Two-dimensional subband transforms: Theory and applications

The one-dimensional subband FFT (SB-FFT) and one-dimensional SB-DCT were extended to the two-dimensional (2-D) case to obtain the 2-D SB-FFT and the 2-D SB-DCT. The two-dimensional subband transforms are based on subband decomposition of the input sequence in both dimensions. They use knowledge about the input signal to obtain an approximation to their transform by discarding the computations in bands that have little energy in both dimensions. Computational savings can be obtained from calculating only the remaining subbands. In many applications the computational speed is so important that some error in the calculated transform can be accepted. In image processing, due to the nature of most natural scenes, most of the energy content of the corresponding digitised images is concentrated predominantly in the low-low spatial frequency domain. The concentration of the energy in a localised region of the transform domain makes the approximate subband transform computation quite suitable for the calculation of the 2-D image spectra. The complexity and accuracy of both 2-D transforms are studied in detail in the paper. The approximation errors in both transforms are derived for a general case, in which any band out of M bands is to be computed. Both transforms are modified to be fully adaptive to select the band of interest to be computed. Image transform application examples are included. Savings in computational complexity of image transforms are shown. The efficiency of subband transforms of different images is indicated by computing the signal-to-noise ratio in the reconstructed images.