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The dual-tree complex wavelet transform (CWT) has recently received significant interest in the wavelet community, owing primarily to its directional selective and near-shift invariant properties. It has been shown that with two separate maximally decimated and dyadic decompositions where filters are offset by a half sample, the resulting CWT wavelet bases form an approximate Hilbert transform pair. In this paper, we present the design, implementation and applications of several families of orthogonal as well as biorthogonal rational-coefficient wavelet filters that satisfy the Hilbert transform pair condition and meet other desirable properties such as high coding gain, good directional sensitivity, and sufficient degree of regularity. The wavelet filters presented here, which confirm to Selesnick's and Kingsbury's design schemes, are designed and implemented directly in the lattice and lifting domain using VLSI-friendly dyadic coefficients. We confirm the fact that rational-coefficient constraint does not impose a significant loss in terms of energy compaction, wavelet smoothness, time-invariance, or directionality. We also propose the time-reversal relationships between the two CWT filter pairs in lattice and lifting domain, thereby facilitating both the design and implementation process. In the end, we present several applications and evaluations to illustrate the performance of the proposed designs.
Date of Publication: Aug. 2008