DCT, DFT, FFT, unitary transform, autocorrelation matrices

It is much more efficient to decorrelate an image in the frequency domain than in the spatial domain.

It is more efficient to capture certain features of an image in the frequency domain for pattern classification and identification purposes than in the pixel domain.

For an image of size N × N pixels, a unitary transform implies that the image can be represented as a linear combination of N^2 basis images. These basis images may be independent of the image being transformed as in DCT, or may be computed from the image itself as in KLT.

If the elements of a unitary transform matrix A are real, then A is called an orthogonal transform and its inverse is its own transpose, that is, inverse(A) = transpose(A). However, a unitary transform needs not be orthogonal.

There are two implementations of the FFT algorithm called decimation in time and decimation in frequency, both of which use the divide and conquer rule. In decimation in time, we divide the input sequence into even and odd indexed sequences.

References
K. S. Thyagarajan, Still image and video compression with MATLAB, John Wiley & Sons, 2011.