;, Part B3. Istanbul 2004 International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
original curve as can be seen in figure 6. Figure 6 shows 
enlargement of a part from the specific curve where the points 
  
  
  
  
  
  
  
  
  
  
: > . : : 3 left image 
(10) indicated by 'o' are the points transferred from right image using es^ igs g 
the computed homography matrix. Initial curve 7 uu 
- zt Aer 
56r transformed ~ a^ 
ai AS 
54} uic "gi 
oints 1,2 of the closest abc d 
3 
ela Heal eas 50 + : us 
Apt agent Qe tod Curve after — 27 
8) for h1..h8 we rewrite OE. dec ser ub asit FU cue 48} ivodteration A 
Ieee : remade CSN da E 45r 
Lr pe iie Es hu du À 44 + Original left image curve 
ets E eu rie: 5 / 
ES : IH 42L 
11) 3* 3 3 = JF 
TE ; A s E - a p 1 1 1 1 1 1 1 
on M 3j gd 9 md q^ EE ua eU 
50 A SO 0 ee Figure 6. Enlargement of the converged curve 
^ T 
p ‚OD get 
D oh X 
i : ; 
(12) Further to the advantages of the homography-based algorithm 
p oD mentioned above, i.e. fewer transformations required and a 
D Oh. Figure 3. 3D model space reduction of the problem dimension from 3-D to 2-D, probably 
l . tp . . . EU 
the most significant one pertains to its insensitivity to the plane 
parameters associated with our planar curves. This unlike the 
algorithm presented in section 2 being subject to singularities 
nir associated with nearly horizontal planar curves. 
ks n: nap 
m d E nou 3.6. Recovering rotation matrix and translation vector 
zl h ed us y. directly from homography matrix. 
I oe E de ne : j 
ar | s ye From equation (9), we need 2 homography matrices to recover 
dA qM the fundamental matrix. As shown in Tsai(1982) the 
E E a e. homography matrix can be decomposed using SVD and the 
"jk 4 NO 08 989 33A 33 rotation matrix, the translation vector and the plane parameters 
; can be recovered. Two possible solutions for the recovering of 
(13) ; 4 if : Y the relative orientation parameters are obtained using Tsai 
Figure 4. left and right images recovering. The possibility of two different solutions using 
direct recovering from the homography matrix is well suited the 
In figure 5 we can see the curve transformed from the right need of two curves providing two homography matrices for the 
image using initial homography matrix. recovery of the fundamental matrix (Shashua 2000). 
Computing the rotation matrix and the translation vector from 
et image the homography matrix is as follows: 
di 
tin NT à yy : [UDV']- H 
met ie i pre 
Li D at s =det(U)det(V) 
5 : ant transformed  / 
Wa =» 7 f : Ny > curve 7 2 2d 
HH £M s A TA, p? 
I ent 6 ue À =+ — = 
La Bu es 2. 2° 
1 p + original Ar 
40 > T c y 2 ae GO» 
+ tg r —" / x = 
rdinates of the specific \ A / a= A +340” 
dep ed A08) (14) 
20r \ xu 2 
E. 
: A B=+1-a 
Ur 2 
N 
a 0 B 
n tested on synthetic data -20} Rall ON 15.0 1: 
n the curves as shown In S. y ss 
AIT . for this -40 | 1 | 1 1 L L J TS Sa 
matrix component ior i 0 20 40 60 80 WW. 7% un 
0,0,1} where f is the foca 2 
, 5 f T “= y 3 : y 
ceep the model scale close to =[-BU, + pud U,] 
the synthetic images of the Figure 5. Curve transformed with initial values 3 
ves is transformed from right 2 t 
values for the homography where: U; U, are the 1? and 3* vector of U 
lgorithm we get close to the 1,..13 are the singular values of H