The possibility of using phase-correlation analysis of an image in optoelectronic devices to measure small
angular and linear displacements of objects is considered. It is shown that the measurement accuracy is
thereby increased.
Key words: optoelectronic system, correlation analysis of the image, accuracy of displacement
measurement.
In geodesy and measurement technique, various phase optoelectronic devices are employed at the present time to
measure the angular and linear displacements of objects, their nonaxiality, to balance the position of individual items of
instruments, etc. [1–3]. These phase devices differ from other similar devices in having a simpler construction, stability of
the characteristics, speed of response, reliability, a wide dynamic range of the reproduced signals and high efficiency of opto-
electronic signal conversion. Sometimes they possess higher sensitivity and higher noise immunity. Compared with ampli-
tude optoelectronic devices they are less sensitive to changes in the radiation flux incident in the image analysis plane.
The drawbacks of phase optoelectronic devices are the low resolving power, due to technical errors in the rasters of
the image analyzers, the low noise immunity from the effect of background and interference, incident in the angular field of
the device, and also from internal noise. The technological errors of image analyzers include deviations of the raster line peri-
ods from their nominal value, the different slope of the lines, and, for rotating analyzers, the eccentricity of their rotation axes
with respect to the center of the raster. In addition, the high resolving power of phase devices may sometimes be limited by
the errors of standard industrial phase-measuring instruments (phasemeters), which are difficult to match with the accuracy
characteristics of image analyzers.
In existing phase optoelectronic devices, the phase difference between the useful signal and a standard voltage is mea-
sured. This is proportional to the displacement of the image of the object or a mark in the analyzer raster plane. One can also
use as a raster certain multielement radiation receivers with signal sampling from their elements in accordance with a certain
rule, as a result of which a virtual displacement of the elements (pixels) of the receiver is produced [2]. Only receivers with
an arbitrary selection of the signal with elements, for example of CMOS type, are suitable for phase analysis [3].
The effect of technical errors of the image analyzer raster on the accuracy of existing phase devices is explained by
the fact that, when measuring the informative parameter of the useful signal, in this case the phase difference, only limited
information on the signal is employed, corresponding to characteristic instants of time during the measurements, and the vol-
ume of information which other signal points carry is not taken into account. Such characteristic points are, for example,
points where the signal passes through zero or its extremal values, points where the signal intersects a chosen threshold level,
etc. If, in a phase device for measuring phase difference, only one characteristic point is used, as often happens in practice,
the result of the measurements will obviously depend to a considerable extent both on the spread in the geometrical param-
eters of the raster and on the signal distortions, due to the actions on it of internal noise, external interference and the nonuni-
form background, and sometimes also vibrations.
Measurement Techniques, Vol. 48, No. 5, 2005
THE USE OF CORRELATION ANALYSIS OF THE
IMAGE IN PHASE OPTOELECTRONIC DEVICES
V. P. Soldatov UDC 621.397.3:621.317.77:621.035
Translated from Izmeritel’naya Tekhnika, No. 5, pp. 37–39, May, 2005. Original article submitted December 20,
2004.
0543-1972/05/4805-0471 ©2005 Springer Science+Business Media, Inc. 471
To reduce the effect of the factors considered above on the accuracy of measurements, one can use the phase-corre-
lation method of analyzing the image, in which information from all points (values) of the signal over a complete measure-
ment period is taken into account. The scheme of the phase-correlation angle gauge, proposed in the Department of
Optoelectronic Instruments at the MIIGAiK is shown in Fig.1 [4].
A bunch of rays from the target is focused by an objective lens 1 in the plane of the raster of a drum image analyzer 3,
which rotates with constant angular velocity, on parts diametrically opposite it. It then passes through a beam-splitting unit 2.
Modulated radiation fluxes Φ1 and Φ2 are incident on photoreceivers 4, in the circuits of which electric signals U1(t) and U2(t)
are produced. After amplification by amplifiers 5, they are incident on the inputs of corresponding similar voltmeters 6, which
measure the effective values of the voltages, and at the inputs of the correlator 7. The outputs of the correlator 7 and of both
voltmeters 6 are connected to a computer unit 8, for example, a microprocessor, which calculates the displacement x of the image
of the target in the plane of the analysis, on which the phase difference ϕ of the signals U1(t) and U2(t) depends.
We will obtain the relation between the cross-correlation function K(τ) of the signals U1(t) and U2(t) and the phase
difference ϕ between them. The function K(τ) is described by the expression
(1)
where τ is the correlation interval and t is the time.
Since U1(t) and U2(T) are sinusoidal signals, between which there is a phase shift ϕ, we can write
(2)
where ω = 2πmn is the angular frequency of the signals, m is the number of periods of the raster of the image analyzer, n is
the rotation frequency of the analyzer, and U1max and U2max are the amplitudes of the signals in the first and second chan-
nels of the angle gauge, respectively.
The signals U1(t) and U2(t) can be considered as individual samples of the corresponding stationary random func-
tions. From (1), taking (2) into account, and fact that the random functions U1(t) and U2(t) are stationary, we obtain
(3)
where T is the period of variation of the functions.
KT
U tU t dtT
( ) sin sin ( ) ,max maxτ ω ω τ= +∫11 2
0
U t U t
U t U t1 1
2 2
( ) sin ;
( ) sin( ) ,max
max
== +
ωω ϕ
K U t U t dt( ) ( ) ( ) ,τ τ= +−∞
∞
∫ 1 2
472
12
34
4
5
5
6
6
7 8
Fig. 1. Sketch of the phase-correlation angle gauge: 1) objective; 2) beam-splitting unit; 3) image
analyzer; 4) photoreceiver; 5) amplifiers; 6) voltmeters; 7) correlator; 8) computer unit.
