International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 2
Number 3
September 2008
279
Estimation of Altitude Accuracy of Punctual
Celestial Bodies Measured with Help of Digital
Still Camera
P. Bobkiewicz
Gdynia Maritime University, Poland
ABSTRACT: Measurement of altitude traditionally made with sextant may be done with help of digital still
camera. Factors influencing accuracy of this measurement done with help of such a device are described in the
paper. Values of errors introduced by each of these factors are estimated basing on example technical data of
typical digital camera. This analysis shows, which factors are the most important and if accuracy of altitude is
sufficient for purposes of celestial navigation.
1 ALTITUDE MEASUREMENT WITH DIGITAL
STILL CAMERA
1.1 Demand accuracy
Demand accuracy of single altitude measurement
may vary in dependence from the assumed number
of measurements to be done. The target is the
suitable accuracy of position, which may be
obtained by increasing the number of measurements,
when single measurement has small accuracy. As a
reference approximately 1′ (one angular minute)
may be assumed as demand accuracy of altitude.
Digital still camera may be used for measurement of
angular distance, but it is necessary to investigate, if
with its technical parameters obtainment of required
accuracy is possible.
1.2 Typical digital still camera
Features of popular digital still cameras:
1 mostly equipped with charge-coupled device
(CCD) image sensor, residual with comple-
mentary metal-oxide-semiconductor (CMOS);
2 color image sensor with Bayer pattern mosaic, 24
bit color depth;
3 image sensors array covered with micro-lens;
4 equipped with mechanical shutter;
5 effective diameter of objective aperture about 15
mm, comparable with diagonal of array;
6 number of active pixels over 3 million with array
format 3:4, dimensions of array for these numbers
in pixels minimum 1500×2000;
7 square pixel, pixel size 5 μm, regarding 6° array
dimensions in mm 10×7.5 and diagonal 12.5;
8 angle of field of view along longer side about 50°,
along shorter 40°; with the usage of zoom 3× it is
adequately 20° and 15°;
9 variable focal length of objective and variable
focusing distance allowing registration of image
from infinity to few centimeters, regarding 6°, 7°
and 8° focal length in pixels is equal
(2000/2)/tan(50°/2)=2145, it is 10.7 mm.
Values of technical parameters of particular
models may differ from listed above. Particularly big
differences are within diameter of objective, number
of pixels and pixel size (features 5°, 6°, 7°).
Increasing number of pixels within the same
dimension of array may be noticed in models
introduced on market. In this situation pixels are
smaller. Maintenance the same level of luminous
energy on single pixel required increasing of light
flow and this may be done by widening objective
280
area. Regarding 8°, measurement of altitude with
help of digital still camera is limited to 50°.
1.3 Calculation of angular distance
Assuming:
− focusing distance is set on infinity – focal plane,
which coincides with image sensor array is in
focus of objective,
− image is formed accordingly to rules of central
projections on plane – plane of array is
perpendicular to optical axis,
− pixels create orthogonal Cartesians coordinate
system,
unit vector of direction p of punctual object P
recorded on array, in orthogonal coordinate system
Oxyz set by main plane of objective and its optical
axis (Figure 1) may be calculated from
d
12
P
1
(x
1
, y
1
)
P
2
(x
2
, y
2
)
x
y
x
y
f
O
z
Fig. 1. Association of image sensor array with coordinate
system Oxyz set by main plane of objective Oxy and its optical
axis Oz
++
++
++
=
=
222
222
22
2
'
'
'
fyx
f
fyx
y
fy
x
x
z
y
x
p
(1)
where x and y are coordinates of P on array plane in
relation to point indicated by optical axis, f focal
length, directions of axis Ox and Oy are equal to
appropriate directions on matrix and direction Oz
agree with optical axis.
