A study of two-channel separable complex-valued wavelets and filter banks (orthogonal and bi-orthogonal).

摘要

The theory and applications of complex-valued wavelets and complex-valued filter banks have received increased amounts of interest over the years. The applications of complex-valued wavelets and filter banks (CVWFB) range from solving partial differential equations, to data compression and image processing, to communications systems. Several methods for designing CVWFBs have been reported in the literature, but require the system designer to solve complex design equations. To date, a general method for the design of CVWFBs derived from existing real-valued wavelets and filter banks (RVWFB) has not been investigated. Such a method would greatly simplify the design of CVWFBs in that no complex design equations would be needed.; In this thesis, a general method for designing separable, complex-valued wavelets and filter banks derived from existing real-valued wavelets and filter banks is proposed. This method covers the construction of both orthogonal and bi-orthogonal complex-valued wavelets and filter banks, and has several advantages over previously reported methods of CVWFB design. With this method, CVWFBs are derived from existing RVWFBs regardless of wavelet family, length, or order of regularity. These CVWFBs inherit their properties from the RVWFB from which they were derived. In addition, the proposed design method produces hybrid complex-valued filter banks in which the filters are no longer purely high-pass or low-pass, as is the case with traditional filter bank design. This method offers the system designer a simple scheme to construct complex-valued wavelets and filter banks.