The inverse function of the geometric model is then
determined in Eqs.(12- 14).
4. Estimation of Geometric Model Parameters
The geometric model described in the previous section
has six unknown or uncertain parameters(01,025,03,0,n,h).
Those parameters can be estimated by using least mean
square method based on the relationships of several
ground control points between satellite images and maps.
The inverse model is defined more simply than the non-
inverse model since the inverse model interpretation
solves a problem to find the intersections between a line
and a plane but the non-inverse model interpretation does
a problem to find those between a line and a spherical
surface. Therefore, the inverse model was used for
parameter estimation instead of the non-inverse model.
Several ground control points are selected by comparing
satellite images with maps and denoted as (xm, ym) and
(Am, Bm) in satellite images and maps, respectively. Using
these ground control points, the parameters
(01,02,05,0,n,h) for the geometric model are to be
estimated. The estimation is performed using a least
mean squares algorithm repeatedly. The inverse
geometric model with uncertain estimated parameters,
(61,02,05,0,n,h) produces (xm, ym) with the input, (am,
Bm) instead of the true value (xm, ym) from GCPs.
(2 »)2 T'(05,07,07,9' m hos, B-) (15)
The error E of the estimation with uncertain parameters is
defined as
1 2 2 2
E =) + O4) (16)
where n is the total number of GCPs used for parameter
estimation.
A least mean squares algorithm is used in order to
determine the six parameters to minimize the error
defined in Eq.(17). When performing parameter prediction
repeatedly, the next parameters, Pn.1, can be estimated
from the current ones, Py, by using the relationship that
dE dE dE dE dE dE
Pau =D EVE EDS ES ESS (17)
d0, d0; dO. dó dy dh
where c is a small converge factor.
The differential values substituted in Eq.(17) are to be
derived in the following equations, Eqs.(18-21). If p. is an
arbitrary component p, the differential value of the error E
about px is described as
dE 1 4 en! dy
—— m cn 2 ee n 2 nm ba ie
= > (x rs +20) 27 (18)
Following equations are the differential values of x,” and
yn' about each model parameter.
432
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
d6: | _ lgsngow 4e A 1 dR= RW
dy! + = d6: |, k do. als e
L d6: |
d6: 1 T T dk 1 dR =" T
= — Rs Ru W-— mj Ra W
dy Lk de: |,, Lk do: 2.3 (m
|. d0: |
deo: 1 T T dk 1 dR" T
ME —R» R« W—— = me R:: W
els e t (19c)
L dO: j
re Lew dd gs RR TC o
dy, L k dó | 23 k dó do 2,3
do.
ja]
a | [Lg row] [Lp wer VIT (de)
dy. dn k dn dn ||,
= 23
dh = RR WA 7 dr nait
dy. E dh | be dh |. d
L dh |
The differential values of k about each parameter
included in Eq.(19) is described in the following equations.
| dk 1 dR»"
A ipl pn RW
E | f 40. | (209)
| dk LR:
— = — eem Rin VW.
E | f de. | (2)
| dk ] AR".
JR d lute RW
5 E 46. | en
| dk 1 dR dW
VAM ao oC ppt W+Ru" (20d)
E 4 | dó dó l
&-- Ia] Wah IV | (20e)
| dn f dn dn } |,
[ dk 1 dW
m= —Ra Rs!
E H dh | Qn
The differential values of the matrix or vector included
Eq.(19-20) are denoted as follows.
: [— $6: z$iC6 85:65 7
[m CC CSS SICH CUuS1C3——S155 (21a)
doitfeso 0 0
= T
JR" — Ci: CiC2C3 — CiC2C3
| = Ë S152 — $1253 SIC2C3 (21b)
: Ÿ C2 $253 -— $263
JR." [0 €152€3 77 5183 C1S253 + S1C3
| = = O0 —5$0c-—05 — S18253 + C1C3 (21c)
LO - C33 zs
= T
dR 1 ——S4€s C4 S485
| EA J- C6 — Si = Cass (21d)
¢ 0 0 0
The m
followin
positior
Cond.
amoun
the sat
Cond.
the ort
longituc
altitude
If the «
mather
The Cc
where
orbital |