Direction of this unit vector is the same as on real
object (opposite turn). Angular distance d
12
between
real objects, recorded on array as points P
1
and P
2
may be obtained from scalar product of their unit
vectors p
1
and p
2
22
2
2
2
22
1
2
1
2
2121
12
21
212112
2112
cos
'''''
'cos
cos
fyxfyx
fyyxx
d
zzyyxx
d
ppd
++++
++
=
++=
=
(2)
2 FACTORS INFLUENCING ACCURACY
2.1 Inaccuracy of lens area, inhomogeneity of
glass and optical aberrations of objective
Distortion of direction of light ray result from these
elements should not exceed the angle of theoretical
resolving power of objective, arising from diffrac-
tion of rays on diaphragm. This may be treated as a
rule during production of optical elements. For
objective with circular aperture this angle is given by
formula
D
u
λ
22.1=
(3)
where λ length of wave, D diameter of objective
aperture (diaphragm), u in radians (Wagnerowski
1959). For yellow-green light λ=0.000556 mm and
objective aperture D=15 mm this angle u=0.15′.
2.2 Inaccuracy of placing of pixels optical centers
on nodes of virtual net with equal mesh
optical centre
of pixel
Fig. 2. Inaccuracy of placing of pixels optical centers on nodes
of virtual net with equal mesh
There is no direct information about this in
technical specifications of digital image sensors. In
case of CMOS, basing on characteristic of this
technology one may conclude that this error is two
orders smaller then pixel size. But regarding feature
3° performance of micro-lens layer has the main
implication here. On microscopic photographs of
micro-lens layer error arising from this is
imperceptible in comparison to pixel size.
281
2.3 Non-perpendicular pixels arrangement
This information may be found in technical
specifications of some manufacturers for example
Dalsa company (Dalsa 2002).
0.04°
0.0
05 mm
image sensor array
1 pixel
2000
1500
Fig. 3. Non-perpendicular pixels arrangement on image sensor
array
In the specification quoted in references (CCD
image sensor, 1024×1024 pixels, pixel size 7.4 μm)
non-perpendicular pixels arrangement allowance is
0.005 mm measuring displacement of parallel edges
perpendicularly to them (Figure 3). For this sensor
the angle between directions of main axis of array
may differ from right angle about 0.0378°. At distant
of 1000 pixels (feature 6°) from the middle of array
it is no more then 1 pixel for this angle. This error
may be neglected if altitude is measured parallel to
one of directional axis (parallel to one of the edges
of array), because it shifts mutually celestial body
and visible horizon parallel to the last one, not
influencing the altitude.
2.4 Inaccuracy of arrangement of pixels at one
plane
According to specification mentioned above
allowance of arrangement of pixels at one plane is
<7μm – about pixel size (Figure 4).
real
surface
50°
1
sensor flatness
O
25°
theoretical
plane
0,5
Fig. 4. Inaccuracy of arrangement of pixels at one plane
Maximal distance between real surface of the
array and theoretical plane is equal half of the
allowance. Error of object location arising from this
increases with angular distance of object from point
on array indicated by optical axis (called later as
focus). For object recorded near boundary of field of
view for data of feature 8° error is maximally a half
of the maximal distance. For example, if value of
allowance is equal two pixel sizes, maximal error of
location may be half of the pixel.
2.5 Non-perpendicular direction of optical axis to
array plane
Array is placed on chip, chip in socket, socket on
electronic board and this is connected with objective
by camera casing. If this construction is not
calibrated to make array and main plane of objective
parallel, then each of the connections carries error in
their parallelism. According to specification men-
tioned above parallelism allowance of array and
plane of chip casing is equal 1.4/100 (0.8°).
α
optical axis
x
“a”
“b”
x′
array
O
main plain of
objective
Fig. 5. Non-perpendicular direction of optical axis to array
plane.
Difference
∆
x between measured distance x′ and
true distance x (for array perpendicular to optical
axis) at part of array open from main plane of
objective (area “a” on Figure 5) is positive and given
by formula
'
1sin
cos
''
xf
xxxΔx
a
+
−=−=
α
α
(4)
and at part open to opposite side (area “b”) negative
fx
xxxΔx
b
α
α
sin
'
1
cos
''
−
−=−=
(5)
where α is angle of array deviation from
perpendicular to optical axis and f focal length. For
focal lengths short in comparison to measured
distance x′ values
∆
x are considerable and regarding
8° they reach 8 pixels for α=1° (Figure 6).
