Therefore the two branches of a wavelet function can be 
described accordingly: 
MCO) = r-?. e 
From the deduction in the previous section, we can obtain a 
group of filter coefficients in they direction: 
Formula (2) shows that the Gaussian normal distribution 
function is a smooth function, and images’ mean peculiarity 
stays unchanged after processing. To facilitate the discussion, 
we first deduce the wavelet function and the wavelet filter from 
the Gaussian function’s one-dimensional representation and 
then extend it to two-dimensions (Lin, 2004). 
According to the peculiarity that the scale exhibits progressional 
variety, we deduced a group of multiscale wavelet filters from 
the two dimensional Gaussian smooth functions and called them 
anti-symmetrical wavelet as follows: 
1 7t 3S 2 0 2 (0 2 
h(s,x) k je 2 e Jka d(o 
fj n 3s 2 g 2 co 2 
g(s,x) t =^~ jj(oe 2 e jk<s> dao 
The lowpass impulse response of the filter is symmetric at the 
origin, the highpass response of the filter is anti-symmetric at 
the origin and the truncation error is very small. They have the 
characteristics of approximation compact support and 
smoothness. Meanwhile nine groups of filter response 
coefficients are given to realize the feature extraction from 
coarse to fine and to provide a useful tool for multiscale edge 
detection. 
2.2 Two-dimensional Wavelet Transformation and Edge 
Detections in Images 
The two-dimensional Gaussian normal function (1) can be 
expressed separately as 
h(s,y) k =h(s,x) k 
g(s,y)t =g( s > x )k 
At last, a wavelet transformation can be implemented with the 
convolution of the image and the corresponding filters: 
~wjf(x,y)' 
f*(G s ,H s ) 
W!'f{x,y)_ 
_f*(H s ,G s ) 
The filters in formula (7) consist of filter response coefficients: 
H s = {h(s,x) k }, G s = {g(s,x) k }. 
From (7), the grads’ module on the pixel (x,y) is: 
mj=Vi »;7(*..rti ! i 2 
The grads’ direction is 
A f = arctan 
W t a f(x,y) 
Wjf{x,y) 
We can use direction profile detection (Chen, 2003) to perform 
extremum detections and acquire the edges in images with the 
grads’ modules and the directions obtained from formula (7), (8) 
and (9). In fact the convolution according to formula (7) can be 
done in a dyadic space, i.e., s=2 J . If we don’t need to generate 
multi-channel images, then there is also no need for 
“extractions”, and even no need for lowpass filters. We can 
reach the destination of multi-channel detection with different 
filter scales(s=1.75- i ) (Cheng, 1998).