International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
Int 
  
2. WAVELET USED IN THE IMAGE FUSION 
2.1 Basic introduction to related theory 
In wavelet transformation, the basis functions are a 
set of dilated and translated scaling functions: 
moe HON 
Qj k(n) = 2/ ! ^o(2/ n — k) (1) 
and a set of dilated and translated wavelet functions: 
12 
y 0 9 27 y n - 4) @) 
where (n) and y(n) are the scaling function and the 
mother wavelet function respectively. One property 
that the basis function must satisfy is that both the 
scaling function and the wavelet function at level j can 
be expressed as a linear combination of the scaling 
functions at the next level j+1: 
@ jk (1) = 3 hûm — 2k)@ j+1,m CM) (3) 
m 
and 
y jk Qn) - $ gm - 2k)o j 1, m Q0) (4) 
m 
where h (m) and g (m) are called the scaling filter and 
the wavelet filter respectively. 
For any continuous function, it can be represented by 
the following expansion, defined in a given scaling 
function and its wavelet derivatives (Burrus, 
et.al.1998): 
oo 
fn) = Se; pj km + X Xdj0Ov;k) 
k ]27]0 k 
(5) 
The fast Discrete Wavelet Transform (DWT) can be 
expressed as follows: 
€ j«1 (A) = X c (mh (n — 2k) (6) 
n 
d 410) 7 X eg(g (n - 2k) (7) 
The scaling filter h (n) is a low pass filter 
extracting the approximate coefficients, C 4100): 
with con) = f (n). while the wavelet filter 
g (m) is a high-pass filter extracting the detail 
coefficients d, (k). The coefficients are 
downsampled (i.e. only every other coefficient is 
taken). 
The reconstruction formulas are given by: 
cj(k) = X (c ja 0h (n - 25) + cj4100g (1 - 2E) 
n 
(8) 
Generally, discrete wavelet is introduced by multi- 
resolution analysis. Let L*(R) be the Hilbert space of 
functions, a multiresolution analysis (MRA) of L*R) 
is a sequence of closed subspaces Vj, j € Z(Z is the 
916 
— 
set of integers), of L(R) satisfying the following six 
properties (Mallat, 1989): 
l^" The subspaces are 
nested: V; c Vi, Viel 
2. Separation: Nez V; = {0} 
3 The union of the subspaces generate 
2 RT 2 
L'(R): ez Vj z LR) 
4. Scale invariance: 
f(t) € V, fO) e Vj41 VieZz 
5. Shift invariance: 
fü)w«Vlg«» fü -k)«e ly Vk eZ 
6. 3$ € Vg,the scaling function, so that 
Jo og | ; j 
9Q 712 k)k € zlis a Riesz basis of 
Vo. 
There is also a related sequence of wavelet 
subspaces W; of L'(R), Vj € Z, where W; is the 
Vj V;.1. Then, 
Via = V; e W; , where CD is the direct sum. 
orthogonal | complement of in 
The above applies to about one-dimension situation; 
for two-dimension situation, the scaling function is 
defined as: 
D(x, y) = 96909(v) (9) 
Vertical wavelet: 
wx, ») = dw) (10) 
Horizontal wavelet: 
V?(x, y) 2 wG)o(y) (11) 
Diagonal wavelet 
V3, y) 2 vG)v) (12) 
d(x, y) can be thought of as a 2-D scaling function, 
3 
vl (x, y). 2 (x, yh ¥(x, vy) are the three 2-D 
wavelet functions. 
For the two-dimension image, the transform can 
be expressed by the follows: 
oc oo 
a(x, WEL v AC 
c=—0 r=—0 
2x)h(r — 2y)f jte r) 
(13) 
SS co 
C=-—00 | =-—00 
di as y) 2x)h(r - 2y)f je. 
(14) 
d? s. y) = 5 S h(c —- 2x)g(r — 2y)f jc. r) 
rm (15) 
45 (y) 7 SY gle - 2x)g(r = 20/567) 
C=—00 y =—0 
(16)