de Carvalho, Luis 
  
Wavelet analysis provides a basis for handling multiresolution data sets, for understanding noisy signals and for 
improving storage efficiency. The technique is already recognised for remote sensing applications such as automatic 
image registration (Djamdji e? al. 1993, Fonseca and Costa 1997, Fonseca et al. 1998), spatial and spectral fusion 
(Garguet-Duport et al. 1996, Blanc et al. 1998, Zhou et al. 1998), feature extraction (Simhadri et al. 1998), speckle 
reduction (Horgan 1998) and texture classification (Zhu and Yang 1998). 
As far as the authors are aware the present contribution is the first attempt to derive a methodology, based on 
multiresolution wavelet analysis, for detecting changes on multidate satellite images with different radiometric and 
geometric characteristics. 
2 MULTIRESOLUTION WAVELET ANALYSIS 
Man-made sensors measure physical quantities and represent these as signals. In most cases the patterns shown by these 
signals are not clear and we try to transform them to extract information of interest. Wavelets are one of the many 
possible ways of transforming a given signal. The principle involved here is the representation of a vector (i.e. digital 
signal) by a linear combination of basis vectors (Unser 1996). The aim is to compare the signal at hand v with a set of 
test vectors v; and their associated parameters b;: 
v=) by, (1) 
In Fourier analysis we represent a given signal with a combination of sines or cosines as the basis, whereas in wavelet 
analysis this is done by dilated and translated versions of wavelets and scaling functions. For digitally sampled signals 
the output y(n), representing the low frequency components, comes from the convolution of the signal at hand x(n) with 
a lowpass filter 4 (scaling function), where k is a positional parameter for the filter coefficients and n is the time step: 
y()="Y" h(k)x(n—k). (2) 
Then, the residual error between this lowpass filter at one level and its dilated version becomes the highpass filter g 
(wavelet function) which, after the convolution of the same input signal x(n), produces the output w(n), representing the 
high frequency components: 
w(n)- > s(k)x(n-k). (3) 
At this point we have two new representations of the original signal: y(n) and w(n). One shows the low frequency 
components and the other the high frequency components that were mixed in one signal before. If we take the y(n) and 
perform the same convolutions again we are moving to a coarser scale and representing this new input with a new set of 
basis vectors. This leads to a natural multiresolution representation of the original signal where the smooth part plus the 
details combine to form the signal at a finer scale level and so on. 
The capacity of perceiving scales can be the key for better understanding the signals around like complex biological 
sensors do. Our eyes, for instance, evaluate the overall picture and afterwards see the details. Remotely sensed images 
are two-dimensional signals which are relatively noisy and provide lots of information at different spatial scales. In this 
context, their analysis can be considerably improved if their transformations show detailed and overall views. 
Multiresolution wavelet analysis in discrete time corresponds to successive band pass filters decomposing the signal at 
each step into details and overall pattern. In a two-channel filter bank (Figure 1) it separates the high from the low 
frequencies recursively using the same transform at a new scale (Strang and Nguyen 1997). 
Lowpass Overall ... 
filter ; 
Lowpass Overall Highpass Detail 
Input filter Hishpass | Detail | filter 
image 
filter 
  
  
  
  
Figure 1. Wavelet tree. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000. 341