The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
As table 1 shown, the measurement length of spectral curve 
decrease with the increase of spectral band width and the trend 
of the decrease is smaller and smaller. And this relationship 
shows the measurement length of spectral curve has the 
exponential relation to the measurement band width. Thus the 
spectral curve can be described with the fractal dimension. Take 
different measurement spectral width as £ , the length of 
spectral curve as N , the statistics curve between log(i') and 
log(A r ) as figure 2: 
Figure 2 Log relation between spectral curve length and width 
As figure 2 shown, the log(£') - log(W) has obvious linear 
relation. The result of line fitting, it can realize the level 
of a < 0.02 , thus the spectral curve has the characteristic of 
fractal and the fractal dimension value can be used to present 
the spectral feature of spectral curve to each pixel. 
3. METHODLOGY 
The dimensionality reduction algorithm can be explained into 
the following three steps. 
Firstly, noise removal processing is done to the hyper spectral 
curve for the dimensionality reduction. Wavelet transformation 
is used to filter the noise of spectral curve of the hyper spectral 
image. With the multiple resolution analysis of wavelet 
transformation, the spectral curve which can be constructed by 
the pixel vector in spectral dimension is decomposed into 
smooth component and noise component. The high frequency 
noise component is removed before the inverse wavelet 
transformation to obtain the noise removal spectral curve. 
Secondly, fractal dimension value is calculated to the noise 
filtering spectral curve. The fractal dimension calculation 
algorithm is designed to the spectral curve. Finally, the 
dimensionality reduction is done with the fractal dimension 
feature of the spectral curve. Spatial spectral data cube of hyper 
spectral remote sensing image is formed by the fractal 
dimension value of spectral value, which can obtain spectral 
distribution image in spatial space. The spatial spectral data 
cube image can combine spectral and spatial texture 
characteristic together. Figure 3 gives the dimensionality 
reduction procedure of hyper spectral image. 
Spatial 
domain 
Fractal feature image of 
Dimensionality reduce result 
Figure 3 Dimensionality reduction with fractal analysis 
3.1 Spectral curve filtering 
Spectral curve noise will affect the result of spectral feature 
analysis. A non-linear strength wavelet filtering algorithm 
(nLWF) is proposed to spectral curve filtering. First the 2 level 
wavelet decomposition is done to the spectral curve with Morlet 
filter. The deviation of low frequency is selected as the noise 
threshold. High frequency coefficient under noise threshold is 
set zero and the coefficient above the noise threshold is non 
linear strength. With the wavelet reconstruction, we can obtain 
the spectral curve after noise removal. Following is the detail 
procedure of spectral curve filtering. 
Step 1 : Determine Morlet filter and filter window size 
Morlet wavelet filter with the window size of 13 is selected as 
the wavelet filter as equation (2) and (3): 
h[] = {- 0.00332761,0.00569791,0.0196637, 
- 0.0482603,-0.0485391,0.2925620.564406,0292562, 
-0.0485391-0.0482602,-0.0196637,0.00569754,-0.0033276} 
g[] = (0.00332761,0.00569794,-0.0196637,.0196637, 
-0.0482603,0.0485391,0.292562,-0.564406,0.292562, 
0.0485391,-0.0482602,0.0196637,0.00569794,0.0033276} 
Where h[] is the low pass filter of Morlet wavelet and g[] is 
the high pass filter of Morlet wavelet. 
Step 2: Cycle expand of spectral curve as the filter window size 
as equation (4). 
(2) 
(3) 
^N-i+i ~ ^N-1- 
(4) 
i = 1,2,3,4,5,6 
Where / ( is the spectral curve and N is the feature point 
number or band number. 
Step 3: Two level wavelet decomposition of spectral curve as 
equation (5). 
{LL h , HL, , LH h , HH, ,1 < i < N} (5) 
Step 4: Non-linear strength of noise removal 
Noise is central at the high frequency coefficient after wavelet 
transformation. The common noise removal methods is to select 
noise threshold and set the coefficient under threshold with zero 
to remove noise from the original signal (Pan Quan,2007,1998; 
Jansen M,2001; Wu C. Q,2004). The noise threshold can be 
determined from the original spectral curve noise level. And the 
noise level of original spectral curve can be calculated from the 
low frequency coefficient thus the deviation of low frequency 
coefficient can be taken as the noise level as equation (6). 
<T 0 = <J{LL, ,1 < / < N} (6) 
Thus the noise level of each decomposition coefficient can be 
calculated as the noise expands theory. 
m—2 
=^0 (n*#/)* G m-i (7) 
1=0 p 
Where H is the Fourier transformation of low pass filter of 
h[]G is the Fourier transformation of high pass filter of 
g[] . * represents convolution, H. is the 2 m ' scale 
expansion of H , G t is the 2™ 1 scale expansion of 
G , is the norm. If the scale is YYl = 2 , the noise level of 
each decomposition coefficient is, 
<* m =(7 o\\ H * G \\ f < 8 > 
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