Wavelets and filter banks: novel approaches for real and complex-valued transform designs

Abstract

Processing of natural images stored as 2-D discrete signals is an inevitable task in almost all areas of image processing. In order to have the desired processing results, directional image representation can be considered as an indispensable step which is similar to what the human visual system (HVS) does. Research in the HVS has confirmed the importance of directional selectivity as a constituent that forms a vital role in visual information processing and its perception. Since, the directional information is often present at various spatial resolutions of an image, one has to capture multiresolution property for its efficient representation. The problem of designing multiresolution transform based image representations having directional selectivity has been a topic of wide interest over many years. In this thesis, using the concepts of wavelets and filter banks, we propose several novel multiresolution transform designs that have finer directional selectivity. We have classified them accordingly, based on, whether the directional decomposition of an input image has real or complex-valued representation. Regarding the real-valued transform representation, we present designs for three transforms with higher directionality by using additional filter bank stages in conjunction with the traditional decimated (subsampled) and undecimated (nonsubsampled) wavelet transforms. All three designs have better adaptability to the oriented features in the underlying image, since the filter bank construction enables us to design the filters with better frequency selectivity. In our next work, we propose two nonsubsampled transform designs belonging to a class of multiresolution directional filter banks. The proposed designs have simple structure but nonuniform frequency partitions which effectively model the irectional frequency distribution of natural images. Also, our designs need reduced computational requirements than the nonsubsampled counterparts obtained using their original subsampled approaches. All the proposed designs are tested for image denoising application using simple thresholding method. In the next part of the thesis, we consider design of complex-valued transforms which offer various advantages over the real-valued transforms. One can have approximate shift-invariance, higher directionality and phase information along with controllable redundancy by using complex-valued representation. In our first work here, we contribute to 1-D filter design aspect of an important complexvalued transform namely dual-tree complex wavelet transform (DTCWT). 2-D DTCWT is obtained using two trees of 2-channel perfect reconstruction filter bank (PRFB) and separable filtering approach similar to discrete wavelet transform (DWT). It offers six directions with redundancy factor of just 4. However, design of 1-D filters used in the DTCWT construction is quite involved. In this work, we propose two new approaches to design the 1-D biorthogonal wavelet filters of DTCWT having near-orthogonal filter response characteristics to get almost tightframe dual-tree complex wavelet transform (DTCWT). The proposed approaches are based on optimization of free variables obtained through factorization of generalized halfband polynomial. Use of unconstrained optimization makes these approaches simple and computationally less taxing. Associated wavelets of the filters obtained using the proposed approaches have better analytic properties leading to improved shift invariance. Also, the wavelets have near-exact symmetry resulting in improved directional feature selectivity for the multidimensional DTCWT extensions which is verified using the image denoising application. Our next work involves design of complex wavelet transform (CWT) which has better directionality and redundancy factor than the 2-D DTCWT. This is achieved by filtering the real-valued subbands of the finer directional wavelet transform using novel complex-valued filter bank stages. Proposed CWT has twelve directional subbands with redundancy factor of just 2. Its undecimated counterpart is also discussed. The generalized separable implementations of the proposed transforms makes them practically tractable. Image denoising using proposed CWTs show promising results for simple subband thresholding approach.