Fast multiscale feature analysis via overcomplete wavelet representations

The lifting scheme is a flexible tool for constructing compactly supported second generation wavelets which are not necessarily translates and dilates of one mother wavelet. The lifting scheme uses a simple basis function and builds a better performing one with desirable properties. Flexibility afforded by the lifting scheme allows basis functions to change their shapes near the boundaries without degrading regularities. The lifting scheme also provides fast processing by making optimal use of similarities between high and low pass filters. The lifting algorithm was originally introduced to construct biorthogonal wavelets associated with interpolating scaling functions, and was generalized later using a finite number of lifting steps by means of the Euclidean algorithm.; The lifting scheme utilizes a classical 2-channel filter bank as a framework for multiresolution analysis. However this traditional framework is not translation invariant. Representations with a translation-invariant characteristic are highly desirable for feature analysis. In this dissertation we address the following question: Can the lifting scheme be used as a framework for overcomplete wavelet representations with multiscale feature analysis in mind? Our work addresses this question by investigating each stage of the multiscale analysis: Split, Dual lifting and Primal lifting. We introduce a smoothing Lazy wavelet in the split stage which does not subsample, but smooths an input image so that the low-pass channel contains some redundant information. Since the new split operation does not use downsampling to make distinct subsets, there is no aliasing. We show that the primal lifting required to compensate for the aliasing by preserving energy between two contiguous approximations needs no longer exist. Only the predict stage, based upon specific properties, indeed leads to a useful multiresolution analysis which is applicable to multiresolution feature analysis. We also show in our work that the proposed scheme achieves better performance without introducing boundary artifacts that exist in the traditional methods. This makes the overcomplete lifted wavelet transform very viable in interactive processing paradigms such as a Web-based client-server model for multiscale feature analysis where fast processing is highly desirable.