ngulation with a
Very hi
It into the ang] SN
Ytical tria.
may be followed, de
'e developed at the
try.
vith respect to à Tectangulay
he X, Y and Z CO-Ordinates
t+1 in a strip are the hag:
a photograph are computed
s of the X, Y and 7 System
Dending
National
orientation, rays from CO
two Corresponding image
ys and must thug Jie in one
m 0
ition equation of this kind,
ue for the base component
graph i+1 are the hag
+1. The co-ordinates X, T
cal length:
a (2)
es of the photograph axes
he sines and cosines of the
ar in the five elements of
tly.
cal operations of addition,
atical operation must be
solve linear equations. The
the elements of relative
OW be substituted by cw
is (2) by substitution oi
> validity of the equation.
ds, y about the Y-axis and
tion then gives:
Z X X; Y; Y, Zi
; ; . byi+t + . bzi+1 + y = (3)
Ba Xa ips Yi, Ya f
Ü
This differentiated equation is a first order approximation. In its co-efficients occur
oximate values of the elements of relative orientation. Five equations (3) obtained
A pairs of corresponding points, are now solved. This results in improved ap-
Em e Two of these: byi+1 an bzi+1 follow immediately from the equations. The
rr dw, dp and dx however are corrections that must be applied about the X, ¥ and
7 axes. After these corrections the direction-cosines l, m and n of the
found by multiplying the determinant of direction-cosines for the a
by the determinant of direction-cosines for the corrections:
photograph axes are
pproximate rotations
Di+1 — peoeorr. 193
(4)
Initial approximations may be chosen arbitrarily such as equal to zero or equal to the
values obtained in the preceding model. Improved approximations are computed with the
equations (2), (3) and (4). These are in their turn substituted into the equ
process is repeated until the corrections obtained are sufficiently small.
The first computed set of approximations of the rotations is generally accurate within
a degree. The second set is then accurate within a few minutes. This accuracy is sufficient
io start the adjustment of the relative orientation.
The relative orientation is adjusted using the condition equations for all pairs of cor-
responding points. The adjustment follows the method of least squares. The points may
be chosen in any desired position in the model. At present up to 15 points may be used.
Equation (1) must now also be differentiated with respect to the pla
This results in additional terms in equation (3)
ations. This
te co-ordinates.
. Putting these in the second part, this reads:
bx by bzi bx byi bi,
=— by, Ma, Ny, . dac, — bs m, ny, .d %
X Y5 | Zi pa Xi; i Yh 2341
bx byi bai ba byi bzi
— X, Y; Z; . dw; 1 T X, Y Z, . dy ia (5)
mi j lj mi ni
Fors A i : l Vi 1 ml, xii
Following the procedure of Professor J. M. TIENSTRA in the paper ‘An Extension
of the Technique of the Method of Least Squares to Correlated Observations” published
in the “Bulletin Géodésique" 1947, 6, pages 301—335, the second part of the equation may
be regarded as a correction to be given to a single observed quantity.
In each equation the co-ordinates of only one pair of corresponding points occur. There-
fore, assuming no correlation between the co-ordinate readings of non-corresponding points,
the “observed” quantity in any one equation is not correlated with the "observed" quantity
in any other equation.
Assuming further non-correlation between the four co-ordinate readings of a pair
of corresponding points, the weight of an "observed" quantity is inversely proportional to
the sum of the squares of the four determinants in the second part of (5). In the case
of nearly vertical photography with one of the photograph axes roughly parallel to the
base, as is nearly always the case, the value of this sum is approximately equal to 2 b2/2
and independent of the position of the points and the orientation of the model. Since
light changes in weight have little or no appreciable effect on the result of an adjustment,
the Weights may here all be chosen equal.
