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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