A SIMPLE CONSTRUCTION FOR THE M-BAND DUAL-TREE COMPLEX WAVELET TRANSFORM ˙ Ilker Bayram and Ivan W. Selesnick Polytechnic University, Brooklyn, NY ibayra01@utopia.poly.edu, selesi@poly.edu ABSTRACT The dual-tree complex wavelet transform (DT-CWT) which utilizes two 2-band discrete wavelet transforms (DWT) was recently extended to M -band by Chaux et al. In this paper, we provide a simple construction method for an M -band DT- CWT, with M = r d where r, d ∈ Z. In particular, we show how to extend a given r-band DT-CWT to an r d -band one. For convenience, the case where r =2, d =2 is consid- ered. However, the scheme can be extended to general {r, d} pairs straightforwardly. The extension to 2-D which achieves a directional analysis is also provided. Index Terms— M-band dual-tree, 2-band dual-tree, Hilbert transform pairs, directional wavelets. 1. INTRODUCTION The discrete dual-tree complex wavelet transform (DT-CWT) [5] provides approximate shift-invariance and directional se- lectivity in 2D (and in higher dimensions). The DT-CWT achieves these properties by employing two discrete wavelet transforms (DWT) with the requirement that the wavelet as- sociated with the second DWT is the Hilbert transform of the wavelet associated with the first DWT. The coefficients of the first and second DWT are then interpreted as the real and imaginary parts of a complex-valued wavelet transform. This scheme was extended to M -band orthonormal wavelet bases recently in [1], and used for image processing in [2]. The transform in [1, 2] employs two M -band discrete wavelet transforms where the wavelets associated with the two trans- forms form Hilbert transform pairs. It can be shown for the 2-band case that, if the Hilbert transform relationship is required to be exact, the filters in one of the trees cannot be FIR if the other tree’s filters are [11]. This is the same for the M -band case. To overcome this problem, in [1], the authors approximate the IIR filters using FIR filters by minimizing the L 2 error of the frequency response and in [2] they perform the filtering operations in the frequency domain. This work was supported by ONR under grant N00014-03-1-0217. It is well known how to extend a 2-channel perfect recon- struction (PR) filter bank (FB) into an M -channel PR FB us- ing a tree-structured FB (with M =2 k ). A tree-structured FB also allows one to extend a 2-band DWT into an M - band DWT. M -band wavelet transforms of that type are of- ten called wavelet packets [7]. However, it is not previously known how to properly extend a 2-band DT-CWT into an M - band one. This paper describes how to use a given r-band DT- CWT to construct an M -band DT-CWT (with M = r d ). In particular, it will be shown how to obtain an FIR 4-band DT- CWT from an FIR 2-band DT-CWT (for which several de- sign methods are known). The construction can be extended to other {r, d} values straightforwardly. 2. THE DUAL-TREE WAVELET PACKET TRANSFORM It is well known that 2-band wavelet bases employ approxi- mation spaces V i which can be decomposed into a higher level approximation space V i+1 and a detail space W i+1 as V i = V i+1 ⊕ W i+1 (1) where ⊕ denotes a direct sum of the vector spaces. The 2- band dual-tree complex wavelet transform asks for a second set of approximation spaces V ′ i and the associated orthogo- nal wavelet spaces W ′ i , such that the wavelets ψ(t) and ψ ′ (t) form a Hilbert transform pair. Similarly, the M -band wavelet transform employs approx- imation spaces V i satisfying V i = V i+1 ⊕W 1 i+1 ⊕... ⊕W M-1 i+1 [13]. The M -band DT-CWT is constructed [1, 2] by finding a second set of approximation spaces V ′ i and wavelet spaces W ′k i such that the associated wavelet functions ψ k (t) and ψ ′k (t) form Hilbert transform pairs, for k ∈{1, 2,...,M − 1}. In the following, we will demonstrate how to construct an r d -band DT-CWT given an r-band DT-CWT. For con- venience we will concentrate on the {r =2,d =2} case, yielding a 4-band DT-CWT, but the procedure can be easily adapted to general {r, d} pairs. Suppose we are given a 2-channel orthonormal filter bank {h (2) 0 (n), h (2) 1 (n)} and its associated scaling function φ (2) (t)