283
( )( ) ( )( )
( ) ( )
( ) ( )
2
2
2
2
2
2
2
1
2
1
2
2121
12
'cos
fΔyyΔxxfΔyyΔxx
fΔyyΔyyΔxxΔxx
d
++++++++
++++++
=
(7)
where x and y with indexes 1, 2 are coordinates of
points 1, 2 in relation to the middle of array.
Value of difference
∆
d in pixels calculated from
(8)
between angular distance d
12
calculated assuming
focus is in the middle of array and true distance d
12
′
in function of
∆
x and
∆
y is presented on Figure 9.
Calculation is made for maximal possible
measurement of distance, parallel to the edge and
through the middle of array, regarding features 6°
and 9° (y
1
=y
2
=0, x
1
=1000, x
2
=-1000, f=2145) –
upper graph – and with distance section
asymmetrical in relation to middle of array – lower
graph. Relevant error may reach significant values
(over ten and so pixels) and may be predominant
among listed above.
-6
-4
-2
0
2
4
6
8
10
12
-100 -75 -50 -25 0 25 50 75 100
-6
-4
-2
0
2
4
6
8
10
12
-100 -75 -50 -25 0 25 50 75 100
Fig. 9. Error ∆d of angular distance resulting from shifting of
focus about values ∆x and ∆y in relation to middle of array; for
symmetrical (upper) and asymmetrical (lower) measurements.
2.9 Summary of factors 1-8
Only error introduced by pixel size has random
characteristic. Factors mentioned in items 1, 2, 3, 4,
5, 8 introduce error with pattern characteristic. It
depends on location of object image on array and is
constant for given device unit during exploitation.
Theoretically it may be determined and then taken
into account during image processing. Assuming that
this is not accomplish and due to discretion of object
location on array it has to be treated as random error.
Factors mentioned in items 1, 2 introduce errors with
specific values spreading on relatively small area
(within 1 pixel) – local range. Factors described in 3,
5, 8 introduce errors with trends spreading on whole
array and factor described in 4 may has such nature
too.
2.10 Inaccuracy of focal length
Camera is to be calibrated before measurement due
to variable focal length of objective (feature 9°).
Aim of calibration is to determine focal length f
from formula (2) basing on measurement of known
angular distance, for instance between two stars.
Proportion of error of distance to distance during
calibration should be as small as possible, because it
determine relative accuracy of focal length.
Condition for calibration:
− stars in vicinity of the same altitude or at very
high altitude (reduced influence of refraction),
− distance between stars as long as possible,
− middle point of array in vicinity of midway point
between stars,
− section connecting stars parallel to one of the
edges of array.
Because the same formula is used for calculation
of altitude, then if celestial body and point below on
visible horizon are situated near points of stars
during calibration, then altitude measurement has
comparative characteristic. In the situation errors
with trends spreading on whole array have only
residual influence on measured distance, resulting
from inaccurate composition of points during
measurement and calibration.
3 CONCLUSIONS
Exact measurement of distance with help of digital
still camera require first of all:
− coordinates of point on array indicated by optical
axis (focus of objective),
− angle of array deviation from perpendicular to
optical axis and its direction on array,