Fig.l The workflow of proposed algorithm 
which is considered as the auxiliary information for the 
following filtering. The non-linear improved local Wavelet 
soft-threshold algorithm is applied to filter the high frequency 
wavelet coefficients respectively with preservation at those 
pixels marked as edge points. Repeat it through the levels and 
after reconstruction and exponential transform, the final 
denoising image is produced.The workflow of proposed 
algorithm is shown in Fig. 1. 
3. IMPLEMENTATION OF ALGORITHM 
3.1 Wavelet Transform Modulus-Maximum Algorithm 
for Edge Detection 
Mallat has established the mathematical relationship between 
wavelet transform and the local singularity of function, and it is 
expressed with the Lipschitz index. According to the study of 
Mallat, for edge points, which performances as step signal, its 
wavelet transform modulus value does not change with the 
scale; otherwise for white noise, which is regarded as random 
process of almost everywhere singularity, its modulus value 
will decrease with the increase of scale (Mallat et al, 1992). 
Therefore, the Wavelet transform modulus maximum algorithm 
can efficiently eliminate the noise disturbance and detect the 
genuine edge points. 
At each step in the decomposition procedure, the signal 
/ 2 y+i(x,y) is decomposed into four independent and spatially 
oriented channels, producing four sub-images 
f 2 j( x ^y) > d X 2 j{x>y) , djj(x,y) , d^j(x,y) , and only a coarse 
approximation image f 2 j(x,y) is decomposed in the next step 
(Gupta, 2007). 
We define the wavelet transform modulus and angle as: 
M s /(Xy) = V"l 2 +n 2 
2 
(1) 
\f( x >y) = artan^-)+• 
0 
(n x > 0, n 2 >0) 
K 
(«1 < 0) 
(2) 
n \ 
2 k 
(«! >0,H 2 <0) 
Where n\ and n 2 are gradient in horizontal and vertical direction 
respectively. M s /(x,y) is the modulus of the wavelet transform at 
the scale s. A s /(x,y) is the modulus angle, which indicates the 
direction where the signal has the sharpest variation. 
After getting the M s /(.x,jp) and A s f (x,y) , the edge point 
is the one that has the locally maximum modulus along the 
direction A s f(x,y). After that, a threshold is applied, the 
edge point modulus below it will be eliminated, and this can 
eliminate many false edge points. 
3.2 Edge Fusion 
To get the robust edge information in each level, it is realized 
by fusion the edge projected from the higher level and the edge 
detected by the Wavelet transform modulus-maximum 
algorithm in current level. Because the edge detection operator 
responds to the same edge differently in different scales, the 
edge position varies with the scale. If we just add the edge of 
two levels together to fuse the edge, this will lead to edge 
redundancy and cannot suppress the noise. In this paper, the 
edge fusion consists of two steps: one is edge correlation and 
the other is edge growth. 
We firstly project the edge points of higher level into the 
current scale level, and for each point a neighborhood is 
defined, which we call it correlation domain. In the correlation 
domain, searching the modulus maxima point in current level 
with similar edge direction, if the edge direction difference is 
below a certain threshold, we assume the edge points of two 
levels are correlated, and the projected edge point’s information 
is updated. Through edge correlation, the edge from higher 
level is transferred to the current level. 
Because the edge correlation is only performed in a small 
correlation domain, this will cause a problem that the edge is 
incomplete. To overcome it, edge growth is used and 
implemented as follows. B s i+1 is the results of edge 
correlation of the level s and s+1, rewritten as Z? 0 5i5+ i . 
I s>s+ \ and is the corresponding modulus and angular 
information, rewritten as / 0 . ? , s +i, v4°i,s+i respectively. M s is 
the modulus maxima image of level s. the £th iteration results 
of edge growth is expressed as B k s ,s+1 (k > 1) . The 
correlation between the pixel (i,j) in M s and pixels of 
5^ _1 i)S+ i is depicted as D k \i,j). If D k s ~ X (i,j) = l , it 
means the pixel (i, j) in M s is correlated with the pixels of 
B k '5,5+1 , and vice versa. The value of D k ~\i,j) is 
determined as follows: