The 2-directional wavelet transform: Theory and implementation

The 2-directional wavelet transform is an image representation which separates the spectrum of an image into a lowpass subband, a horizontal highpass subband and a vertical high-pass subband. Each of the decomposed highpass subbands, corresponding to pairwise trapezoidal regions of the spectrum of an original image, is critically sampled to a rectangular shaped coefficient image in the spatial domain. This paper illustrates how an innovative method can be generated by the creative combination of filter banks, integer resampling, and frequency shifting. The proposed 2-directional wavelet transform achieves critical sampling and perfect reconstruction properties. The new transform is an extension of the traditional wavelet transform, and is also the fundamental stage of a new method called the nonredundant contourlet transform (NRCT) developed in a companion paper. Experimental results are presented to demonstrate the potential of the 2- directional wavelet transform.