1 Po) + 
(8) 
d during the 
1aximum al- 
seed points 
O eliminate 
tion of mis- 
» Pig) ex 
connecting 
nection. 
of the asso- 
servations: 
ors on both 
d curvature, 
not change 
| 
intl y. 
d combined 
+BT(Py] 
(9) 
cal texture, 
, and a,Bx 
jyved by dy- 
he modified 
c program- 
and 
S 
nakes”, are 
pe and radi- 
dges in dig- 
lementation 
jally provid- 
tions in the 
ge behavior 
1g that local 
boundaries, 
edges are identified by minimizing an energy function 
E. This function comprises of two terms, one radio- 
metric ( E. p? and another geometric ( E 9) 
Es EAE, (10) 
Parameter A expresses a measure of the roughness 
of the initial edge estimate. The radiometric energy 
part assumes that the first derivative of image inten- 
sity in the direction of the gradient is extremal at step 
edge points. The geometric part defines the form of 
the edge contour to be a cubic spline function, contin- 
uous in first and second derivative and thus enforcing 
its smoothness. The total energy is minimized 
through an optimization procedure which forces the 
snake to approach the actual edge contour (Fig. 2). 
Iterative convergence is achieved by mathematically 
simulating the behavior of a deformable body embed- 
ded in a viscous medium and solving the correspond- 
ing dynamics equation [Fua & Leclerc, 1990]. 
  
Fig. 2: Edge contour detected by snakes 
Active contour models present several advantages: 
0 They are able to bridge radiometric gaps and 
weak regions because they are using global infor- 
mation. 
0 They can be applied for the extraction of both 
open and closed edge contours. 
0 They can successfully follow the contour around 
corners by relaxing the second derivative continu- 
ity constraint of their geometric energy part. 
Q Their mathematical foundation can be extended 
and customized by the use of additional geometric 
constraints to fit specific and various object types 
(e.g. parallel curves for road extraction). 
Q Snakes can be extended towards 3-D object ex- 
traction by using an underlying DTM or by includ- 
ing camera models. 
9. LEAST SQUARES TEMPLATE MATCHING 
This modified matching method, used for edge detec- 
tion and tracking, is based on /east squares match- 
Ing. A synthetic edge pattern is introduced as the 
reference template which is to be subsequenlty 
matched with image patches containing actual edge 
segments. Assuming f(x, y) to be the synthetic edge 
template and g(x,y) to be the actual image patch, ob- 
servation equations are formed between conjugate 
pixels as 
f(x, y) - g(x y) » e(x y) (11) 
By relating template and image patch through an af- 
fine transformation 
X; = 844 + 9,2X+ 8,4) (12) 
y; = b, + D,9x+ by (13) 
and linearizing with respect to the affine transforma- 
tion parameters, the observation equations for all in- 
volved pixels can be written in matrix form as 
—-e = Ax-! (14) 
where / is the observation vector containing gray 
value differences of conjugate pixels, x is the vector 
of unknowns consisting of the affine transformation 
parameters, and A is the associated design matrix in- 
cluding the derivatives of the observation equations 
with respect to the unknowns [Gruen & Baltsavias, 
1988]. The least squares matching solution is then 
obtained by minimizing the squared sum of gray 
value differences 
—1 
X =. (A) PA) „A’PI (15) 
and a new position of the image window is deter- 
mined as the conjugate of the template through the 
updated affine transformation parameters (Fig. 3). 
initial 
position 
final 
position 
  
Fig. 3: Visualization of least squares template 
matching 
However, due to the particular gray value distribution 
of the edge patches, a full set of affine transformation 
parameters cannot be obtained [Gruen & Stallmann, 
1992]. Instead, only two shift and one rotation param- 
eter relating template and image patch can be deter- 
149