The International Archives of the Photogrammetry, Remote Sensing andSpatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
Considering ||H * G|| is tend to 1, the noise level of low 
frequency can approximately present noise level of high 
frequency coefficient thus it can be taken as the noise threshold. 
If the coefficient of high frequency is under 3cf^ , it can be 
done as equation (9). 
{HL,, LH,, Hp Ì = 0 J {H^, Lp, HH h }|| 2 < 3oi (9) 
If the coefficient of high frequency is above 3<T^ , it should be 
non-linear strength as equation (10) and (11). 
a si 8m(c * (yfy m ax ~ b)) - sigm(-c * (y/y mM + b)) „ (10) 
sigm(c * (1 - bj) - sigm{-c * (1 + bj) 
f/..\_ si Sf<c*(yly mM -b))-signi-c*(y/y tmx +b)) ± ( U ) 
sign(c*(i-b))-sign(~c*(i + b)) P 
Where 
sigm(t) = -—!—- (12) 
1 + exp 
y is the high frequency coefficient of wavelet decomposition. 
C, d is to adjust the strength coefficient, b is to adjust the 
initial value of the strength detail and it is the threshold of the 
strength processing. In this paper, b is taken as the threshold 
of filtering that means b = 3cr^ . C is taken as the 
characteristic of image between 20 and 40. In this paper, C is 
30. d is ranged among 1 and 0.05. 
Step 5: Wavelet reconstruction 
Mallat reconstruction is done to wavelet coefficient after 
filtering and strength to obtain the filtered spectral curve with 
noise removal. 
A spectral curve filtering processing experiment is done with 
mean filter smoothing, least squares smoothing and the non 
linear strength wavelet filtering algorithm (nLWF). Figure 4 
gives the different curve filtering result. 
Figure 4 Different filtering result of spectral curve 
As figure 4 shown, nLWF filtering algorithm of spectral curve 
can obtain reasonable result in two typical noise cases (as the 
rectangle in figure 4) compared with mean filter filtering, least 
squares filtering. Mean filtering is over smoothing and some 
detail is lost while the least square filtering is coordinate to the 
original spectral curve. 
Signal noise ratio and noise level of spectral curve are selected 
to further assess different spectral curve filtering algorithms. 
Table 2 gives the different spectral curve filtering algorithms to 
different objects of the 30 bands MAIS images and 128 bands 
OMIS images. 
Sensor 
Object 
Noise 
parameter 
Original 
Spectral 
Curve 
Mean 
filtering 
Least 
square 
filtering 
nLWF 
MAIS 
Water 
Noise level 
0.996 
1.011 
1.006 
1.05 
SNR 
10.129 
7.432 
8.192 
3.829 
Tree 
Noise level 
1.312 
1.332 
1.324 
1.338 
SNR 
9.061 
6.341 
8.929 
3.513 
Resident 
Area 
Noise level 
2.306 
2.438 
2.387 
2.445 
SNR 
13.072 
6.713 
5.22 
3.798 
Spare 
Area 
Noise level 
9.418 
9.939 
9.756 
9.953 
SNR 
5.428 
2.344 
2.202 
1.656 
OMIS 
Road 
Noise level 
3.576 
3.948 
3.867 
3.955 
SNR 
18.962 
7.274 
4.948 
4.673 
Tree 
Noise level 
0.585 
0.783 
0.691 
0.794 
SNR 
28.386 
8.179 
9.66 
6.03 
Resident 
Area 
Noise level 
1.778 
1.786 
1.845 
1.873 
SNR 
24.049 
11.365 
7.536 
6.887 
Field 
Noise level 
3.054 
3.48 
3.674 
3.674 
SNR 
27.43 
7.899 
11.309 
7.316 
Table 2 Different spectral curve filtering results 
As table 2 shown, different object in different sensor hyper 
spectral image has different noise level and signal noise ratio. 
nLWF algorithm obtains higher SNR and lower noise level than 
original spectral curve. And its result is better than the mean 
filtering and least square filtering algorithm. 
3.2 Fractal dimension calculation algorithm 
In this paper, a step measurement method of fractal dimension 
calculation algorithm considering the spectral curve 
characteristic is proposed. Curve length Z,(r) under different 
step measurement units is determined by the step length N(r) 
and steps T as equation (13). 
L(r) = N(r)-r (13) 
And the curve length L(r) can be represented as equation (14) 
under the definition of fractal. 
L(r) = q-r M (14) 
Where T is the step number under different step measurement 
unit, fi = 1 — D is the reminder dimension, D is the fractal 
dimension value of the spectral curve, q is the coefficient to 
determine. Thus we can obtain, 
log(L(r)) = (l-D)logr + C (15) 
Where C is the coefficient to determine. With the different step 
measurement unit, we can obtain point array of 
(log(L(r)), log r) . Thus we can calculate the slope of the 
line K( or u ) which is the fractal dimension value of spectral 
curve. 
D = \-K (16) 
The detail procedure of step measurement fractal dimension 
calculation can be described as following steps:® Take d { as 
the initial step unit, calculate the distance d, 2 between the first 
point and the second point P 2 of spectral curve. © if 
dyi > d\ , interpolate one point p between P ] and P 1 to 
make the distance from p to P x as d x . G) if d X2 < d x , to 
calculate the distance d u between P x and P 3 , if d l2 >dj, 
to interpolate one point p between P 2 and P 2 to make the