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ted in the foreground. Behind, the HR lens is visible, 
arranged between two additional lenses for multispectral 
data acquisition. 
  
  
  
  
  
Fig. 1: Optical system of MOMS-02 
For the illustration of the functional model, the object 
coordinate system XYZ and the image coordinate systems 
of the HR lens xy, yy, Zu and the forward looking lens x; Yr, 
z, are represented in Figure 2. All coordinate systems are 
right handed Cartesian, at which the image coordinate sy- 
stems are defined as follows: 
>» origin: centre of the first active pixel of the sensor line, 
» direction of z-axis: parallel to the optical axis, pointing 
downward, 
> direction of y-axis: straight line through all pixel 
centres, 
» direction of x-axis: orthogonally completed, close to 
direction of flight. 
For each inclined lens an additional 6 parameter trans- 
formation was introduced to rigorously model the displace- 
ments Ax,Ay,Az of the projection centres (PC) and the 
rotations Ag,Aw,Ax of the image coordinate system with 
respect to the image coordinate system of the HR lens. 
Thus, the MOMS-02 camera geometry is described by 21 
parameters: 2 x 9 parameters (A @gp, A pp, App, AXpp, AYig, 
AZgg, Xorps Yors Cp) for the forward and the backward (EB) 
looking lens each, and 3 parameters (Xu, Yon, cy) for the 
nadir looking lens. In principle, these parameters can simul- 
taneously be estimated by the bundle adjustment. In 
practice they will be treated as constant values, determined 
by camera calibration previously. The extended collinearity 
equations (2) are derived from the general approach (1) 
and are applied for the inclined lenses. In case of the nadir 
looking lens (Ax,=Ay,=Az,=0, M=I) the classical 
collinearity equations are obtained from (2). 
X, Y : image coordinates of the object point 
Xo, Yo : image coordinates of the principal point 
k : scale factor 
M, D, R : rotation matrices (M" D" — R7?) 
X, Y,Z : ground coordinates of the object point 
459 
X, Y» Z, : ground coordinates of the projection centre of 
the nadir looking HR lens (position) 
Po, 9,» K : rotation angles of exterior orientation of the 
nadir looking HR lens (attitude) 
  
  
  
  
  
Fig. 2: Coordinate systems 
  
  
X-Xo X - (X,* AX) 
y-yxy| = k MT(A@,Aw,Ax) D'(gy ok) | Y - (Y,*AY) 
18 Z - (Z,+AZ) 
AX, Ax 
with |AY,| - D(os, ek) |Ay (1) 
AZ, Az 
ax REX) RAT Y) +R, (2-25) - (M, Ax+M, Ay+M, A7] 
Tw Ry(X-X) *R4(-Y) *R4(Z-Z) - [MjAx*MjAy*MjAz] 
(2) 
jon RyC-X) «RJ(O-Y) *R(Z-Z) - [Mj Ax*M, Ay «MyAz] 
Ry(X-X) *RÁQ-Y) *RAG-Z) - [Mj,Ax*«MAAy *MaAz] 
This general approach allows for processing randomly orien- 
tated image coordinate systems of lenses with different inte- 
rior orientations (X, y, c). Self-calibration using additional 
parameters for the correction of systematic image errors can 
be applied as usual. 
To demonstrate the influence of different focal lengths c, 
and cz, the standard deviations oy, oy, 0, of a symmetrical 
3-ray forward intersection (standard deviations of image 
coordinates o, flying height h, baselength b, @,=w,=x,=0) 
are listed (3). It is conspicuous, that the height accuracy 
does not depend on cy, whereas all 3 rays contribute to the 
planimetric accuracy. The MOMS-02 geometry (c:c;4 — 
3:1) improves the accuracy in planimetry by a factor of