPEG and digital image format.

| 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, and supports animation with much more efficient compression than the animated GIF format.

| BPG ||2014||Based on HEVC compression

| JPEG XL ||2020|| A royalty-free raster-graphics file format that supports both lossy and lossless compression.

Video formats

{| class="wikitable"

! Video coding standard !! Year !! Common applications

| ||1988|| First of a family of video coding standards. Used primarily in older video conferencing and video telephone products.

| Motion JPEG (MJPEG) ||1992|| QuickTime, video editing, non-linear editing, digital cameras

| MPEG-1 Video ||1993|| Digital video distribution on CD or Internet video

| MPEG-2 Video ()

| Advanced Video Coding (AVC, , MPEG-4) video telephony, FaceTime

| Theora ||2004|| Internet video, web browsers

| VC-1 ||2006|| Windows media, Blu-ray Discs

| Apple ProRes ||2007|| Professional video production.

| VP9||2010|| A video codec developed by Google used in the WebM container format with HTML5.

| High Efficiency Video Coding (HEVC, ) ||2018|| An open source format based on VP10 (VP9's internal successor), Daala and Thor; used by content providers such as YouTube and Netflix.

MDCT audio standards

General audio

{| class="wikitable"

! Audio compression standard

! Year

! Common applications

| Dolby Digital (AC-3) 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, lapped orthogonal transform and cosine-modulated wavelet bases.

Digital signal processing

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. caused by DCT blocks. 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 are quantized. This process can cause blocking artifacts, primarily at high data compression ratios.

DCT blocks are often used in glitch art. particularly the DCT blocks found in most digital media formats such as JPEG digital images and MP3 audio.

Informal overview

Like any Fourier-related transform, DCTs express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes. Like the 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 <math>f(x)</math> as a sum of sinusoids, you can evaluate that sum at any <math>x</math>, even for <math>x</math> where the original <math>f(x)</math> 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.

thumb|right|350px|Illustration of the implicit even/odd extensions of DCT input data, for N=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 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).

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 also discrete sine transforms (DSTs). 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. 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. 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.

Formal definition

Formally, the discrete cosine transform is a linear, invertible function <math> f : \R^{N} \to \R^{N} </math> (where <math> \R</math> denotes the set of real numbers), or equivalently an invertible × square matrix. There are several variants of the DCT with slightly modified definitions. The real numbers <math>~ x_0,\ \ldots\ x_{N - 1} ~</math> are transformed into the real numbers <math> X_0,\, \ldots,\, X_{N - 1} </math> according to one of the formulas:

DCT-I

:<math>X_k

= \frac{1}{2} (x_0 + (-1)^k x_{N-1})

+ \sum_{n=1}^{N-2} x_n \cos \left[\, \tfrac{\ \pi}{\,N-1\,} \, n \, k \,\right]

\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 but breaks the direct correspondence with a real-even DFT.

The DCT-I is exactly equivalent (up to an overall scale factor of 2), to a 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 (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

:<math>X_k =

\sum_{n=0}^{N-1} x_n \cos \left[\, \tfrac{\,\pi\,}{N} \left( n + \tfrac{1}{2} \right) k \, \right]

\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. 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), and a scaling can be chosen that allows the DCT to be computed with fewer multiplications.

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

:<math> X_k =

\tfrac{1}{2} x_0 +

\sum_{n=1}^{N-1} x_n \cos \left[\, \tfrac{\,\pi\,}{N} \left( k + \tfrac{1}{2} \right) n \,\right]

\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).

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>.

DCT V-VIII

DCTs of types I–IV treat both boundaries consistently regarding the point of symmetry: they are even or 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 or 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 <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 and III) or <math>8 N</math> (for DCT-IV). The four additional types of discrete cosine transform 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 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. Not that the trivial real-even array, a length-one DFT (odd length) of a single number <math>a</math>, corresponds to a DCT-V of length <math>N = 1</math>.

Inverse transforms

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. The transform size N × N × N is assumed to be 2.

thumb|200px|The four basic stages of computing 3-D DCT-II using the VR DIF Algorithm.

