Multiresolution direction filterbanks: theory, design, and applications
This paper presents new developments of directional filterbanks (DFBs). The motivation for the paper is the existence of multiresolution and multidirection orthogonal transform for two-dimensional (2-D) discrete signals. Based on the frequency supports of the ideal transform, a new uniformly, maximally decimated DFB with six highpass directional subbands and two lowpass subands is introduced. The uniform DFB (uDFB) can be implemented by a binary tree structure of two-channel filterbanks. The filterbank employed in the tree is shown to be alias-free decimation and permissible. The uDFB is then extended to a nonuniform case (nuDFB), which is still maximally decimated, by combining the two lowpass subbands. The nuDFB yields nonuniform frequency division, which composes of one lowpass filter with a decimation factor of one fourth and six highpass directional filters with a decimation factor of one eighth. The new DFBs offer alternative image decompositions, which overcome the limited directional selectivity of the separable wavelets and the limited multiresolution of the conventional DFB. The lowpass subband of the nuDFB can be used to obtain a multiresolution representation by simply reiterating the same nuDFB decomposition. On the other hand, the directional subbands can also be further refined by simply applying a two-channel conventional DFB at each highpass component. A simple design method yielding near orthogonal uniform and nonuniform multidimensional filterbanks is presented. Finally, the performances of the newly proposed nuDFB are compared with other conventional transforms in nonlinear approximation, image denoising, and texture classification to demonstrate its potential.