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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