:<math>

\begin{array}{lcl}\tilde{x}(n_1,n_2,n_3) =x(2n_1,2n_2,2n_3)\\

\tilde{x}(n_1,n_2,N-n_3-1)=x(2n_1,2n_2,2n_3+1)\\

\tilde{x}(n_1,N-n_2-1,n_3)=x(2n_1,2n_2+1,2n_3)\\

\tilde{x}(n_1,N-n_2-1,N-n_3-1)=x(2n_1,2n_2+1,2n_3+1)\\

\tilde{x}(N-n_1-1,n_2,n_3)=x(2n_1+1,2n_2,2n_3)\\

\tilde{x}(N-n_1-1,n_2,N-n_3-1)=x(2n_1+1,2n_2,2n_3+1)\\

\tilde{x}(N-n_1-1,N-n_2-1,n_3)=x(2n_1+1,2n_2+1,2n_3)\\

\tilde{x}(N-n_1-1,N-n_2-1,N-n_3-1)=x(2n_1+1,2n_2+1,2n_3+1)\\

\end{array}

</math>

:where <math>0\leq n_1,n_2,n_3 \leq \frac{N}{2} -1</math>

The adjacent figure 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 <math>c(\varphi_i)=\cos(\varphi_i)</math>.

The original 3-D DCT-II can now be written as

:<math>X(k_1,k_2,k_3)=\sum_{n_1=1}^{N-1}\sum_{n_2=1}^{N-1}\sum_{n_3=1}^{N-1}\tilde{x}(n_1,n_2,n_3) \cos(\varphi k_1)\cos(\varphi k_2)\cos(\varphi k_3)

</math>

:where <math>\varphi_i= \frac{\pi}{2N}(4N_i+1),\text{ and } i= 1,2,3.</math>

If the even and the odd parts of <math>k_1,k_2</math> and <math>k_3</math> are considered, the general formula for the calculation of the 3-D DCT-II can be expressed as

thumb|900px|The single butterfly stage of the VR DIF algorithm.

:<math>X(k_1,k_2,k_3)=\sum_{n_1=1}^{\tfrac N 2 -1}\sum_{n_2=1}^{\tfrac N 2 -1}\sum_{n_1=1}^{\tfrac N 2 -1}\tilde{x}_{ijl}(n_1,n_2,n_3) \cos(\varphi (2k_1+i)\cos(\varphi (2k_2+j)

\cos(\varphi (2k_3+l))</math>

: where

:: <math>\tilde{x}_{ijl}(n_1,n_2,n_3)=\tilde{x}(n_1,n_2,n_3)+(-1)^l\tilde{x}\left(n_1,n_2,n_3+\frac{n}{2}\right) </math>

:: <math>+(-1)^j\tilde{x}\left(n_1,n_2+\frac{n}{2},n_3\right)+(-1)^{j+l}\tilde{x}\left(n_1,n_2+\frac{n}{2},n_3+\frac{n}{2}\right) </math>

:: <math>+(-1)^i\tilde{x}\left(n_1+\frac{n}{2},n_2,n_3\right)+(-1)^{i+j}\tilde{x}\left(n_1+\frac{n}{2}+\frac{n}{2},n_2,n_3\right) </math>

:: <math>+(-1)^{i+l}\tilde{x}\left(n_1+\frac{n}{2},n_2,n_3+\frac{n}{3}\right)</math>

:: <math>+(-1)^{i+j+l}\tilde{x}\left(n_1+\frac{n}{2},n_2+\frac{n}{2},n_3+\frac{n}{2}\right) \text{ where } i,j,l= 0 \text{ or } 1.</math>

Arithmetic complexity

The whole 3-D DCT calculation needs <math>~ [\log_2 N] ~</math> stages, and each stage involves <math>~ \tfrac{1}{8}\ N^3 ~</math> butterflies. The whole 3-D DCT requires <math>~ \left[ \tfrac{1}{8}\ N^3 \log_2 N \right] ~</math> 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 <math>~ \left[ \tfrac{7}{8}\ N^3\ \log_2 N \right] ~,</math> 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 Therefore, although the above proposed 3-D VR algorithm does not achieve the theoretical lower bound on the number of multiplications, 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 characterizes butterfly-style Cooley–Tukey FFT algorithms.

thumb|250px|Two-dimensional DCT frequencies from the [[JPEG#Discrete cosine transform|JPEG DCT]]

The image to the right shows a combination of horizontal and vertical frequencies for an <math>(~ N_1 = N_2 = 8 ~)</math> two-dimensional DCT. Each step from left to right and top to bottom is an increase in frequency by a half-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 () is transformed to a linear combination of these 64 frequency squares.

