Everything about Discrete Cosine Transform totally explained
A
discrete cosine transform (DCT) expresses a sequence of finitely many data points in terms of a sum of
cosine functions oscillating at different
frequencies. DCTs are important to numerous applications in science and engineering, from
lossy compression of
audio and
images (where small high-frequency components can be discarded), to
spectral methods for the numerical solution of
partial differential equations. The use of cosine rather than
sine functions is critical in these applications: for compression, it turns out that cosine functions are much more efficient (as explained below, fewer are needed to approximate a typical
signal), whereas for differential equations the cosines express a particular choice of
boundary conditions.
In particular, a DCT is a
Fourier-related transform similar to the
discrete Fourier transform (DFT), but using only
real numbers. 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), where in some variants the input and/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"; 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.
Applications
The DCT, and in particular the DCT-II, is often used in signal and image processing, especially for lossy data compression, because it has a strong "energy compaction" property (Rao and Yip, 1990): most of the signal information tends to be concentrated in a few low-frequency components of the DCT, approaching the
Karhunen-Loève transform (which is optimal in the decorrelation sense) for signals based on certain limits of
Markov processes. As explained below, this stems from the boundary conditions implicit in the cosine functions.
A related transform, the
modified discrete cosine transform, or MDCT (based on the DCT-IV), is used in
AAC,
Vorbis,
WMA, and
MP3 audio compression.
DCTs are also widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even/odd boundary conditions at the two ends of the array.
DCTs are also 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.
JPEG
The DCT is used in
JPEG image compression,
MJPEG,
MPEG, and
DV video compression. There, the two-dimensional DCT-II of
blocks are computed and the results are
quantized and
entropy coded. In this case,
is typically 8 and the DCT-II formula is applied to each row and column of the block. The result is an 8 × 8 transform coefficient array in which the
element (top-left) is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies.
Informal overview
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 than 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
as a sum of sinusoids, you can evaluate that sum at any
, even for
where the original
wasn't 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 don't 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 (for example 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 is even about the sample
a, in which case the even extension is
dcbabcd, or the data is 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
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 compaction" 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's 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 doesn't 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.
Formal definition
Formally, the discrete cosine transform is a
linear, invertible
function F :
RN -> RN (where
R denotes the set of
real numbers), or equivalently an invertible
N ×
N square matrix. There are several variants of the DCT with slightly modified definitions. The
N real numbers
x0, ...,
xN-1 are transformed into the
N real numbers
X0, ...,
XN-1 according to one of the formulas:
DCT-I
»
Technically, computing a two- (or multi-) dimensional DCT by sequences of one-dimensional DCTs along each dimension is known as a
row-column algorithm (after the two-dimensional case). As with
multidimensional FFT algorithms, however, there exist other methods to compute the same thing while performing the computations in a different order (for example interleaving/combining the algorithms for the different dimensions).``
The image to the right shows combination of horizontal and vertical frequencies for an 8 x 8 (
) two-dimensional DCT.
Each step from left to right and top to bottom is an increase in frequency by 1/2 cycle.
For example, moving right one from the top-left square yields a half-cycle increase in the horizontal frequency (goes from white to black).
Another move to the right yields two half-cycles (white to black to white).
A move down yields two half-cycles horizontally and a half-cycle vertically.
The source data (8x8) is transformed to a
linear combination of these 64 frequency squares.
Computation
Although the direct application of these formulas would require O(
N2) operations, it's possible to compute the same thing with only O(
N log
N) complexity by factorizing the computation similar to the
fast Fourier transform (FFT). One can also compute DCTs via FFTs combined with O(
N) pre- and post-processing steps.
The most efficient algorithms, in principle, are usually those that are specialized directly for the DCT, as opposed to using an ordinary FFT plus 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 (Frigo & Johnson, 2005). 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 Feig & Winograd (1992) 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 (Duhamel & Vetterli, 1990).
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's often easier to obtain high performance for general lengths
N with FFT-based algorithms. (Performance on modern hardware is typically not dominated simply by arithmetic counts, and optimization requires substantial engineering effort.) 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
with real-even symmetry whose even-indexed elements are zero. One of the most common methods for computing this via an FFT (for example the method used in
FFTPACK and
FFTW) is due to 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. (The radix-4 step reduces the size
DFT to four size-
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-
FFT of real data plus
butterflies.) Because the even-indexed elements are zero, this radix-4 step is exactly the same as a split-radix step; if the subsequent size-
real-data FFT is also performed by a real-data
split-radix algorithm (as in Sorensen et al., 1987), then the resulting algorithm actually matches the lowest published arithmetic count for the power-of-two DCT-II (
real-arithmetic operations). So, there's 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
, but this is an implementation rather than an algorithmic question since it can be solved by unrolling/inlining.)
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