Tunable halfband-pair wavelet filter banks and application to multifocus image fusion

Traditional maximally flat wavelet filters are highly regular but suffer from poor frequency-selectivity because of their wide transition band. In this paper, an efficient method is proposed for the design of biorthogonal perfect reconstruction wavelet filter banks, known as halfband-pair filter banks (HPFB), to be used in several applications in image processing and pattern recognition. The formulation is based on representation of a general halfband polynomial in the variable x. We first derive filter coefficients in the polynomial domain (in the variable x) in terms of the coefficients of the corresponding function in z-domain. Using convex optimization techniques, and due to the simple structure of a parametric polynomial in general, we can impose some free parameters to provide a tuning opportunity to optimize and control the wavelet filter characteristics. Perfect reconstruction and desired number of vanishing moments (NVM) are incorporated into the design procedure. The method is systematic, renders a reasonable optimization problem, and it offers wavelet filters ranging from the maximally flat to the sharpest transition band. Therefore, it can provide a useful design tool, with a fine-tuning option, which is required in many applications such as watermarking, detection, segmentation, fusion, denoising, and feature extraction. The application of the wavelet pairs, which have sharper transition band and better frequency-selectivity, is shown in multifocus imaging to obtain a fully focused image from a set of registered input images at varying foci by employing the distance transform and exponentially decaying function on the subbands in the wavelet domain. Various images are tested and experimental results compare favorably to the results in the literature.