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3.6 Bridge image simulation 
3.6.1 Rio-Niteroi bridge image simulation 
The target image simulation of the Rio-Niteroi bridge uses the 
bridge center estimation of the Section 3.4 and a geometric 
transformation from real coordinate to discrete coordinate. 
A 
For a given column j € n, let c = a.j + b : 
Let G,, be a finite interval of Z with an odd number of 
elements, denoted p, 
Let v = (p + 1) / 2 be the center of Gy , 
Let assume that G + ed —vcn 
D 
Let T, bea geometric transformation from G; to F, given by 
I 
Ty, (y) = 20 (y = v) + i = Uy + ki Ve Gi 
ky = | 20 pae. M = 
| 2 2 
In the above definition, v, is the center of the bridge, and A; 
represents how far the transformation of v is from u,. Figure 7 
shows the Rio-Niteroi Bridge model. 
where 
3.6.2 Causeway bridge image simulation 
In this simulation, the method of Rio-Niteroi image simulation 
was used also to estimate the distance between the true bridge 
center and the estimated bridge center. 
^ 
For a given row / € m, let c2 = aii 4 b 
Let G,, be a finite interval of Z with an even number of 
elements, denoted p. 
Let assume that G_+| ec, | — Zz cn: 
: i 2 
Let 7, be a geometric transformation from G; to £7 given by 
T. (y)= 20 (y — v)+ un, + kr ve G,, 
where v 7 (p * 1) / 2 is the “center”of G, and 
In the above definition > is the center of the bridge, and k, 
represents how far the transformation of v is from u,. 
^ 
Kk. zc 20 C + 
D|- 
bas — 
In the Causeway bridge image simulation, the bridge center 
A 
estimation C, is actually biased due to the different radiometry 
of the two decks. Accordingly, 4, is expressed as 
bk, 2k, +A 
where 
  
and A is a corrective term that takes into account the bridge 
center estimation bias. Since A assumes only a few integer 
values its estimation can be based on an exhaustive search. 
Figure 8 shows the Causeway Bridge model. 
3.7 PSF estimation 
j 4T . - * : 
Let g / be the /" target image column defined on Gi given by 
: ^ 
2j (y) = g||er+ SA VER 7 
j 4h : A S. 
andlet g; be the /"' target image row defined on G» given by 
: ^ 
gilyy= gli lca ~ 
P 
—— y 
2 : 
The PSF estimation in the along-track direction consists of 
finding o; such that 2; and Ch LE. )o I, best fits under 
the root mean square criteria. The PSF estimation in the across- 
track direction consists of finding o> such that a! 
and Ch * h... Jo 7 best fits under the root mean square 
criteria. 
Let RMS, be the real number given by 
53/2 
RMS, = *} TE NY) 
JU Un nO 
The along-track estimation procedure is performed in two 
steps. Firstly, we look for /, s and o; that minimize RMS. 
Afterwards, using their mean values obtained from the 
first step, one looks for o; that minimizes RMS,. 
Let RMS; be the real number given by 
1/2 
RMS, =| X (Ate, ) (76,0) -£2) 
yea, = 
The across-track estimation procedure is also performed in two 
steps. First of all, one looks for A, #,, ^», s and o» that minimize 
RMS-. At the second step, using their mean values obtained 
from the first step, we look for o» that minimizes RMS;. For 
both simulations (in the along-track and across-track-directions) 
the optimum values of o; and o» have been obtained by 
nonlinear programming (Himmelblau, 1972). Results of the best 
fitting between the image measurements and the simulated data, 
for band 3 in along-track and across-track directions, are shown 
in Figure 9 and Figure 10, respectively. Table 2 and Table 3 
present the EIFOV values obtained by the method proposed in 
this work. EIFOV is related to the standard deviation o so that 
EIFOV - 2.666 (Slater, 1980).