International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
7
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Figure 2. Parallel projection parameters
Equations | represent a non-linear form of the parallel
projection model. They can be reduced to a linear form,
Equations 2:
Xze4XE4Y44.7-t4, Q)
ys X AQ Zt A,
where A; to Ag are the linear parallel projection parameters, and
will be referred to as “2-D Affine parameters” (Ono et al.,
1999). It is important to mention that Equations 1 are useful if
scanner navigation data are available (the derivation of scene
parallel projection parameters from the navigation data is
presented in Section 4). On the other hand, if GCP are available,
the linear model in Equations 2 becomes easier to use. The
relationship between the linear and non-linear forms is
summarized in the following subsection.
3.3 Transformation between Parallel Projection Forms
3.3.4 Transformation from the Non-Linear to the Linear Form
Knowing the scene parallel projection parameters, intermediate
parameters (U, V) can be computed as expressed in Equations 3.
Afterwards, the 2-D Affine parameters are computed.
U- 1h trader 8
rh brad N. (3)
Fb MAIN
bade t tn M tr NX
4, = se, = nU) À, = s(n; T nV)
4, = slry = r,U) dm sin = rV)
sir. E SU) À, = is V)
A, = Ax
|
>
"cm
™
va
I]
A,
where r;, to 73; are the elements of the rotation matrix R. The
derivation of this transformation can be found in (Morgan,
2004). Next subsection deals with the reverse of such
transformation.
3.3.2. Transformation from the Linear to the Non-Linear Form
Knowing the 2-D Affine parameters, scene parallel projection
parameters can be derived in three steps:
Step 1: Components of the projection vector have to be
computed as expressed in Equations 4.
A;
L'= = 3
A rud. ur. oN al Ad ed A) aE
ArT Alli YH A lM AS
dde {jar Ven Ada rit dant c }
MA A
m
vus dos Mta
A,
N°
C=t
DA
Me
[N
N = [A
Step 2: The scale factor can be computed as expressed in
Equations 5, by first computing intermediate parameters (7, 75,
T. A.B. C. U and D).
T aded. ud
T, =A; + A; + 4;
BL, = À 4. + 4. 4, + 4.4,
A=F Tl,
B= 27, +1, -TT,
Cal;
sfr BEN HACL
24 n
pub fry’)
f uU
snb, (5)
]12-U*
Step 3: Finally, the scene orientation angles can be computed as
expressed in Equations 6, by first computing intermediate
parameters (D, E, and F).
D=U FRI
A:
Z = gu 4. 2V —
S S
A dz
E : >—]
Sf S
E+ VE -4DF
@ = arcsin =
: A;
#4 sing+->
K = arctan Sos A
U sin @ + —
Sn
cos @
International Arc
CERO ne
À,
: S
e zarcsmi--—-
‘
A
Readers interestei
refer to (Morgan,
non-linear and lir
established. It is i
preferred in cases
model is preferr
Section 4 deals \
transformation pr
parallel projection
3.4 Perspective
Original scenes c
the rigorous pe
Therefore, before
Perspective To I
coordinates are re
coordinates along
comply with pa
mathematical m«
knowledge of the
where:
c is the scan
yv are tle. cc
parallel an:
Therefore, any co
transformed from
coordinate accord
that the mathem:
terrain. Morgan (2
angle together wit
projection. This m
x = A, X -
ta
1+ —
In the above mode
are used directly ti
with the scanner
alternative approa
parameters — that i
4. THE RI
NAVIGATIC
PROJECTIC
Referring to the di
assumed to travel :
velocity V.CAX, A1