Yunham Dong 
  
2 ALGORITHM 
2.1 One Dimensional Algorithm 
Let us consider a one-dimensional signal, f(x). Fluctuations in f(x) can be due to either local oscillations or edge 
crossings. An appropriate filtering algorithm should react differently to these two types of fluctuations: to smooth the 
local oscillations to reduce the speckle (noise) level and to enhance edge crossings to avoid blurring. Figure 1 depicts 
the ideally expected filtering results for these two types of fluctuations. 
  
(a) (b) 
Figure 1. The ideal filtering results for (a) local oscillations and (b) edge crossings. 
Wavelet transform techniques have been used extensively in data processing, where wavelet function can be chosen for 
different purposes, such as data compression, noise smoothing and edge detection (Mallat and Zhong, 1992). One of the 
distinct advantages of wavelet transforms lies in the adaptive dilation of the wavelet. As a result, the wavelet transform 
can capture rapid changes in the signal at small dilation factors, or be not so sensitive to rapid but small changes at large 
dilation scales. Denoting 
1 2 
sta (1) 
the dilation by a scaling factor s of any function g(x) in the space domain, the Fourier transforms of g(x) and g,(x) 
are, 
G(w) = [Tew exp(—iax)dx (2) 
"Foo . 
G,(w)= [^ g(x) exp(—iax)dx =G(sw) (3) 
The second order derivative of Gaussian function, which has been shown to be a good edge detector (Canny, 1986), is 
selected in this paper as the wavelet function for detecting edge crossings. Local oscillations, which are often faster in 
frequency and smaller in amplitude, compared to edge crossings, can be suppressed using a proper scaling factor, s. Let 
q,(x) be the edge crossing detector defined by the wavelet transform as, 
q,(X) 2 f X)* g,(s) (4) 
Where * denotes one-dimensional convolution and g(x) is the second order derivative of Gaussian function with a 
scale factor of s. 
d 
g(x)e-— 
dx 216 
  
exp(-x" /260*) | (5) 
where o is the standard deviation of the Gaussian function (6 =1 has been chosen for use in the paper). The 
computation of (4) can be easily implemented by use of the DFFT (discrete fast Fourier transform) technique in the 
frequency domain. The computation can also be achieved by direct convolution in space domain, as g(x) rapidly 
approaches to zero at both ends. 
The detected edge crossing points correspond to q,(x) — 0. A region between two consecutive edge crossing points is 
either a peak ( q, (x) « 0) or a valley ( q,(x) » O0 ). With the assistance of such information, and using a moving window, 
the new value f(x;) for pixel x; is determined as, 
1. the mean value of the original signal if there is no any edge crossing point in the moving window; or 
2. the mean value of the original signal in the peak or valley region where the current pixel x; is located if there 
exits one or more than one edge crossing point in the moving window. 
  
90 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part Bl. Amsterdam 2000. 
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