Wavelet-like filter banks for multi-dimensional and multi-resolution signal processing

The aim of this research project is to investigate the possibility of establishing a general framework for the implementation of the wavelet transform by numerical optimisation of various constraints associated with the wavelet basis function, thereby enabling various multi-dimensional and multi-resolution signal processing operations to be performed efficiently. Many efforts, including theoretical investigation, numerical simulation and application study were made to achieve the project aim. The multi-resolution theory was investigated to provide the fundamentals of the basic idea of the wavelet transform in the signal processing field. The wavelets and wavelet transform were studied in detail, particularly their relationship to the filter bank. The investigations on the properties of the wavelet revealed that the advantages of the wavelet filter banks over the conventional perfect reconstruction filter banks lie in the extra constraints imposed on the wavelet filter banks, and that the properties of a wavelet filter bank have a different level impact on its application areas. A new approach to implement the wavelet transform, namely, wavelet-like filter bank, was proposed as an alternative to the conventional wavelet transform. By using the proposed method it is possible to design a filter bank with some desired properties for certain applications. Starting from the classical windowing method to generate a linearphase lowpass filter the author derived a set of wavelet-like filter coefficients by optimising a set of core properties of a wavelet filter bank. The resulting wavelet-like filter not only approximates the properties of a wavelet filter bank, but also possesses properties which are mutually exclusive, e.g. linear phase and orthogonality. Through the simulation it is shown that the proposed wavelet-like filter banks have superior reconstruction performance over the classical wavelet filter bank in terms of signal-tonoise ratio. The wavelet-like filters were applied to two signal processing tasks, image compression and signal restoration. For image compression, the performance of the wavelet-like filter bank is comparable to the standard wavelet filter banks. For signal restoration, the proposed wavelet-like filter bank is shown to offer better performance in terms of SNR and visual quality, when compared with the classical wavelet filter banks. These simulation results validated the original idea: the wavelet transform can form a nearoptimal base for the representation of the signal therefore enabling efficient processing of the signal for certain tasks.