(2-9) 
s formed for 
mx and dy 
P... arethe 
my 
| as diagonal 
set of seed 
1 operator or 
ast squares 
the control 
9e extracted. 
in has to be 
bservations. 
> established 
(2-10) 
ordinates of 
cx and P "m 
as diagonal 
'spect to the 
rix form as 
(2-11) 
(2-12) 
| first shape 
e the local 
following 
xpressed by 
ybservation 
the curve as 
(2-13) 
(2-14) 
he B-spline 
me x = NAX -t $ P. > (2-15) 
TE sy = N AY t. > P , 
CÉssx 7 N AK rl > P. , (2-16) 
7E ssy = N SAY - ts , $sv 
Where N, and N,, are the first and second derivatives of N 
defined in equation (2-8), and the terms £ are given by 
Om N.X0-x9 , 
E (2-17) 
S 
zZ 
Nl 
e 
| 
= 
, 
ssx ss SS. 2 (2-18) 
Any other a priori geometric information of the feature can be 
formulated in this manner. A joint system is formed by all of 
these observation equations (2-9), (2-11), (2-15) and (2-16). 
2.3 Solution of LSB-Snakes 
In our least squares approach linear feature extraction is treated 
as the problem of estimation of the unknown coefficients X and 
Y of the B-spline curve. This is achieved by minimizing a goal 
function which measures the differences between the template 
and the image patch and which includes the geometrical 
constraints. The goal function to be minimized in this approach is 
the L, — norm of the residuals of least squares estimation. It is 
equivalent to the total energy of snakes and can be written as 
T 
T T 7 
(Y Py, * pP sn ss) * Gr t m rv, Pv, E 
T. 
y Pv 
(2-19) 
E, * Ey * Ec => Minimum : 
In terms of snakes, E, denotes the internal (geometric) energy of 
the snakes derived from smoothness constraints, Ey denotes the 
external (photometric) energy derived from the object model and 
the image data, and E represents the control energy which 
constrains the distance between the solution and its initial 
location. 
To minimize this goal function (total energy of snakes), we have 
the following necessary conditions 
  
  
J vIPy = vIPy = 0. (2-20) 
JAX oAY 
A further development of these formulae will result in a pair of 
normal equations used for estimation of AX and AY 
respectively. Because of the local support property of B-splines, 
it can be shown that the normal equations are banded (bandwidth 
b = m+ 1) and the solution can be efficiently computed. The 
various tools of least squares estimation with their familiar and 
well established mathematical formulations can be favourably 
utilized for the statistical analysis of the obtained results and the 
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realistic evaluation of the algorithmic performance. So one can 
evaluate the covariance matrix of the estimated parameters and 
derived quantities therefrom. Also, one obtains an estimate of the 
system noise. In addition to the traditional least squares 
estimation, a robust estimation can be efficiently computed too, if 
required. 
3. LSB-Snakes with multiple images 
If a feature is extracted from more than one image, its coordinates 
in 3-D object space can be derived. The 3-D coordinates of the 
feature point in object space can be directly obtained by the 
MPGC matching technique (Gruen, 1985, Gruen, Baltsavias, 
1985) or an object-space correlation algorithm (Wrobel, 1987, 
Helava, 1988). 
Suppose a linear feature in 3-D object space can be approximated 
by a spline curve and represented in B-spline parametric form as 
Xps) — NX. 
Y7(s) = NY, (3-1) 
Zs) = NZ. 
where N is defined in (2-8), X, Y and Z are the coefficient 
vectors of the B-spline curve in 3-D object space and X,, Y7 
and Z; are the object space coordinates of the feature. If 
multiple images are available, there are two main ways to 
perform the multiphoto matching. The first method is to connect 
the photometric observation equations of every image by means 
of external geometrical constraints. One class of the most 
important constraints is generated by the imaging rays inter- 
section conditions (Gruen, 1985). The second method is object- 
space correlation, which is performed in object space by 
matching densities assigned to “groundels” (ground elements). 
Both methods can be applied for LSB-Snakes. Since LSB- 
Snakes deal with a curve instead of an individual point, direct use 
of the MPGC algorithm will introduce much more unknowns 
than necessary. The method of object-space correlation is 
definitely of theoretical interest, however, it cannot be easily 
applied to LSB-Snakes without extension of the algorithm, since 
we are facing here a truly 3-D problem. Our 3-D LSB-Snakes can 
be interpreted as the object-space analogy of MPGC for multiple 
points defined on a deformable spline curve. 
Assume patches are used from k>1 images. If the image 
forming process followed the law of perspective projection, a 
pair of collinearity conditions in parametric form can be 
formulated for each of the image patches as 
Y ar Au en pe Wo rer 29) ‘ 
aig(Xr- X9) t aj(Yr - Yg) t a3(Zr - Zo) 
y Oe Xo ean Ya) Harz Dal 
ay3(X7 — X0) + 23(Ÿ1 = Yo) + a33(Zz - Zo) 
  
(3-2) 
  
If the interior and exterior orientation parameters of each image 
are given or can be derived, the unknowns in equation (3-2) to be 
estimated are the coefficient vectors of a B-spline curve. The first 
order differentials can be obtained as 
269