The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
1180
paper. The second section discusses the theoretical basis and
transformation characteristics of multi-band and biorthgonal
wavelet. Then the fusion method based on multi-band and
biorthgonal wavelet is implemented to fuse 10-m SPOT
panchromatic and 30-m multispectral TM images Next, the
experimental result is compared with previous methods
developed for image fusion, such as IHS ,PCA and two-band
wavelet
coefficients HL (variations along the columns), vertical
coefficients LH (variations along the rows), diagonal
coefficients LL (variations along the diagonals) (Gonzalez and
Woods, 2001). The three high frequency image is called detail
image, which contain information of local details.
When a = rf and b= rfkdhe resulting equation is
2. BIORTHOGONAL AND MULTI-BAND WAVELET
2.1 MItiresolution Analysis
Wavelets are functions in //^determined from a basic wavelet
function by dilations and translations.They are used for
representing the local frequency content of functions. The basic
wavelet should be well localized in general, and the wavelet
should have zero mean( Daubechics, I., 1992.). The basic
method to construct a wavelet is Multiresolution Analysis.A
Multiresolution Analysis (MRA) is defined by a sequence of
closed subspaces (Vj ) j EZ ,which approximates L 2 R, and a
function <£> EL]_ R s an orthonormal basis for V 0 .
V'j.k(x) =
ip (n j x - k)
(4)
WT,U,k) = {f(x),<P i M = X f f(x) V \ri"-k)dx (5)
v« J R
We denoted it as n-band wavelet. After n-band discrete wavelet
transform, an image yields n2 images: one low-pass image and
n2-l high-pass The high frequency image is called detail image,
which contain information of local details and low frequency is
approximate. image.
{0}--- e V_ t e V 0 c F, c •• L 2 (R)
{$ o.«;® <>,„(*) = - «)> n e z }
Where O(x) is a scaling function
2.2 Multi-band wavelet
Generally, a wavelet family is described in terms of its mother
wavelet, denoted as v|/(x). A daughter wavelet \|/ ab (x) is defined
by the equation
l / / aA x )= a)
va
Figure 1 two-band wavelet and three-band wavelet transform
2.3 Biorthogonal wavelet
Generally a function f(x) can be decomposed as a superposition
of the orthogonal basisv|/j k (x), But to biorthogonal wavelet, a
function f(x) can be written (7)
where a, b e R and a^O;
a is called the scaling or dilation factor and b is called the
translation factor.
A common choice is a=2 J and b=2 J k, where j and k are integers.
The resulting equation is
¥j,ki x ) = —(2 x - k) l (2)
V2'
WT f U,*) = (/№, k (tj)=-j= J^/(0i^*(2’ j x- k)dx (3)
This equation(2) is two-band orthogonal wavelet. Accordingly
the wavelet transform of function f(x) is (3)
After 2-band discrete wavelet transform, an image yields four
images: one low-pass image and three high-pass images.
Namely, approximation coefficients (labeled LL), horizontal
fix) = (WT f ,<p jtk (x)) = \(f(x),<p ik (x))<p jk (x)dx (6)
fix) = (WT f , (pj, k (*)) = ¡(fix)jp* j,k {x))(p jk (x)dx (7)
Where {(p jk O), (p* j,10)> - ô jr ô kl (8)
When applying the biorthogonal wavelet to decompose function
or images, the function (p j,k (x) is used, while the
function (p . k (x) is applied to reconstrut function f(x) The
biorthogonal wavelet systems generalize the classical
orthogonal wavelet systems. They are more flexible and
generally easy to design. One of the main reasons to choose
biorthogonal wavelets over the orthogonal ones is symmetric.
Symmetric wavelets and scaling functions are possible in the
framework of biorthogonal wavelets. However, the
orthogonality no longer holds in biorthogonal wavelet systems.
However,that is the near orthogonal system.