In spatial domain, the corresponding wavelet will be 
w(x). The dilation of this function by s 
v (0) = sy (sx) (5) 
is used for the wavelet transformation of the image 
through its application along each image coordinate 
direction. The wavelet w(x) can be interpreted as the 
impulse response of a band-pass filter. Therefore, a 
wavelet transformation is a convolution with a dilated 
band-pass filter. The scale parameter s defines the 
size of the features to be enhanced by this transfor- 
mation. When scale decreases, finer details are em- 
phasized. Various types of y(x) (or correspondingly 
of ¥(w)) can be used, according to the characteristics 
of the sought features. In our experiments, road net- 
works in SPOT images were enhanced using 
di 2 
3 [sin | 
H(o) = (cos 5) and ¥ (0) = we (6) 
4 
The wavelet transformation of an image generates a 
new image version, in which road networks are more 
prominent. These networks are then precisely identi- 
fied through dynamic programming, which is a gen- 
eral multistage optimization technique to solve 
problems by maximizing a merit function (or minimiz- 
ing a cost function) [Ballard & Brown, 1982]. Its appli- 
cation to edge extraction involves the definition of a 
function which embodies the notion of "best edge", 
and the solution of an optimization problem to extract 
the actual edge. A such function can be defined as 
n n-1 
hx, ux) m s s(x) +a Y QOO Xr +1) (7) 
k=1 k=1 
where s(x,) expresses edge strength, q (X, X, , 4) 
expresses change in edge direction between two 
successive edge pixels, and a is a negative constant. 
By evaluating the above function for possible edge 
paths, we select as edge the sequence of eight-con- 
nected pixels which maximizes the merit function. 
The disadvantages of this approach can be synop- 
sized as follows: 
Q time consuming due to pixel-by-pixel tracking, 
Q sensitive to noise, 
a difficulty in bridging edge gaps, and 
Q appropriate only for low curvature edges. 
To overcome the above disadvantages, an edge can 
be extracted as a set of n smaller segments defined 
by the seed points Poy P»,,..,P? which are man- 
ually provided on-screen. The merit function is then 
expressed as the summation of smaller terms 
h(P Puy = Ho (Py, PY + 1, (Py Po) + 
Fon Py Py) (8) 
The original seed positions can be altered during the 
solution. A preset value 5 defines the maximum al- 
lowable change in the position of seed points 
(|Pj- P*| = |V\<5) and it is used to eliminate 
tracking errors associated with the selection of mis- 
leading local maxima. Each term hi( PP; 0) 9* 
presses how well a path in the image connecting 
points P; and P, , , satisfies the merit function. 
Applied to road extraction, the definition of the asso- 
ciated merit function is based on four observations: 
0 a road pixel is lighter than its neighbors on both 
road sides, 
0 a road is usually smooth and of limited curvature, 
Q gray values along a road usually do not change 
very much within a short distance, and 
Q road width does not change significantly. 
These four conditions are formulated and combined 
as a merit function 
h(P P, a) = Y [S(P) +aD(Py +BT(P,)] 
P,€ C, 
+KQ(P, Pi. (9) 
where S(P, denotes road strength, D(P,) the lo- 
cal variance of strength, T(P,) the local texture, 
Q(P; P;,1) the difference of directions, and a.Bx 
are constants. 
This multistage optimization problem is solved by dy- 
namic programming. The advantages of the modified 
optimization method (improved dynamic program- 
ming) are: 
2 efficient handling of long curves, 
Q robustness in the presence of noise, and 
Q ability to bridge edge gaps. 
4. ACTIVE CONTOUR MODELS 
Active contour models, also known as "snakes", are 
energy minimizing splines, guided by shape and radi- 
ometry forces to fit to, and thus identify, edges in dig- 
ital images [Kass et al., 1988]. Their implementation 
is semi-automatic, with an operator manually provid- 
ing the necessary initial edge approximations in the 
form of a few seed points. The desired edge behavior 
is expressed in energy terms and assuming that local 
energy minima correspond to object boundaries, 
148 
edges are ic 
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Fig. 2: 
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