DCT - Discrete Cosine Transform

By: Preethi Ramkumar

Meaning of DCT – “Discrete Cosine Transform”, a mathematical transform that can provide aliasing cancellation and good frequency resolution, used in some codecs to convert the audio or video signal from the time domain to the frequency domain. In the decoder, an inverse discrete transform is used to reverse the process. Discrete Cosine Transform is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. It is equivalent to a DFT 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 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.

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: 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. For example, the DCT is used in (JPEG) image compression, MJPEG video compression, and (MPEG) video compression. There, the two-dimensional DCT-II of n × n blocks is computed and the results are quantized and entropy coded. In this case, n 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 (0,0) element is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies.

Other Related Definitions:

“…The discrete cosine transform (DCT) is a technique for converting a signal into elementary frequency components. It is widely used in image compression. Here we develop some simple functions to compute the DCT and to compress images. These functions illustrate the power of Mathematica in the prototyping of image processing algorithms.” [Nasa]

“…The DCT does a better job of concentrating energy into lower order coefficients than does the DFT for image data.The DCT is purely real, the DFT is complex (magnitude and phase).The DCT is purely real, the DFT is complex (magnitude and phase).A DCT operation on a block of pixels produces coefficients that are similar to the frequency domain coefficients produced by a DFT operation. An N-point DCT has the same frequency resolution as and is closely related to a 2N-point DFT. The N frequencies of a 2N point DFT correspond to N points on the upper half of the unit circle in the complex frequency plane.Assuming a periodic input, the magnitude of the DFT coefficients is spatially invariant (phase of the input does not matter). This is not true for the DCT. For most images, after transformation the majority of signal energy is carried by just a few of the low order DCT coefficients. These coefficients can be more finely quantized than the higher order coefficients. Many higher order coefficients may be quantized to 0 (this allows for very efficient run-level coding). ” [Bretl]

“…Similar to discrete fourier transform (DFT), discrete cosine transform (DCT) is a function that maps the input signal or image from spatial domain to frequency domain. DCT transforms the input into a linear combination of weighted basis functions. These basis functions are the frequency component of the input data. The two-dimensional DCT is just a one-dimensional DCT applied twice, once in the x direction, and again in the y direction. When you apply the DCT to an input image, it yields a matrix of weighted values corresponding to how much of each basis function is present in the original image. For most images, much of the signal energy lies at low frequencies; these appear in the upper-left corner of the DCT. The lower-right values represent higher frequencies, and are often small - small enough to be neglected with little visible distortion.” [Altera]

“…The DCT is closely related to the discrete Fourier transform; the DFT is actually one step in the computation of the DCT for a sequence. The DCT, however, has better energy compaction properties, with just a few of the transform coefficients representing the majority of the energy in the sequence. The energy compaction properties of the DCT make it useful in applications such as data communications. The function idct computes the inverse DCT for an input sequence, reconstructing a signal from a complete or partial set of DCT coefficients. ” [The MathWorks]

“…The discrete cosine transform (DCT) is an important transform in 2D signal processing. It is known to be close to optimal in terms of its energy compaction capabilities and can be computed via a fast algorithm. The DCT is used in two international image/video compression standards, Joint Photographic Experts Group (JPEG), and Motion Picture Experts Group (MPEG).” [Wolfram Research]

Related Links:

Nasa - Image Compression Using the Discrete Cosine Transform
Connexions - The Discrete Cosine Transform (DCT)
Mathworks - Discrete Cosine Transform
Wolfram - (DCT) is an important transform in 2D signal processing
Redwoods - The Discrete Cosine Transform (DCT)
GTAV - Discrete Cosine Transform (DCT)
Cmlab - Two-Dimensional DCT
Uiowa - Digital Image Processing

Technical Resources:

Multimedia - The Discrete Cosine Transform (DCT)
Bretl - Discrete Cosine Transform

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