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Image Transforms

8.4 Discrete Cosine Transform

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) [Pen90, Pen93].

Transform	Short
DiscreteCosineTransform[img]	DCT[img]	discrete cosine transform of img
InverseDiscreteCosineTransform[coef]	IDCT[coef]	inverse discrete cosine transform of coef
CosineMatrix[n]		returns an n × n matrix of n-point cosine basis sequences

Cosine transform functions.

The 1D discrete cosine transform (1D DCT) X[k] of a sequence x[n] of length N is defined as

The inverse DCT is defined as

where in both Equations (8.4.1) and (8.4.2)

[k] is defined as

The basis sequences of the 1D DCT are real, discrete-time sinusoids defined by

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The series form of the 2D discrete cosine transform (2D DCT) pair of formulas is

for k₁= 0,1,...,N₁-1 and k₂ = 0,1,...,N₂-1 and

for n₁ = 0,1,...,N₁-1 and n₂ = 0,1,...,N₂-1, with

[k] defined as in Equation (8.4.3). Equation (8.4.6) is called the analysis formula or the forward transform, while Equation (8.4.7) is the synthesis formula or inverse transform. The DCT is separable and so may be computed using 1D DCT row and column operations.

We conclude this section with examples of 1D and 2D DCTs of some simple signals. Consider the following sequence x[n] :

Here is the N=32 point DCT of this sequence.

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Next, we show the 2D DCT of a box sequence.

This plots the result.

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Here we demonstrate the DCT magnitude spectrum of an example image.

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Here is the original image and its discrete cosine transform.

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Particularly note the concentration of large DCT coefficients in the low-frequency zone. The DCT is known to have excellent energy compaction properties.

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