In: Paparoditis N., Pierrot-Deseilligny M., Mallet C. Tournaire O. (Eds). 1APRS. Vol. XXXVIII, Part ЗА - Saint-Mandé, France. September 1-3. 2010 
I(x 1 ,y 1 ,...,x n ,y n )=L(oX 1 +by i +c,dX 1 +ey i + (9) 
aX m + bY m +c,dX m +eY m +f) 
Equation (9) can be written as: 
Z[(x-Xi-i) +(y,-y,-i) ] = 
/=2 
m 
I {[(a X,+ b Yj+c)-(oXr J _,+c)] ! + 
{(dX^eYj+n-idX^+eY^+nf}' 1 ^ (10) 
m 
=£ {[o(X r X,J+b(Y -Y,_,)?+ 
J=2 
m 
{d(X r X l _ t )+e(,Y l -Y hl )] 2 } m = Y,Lj 
j* 2 
Results derived from various tests made for this research, 
proved that the equations of 3 moments combined with the 
equation of the lengths greatly enhance the results. If 4 
moments are used the results are slightly better, but 5 or more 
moments give no noticeable improvements. 
3.LEAST SQUARES ADJUSTMENT 
As there are at least 7 non-linear equations for 6 unknowns, 
the system of equations must be solved using the non-linear 
LSM. This implies that the equations must be linearized and 
initial values for the unknown coefficients must be computed. 
The partial derivatives of Equations (6), (8) and (10) with 
respect to the unknown parameters are shown in the Appendix. 
The matrix equation of the iterative non-linear LSM is 
[A][dx]=[B] , where [A] is the design matrix which contains 
the partial derivatives of each equation with respect to each 
unknown parameter, [dx] is the vector of the adjustment 
values of the unknown parameters. The vector [dx] which is 
computed by the first iteration of LSM, is used to update the 
initial values of the unknown parameters [2C°] : 
[A 1 ]=[A°]+[dx] . The procedure is repeated until conver 
gence ([dx]«*). 
The initial values [X°] of the unknown parameters can be 
found indirectly using the similarity transformation: 
x=X 0 +n(X cos0 + y sincoscf))X +{^i sin 4>)Y + A 0 (li) 
y = Y 0 +n(-X sin cf)+Y cos d>) = (-/j sin</>)X+(/JCOS</>)y +y o 
where X. Y, is the translation, p is the scale and <p is the 
rotation angle. It must be noted that in this case the similarity 
transformation is not used as a first approximation to the ICP 
algorithm, but only to compute the initial values for the 
application of the non-linear LSM. Comparing the similarity 
and the polynomial transformations (Equations (1) and (11)), 
the following initial values can be found: 
o=pcosd> , b=psind> , c=A 0 
d=-ysin(f) , e=pcosd> , f = Y 0 
(12) 
The parameters A c , y o , p, 0 of the similarity transformation 
can be determined as described in Vassilaki et al. (2008c). 
Specifically, the translation X 0 Y 0 is computed as the distance 
of the centroids of SAR curve (B) and the object curve (A): 
*0 X B X A > ^0 — У В У f 
J s x(s)ds 
(13) 
where x,= 
J s y(s)ds 
Js, ds 
Js, x ( s )ds 
JU 
" A= ' /<* 
J Sn y(s)ds 
L ds 
Ув-- 
The scale p is computed as the ratio of the length of the two 
curves: 
£ 
'=7f » where S A =f s ds , S B =J s ds 
(14) 
The rotation <p between the two curves is computed by trying 
all the possible rotation angles between 0 s and 400 g in small 
steps (for example 3 g ). For each angle step the error is 
computed employing the ICP algorithm (as described later) to 
find homologous points. The rotation angle with the least error 
is chosen. 
4.ICP BASED MATCHING FOR FREEFORM CURVES 
The method used in this paper matches ffee-form curves using 
the ICP algorithm as summarized below: 
1. Compute the polynomial first approximation using moments 
and length equations. 
2. Using the polynomial first approximation, project the 3D 
nodes of the object curve, remembering which 3D node is 
projected to each of the 2D nodes. The projection is 
sufficiently near to the SAR curve. 
3. For each 2D node of the projected object curve, find its 
closest point in the SAR curve, thus producing homologous 
3D object curve nodes and (2D) SAR curve points. 
4. Using the homologous pairs compute the parameters of the 
arbitrary projection selected (DLT, rational function, etc), 
using the LSM. 
5. Compute the error of the projection. 
6. Using the computed parameters of the transformation make a 
new projection of the project curve, remembering which 3D 
nodes is projected to each of the 2D nodes. 
7.Steps 3 to 6 are repeated until the error stabilizes. 
From the above mentioned steps, the operation of step 3 needs 
further explanation. If the SAR curve were a straight line, its 
closest point to a node could be found analytically, by 
minimizing the function which gives the distance between the 
node and a point on the line. In reality though, the SAR curve 
is a ffee-form curve which is described by nodes linked with 
straight line segments, in the simplest case. This means that 
the analytical minimum may correspond to a point outside the 
line segment, and this may be true for all line segments. This 
leads to complexity and computational cost. If the nodes are 
linked with higher order curves (for example cubic splines) the 
analytical computation of the minimum will be iterative, which 
leads to more complexity, more computational cost and the 
need to ensure convergence. 
To overcome the problems, it is proposed by the authors to 
split the SAR curve to a large number of consecutive 
interpolated points using the arbitrary type of curve that links 
the nodes (linear, cubic splines, etc). The closest point of the 
SAR curve to a node is found by computing the distance of all 
the points to the node, and choosing the one with the least 
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