Two-dimensional, two-channel signal-adapted filter banks

The existence of principal component filter banks (PCFBs) has been proved for the class of one-dimensional (1D), two-channel finite impulse response (FIR) filter banks. The existence issues of two-dimensional (2D), two-channel PCFBs are studied. The support of the filters will vary with the subsampling matrix used. It is seen that the quincunx matrix is optimum in most of the natural images. Moreover, the existence of non-uniform 2D PCFBs, in which the PCFB is defined by normalised subband variances, for dyadic splitting of the frequency spectrum, in the case of natural images is proved. It is also shown that 2D PCFBs, defined by equivalent uniform variances, can exist in the case of most of the natural images, for octave band filter banks. In addition, 2D FIR signal-adapted filter banks from 1D filters, for separable subsampling, are designed. These 1D filters are designed from the separable components of the 2D power spectral density, using iterative greedy algorithm. The iterative algorithm is extended to two dimensions in order to design non-separable FIR filter banks for different subsampling matrices. The simulated FIR responses match the ideal responses very closely.