=Paper=
{{Paper
|id=Vol-2590/short27
|storemode=property
|title=Computer Modeling of an Image of the Optical-Electronic System for Reference Mark Position Control
|pdfUrl=https://ceur-ws.org/Vol-2590/short27.pdf
|volume=Vol-2590
|authors=Tuan Pham Ngoc,Aleksandr Vasilev,Alexander Timofeev,Valery Korotaev,Anton Maraev
|dblpUrl=https://dblp.org/rec/conf/micsecs/NgocVTKM19
}}
==Computer Modeling of an Image of the Optical-Electronic System for Reference Mark Position Control==
https://ceur-ws.org/Vol-2590/short27.pdf
Computer Modeling of an Image of the
Optical-Electronic System for Reference Mark
Position Control
Tuan Pham Ngoc, Aleksandr Vasilev, Alexander Timofeev, Valery Korotaev,
and Anton Maraev
ITMO University, Saint Petersburg, Russia
{ngoctuan,a-s-vasilev,timofeev,vvkorotaev,aamaraev}@itmo.ru
Abstract. An impact of evaluation error of reference mark image en-
ergy center on accuracy of optical-electronic system for reference mark
position control (OES RMPC) is considered. Basic problems related to
computer modelling of reference mark image, wherein reference mark is
a light source, and needed for OES RMPC accuracy estimation are an-
alyzed. A general algorithm for reference mark image description taking
into account its relative motion is presented. A numerical experiment of
image restoration using Matlab is shown. It is demonstrated that, when
information is processed by OES RMPC, image restoration algorithm by
Wiener parametric filtration and Tikhonov regularization are the most
effective.
Keywords: Computer modeling · Image blur · Position control · Refer-
ence mark· deconvolution image· Matlab
1 Introduction
When a railway track is built or repaired with modern high-performance track
machines, definition of the actual railway track position and estimation of the re-
sult with technical means are important aspects for improvement of an operation
of railway track placing in a required position [1, 2, 9]. An optical-electronic sys-
tem for reference mark position control (hereinafter referred to as OES RMPC)
developed at ITMO (St. Petersburg, Russia) enables to define the railway track
position in the vertical plane (surfacing), vertical relative position of track rails
(transverse gradient) and horizontal position (realigning) relative to reference
marks conjugated to geodetic network coordinates, as the track renewal train
runs. The minimal error is limited by the accuracy of reference mark image
energy center definition [3, 10, 11] on the receiving senor array. This error de-
pends on the number of parameters, such as video camera pixel size, optical
system aberration characteristics, background luminance distribution and refer-
ence mark (RM) image smear. The objective of the paper is to form a computer
Copyright c 2019 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
2 Pham N.T et al.
model, which synthesizes a smeared RM image in different background lumi-
nance distributions, and to estimate the error of coordinate definition in the
restored smear-free image, considering RM relative motion.
2 Impact of evaluation error of RM image energy center
on OES RMPC accuracy
In the category of systems we are considering [12, 4, 5], to define the energy center
(EC) of the RM image the weighted summation method is used, the method is
expressed with formulas (1), (2) and provides an accuracy of less than 0.1–0.01
pixel [6, 7]. ,
M X
X N XM X N
XEC = (xi,j · Qi,j ) Qi,j (1)
i=1 j=1 i=1 j=1
,
X N
M X XM X
N
YEC = (yi,j · Qi,j ) Qi,j (2)
i=1 j=1 i=1 j=1
where XEC ; YEC are coordinates of the RM energy center on the sensor array,
Q is a total signal from array elements.
However, the error of RM image coordinate evaluation, when the receiver is
exposed to the background radiation, proper detector noise and camera motion,
rises dramatically, thus increasing the total error of the system.
Attenuation of image distortion influence generally consists in using special-
ized optical systems or high-speed cameras, which makes implementation of the
system more complex. Due to use of software it is possible to eliminate a number
of optical distortions by means of mathematical processing of the images, it is
possible to lessen requirements to imaging system hardware [13].
3 Restoration methods of the distorted image
For an effective energy center coordinates definition, the following known meth-
ods for distorted RM image restoration can be used [17, 14, 18]:
• inverse filtering
_ N (u, v)
F (u, v) = F (u, v) + (3)
H(u, v)
• optimal Wiener filtration
2
_ 1 |H(u, v)|
F (u, v) = .G(u, v) (4)
H(u, v) |H(u, v)|2 + Sn (u,v)
Sf (u,v)
• smoothing functional method (Tikhonov method)
H ∗ (u, v)
_
F (u, v) = .G(u, v) (5)
|H(u, v)|2 + γ|P (u, v)|2
Title Suppressed Due to Excessive Length 3
• Lucy-Richardson method
!
