The Discrete Cosine Transform

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Introduction
Transform coding constitutes an integral component of contemporary image/video processing
applications. Transform coding relies on the premise that pixels in an image exhibit a certain
level of correlation with their neighboring pixels. Similarly in a video transmission system,
adjacent pixels in consecutive frames2 show very high correlation. Consequently, these
correlations can be exploited to predict the value of a pixel from its respective neighbors. A
transformation is, therefore, defined to map this spatial (correlated) data into transformed
(uncorrelated) coefficients. Clearly, the transformation should utilize the fact that the information
content of an individual pixel is relatively small i.e., to a large extent visual contribution of a
pixel can be predicted using its neighbors.


The Discrete Cosine Transform
Like other transforms, the Discrete Cosine Transform (DCT) attempts to decorrelate the image
data. After decorrelation each transform coefficient can be encoded independently without losing
compression efficiency. This section describes the DCT and some of its important properties.



The Two-Dimensional DCT
The objective of this document is to study the efficacy of DCT on images. This necessitates the
extension of ideas presented in the last section to a two-dimensional space. The 2-D DCT is a
direct extension of the 1-D case and is given by


Properties of DCT
Discussions in the preceding sections have developed a mathematical foundation for DCT.
However, the intuitive insight into its image processing application has not been presented. This
section outlines (with examples) some properties of the DCT which are of particular value to
image processing applications.