86 
c = F(x', d) = D„'d (6) 
where matrix Dy' is a suitable function of the 
point position x' in the image. Depending on 
the type of imagery, matrix D, assumes differ- 
ent forms to express the effect of affine, ra- 
dial, tangential, spiral, or other distortions. If 
possible the distortion can be regarded as 
radial and then computed as a function of the 
radial distance. Distortion parameters d are 
usually known from separate calibrations and 
available for on-line operations as constants 
to be used in Equation 6. Another possible 
way of applying corrections is based on an 
interpolation from a computer-stored lookup 
table. 
MODEL RECONSTRUCTION 
This chapter deals with the geometry rep- 
resented by the central projection. Typical 
modifications for parallel projection can be 
obtained from derivations given by Kratky 
(1975b). 
Basic relations. In any off-line computa- 
tion the photogrammetric model can be re- 
constructed by matching corresponding sets 
of photo coordinates x',x" with control coor- 
dinates X as expressed in Equation 2, using a 
suitable mathematical model for the ex- 
pected relationship. In on-line analytical sys- 
tems the same relationship is expressed in a 
slightly different form. In accordance with 
Figure 1 the communication between photo 
and object coordinates is mediated by virtual 
model coordinates x = (x,y,z)T 
g 
(x',x",d) Xx 
2 gi 1 82 
x 
(2a) 
The coordinate system of the virtual model 
becomes a master for the remaining systems, 
which now also include the graphical output 
x. 
Before the feedback link (x — x') is estab- 
lished photo-coordinates x',x" are measured 
in on-line systems in the same way as in off- 
line systems. Derived parameters g can then 
be used in the on-line mode with an arbitrary 
decomposition gT z(glgf) where g, repre- 
sents the return in the feedback link (x — x’). 
In general, the model coordinate system 
can be defined in any arbitrary manner with 
respect to the object, but it is advantageous to 
assume equal photo and model scales M = 1/m 
so that the flying altitude is equal to the nega- 
tive principal distance f. Then it holds 
AX = mAx (7) 
where Ax =x — x,, and the projection centers 
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1976 
in a steremodel can be assigned special val- 
ues, e.g., x, = 0 and x} = (b 0 0) where b is 
the photogrammetric base in the photo scale. 
The computation of parameters g is then 
based on the relation x €» x' rather than on 
X x'. Forthat purpose X is converted into x 
in accordance with Equation 7. This is done 
with the use of suitable estimates of coordi- 
nates for the first projection center X«, = C 
and with the use of x«, = 0, Ax = x so that 
x — (X-C)Im (8) 
Here, vector C is derived from an arbitrary 
given single pair of X and x according to C 
— X — mx. A successful reconstruction ofthe 
model ultimately yields the rotation matrices 
P and the vectors x, for both images. 
Now the original Equation 4 is modified, 
by substitution of Equation 7, into 
x' + c = AP'Ax (9) 
where 
A= "n and y = - fpr (9a) 
The value of À is always very close to unity. 
Here, matrix P and vector p are defined by 
column partitioning of the rotation matrix P 
P [Pp]. (10) 
Thus, the working equations of an on-line 
analytical system which is physically driven 
via the virtual model, can be given by 
  
x 
FULL RETURN «XC (11) 
X t 
  
  
  
where v is an arbitrary scale factor for 
generating a graphical plot. The first formula 
in Equation 11 represents the transforma- 
tions for both images in a stereopair. In this 
system, the exterior orientation is fully re- 
turned to the (x > x') link except for the photo 
scale factor m which is used in the (x—X) 
computation. In operations, the floating mark 
is driven in the directions of the object coor- 
dinate system. 
In photogrammetric compilations, it is al- 
ways necessary to establish a horizontal x,y 
plane, but in some instances it may be incon- 
venient to fit the control drive with the object 
X,Y axes especially if the photogrammetric 
base is azimuthally rotated. Rotation matrices 
P can then be factorized as P = KT where any 
arbitrary rotation K around the Z axis defines 
a new orientation T = K'P to be returned to 
the photo feedback(x >x') whileK is applied 
in the (x — X) conversion