A set of linearly independent vectors whose linear combination can be used to express any vector in a given vector space (In our case, the vector space is the 8 x 8 matrix, X). There can be infinitely many bases for a given vector space. So, for representing our image, we are free to choose any basis that is convenient to us. The coefficient matrix will vary accordingly. ( Since X = BC and C = B-1X)
B = [ b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 ] where b0, b1,.., b7 are 8 x 1 linearly independent vectors.
A “Good” basis should have more of low frequency vectors or bis (ideal: all ones in the column; imply less variation of pixel values in space) and very few high frequency vectors (alternate +1s and -1s; imply maximum variation of pixel values in space) in order to account for the general smoothness of images.
Bases To Choose From ‘w’ in Fourier Basis is the nth root of unity for a basis of dimension n x n.