Discrete Cosine Transform

  Also abbreviated DCT, the transform is closely related to the fast Fourier transform; it plays a role in coding signals and images [Jain89], e.g. in the widely used standard JPEG compression. The one-dimensional transform is defined by

where s is the array of N original values, t is the array of N transformed values, and the coefficients c are given by

for .

The discrete cosine transform in two dimensions, for a square matrix, can be written as

with an analogous notation for N, s, t, and the c(i,j) given by c (0,j) = 1/N, c (i,0) = 1/N, and c (i,j) = 2/N for both i and .

The DCT has an inverse, defined by

for the one-dimensional case, and

for two dimensions.

The DCT is included in commercial image processing packages, e.g. in Matlab (see [MATLAB97]).