(leading to linear phase). Orthogonality mostly cannot be 
strictly emphasized in practice. 
For the particular complex wavelets used in this work, the im- 
pluse reponses g of the wavelet function and A of the scaling 
function are a complex pair of even-length modulated win- 
dows. 
g(k) = by w1(k + 0.5) 05) (28) 
h(k) = bi wi (k + 0.5)e'#1(R+0,5) (29) 
(k = —nu, —ny t 1,..., ny — 1) 
where à and à are complex constants, w; and $4 are a pair 
of real-valued low-pass windows of width 2n,, symmetric 
about k — 0 and decaying to zero at each end. A commonly 
used one of this type is Gaussian 
w1(k) = exp (-&) ;, Ou(k) m exp (5) (30) 
Due to the compromise between the good locality of match- 
-ing and information sufficiency, the minimum width of win- 
dow functions should be 4, thus nu = 2. The modulation 
frequencies wy and @; should be complementary 
w +0 =T (31) 
in order to cover the frequency range [0,7]. Because ¢ and 
V are a pair of low- and high-pass filters, we have w; > &1. 
With the Gaussian window functions defined in (30), the 
Fourier transforms of g and h have conjugate symmetry about 
the modulation frequencies wy and &;. Since real 1-D signals 
have conjugate symmetric spectra, the neglect of the nega- 
tive half spectrum [—7,0] does not exclude any significant 
information about a real 1-D input. Ideally, a maximum cov- 
erage on the frequency range [0, 7] without significant gaps 
and with minimal overlap can be effective achieved if on each 
level 7, 
0j = 30; (32) 
thus, by (31), we have the modulation frequencies on the first 
level (bottom-up) 
Qj -— —, i 
c (33) 
e 
Ideally, the modulation frequencies are to be decomposed 
through levels 
Q3—1 
2 
  
ics (34) 
In practice, if Gaussian windows of (30) are used, the fre- 
quency decomposition through levels approximates asymp- 
totically to (34). 
Using the 1-D complex wavelet and scaling filters defined 
by formulas (28)-(29), we can implement the 2-D complex 
wavelet analysis in the same separable way as described in 
section (3.1). The 2-D wavelet filters so formed will be pre- 
dominantly first quadrant filters in the frequency domain. As 
real discrete images contain significant information in the first 
and second quadrant of the unit frequency cell, we need to 
use the complex conjugate filters g and h in addition to g 
and h in order to produce a mirror set of difference coeffi- 
cient matrices D;p, p = 1,2, 3, for each j-th level, containing 
the second quadrant information. 
622 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
The algorithm for the complex conjugate wavelet analysis of 
an image is similar to the one shown in Fig.1. The wavelet 
analysis from level 7 — 1 to level j correspond to transform- 
ing two complex approximation submatrices to eight complex 
approximation and difference submatrices 
TA $1; A;—1} res {A À; Din Dip? = 1,2,3} (35) 
where A; is the mirror of A;, and D; , is the mirror of D; ,. 
The algorithm is illustrated in Fig.2 - 3. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
D, | A As | Ds, 
B, Ds D, Ds, 
Dz1 A, A2 D, 
B5 B5, D,, D23 
A 
D, ,1 A 1 A 1 D 1 ; 1 
Ds Di; Dis Dis 
à 
  
  
  
  
  
  
  
  
  
f(x.y) 
  
  
  
Figure 2: Complex conjugate wavelet pyramid of an image 
f(x. y) 
  
Figure 3: Flowchart of complex conjugate wavelet analysis 
   
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