The dual-tree complex wavelet transform

TLDR: Several methods for filter design are described for dual-tree CWT that demonstrates with relatively short filters, an effective invertible approximately analytic wavelet transform can indeed be implemented using the dual- tree approach.

Abstract: The paper discusses the theory behind the dual-tree transform, shows how complex wavelets with good properties can be designed, and illustrates a range of applications in signal and image processing The authors use the complex number symbol C in CWT to avoid confusion with the often-used acronym CWT for the (different) continuous wavelet transform The four fundamentals, intertwined shortcomings of wavelet transform and some solutions are also discussed Several methods for filter design are described for dual-tree CWT that demonstrates with relatively short filters, an effective invertible approximately analytic wavelet transform can indeed be implemented using the dual-tree approach

Figures

Figure 5. Notice that the transform is critically sampled—the total data rate of the subband signals is equal to the input data rate (although the outputs are now complex).

Figures 19, 21, and 22. Ivan Selesnick thanks ONR for support of this work under grant N00014-03-1-0217. Richard Baraniuk thanks NSF grant FMF 04-520, ONR grant N0001402-1-0353, AFOSR grant FA9550-04-1-0148, and the Texas

Figure 12 illustrates the frequency responses of stages 1–4 of the dual-tree CWT. The first stage is quite far from being analytic, but the later stages are quite close to being analytic. For every stage after the first stage, the frequency responses of the complex filters are close to being single sided and are free of the unwanted lobes on the opposite side of the frequency axis that are present in Figure 6. In this example , h(1)0 (n) i s a Daubechies length-10 filter, g(1)0 (n) = h(1)0 (n − 1), and gi(n), hi(n) are orthonormal solutions of length 12 designed according to the algorithm of the “Common Factor Solution” section.

Figure 15 illustrates the six real oriented wavelets derived from a pair of typical wavelets satisfying ψg(t) ≈ H{ψh(t)} . Compared with separable wavelets (see Figure 14), these six wavelets (which are strictly nonseparable) succeed in isolating different orientations—each of the six wavelets are aligned along a specific direction and no checkerboard effect appears. Moreover, they cover more distinct orientations than the separable DWT wavelets.

Frequently asked questions

Q1. What are the contributions in this paper?

The dual-tree complex wavelet transform ( CWT ) this paper is an extension of the DWT that is nearly shift invariant and directionally selective. 

Q2. What can be done to estimate the geometrical structure of an image?

The shift and rotation invariance properties of the CWT can also be harnessed to compute accurate and efficient estimates of the geometrical structure in images, namely the strength, orientation, and offset of image edges, ridges, and other singularities. 

Q3. What is the property of a truly shift invariant transform?

A truly shift invariant transform has the property that the signal path through any single subband of the transform and its inverse may be characterized by a unique z transfer function, which is unaffected by the down and up sampling within the transform. 

Q4. What is the use of finite support for wavelet-based signal processing?

Finite support is extremely useful for wavelet-based signal processing, since it limits the extent to which a signal feature can affect the wavelet coefficients. 

Q5. How many complex wavelets can be formed by using the real parts?

a set of six complex wavelet can be formed by using wavelets (43)–(44) as the real parts and wavelets (49)–(50) as the imaginary parts. 

Q6. What is the perfect reconstruction condition for the analysis and synthesis FBs?

For the analysis and synthesis FBs to represent a forward and inverse wavelet transform, it is necessary that the perfect reconstruction (PR) condition be satisfied: y(n) = x(n), or more generally y(n) = x(n − no). 

Q7. How can the displacement between two images be estimated?

Local displacement (motion) between two images can be estimated from the change of phase of CWT coefficients from one image to the next. 

Q8. What is the advantage of the oriented complex 2-D dual-tree wavelet transform?

The oriented complex 2-D dual-tree wavelet transform is four-times expansive, but it has the benefit of being both oriented and approximately analytic. 

Q9. Why are the filters for the dual-tree CWT generally longer than real?

It should be noted however, that filters for the dual-tree CWT are generally somewhat longer than filters for real wavelet transforms with similar numbers of vanishing moments, because of the additional constraints (10)–(11) that the filters must approximately satisfy. 

Q10. What is the oriented complex 2-D dual-tree wavelet transform?

Note that the oriented 2-D dual-tree CWT (applied to real or complex data) requires four separable wavelet transforms in parallel, and so it is no longer strictly a dual-tree wavelet transform. 

Q11. What are the main shortcomings of the wavelet transform?

In spite of its efficient computational algorithm and sparse representation, the wavelet transform suffers from four fundamental, intertwined shortcomings. 

Q12. What is the way to illustrate the near shift invariance of the CWT?

One way to illustrate the near shift invariance of the dual-tree CWT is to observe how the projection of a signal onto a certain scale varies as the signal translates. 

Q13. Why is the 2-D wavelet transform less efficient for line-and-curve-s?

The reason for this is that while the separable 2-D wavelet transform represents point-singularities efficiently, it is less efficient for line-and curve-singularities (edges). 

Q14. What is the oriented real 2-D dual-tree wavelet transform?

Like the 1-D dual-tree CWT, the oriented real 2-D dual-tree wavelet transform is still a dual-tree wavelet transform and is also two-times expansive. 

Q15. What is the inverse of the oriented real 2-D dual-tree wavelet transform?

If the two real separable 2-D wavelet transforms are orthonormal transforms, then the transpose of Fhh is its inverse: Fthh · Fhh = I, and similarly Ftgg · Fgg = I. 

Q16. How can the authors reduce the width of the bump?

It is possible to reduce the width of the bump by designing H1(z) and Hn(z) so that they have narrower transitions bands, how-[FIG4] 

Q17. How can the authors design a pair of filters of equal length?

By jointly designing h0(n) and g0(n), the authors can obtain a pair of filters of equal (or near-equal) length, where both are relatively short. 

Q18. What is the spectrum of the complex wavelet h(t) + j?

The spectrum of the complex wavelet ψh(t) + jψg(t) is shown in the figure, and it is clearly nearly analytic (approximately zero on the negative frequency axis).