The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B6b. Beijing 2008 
174 
Figure 9. Grid figure of approximate coefficients of 1-level 
wavelet transform of uniform illumination image 
Three conclusions can be deduced after analysis of these figures 
(1) Illumination attenuating model corresponding to the low 
frequency signal; 
(2) Approximate coefficients are most sensitive to the varying 
of illumination; 
(3) Detail coefficients corresponding to the detail information 
of the image; 
Based on these conclusions, we can believe that the 
illumination equalization can be achieved by adjusting the 
approximate coefficients, and the detail information of the 
image can be enhanced by adjusting the detail coefficients. 
4. WAVELET COEFFICIENTS ADJUSTING 
SCENARIO 
4.1 Approximate Coefficient Adjusting Method 
One attenuation operator is applied to the approximate 
coefficients which are to mitigate the big coefficients and 
augment the small ones. This operation can make the 
approximate coefficients much smooth. 
Experiment results show that if only adjust the highest level 
approximate coefficients, the image visual effect will be very 
abruptly. Adjusting the approximate coefficients level by level 
and reconstructing the image can get better result. 
The original image is first decomposed with four level wavelet 
transform, we choose db4 wavelet in experiment. Next, the 
natural logarithms of approximate coefficients of each level are 
A ( JC V ) 
calculated, noted as v ’ * ' and then an appropriate 
attenuating operator ^ ( x ’ ^ ) [11] is applied to implement 
the non-uniformity correction of approximate coefficients. 
«>(*,,)=—2—• ( 4<£>ZV 
A(x,y) 
(12) 
where a is the average value of the approximate coefficients of 
that level, ^ is a parameter which is used to adjust the contrast 
of the image, it’s value is usually between 0.9 to 1.0. 
Detail information contained in different level is different, high 
level approximate coefficients contain less detail information, 
the attenuation degree is much less, and ^ is a bit smaller, 
moreover, the small scale approximate coefficients contain 
more, and ^ is a little big. through this process the whole 
illumination is balanced. 
By attenuating the big coefficients and enhancing the small 
coefficients, we adjust the approximate coefficients in different 
scales, and then a linear stretch is applied to all the approximate 
coefficients, so as to adjust the whole illumination. From these 
steps the illumination of image is more balanced. 
4.2 Detail Coefficient Adjusting Method 
Detail information is contained in the detail coefficients; by 
enhancing these coefficients the detail information of the image 
can be enhanced. We suppress coefficients of very small 
amplitude and enhance only those coefficients that are large 
within each level of transform coefficients. Equation (13) is 
used to accomplish this nonlinear operation [12] : 
/ (y) = a[sigm(c(y - b)) - sigm(-c(y + 6))] (13) 
Where 
1 
sigm[c{ 1 - 6)] - sigm[-c( 1 + b)] 
0<b<l 
sigm(x) is defined by 
sigm (x) = - — 
1 + e 
b and c control the threshold and rate of enhancement, 
respectively. It can be easily shown that f (jp) is continuous 
and monotonically increasing in interval [-1, 1]. Furthermore, 
any order derivative of f(y) of exists and is continuous. 
Therefore, enhancement using f (y) will not introduce any new 
discontinuities. In addition, f{y) satisfies the conditions 
/(0) = 0and/(l) = l. 
For the input detail coefficient y with maximum absolute 
amplitude _y max » w e map the coefficients range[~y mia ,y miix ] 
onto the interval [—1,1] . This is accomplished by using 
•Ушах as a normalizing factor in (14). Thus, f(y) may be written 
as f(y) = яу тах [sigm(c{y / y m - b)) - sigm(-c(y / у ж + 6))] ( 
14) 
The benefit of the normalization is that a, b, and c can be set 
independently of the dynamic range of the input coefficients. 
And last linear stretch is also applied to all the detail 
coefficients.