After integrating in (3), we obtain
K(τ) = 0.5U1maxU2maxcosωτ. (4)
Expression (4) shows that the argument of the cross-correlation function K(ϕ) is the phase shift ϕ = ωτ between the
random functions U1(t) and U2(t).
Consequently, from (4) we obtain an expression which describes the change in the phase shift ϕ of the signals U1(t)
and U2(t) as a function of the change in the cross-correlation function K(τ) of these signals in the form
(5)
We will convert this formula as follows. Assuming the random functions U1(t) and U2(t) are stationary, and their
mathematical expectations m1(t) and m2(t) are equal to zero, taking (2) into account we can write the following expressions
for the variances D1(t) and D2(t) of these functions:
(6)
Substituting the values of U1max and U2max from (6) into (5), we obtain
Since the expression is the normalized cross-correlation function Kn(ϕ) of the signals U1(t) and U2(t), we
can write the following simpler relation for the required phase difference:
ϕ = arccosKn(ϕ). (7)
We will replace ϕ in this formula by the shift x of the position of the target in the plane of the drum image analyz-
er, corresponding to the conversion function of this analyzer. As follows from [1], the conversion function has the form
ϕ = marcsin(x /R),
where R is the radius of the drum image analyzer.
Taking this formula into account, after simple reduction we obtain from (7) the following formula describing the
relation between the shift x of the image of the target in the analysis plane and the cross-correlation function of the signals
U1(t) and U2(t):
x = Rsin[m–1arccosKn(ϕ)].
Hence the value of the mismatch angle α, measured by the instrument, is expressed in terms of Kn(ϕ) in the form
α = arctan{(R /ƒ′)sin[m–1arccosKn(ϕ)]}, (8)
where ƒ′ is the equivalent focal length of the objective of the angle gauge.
K
D D
( )ϕ
1 2
ϕϕ
=
arccos( )
.K
D D1 2
D tT
U t m t dt U
D tT
U t m t dt U
T
T
1 1 12
012
2 2 22
022
10 5
10 5
( ) [ ( ) ( )] . ;
( ) [ ( ) ( )] . .
max
max
= − =
= − =
∫
∫
ϕϕ
=
arccos
( ).
max max
2
1 2
K
U U
473
When the angle gauge is operating, its computer unit 8 calculates the required angle α from formula (8) in accor-
dance with the values of the correlation function K(ϕ), obtained by the correlator, the values of D1(t) and D2(t), determined
by the voltmeters, and the constant parameters R, m, and ƒ′.As follows from (8), the values of the angles α theoretically do not depend on the period 2a of the image analyzer
raster, while the resolving power ∆αp of the angle gauge increases as the number of periods m of the raster increases, i.e., as
the periods decrease. Consequently, the technical deviation of the period 2a of the raster from the nominal value also does
not give rise to any corresponding error ∆α when measuring the angle. It also follows from (8) that the errors in the mea-
sured angles α are also independent of the instability of the amplitudes of the signals U1(t) and U2(t) in both channels of the
instrument, since the normalized correlation function Kn is invariant to this instability.
Moreover, since the correlation method of processing data is optimum from the point of view of obtaining the max-
imum signal/noise ratio [5], in the proposed angle gauge the random partial error due to noise is considerably attenuated com-
pared with the existing phase instruments, described in [1–3], and the noise protection when operating in the presence of
background and interference is also improved.
As regards the effect of the eccentricity of the axis of rotation of the analyzer with respect to the center of the raster
on the accuracy of measurements, since this eccentricity causes a periodic variation in the raster period when the analyzer
rotates, this has no effect, as was noted above.
It can similarly be shown that when using a disk image analyzer in a phase-correlation angle gauge, taking its con-
version function ϕ = marctanx/R into account [1], the angle is found from the following relation:
α = arctan{(R /ƒ′) tan[m–1arccosKn(ϕ)]}.
Note also that when using a line image analyzer raster with conversion function ϕ = (π/a)x [1], the working formu-
la for the phase-correlation angle gauge takes the following form:
α = arctan[(R /ƒ′)m–1arccosKn(ϕ)].
The proposed phase-correlation method of analyzing an image can be used in the construction of various instruments
for measuring angular and linear displacements and in many other of the instruments mentioned above, thereby improving
their metrological characteristics. A drawback of this method of analysis is that it is not possible to distinguish the sign of
the target displacement. However, this drawback is easily removed, for example, by a preliminary mutual displacement x0 of
the images of a target in the analysis plane by a distance equal to a quarter of a raster period, i.e., x0 = a/2. In this case, in
particular when using line rasters, the conversion function can be reduced to the form
Kx
an =π
sin .
474
n
Fig. 2. Form of the conversion function Kn in the working region
of image displacements –a/2 ≤ x ≤ a/2.
This function is shown in Fig. 2 in the working region of image displacements –a/2 ≤ x ≤ a/2.
REFERENCES
1. D. A. Anikst et al., High-Accuracy Angle Measurements [in Russian], Yu. G. Yakushenkov (ed.), Mashinostroenie,
Moscow (1987).
2. V. A. Solomatin, Systems for Monitoring and Measuring with Multielement Receivers [in Russian],
Mashinostroenie, Moscow (1992).
3. Yu. B. Parvulyusov et al., The Design of Optoelectronic Devices [in Russian], Yu. G. Yakushenkov (ed.), Logos,
Moscow (2000).
4. V. P. Soldatov, Inventor’s Certificate No. 1733923 USSR, Izobret., No. 19 (1992).
5. G. M. Mosyagin et al., Theory of Optoelectronic Systems [in Russian], Mashinostroenie, Moscow (1990).
475
