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.

REFERENCES

Bobkiewicz P., 2006. Zastosowanie monolitycznych analizato-

rów obrazu do pomiaru wysokości ciał niebieskich na

morzu. Proceedings of the XV-th International Scientific

and Technical Conference The Role of Navigation in

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

Sensors, course notes. University of Waterloo, may.

Wagnerowski T., 1959. Optyka Praktyczna. Warszawa: PWT.