The multiresolution directional filter banks

Multiscale and multidirectional are desirable properties for image decomposition. A multiresolution discrete transform represent image data in a hierarchical manner, each added level contributes to a finer representation of the image. The motivations for the thesis to develop multidirectional transforms are two fold. Natural images typically contain many geometrical features, such as edges and textures, and it is believed that directional bases would better represent those features. On the other hand, research in human visual system shows that directional information plays an important role in visual perception. Therefore, the transformation of image into directional representations should be the first step in the extraction and understanding of visual information. The thesis presents the theory and implementation of several multiscale and multidirectional transforms. First, based on the frequency supports of a multiresolution and multidirection orthogonal transform for two-dimensional discrete signals, a new uniformly, maximally decimated Directional Filter Bank (DFB) with six highpass directional subbands and two lowpass subbands is introduced. The uniform DFB (uDFB) can be implemented by a binary tree structure of two-channel filter banks. The uDFB is then extended to a non-uniform case (nuDFB), which is still maximally decimated, by combining the two lowpass subbands. 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 uDFB is then generalized to a family of multiresolution DFBs. Careful investigation proves that DFBs belong to the class do not have permissibility, except one special case, which is called the quincunx uDFB. This special filter bank has many attractive properties, such as orientation selectivity, multiresolution capability, fast implementation and good filter designs. An image transform implemented by a critically sampled filter bank may not be suitable for image analysis tasks, because the representation is not stable when the input signals are shifted. This thesis presents a shift-invariant complex directional pyramid transform, which is implemented by a pyramidal dual-tree directional filter banks structure. The double filter bank framework consists of a shift-invariant Laplacian pyramid and a dual-tree DFB. It is proven analytically and experimentally that the two corresponding directional filters produced by the primal and dual filter banks are symmetric and anti-symmetric, and they are the real and imaginary parts of a complex filter. Besides other advantages, the novel shift-invariant, multiscale, multidirectional image decomposition has two unique characteristics that other shift-invariant decompositions do not possess. First, the directional resolution of the image transform is not fixed, and the number of directional subbands can be arbitrarily high. Secondly, the two-dimensional filter bank is implemented in a separable fashion, which makes the entire structure very computational efficient. Various design and implementation issues are also discussed. Several issues related to the realization of the two-dimensional filter banks associated with the proposed transforms in the thesis are also discussed, and a number of new findings and results are described. In particular, the thesis presents a novel algorithm to design two-dimensional filter banks and an efficient structure to implement undecimated DFB. Finally, promising results of the new filter banks in image coding, texture image classification and retrieval are demonstrated.