2.3 The Functional as well as the Stochastic Model of the Surface Constraint
The symbols in equation (2.3) are classified into two groups, namely the observations and the
unknowns. For this study, what is known are the surface points while those points
P
i(X
i, Y
i, Z
i)
considered to lie on the planes are the unknowns through the photogrammetric measurements.
One, therefore, is able to formulate functional model as well as stochastic model of the control
surface constraint shown as equation (2.4)
 |
(2.4) |
 |
where
n : the number of object points from the photo measurements;
m : the number of surface constraints;
q :the number of registered surface points;
w: the discrepancy ((l
io derived with the approximations of the unknowns and the
observations;
y : the observations of the known surface points;
B : the coefficients of partial derivatives with respect to the surface points;
A : the coefficients of partial derivatives with respect to the unknowns of the object points( may
include 0
mx3(n-m) for those object points unable to find registered surface planes);
x :the unknowns of the object points ;
e :the random errors of the surface points;
S0 : the dispersion matrix of the surface points;
Po : the weight matrix of the surface points;
s02 : the variance component of the surface points;
Yet from the prediction point of view, the closer the unknown object point is to the registered
surface point, the more reliable the constraint information would become. The error-propagated
uncertainty assessment of the surface constraint based on the aforementioned model does not
find itself the same conclusion. Therefore a modification of the stochastic model compromising
the likely conflict due to the imperfect analytical model is needed [Jaw, 1999]. The modification
performs in a way such that the
x00, as shown below, reflects the deviation of the plane
formation by three surface points from the fitting plane by all the neighboring surface points. We
therefore come to the modified model as follows,
 |
(2.5) |
 |
where
P
oo : the weight matrix of the added uncertainty;
stands for the weight matrix of surface constraint after the modification;
Due to the restricted space in this paper, the author encourages the interested readers refer [Jaw,
1999] for more detailed explanation of the model modification.
3. Control Surface in Aerial Triangulation
With the formulation of surface constraint model, the photogrammetric measurements can be
joined into the system performing the aerial triangulation task. The combined model is
expressed as below,
 |
(3.1) |
 |
where
y
1 : increments of the photo observations, namely the difference between the original
observations and the derived observation via the approximations;
k : number of the photo point measurements;
p : number of the photos ;
n : number of the object points to be determined;
A
11 : the coefficient matrix derived from taking partial derivatives with
respect to the exterior orientation parameters;
A
12 : the coefficient matrix derived from taking partial derivatives with respect to the object point
coordinates;
x=[
DXol,
DYol,
DZol,
Dwol,
Djol,
DKol,
...,
DXop,
DYop,
DZop,
Dwop,
Djop,
DKop]T :
unknowns of exterior
orientation. D indicates the increment of the parameters;
x=[
DX1,
DY1,
DZ1,
DX2,
DY2,
DZ2, ...,
DX3,
DY3,
DZ3,
]T
: unknowns of object points;
e1 : the random errors of the photo point measurements;
s012 : the variance component of the photo point measurements;
P1 : the weight matrix of the photo point measurements;
w : the discrepancy of the surface
constraint;
A22 is A in equation (2.4) plus 0mx3(n-m) for those object points unable to find registered surface
planes;
m : number of the surface constraints;
e2 : the random errors of the registered surface points;
s012 : the variance component of registered surface points;
: the weight matrix of the modified surface constraints;
Thus the redundancy of the system is calculated as
r=(2k+m)-(6p+3n) (3.2)
