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This formulation is commonly referred to as the inverse sensor
model. As described in the SPOT Satellite Geometry
Handbook, f and g can not be directly derived from the SPOT
header information, but is obtained with an iteratively algorithm
based on the direct model F and G.
3.3 XS Image Sensor Model refinement
From the PAN and XS image header metadata files, we will be
able to establish the respective sensor model independently. To
co-register the images, we choose to refine the XS image sensor
model relative to the PAN image by introducing 3 parameters,
Coys C0 ps Cor » tO correct for the biases differences in yaw, pitch
and roll respectively. Such that:
yaw = yaw, + cy,
pitch = pitchy +c p (4)
roll = rolly + cy,
where yawy, pitchy and roll, = initial yaw, pitch and roll
value calculated from sensor model of XS image,
respectively
yaw, pitch and roll = refined yaw, pitch and roll,
respectively
Coy Cop>Cor = correction parameters.
The correction parameters, c9,,c9,,co, can be expanded into
higher order polynomials, but for this work, the zeroth order
constant term corrections is found to yield sufficient accuracy.
34 Height Constrain
In the solution for the model corrections parameters, the
geographical coordinates of the tie points were also solved
simultaneously. Since the angle subtended between the PAN
and XS is only 1.06°, the least squares solution was found to be
unstable. This stability problem can be remedied by
introducing a height control value to one of the tie points. This
height control need not be accurate. We have conveniently
chose to extract it from the 1km gridded GLOBE or SRTM
DEM (same DEM will be used later for orthorectification of the
images).
3.5 Procedures for refinement
By automatic selecting about 20 tie points or more, and one
ground point for height control, the refined parameters for yaw,
pitch and roll correction can be derived from Equations 3 and 4
by using least mean square method as following:
M; = sampy; — filon; lat; by)
V; 7 liney; — g, (lon; lat;,h;) (5)
4; 7 samp», — f»(lon;,lat;, h;)
Vo; = line»; — g(lon;,lat;,h;)
where 1 = Pan image
941
-* rammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
2 — Xs image
i e[0,n)
n = total number of tie points, and one ground point
Using the first order Taylor expansion and least-square
estimation method, the ground coordinates of tie points and the
refined parameters Coys Cop andco, can be computed as
follows :
ME
o - don; — N
4j & sampy; — fV (lonjo,lat;o, hj) — iz da
Ölon; Ölat;
af
P
= e - ON,
oh;
1 = liney; - gy om ae Aa) ZZ Slon, - EL. à
Vi; & Finey; — gi (onjo,lat;g, Io) — don -olon; — al -olat;
i var,
M
0 .
08. 2
Oh;
onis o of, i,
4b; & samp», — f»(lonjg,lat;o,h;) — 72 - dion; — N ‚Ölat;
Olon; 0
Vn hy ged
D ru gy -— : co, = 5
on; CC0 y, OC 0p € Cor
: Bgy = m
V5; & line»; — g5(lonjo,lat;o, hg) — 3 £2 -Olon; — -52 - Olat;
lon; Olat,
082 Og» 082 og
2 e d 3 2 >
en 08 es a Cop mer Re
on; Coy c Cop CC,
For izn-],i.e. ground point with height constrain, there is no
oh term in equation 6.
The above equation 6 can be grouped as below:
2 2 2 2
E= (nj TM + 4h; +V9; ) (7)
Vigan + A(4n)x(3n-1+3) . X n-br3pd = Lan (8)
where V stand for 44;, vi;, 4/5; and v5; ,
L4;o = -samp;; * fy (lonjo,latjo, hio)
La;41.0 = —liney; + g (long, lat;y, hyp)
L4j45,9 = -samp»; * f» (lonjg,lat;g, hio)
L4i43,0 = -line»; + g» (lon, lat;o, Trjg)
3x(i-1) - a A 3x(m-iy-l 4
Ay; = 0,0,A 0,- N 37 N — e ,0,0,A ,0,0,0,0
Olon; Olat; Oh;
1
3x(i-l) 3x(n=0)=1" 3
de. Om Gg
4. = [GA ati "8 EL 00,4 0,600
no Olon;' ólat; oh;
3x(n-i)-164444744448
Ix(i gh A ; of of of of
N 4 ff N Cf? en
Aaj+2 =| 0,0,A ‚0,- N es 72 EE DOA emt fe
€ élon; êlat; Oh; Op OP OL