282
-10
-8
-6
-4
-2
0
2
4
6
8
10
-1000 -750 -500 -250 0 250 500 750 1000
x' [pixels]
∆
x [pixels]
Fig. 6. Difference ∆x between distance of object from focus x′
measured on array non-perpendicular to optical axis and true
distance x on perpendicular array in function of x′.
Absolute values of differences in both areas for
the same values of x′ are almost equal. Therefore if
focus is lying exactly at half a way on straight line
between both objects, then this error compensate
itself. In practice such a measurement is difficult to
be done, but one may assume certain allowance in
distance, with which observer is able to divide
section between to points on two even parts.
Assuming, that after such a division one part may be
shorter then another maximally about 1/3, in extreme
case error of distance is 3 pixels.
2.6 Pixel size
Location of punctual object on image sensor array
may be obtained with accuracy of half of the pixel
size.
m=0.8′
f
O
pixel
Fig. 7. Accuracy of location of punctual object arising from
pixel size
Regarding 9° angular resolving power m near
focus is equal arctan(0.5
ּ
◌1/2145) =0.8′.
2.7 Variable angular resolving power
Scale of the image increases with the distance from
focus. With constant spatial resolving power of
array, angular resolving power m in radians is given
by equation
⋅=
β
2
cos
1
5,0arctan
f
m
(6)
where
β
is angular distance from focus and f focal
length in pixels.
Accuracy of location in spherical coordinate
system calculated on location on array is minimal for
objects imaged in focus and increases with distance
from this point. Regarding 8° and 9° angular
resolving power near boundary of field of view is
equal 0.66′.
Assuming, that interior spot and the first ring of
diffractive image of star projects on array, diameter
of star image is equal 4×u=0.6′. It may be contained
within one pixel and may occupy no more then 4
pixels. For single-bit depth of colors accuracy of
location increases with the square root of number of
pixels affected – for 4 pixels – twice. Image of star
recorded on array has in fact diameter about 3-4
pixels. It is the result of algorithm of obtaining
colors (feature 2°) and for this reason the image is
enlarged 1 pixel to each direction. It not improves
accuracy anyway.
2.8 Differentiation in location of middle of array
and focus
Coordinates of focus are necessary for calculation of
altitude. With regard on lack of these data coordinate
of middle of array are accepted in turn. If accepted
coordinates differ from focus (Figure 8)
middle of array
d
12
d
12
′
focus
O
Fig. 8. Focus is not in the middle of array
about shifts
∆
x and
∆
y along appropriate axis, then
true distance d
12
′ is given by formula
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
( )
'
1212
ddfΔd −=
(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.
x1
=1000
x
2=-1000
∆
y =0
∆
y =100
-6
-4
-2
0
2
4
6
8
10
12
-100 -75 -50 -25 0 25 50 75 100
∆
x[pixels]
∆
d
[pixels]
x
1
=700
x
2
=-1000
∆
y =0
∆
y =100
-6
-4
-2
0
2
4
6
8
10
12
-100 -75 -50 -25 0 25 50 75 100
∆
x[pixels]
∆
d
[pixels]
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,
284
− focal length.
Except focal length (published anyway with
insufficient accuracy), these data do not appear in
technical information of user manuals. Therefore
error of measurement may exceed the acceptable
level even several times. So calibration of camera
aiming calculation of required data is essential.
At comparative measurement error compose from
local range errors and predominant error arising
form pixel size. In this case error of measurement is
about 1′ but fitting of measurement of altitude in
place, where stars were during calibration is
difficult. At high stability of focal length during
exploitation of camera (between switching on/off) it
is worth to do many calibrations, with keeping only
first condition from mentioned in 2.10. Then more of
measurements of altitude may be treated as
comparative.
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Bobkiewicz P., 2006. Zastosowanie monolitycznych analizato-
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Support of Human Activity on the Sea, Naval University of
Gdynia Institute of Navigation and Hydrography, Gdynia.
Dalsa. 2002. Data Sheet FT 18 Frame Transfer CCD Image
Sensor, Product specification.
Hornsey R., 1999. Design and Fabrication of Integrated Image
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Wagnerowski T., 1959. Optyka Praktyczna. Warszawa: PWT.