MD DCT-IV

The MD DCT-IV is an extension of 1-D DCT-IV onto an -dimensional domain. The 2-D DCT-IV of a matrix or an image is given by

:<math> X_{k,\ell} =

\sum_{n=0}^{N-1} \; \sum_{m=0}^{M-1} \ x_{n,m} \cos\left(\ \frac{\,( 2 m + 1 )( 2 k + 1 )\ \pi \,}{4N} \ \right) \cos\left(\ \frac{\, ( 2n + 1 )( 2 \ell + 1 )\ \pi \,}{4M} \ \right) ~,</math>

: for <math>~~ k = 0,\ 1,\ 2\ \ldots\ N-1 ~~</math> and <math>~~ \ell= 0,\ 1,\ 2,\ \ldots\ M-1 ~.</math>

We can compute the MD DCT-IV using the regular row-column method, or we can use the polynomial transform method for fast and efficient computation. The main idea of this algorithm is to use the polynomial transform to directly convert the multidimensional DCT into a series of 1-D DCTs.

Computation

Although the direct application of these formulas would require <math>~ \mathcal{O}(N^2) ~</math> operations, it is possible to compute the same thing with only <math>~ \mathcal{O}(N \log N ) ~</math> complexity by factorizing the computation similarly to the fast Fourier transform (FFT). One can also compute DCTs via FFTs combined with <math>~\mathcal{O}(N)~</math> pre- and post-processing steps. In general, <math>~\mathcal{O}(N \log N )~</math> 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 <math>~ \mathcal{O}(N) ~</math> 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 power-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 . 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 minimal-multiplication algorithms for the DFT, albeit generally at the cost of more additions, and a similar algorithm was proposed by 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 .

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 <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 FFTW) was described by and , 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.

Because the even-indexed elements are zero, this radix-4 step is exactly the same as a split-radix step. If the subsequent size <math>~ N ~</math> real-data FFT is also performed by a real-data split-radix algorithm (as in ), then the resulting algorithm actually matches what was long the lowest published arithmetic count for the power-of-two DCT-II (<math>~ 2 N \log_2 N - N + 2 ~</math> real-arithmetic operations).

A recent reduction in the operation count to <math>~ \tfrac{17}{9} N \log_2 N + \mathcal{O}(N)</math> also uses a real-data FFT. 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 <math>~ N ~,</math> but this is an implementation rather than an algorithmic question since it can be solved by unrolling or inlining.)

Example of IDCT

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 grayscale image of capital letter A.

frame|center|Original size, scaled 10x (nearest neighbor), scaled 10x (bilinear).

[[File:dct-table.png|frame|center|Basis functions of the discrete cosine transformation with corresponding coefficients (specific for our image).

<br/>DCT of the image =

<math>

\begin{bmatrix}

6.1917 & -0.3411 & 1.2418 & 0.1492 & 0.1583 & 0.2742 & -0.0724 & 0.0561 \\

0.2205 & 0.0214 & 0.4503 & 0.3947 & -0.7846 & -0.4391 & 0.1001 & -0.2554 \\

1.0423 & 0.2214 & -1.0017 & -0.2720 & 0.0789 & -0.1952 & 0.2801 & 0.4713 \\

-0.2340 & -0.0392 & -0.2617 & -0.2866 & 0.6351 & 0.3501 & -0.1433 & 0.3550 \\

0.2750 & 0.0226 & 0.1229 & 0.2183 & -0.2583 & -0.0742 & -0.2042 & -0.5906 \\

0.0653 & 0.0428 & -0.4721 & -0.2905 & 0.4745 & 0.2875 & -0.0284 & -0.1311 \\

0.3169 & 0.0541 & -0.1033 & -0.0225 & -0.0056 & 0.1017 & -0.1650 & -0.1500 \\

-0.2970 & -0.0627 & 0.1960 & 0.0644 & -0.1136 & -0.1031 & 0.1887 & 0.1444 \\

\end{bmatrix}

</math>.]]

Each basis function is multiplied by its coefficient and then this product is added to the final image.

frame|center|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×.

Notes

References

</references>

External links

Syed Ali Khayam: The Discrete Cosine Transform (DCT): Theory and Application
Implementation of MPEG integer approximation of 8x8 IDCT (ISO/IEC 23002-2)
Matteo Frigo and Steven 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.

Discrete cosine transform

History

Applications

Image formats

Video formats

MDCT audio standards

General audio

Digital signal processing

Informal overview

Formal definition

DCT-I

DCT-II

DCT-III

DCT V-VIII

Inverse transforms

Arithmetic complexity

MD DCT-IV

Computation

Example of IDCT

See also

Notes

References

Further reading

External links