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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
curvature k such that it is defined as F c = -sk , where £ is a 
coefficient. Finally, F a represents the speed moving passively 
by an underlying velocity field U{x,y,t)-N , in which 
N = V^/| V<j)\, and thus F a =U(x,y,t)- N . Plugging this speed 
function rewrites the evolution equation as equation (9). 
</> l +F 0 \V</>\+U(x,y,t)-V</> = -£k\V4>\ (9) 
The first term after the time derivative on the left is concerned 
with the propagation expansion speed and should be 
approximated through the entropy satisfying schemes. The 
second term is related with the advection speed and can be 
simply approximated by upwind scheme with the appropriate 
direction. The third term is curvature speed alike a non-linear 
heat equation, to which an appropriate solution approach is the 
central difference scheme since the information propagates in 
both directions. The following paragraphs will present details. 
Edge flow. In this study the active contour is deformed by the 
edge flow towards the image pixels that have high probability 
to be the segment boundaries (Ma and Majunath, 2000). The 
method was originally designed for boundary detection or 
image segmentation considering regional image attributes.. 
Figure 3 illustrates edge flows generated from the image that 
move towards an expected boundary edge. Each flow vector 
indicates direction towards the closest edge and an edge can be 
found at locations where the flow vectors meet from opposite 
directions. This edge flow method requires edge linking step for 
a proper image segmentation result, which can be done through 
active contour since it propagates the closed polygons. 
Figure 3. Boundary detection using edge-flow 
The general form of an edge flow vector r at an image location 
s with an orientation <9 is defined in equation (10) as a function 
of the edge energy E(s, 9), the probability P(s,#) of finding the 
image boundary in the direction 0 , and the probability 
P(s,0 + ri) in the opposite direction# + k 
r (5,9) = T(E(s, 9), P(s, 9), P(s, 9 + n)) (10) 
The first component measures the energy of local image 
information change and the rest two components determine the 
contour flow direction. The prediction error Err{s,6) at pixel 
location 5 = ( x ,y) is defined as equation (11) using the 
smoothed image I a (x,y) obtained by applying the Gaussian 
kernel G a (x,y) with a variance a 1 . The error function 
essentially estimates the probability of finding the nearest 
boundary in two possible flow directions 
Err(s, 9) = 11 a (x + d cos 9, y + d sin 9) -1 a (x, >>)| ( 11 ) 
From these prediction errors, an edge likelihood P(s,0) using 
relative error is obtained 
P(s,0) = 
Err(s,9) + Err(s,9 + Tr) 
The probable edge direction is then estimated by 
(12) 
Ut 71 / X. 
9'=argmax jp(s,9')d9'■ 
(13) 
On the other hand, the edge flow energy E(s,0) at scale a is 
defined as the magnitude of the gradient of the smoothed image 
I a (x,y) along the direction 6'. 
E(s,9) = 
l—K^y) 
on 
I(x,y)* — G a (x,y) 
on 
(14) 
= \l(x,y)*GD a0 (x,y) 
where h represents the unit vector in the gradient direction, 
GD a (x,y) is the first derivative of the Gaussian along the x-axis, 
and GD a0 (x,y) is the first derivative of the Gaussian along 
orientation 0 
GD, e (x,y) = GD,(,x\y’) (15) 
where 
~x r 
cos# 
sin# 
X 
X 
-sin# 
cos# 
_y_ 
Once the flow direction and the edge energy are computed, the 
“edge flow” field is computed as the vector sum in equation (16) 
0+x/2 
f(s) = i[£(s,#')cos#' E(s,9')sin9'] T d9' ( 16 ) 
e-ni 2 
Boundary refinement using active contours. The edge flow 
vector field computed in the aforementioned steps is used as the 
external force to enforce the contour move towards edges. The 
contour curve evolution can be formulated as equation (17) 
where f is the edge flow vector field and n = yj,/1 y^ | 
C,=(f-N)N + kgN-F 0 gN (17) 
The edge penalty function g attracts the contour towards the 
boundary and has a stabilizing effect when there is a large 
variation in the image attribute value. It is produced from the 
edge-flow vector field f by solving the Poisson equation as 
equation (18), where A is the Laplacian (Sumengen et al.,2002). 
V • f = —Ag (18) 
Comparing with the traditional gradient edge penalty function, 
edge penalty function derived from edge flow is more rigid to 
noise as shown in figure 4. 
Figure 4. Comparison of edge penalty functions