(6)
sed
Xc,
der
7)
nal
fts
NS
he
ES
he
8)
ts,
Ire
of
its
nd
ng
2)
ISPRS Commission III, Vol.34, Part 3A »Photogrammetric Computer Vision“, Graz, 2002
In the same way, we get:
WA =
Zivzi«zi zzi
0*0 0, -oj! (11)
9,*91*95-9.
Kot oo x)
Similarly, imposing the first and second order continuity
constraints for X function, we have:
axis ain
Eje U2
"ES t=0
and
d?xi "doin (^
dildo van? am
t=1 t=0
In the same way, Equations 12 and 13 are written for the other
external orientation functions, giving:
AA ZN aid
Y 32yi s» y]!
Zi 4221 - zi!
@i +20 = wi!
9j 29) - gr^
ki x 2k) = kit
(14)
for the first order derivative and:
Ais
Yir"
Do "ubt
Zi=Z5
i itl (15)
i+l
-0»
zx
9
i
2
Ki
for the second order one.
Equations 10, 11, 14 and 15 are treated as soft (weighted)
constraints.
3.0 Mathematical solution
The functions modelling the external orientation (Equation 7)
are integrated into the collinearity equations (Equations 2 or 4),
resulting in an indirect georeferencing model.
The collinearity equations are linearized with the first-order
Taylor decomposition with respect to the unknown parameters
modelling the sensor external orientation (xzo), according to
Equations 2 (or 4) and 7, and with respect to the unknown
ground coordinates of the TPs (x7p), according to Equation 7.
As initial approximations, the parameters modelling the sensor
external orientation (329) are set equal to zero and the
approximative ground coordinates of the TPs (32,) are
estimated with forward intersection, using (Xinstre Yinstre Zinstr»
Ojnstr» Pinstr» Knsrr) AS external orientation.
Combining the observations equations and the constraints, the
system:
—e€gcp * Agcp Xo -Igcp;PGcp
— erp = Arp Xgg t Brp Xpp — lzp > Prp
- eco 9 C$ Xgo - leg ; Peg (16)
- 6c; 9 C; Xygo ieri P
—eco = C)? Xgo —lez + Pos
is built, where:
Xgo: Vector containing increments to x05 ;
Xrp: Vector containing increments to;
Agcp: design matrix for xgo for GCPs observations;
Arp: design matrix for xp, for TPs observations;
Brp: design matrix for xrp for TPs observations;
Co, C;, C3: design matrices for constraints on zero, first
and second order continuity;
e: errors of measurements;
[: discrepancy vectors;
P: weight matrices for each group of observations.
GCPs and TPs are required in order to solve the system and
estimate the unknown external orientation parameters and TPs
ground coordinates.
Considering a sensor with S linear CCD arrays, Ngcp GCPs,
Np TPs and n, trajectory segments, the complete system
contains 2xSX(Ngcp+Nrp) collinearity equations, together with
6x(n,-l) equations for each group of constraints described in
Equations 10, 11, 14 and 15. The unknowns are 18xn, for the
external orientation and 3x7» for the TP ground coordinates.
The vectors xzo and x7p are estimated with least-squares
adjustment and added to x0 and x. in the next iteration.
The process stops when xgo and xz» are smaller than suitable
thresholds.
4. DATA SIMULATION
Simulated data were used in order to test the indirect
georeferencing algorithm.
We assumed an airborne sensor, with the optical system
consisting of one lens with focal length equal to 60.36 mm. The
stereo viewing is achieved along-track with 3 linear CCD arrays
scanning in forward (+21.2 deg), nadir and backward (-21.2
deg) directions (Figure 3).
Each linear CCD array consists of 10200 squared elements of
dimension 7 um.
Supposing 40832 exposures, the sensor position and attitude at
each exposure (40832 x 6 data) were generated in a tangent
local system with origin in the mean of the trajectory at null
height, X- axis in East direction, Y- axis in North direction and
Z- axis directed upwards. The aircraft was supposed to flight at
a mean height of 500 m, along the trajectory shown in Figure 4,
with a resulting ground pixel size of about 6cm.
40 GCPs were chosen in a 400 m large and 1600 m wide field,
with the height in the range between 50 and 80 m. The
distribution and height of the GCPs is shown in Figure 5. The
resulting base over height ratio was about 0.7.
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