Discrete cosine transform

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. Image formats Video formats MDCT audio standards General audio Speech coding Multidimensional DCT 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, 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 f(x) as a sum of sinusoids, you can evaluate that sum at any x, even for x where the original f(x) 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. 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 f : \R^{N} \to \R^{N} (where \R 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 ~ x_0,\ \ldots\ x_{N - 1} ~ are transformed into the real numbers X_0,\, \ldots,\, X_{N - 1} according to one of the formulas: DCT-I :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 ~. Some authors further multiply the x_0 and x_{N-1} terms by \sqrt{2\,}\, and correspondingly multiply the X_0 and X_{N-1} terms by 1/\sqrt{2\,} \, which, if one further multiplies by an overall scale factor of \sqrt{\tfrac{2}{N-1\,}\,}, 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 2(N-1) real numbers with even symmetry. For example, a DCT-I of N = 5 real numbers a\ b\ c\ d\ e 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 N less than 2, while all other DCT types are defined for any positive N. Thus, the DCT-I corresponds to the boundary conditions: x_n is even around n = 0 and even around n = N - 1; similarly for X_k. DCT-II :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 ~. 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: x_n is even around n = -1/2 and even around n = N - 1/2 \,; X_k is even around k = 0 and odd around k = N. DCT-III : 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 ~. 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: x_n is even around n = -1/2 and odd around n = N - 1/2; similarly for X_k. 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 N is even or odd), since the corresponding DFT is of length 2(N-1) (for DCT-I) or 4 N (for DCT-II and III) or 8 N (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 N \pm {1}/{2} 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 a, corresponds to a DCT-V of length N = 1. == 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. 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 \sqrt{2/N} 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 DCTs ==

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. M-D DCT-II 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). 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 2D DCT-II is given by the formula (omitting normalization and other scale factors, as above) : \begin{align} X_{k_1,k_2} &= \sum_{n_1=0}^{N_1-1} \left( \sum_{n_2=0}^{N_2-1} x_{n_1,n_2} \cos \left[\frac{\pi}{N_2} \left(n_2+\frac{1}{2}\right) k_2 \right]\right) \cos \left[\frac{\pi}{N_1} \left(n_1+\frac{1}{2}\right) k_1 \right]\\ &= \sum_{n_1=0}^{N_1-1} \sum_{n_2=0}^{N_2-1} x_{n_1,n_2} \cos \left[\frac{\pi}{N_1} \left(n_1+\frac{1}{2}\right) k_1 \right] \cos \left[\frac{\pi}{N_2} \left(n_2+\frac{1}{2}\right) k_2 \right] . \end{align} 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_{k_1,k_2,k_3} = \sum_{n_1=0}^{N_1-1} \sum_{n_2=0}^{N_2-1} \sum_{n_3=0}^{N_3-1} x_{n_1,n_2,n_3} \cos \left[\frac{\pi}{N_1} \left(n_1+\frac{1}{2}\right) k_1 \right] \cos \left[\frac{\pi}{N_2} \left(n_2+\frac{1}{2}\right) k_2 \right] \cos \left[\frac{\pi}{N_3} \left(n_3+\frac{1}{2}\right) k_3 \right],\quad \text{for } k_i = 0,1,2,\dots,N_i-1. The inverse of 3-D DCT-II is 3-D DCT-III and can be computed from the formula : x_{n_1,n_2,n_3} = \sum_{k_1=0}^{N_1-1} \sum_{k_2=0}^{N_2-1} \sum_{k_3=0}^{N_3-1} X_{k_1,k_2,k_3} \cos \left[\frac{\pi}{N_1} \left(n_1+\frac{1}{2}\right) k_1 \right] \cos \left[\frac{\pi}{N_2} \left(n_2+\frac{1}{2}\right) k_2 \right] \cos \left[\frac{\pi}{N_3} \left(n_3+\frac{1}{2}\right) k_3 \right],\quad \text{for } n_i=0,1,2,\dots,N_i-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 or combining the algorithms for the different dimensions). Owing to the rapid growth in the applications based on the 3-D DCT, several fast algorithms have been 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. Vector-Radix Decimation in Frequency (VR DIF) is an example of an algorithm to compute a 3-D DCT-II efficiently. 3-D DCT-II VR DIF In order to apply the VR DIF algorithm, the input data must be formulated and rearranged as follows. The transform size N × N × N is assumed to be 2. : \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} :where 0\leq n_1,n_2,n_3 \leq \frac{N}{2} -1 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 c(\varphi_i)=\cos(\varphi_i). The original 3-D DCT-II can now be written as :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) :where \varphi_i= \frac{\pi}{2N}(4N_i+1),\text{ and } i= 1,2,3. If the even and the odd parts of k_1,k_2 and k_3 are considered, the general formula for the calculation of the 3-D DCT-II can be expressed as :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)) : where :: \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) :: +(-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) :: +(-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) :: +(-1)^{i+l}\tilde{x}\left(n_1+\frac{n}{2},n_2,n_3+\frac{n}{3}\right) :: +(-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. Arithmetic complexity The whole 3-D DCT calculation needs ~ [\log_2 N] ~ stages, and each stage involves ~ \tfrac{1}{8}\ N^3 ~ butterflies. The whole 3-D DCT requires ~ \left[ \tfrac{1}{8}\ N^3 \log_2 N \right] ~ 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 ~ \left[ \tfrac{7}{8}\ N^3\ \log_2 N \right] ~, 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. The image to the right shows a combination of horizontal and vertical frequencies for an (~ N_1 = N_2 = 8 ~) 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 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 : 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) ~, : for ~~ k = 0,\ 1,\ 2\ \ldots\ N-1 ~~ and ~~ \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 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. == Computation ==

Although the direct application of these formulas would require ~ \mathcal{O}(N^2) ~ operations, it is possible to compute the same thing with only ~ \mathcal{O}(N \log N ) ~ complexity by factorizing the computation similarly to the fast Fourier transform (FFT). One can also compute DCTs via FFTs combined with ~\mathcal{O}(N)~ pre- and post-processing steps. In general, ~\mathcal{O}(N \log N )~ 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 ~ \mathcal{O}(N) ~ 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 ~ 4N ~ 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.{{efn| The radix-4 step reduces the size ~ 4N ~ DFT to four size ~ N ~ 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 ~ N ~ FFT of real data plus ~ \mathcal{O}(N) ~ butterflies, once the trivial or duplicate parts are eliminated or merged. }} Because the even-indexed elements are zero, this radix-4 step is exactly the same as a split-radix step. If the subsequent size ~ N ~ 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 (~ 2 N \log_2 N - N + 2 ~ real-arithmetic operations). A recent reduction in the operation count to ~ \tfrac{17}{9} N \log_2 N + \mathcal{O}(N) 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 ~ N ~, but this is an implementation rather than an algorithmic question since it can be solved by unrolling or inlining.) ==Example of IDCT==

Consider this grayscale image of capital letter A. Each basis function is multiplied by its coefficient and then this product is added to the final image. ==See also==