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The Discrete Cosine Transform (DCT)

  The discrete cosine transform (DCT) helps separate the image into parts (or spectral sub-bands) of differing importance (with respect to the image's visual quality). The DCT is similar to the discrete Fourier transform: it transforms a signal or image from the spatial domain to the frequency domain (Fig 7.8).

DCT Encoding

The general equation for a 1D (N data items) DCT is defined by the following equation:

\begin{displaymath}
F(u) = \left(\frac{2}{N}\right)^{\frac{1}{2}} \sum_{i=0}^{N-1}
\Lambda(i).cos\left[
\frac{\pi.u}{2.N}(2i+1)
\right]f(i)\end{displaymath}

and the corresponding inverse 1D DCT transform is simple F-1(u), i.e.:

where

\begin{displaymath}
\Lambda(i) = \left\{ \begin{array}
{ll} \frac{1}{\sqrt{2}} & {\rm
for}
\xi = 0\ 1 & {\rm otherwise}\end{array} \right.\end{displaymath}

The general equation for a 2D (N by M image) DCT is defined by the following equation:

\begin{displaymath}
F(u,v) = \left(\frac{2}{N}\right)^{\frac{1}{2}}
\left(\frac{...
 ...}(2i+1)
\right]cos\left[ \frac{\pi.v}{2.M}(2j+1) \right].f(i,j)\end{displaymath}

and the corresponding inverse 2D DCT transform is simple F-1(u,v), i.e.:

where

\begin{displaymath}
\Lambda(\xi) = \left\{ \begin{array}
{ll} \frac{1}{\sqrt{2}} & {\rm
for}
\xi = 0 \ 1 & {\rm otherwise}\end{array} \right.\end{displaymath}

The basic operation of the DCT is as follows:


next up previous
Next: Differential Encoding Up: Source Coding Techniques Previous: Relationship between DCT and
Dave Marshall
10/4/2001