International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
2. THE 1-D DUAL-TREE COMPLEX WAVELET 
TRANSFORM 
It is well-known that the real biorthogonal wavelet transform 
can provide PR and no redundancy, but it is lack of shift variant. 
Then Kingsbury (Julian Magarey! 1 1998;N. Kingsbury, 1998a ; 
Nick Kingsbury! 1998b; Serkan Hatipoglu, 1999) has developed 
a dual-tree algorithm with a real biorthogonal transform, and an 
approximate shift invariance can be obtained by doubling the 
sampling rate at each scale, which is achieved by computing 
two parallel subsampled wavelet trees respectively. 
For one dimension signal, we can compute two parallel wavelet 
trees. There is one sample offset delay between two trees at 
level 1, which is achieved by doubling all the sample rates. The 
shift invariance is perfect at level 1, since the two trees are fully 
decimated. To get uniform intervals between two trees beyond 
level 1, there have to be half a sample delay. The term will be 
satisfied using odd-length and even-length filters alternatively 
from level to level in each tree. Because we use the decimated 
form of a real discrete wavelet transform beyond level 1, the 
shift invariance is approximate. 
The transform algorithm is described as following. Its process is 
illustrated by fig.1. 
1D COMPLEX WAVELET TRANSFORM: 
>  Atlevel 1, there is one sample offset between the trees. 
(at), (a * a7). dy, =" zg"), 
Gi an], O 1 rase 
(aj), *(a * h^) (di), 2 (a * g^, 
> Beyond level 1, there must be half a sample difference 
between the trees. 
  
ju > e +1 j e 
(a; Yan = (a) *h Yan (d) > = (a; "o )an 
je = | 3 o +1 j o | | 
(a, I? =z (ay + h dant (dj, ). = (aj > 8 )»a 2 
: — Ah“ | > di} 
dh Ja —a— E 
Tree A j T 2 , 
pi» a +. g | 2 ——d; 
— 8 | 2 | d', > 
qi up on mer 
is — hl | | 2° Ld. 
E h ]2 — a,— 
— ; bend — g^ | 2" dl 
Tree B . ud ! L i 
pps | +de 
Level 1£ j z16 Level 2f j=2© 
Figure 1. The unidimensional dual tree complex wavelet 
transform 
The details 4, and g, can be interpreted as the real and 
imaginary parts of a complex process z = d, +id,- The essential 
property of this transform is that the magnitude of the step 
response is approximately invariant with the input shift, while 
only the phase varies rapidly (see (Nick Kingsbury! 1998a) for a 
good illustration).(A. Jalobeanu , 2000) 
It is not really a complex wavelet transform, since it does not 
use any complex wavelet. It is implemented with real wavelets. 
Classical complex-valued wavelet transforms can provide appr- 
oximate shift invariance and good directionality, but PR and 
good frequency characteristics cannot be obtained using compl- 
ex filters in a single tree. At the same time, it is different from a 
real wavelet transform because of the variety of filters. At level 
1, the filter in tree A are odd-length filter, is same to tree B. 
Beyond level 1, the filters in two trees are different, and they are 
different between different levels in each tree. Hence the 
wavelet functions varies continuously from level to level, whic- 
h is quite different from the classical multi-resolution analysis. 
Reconstruction is performed independently in each tree, and the 
results are averaged to obtain 4° at level 1, for symmetry 
between the two trees. This is illustrated by the following 
algorithm and fig 2. 
1D INVERSE COMPLEX WAVELET TRANSFORM: 
€ Level; (j>0) 
(a^), sb! * 5), «di *g*), 
  
i m T. o ) + 0 7 f 
(ag), = (; rot rudi tg), 3 
e At j 20: 
E ; C - 3. 141.1 
atis E * h^), tr (d'* g^), + *h^), (dL * 8"),) 
where { ER yg n-2p , x, if n=2p+l 
, |o i nz2pil ^" 0 if n=2p 
a, —- 21 h — 
; $—a,, — 2 | nn 
Dp | 4| i « bes j i | Tree A 
€ RD 24 r g —— 3 [em 
d'—5 2155 8$" — 
ewe Sy 
itn STE rir 
ay—s 27 pe Mt ui | 
i j | | | 
; n ga. —— 21 LE nma 
did 21 g — = LE 
- 1 | | 
d, 21 Eug Tree B 
Level 2£ j 26 Level 1£ j=10 
Figure 2. The unidimensional dual tree inverse complex wavelet 
transform 
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