Overcomplete lifted wavelet representations for multiscale feature analysis

The lifting scheme was introduced as a flexible tool to construct compactly supported second generation wavelets and the wavelet transform. However because it is not translation invariant, the traditional lifting framework may not be good for multiscale feature analysis where translation-invariant characteristics are highly desirable. In this paper we address the following question: can the lifting scheme be used as a framework for overcomplete wavelet representations with multiscale feature analysis in mind? We address 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. We then show that only the dual lifting is necessary since the primal lifting required to compensate for the aliasing needs no longer exist. We also demonstrate that the proposed scheme achieves better performance without introducing boundary artifacts that exist in the traditional methods.