# discrete cosine transform

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## discrete cosine transform

(mathematics)
(DCT) A technique for expressing a waveform as a weighted sum of cosines.

The DCT is central to many kinds of signal processing, especially video compression.

Given data A(i), where i is an integer in the range 0 to N-1, the forward DCT (which would be used e.g. by an encoder) is:

B(k) = sum A(i) cos((pi k/N) (2 i + 1)/2) i=0 to N-1

B(k) is defined for all values of the frequency-space variable k, but we only care about integer k in the range 0 to N-1. The inverse DCT (which would be used e.g. by a decoder) is:

AA(i)= sum B(k) (2-delta(k-0)) cos((pi k/N)(2 i + 1)/2) k=0 to N-1

where delta(k) is the Kronecker delta.

The main difference between this and a discrete Fourier transform (DFT) is that the DFT traditionally assumes that the data A(i) is periodically continued with a period of N, whereas the DCT assumes that the data is continued with its mirror image, then periodically continued with a period of 2N.

Mathematically, this transform pair is exact, i.e. AA(i) == A(i), resulting in lossless coding; only when some of the coefficients are approximated does compression occur.

There exist fast DCT algorithms in analogy to the Fast Fourier Transform.

## DCT

(Discrete Cosine Transform) An algorithm that is widely used for data compression. Similar to Fast Fourier Transform, DCT converts data (pixels, waveforms, etc.) into sets of frequencies. The first frequencies in the set are the most meaningful; the latter, the least. To compress data, the least meaningful frequencies are stripped away based on allowable resolution loss. DCT is used to compress JPEG, MPEG, DV and H.263 frames.
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References in periodicals archive ?
Various applications are employed directly and indirectly to serve this purpose, some of them are: Discrete Fourier Transform (DFT), Short Fourier Transform (SFT), Discrete Cosine Transform (DCT), Hadamard-Haar Transform (HHT), Slant-Haar Transform (SHT), Karhune-Loeve Transform (KLT), and Discrete Wavelet Transform (DWT) (12).
Feature extraction using discrete cosine transform and discrimination power analysis with a face recognition technology.
Extraction of Geologic Features Using Discrete Cosine Transform. Discrete cosine transform (DCT) has been utilized as an image-processing tool for characterization of channelized reservoirs due to the periodicity of cosine functions .
The discrete wavelet transform (DWT) method is favored much than discrete cosine transform (DCT) method, owing to the resolution, WT provides to the image at various levels.
The signal processing transforms such as fast Fourier transform (FFT), discrete cosine transform (DCT), discrete wavelet transform (DWT), redundant discrete wavelet transform (RDWT) and singular value decomposition (SVD) are used in transform domain based watermarking algorithms.
Due to its important properties, quaternion discrete Fourier transform (QDFT), its counterparts quaternion discrete cosine transform (QDCT) and quaternion wavelet transform have been widely used and applied to both single and two dimensional signals in the fields of image processing, radar, robotics and cryptographic.
The Integer Modified Discrete Cosine Transform (Int-MDCT) approximates the MDCT while producing integer output values.
Gottlieb, "VLSI implementation of a 16 x 16 discrete cosine transform," IEEE Transaction on Circuits and Systems, vol.
for k [member of] K do > Calculate the Discrete Cosine Transform (DCT) coefficients [B.sup.k.sub.u,v] for each 8 x 8 byte matrix ([B.sup.k.sub.i,j]).
The orthogonal transform is selected to be the discrete Hartley transform (DHT) in , discrete cosine transform in , and the Walsh-Hadamardtransform (WHT) in ,respectively.
In comparison in the simulated experiment, we used the discrete cosine transform (DCT), discrete sine transform (DST) [56,57], and local discrete cosine transform (LDCT) dictionaries for major EEG parts.
A novel robust color image digital watermarking algorithm based on discrete cosine transform. Journal of Computers, 8(10), 2507-2511.

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