Complex Wavelet Transforms

Standard DWT (Discrete Wavelet Transform), being non-redundant, is a very powerful tool for many non-stationary Signal Processing applications, but it suffers from three major limitations; 1) shift sensitivity, 2) poor directionality, and 3) absence of phase information. To reduce these limitations, many researchers developed real-valued extensions to the standard DWT such as WP (Wavelet Packet Transform), and SWT (Stationary Wavelet Transform). These extensions are highly redundant and computationally intensive. Complex Wavelet Transform (CWT) is also an alternate, complex-valued extension to the standard DWT. The initial motivation behind the development of CWT was to avail explicitly both magnitude and phase information. This thesis presents a detailed review of Wavelet Transforms (WT) including standard DWT and its extensions. Important forms of CWTs; their theory, properties, implementation, and potential applications are investigated in this thesis. Recent developments in CWTs are classified into two important classes first is, Redundant CWT (RCWT), and second is Non-Redundant CWT (NRCWT). The important forms of RCWT include Kingsbury’s and Selesnick’s Dual-Tree DWT (DT-DWT), whereas the important forms of NRCWT include Fernandes’s and Spaendonck’s Projection based CWT (PCWT), and Orthogonal Hilbert transform filterbank based CWT (OHCWT) respectively. All recent forms of CWTs try to reduce two or more limitations of standard DWT with limited (or controllable) redundancy, or without any redundancy. Potential applications such as Motion estimation, Image fusion/registration, Denoising, Edge detection, and Texture analysis are suggested for further investigation with RCWT. Directional and phase based Compression is suggested for investigation with NRCWT. Denoising and Edge detection applications are investigated with DT-DWTs. Promising results are compared with other DWT extensions, and with the classical approaches. After thorough investigations, it is proposed that by employing DTDWT for Motion estimation and NRCWT for Compression might significantly improve the performance of the next generation video codecs. vii