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Title: Lifting based direction adaptive two-dimensional wavelet transforms
Authors: Dakala Jayachandra
Keywords: DRNTU::Engineering::Electrical and electronic engineering::Electronic systems::Signal processing
DRNTU::Science::Mathematics::Applied mathematics::Signal processing
Issue Date: 2015
Source: Dakala Jayachandra. (2015). Lifting based direction adaptive two-dimensional wavelet transforms. Doctoral thesis, Nanyang Technological University, Singapore.
Abstract: Standard separable wavelets are very successful for image representation over Fourier basis. However, due to its limited directionality and isotropic nature of the basis, the standard wavelet transform (WT) fails to exploit the correlation along the edges. Many directional wavelet transforms, both signal adaptive and non-adaptive, have been developed with an aim to exploit the correlation along the edges. Lifting factorization of filter banks (FB) has been at the heart of many adaptive directional transforms like adaptive directional lifting (ADL), curved WT, and many more. Lifting factorization is flexible in building different adaptive transforms due to its structural perfect reconstruction property. To further enhance the flexibility to construct adaptive wavelet transforms, we work on lifting based construction of adaptive wavelet transforms either with spatially varying filters or with spatially varying downsampling, and also work on efficiently finding local content directionality. Adaptive directional wavelet transforms, like directionlets, ADL, etc., try to locally approximate the image regions with smooth polynomials in a locally appropriate direction. Finding the local content direction is an essential and computationally intensive step of adaptive directional wavelet transforms. We propose a new directional metric that we define as Directional Variance (DirVar). DirVar is based on variance and mean values of directional lines. We show its relation with isotropic variance and also analyze its characteristics for step edges both with and without noise. Overall, we show that DirVar is sensitive to edge direction, robust to noise, and also computationally efficient. Compared to rate distortion based direction selection, DirVar based direction selection methods for adaptive directional transforms, such as directionlets, ADL with subpel directions and ADL with full pel directions, show only negligible loss in performance for considerable gain in computation. Next, we study lifting based switching of two-channel filter banks with fixed downsampling. We show that such switching between arbitrary filter banks induces transients, and then we propose two solutions to overcome the transients. One solution consists of a boundary handling mechanism to switch between any arbitrarily designed FBs, while the other solution proposes to design the FBs with a set of conditions applied on lifting steps. We show that both solutions maintain good frequency response during the transition and eliminate the transients due to the filters response at alias frequencies. Using the proposed methods, we develop a spatial adaptive transform by switching between the long length FBs (either the JPEG2000 9/7 FB or the newly designed 13/11 FB) and the short length FBs (JPEG2000 5/3 FB). This adaptive transform shows reduced ringing artifacts for images over JPEG2000 9/7 FB. We then study lifting based construction of adaptive transforms with spatially varying downsampling, particularly directionlets. Directionlets allow construction of perfect reconstruction and critically sampled multi directional anisotropic basis, yet retaining the separable filtering of standard wavelet transform. However, due to the spatially varying downsampling direction, it is forced to apply spatial segmentation and process each segment independently. We show that, with simple modifications in the block boundaries, we can overcome these limitations by, what we call in-phase lifting implementation of directionlets. In the context of directionlets using in-phase lifting, we identify different possible groups of downsampling matrices that would allow construction of multi-level transform without forcing independent processing of segments. In-phase lifting implementation of directionlets shows improved image coding results and also eliminates blocking artifacts inherent in the directionlets.
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