_ _ g(x, y)
f k+1 (x, y) = f k (x, y) h(−x, −y) ∗ _
(6)
h(x, y) ∗ f k (x, y)
_
where F (u, v) is a Fourier transform (FT) of the original RM image; N (u, v) is a
FT of random value of noise; H(u, v) is a FT of a distorting operator; G(u, v) is
a FT of RM image; Sn (u, v) is a noise energy spectrum n(x, y); XEC is an energy
spectrum of the original image f(x,y); XEC is a FT of the Laplacian operator;
Sf (u, v) - is a regularization parameter; f, g, h are F, G, H function in the spatial
domain, respectively.
The methods mentioned above are based on an a priory defined distortion
operator H(u, v), however, when real RM images are processed, an exact point
spread function (PSF) is not known or known approximately as a result of image
analysis by distinct fragments [15]. For RM image processing in case the PSF is
unknown a range of blind deconvolution methods can be applied [19].
Restoration of distorted images with the methods considered doesn’t prevent
emergence of edge effects in restored images (e.g., false waves effect, Gibbs effect),
which require additional solutions to eliminate them. The majority of distorted
RM image reconstruction methods are not adequate to physical essence of the
smear phenomenon and don’t take into account motion speed and RM position
up to a fraction of a pixel. In the present work we consider a mathematical
modelling of the distortion function, which allows to understand the essence
of distortion effect, which is necessary for estimation of RM image restoration
algorithms, when image coordinates are defined.
4 Computer model of the reference mark image
To develop and study an image model of the RM as a light source, a mathematical
model has been created, its general structure is shown in Fig. 1. To set the RM
spatial position on the image with a precision up to a hundredth of a pixel, it
is necessary to apply the shift property of the Fourier transform to the original
object form. To do this, the original spectrum must by multiplied by the phase
component with shift parameters x0 , y0 . The transfer function for RM spatial
position setting can be defined as:
Hd (u, v) = e−i2π(ux0 +vy0 ) (7)
Results of accuracy estimation of RM position setting obtained with weighted
summation using Matlab are presented in table 1. It should be understood, that
when this method is used, RM image mustn’t be converted in a binary image
(this procedure decreases measurement accuracy). From the results in the table
above we see that accuracy of setting RM shifts using transfer function H(u, v)
(7) provides an error less than a hundredth of sensor array pixel. The transfer
4 Pham N.T et al.
Fig. 1. Basic information transformations in the mathematical model of RM image
formation in the OES RMPC
Table 1. Results of accuracy estimation of RM position setting obtained with weighted
summation.
Actual RM image shift, px Average measured shift, px
0.1 0.1094
0.01 0.0117
0.001 0.0016
function of the optical system, in turn, can be found through FT of function [8]:
+∞ Z
Z +∞
1 − x22 − y22 −i2π(ux+uv)
Hop (u, v) = e a e b e dxdy (8)
2πab
−∞ −∞
When RM image f(x,y) moves relative to OES RMPC according to functions
x0 (t) and y0 (t) (along spatial axes x and y respectively), the total exposure when
RM image is acquired will be defined as time integral of instant exposure T. In
this case, the image smear model transfer function can be found through FT as:
T
Hl (u, v) = sin (π(ua + vb)) e−iπ(ua+vb) (9)
π(ua + vb)
where a, b - are RM image shifts during exposure time T. Besides the functions
shown above, RM image formation model also considers the following parame-
ters: RM type, its shape and size (see Fig. 2a), background radiation distribution
(see Fig. 2b), observation conditions (see Fig. 2c), properties of a detector of op-
tical radiation.
Based on proposed elements of generalized OES RMPC imitation model a
synthesis function of RM digital image is obtained [16]:
Title Suppressed Due to Excessive Length 5
Fig. 2. Examples of an object (a), background radiation (b) and radiation attenuation
by the air path (c), generated by the developed model.
Φ−1
n o
F (x, y) = f (x, y)M ∗ [Hop (u, v).Hc (u, v).Hd (u, v)] + ηb (x, y) .
(10)
.ηw (x, y) + ηe (x, y)
where f (x, y) - the function of the object form; M - is the function of the
object scaling taking into account the distance H and focal distance f’ of the
optical system; Hd (u, v) - transfer function for determining the spatial position
of an object in an image with the translational property of the time delay of
the Fourier transform, which ensures the spatial position of the object with
the highest possible accuracy; Hop (u, v) - is the weight function of the optical
system; Hc (u, v) - transfer function image blur; Φ−1 – Inverse Fourier transform;
ηb (x, y) – the function of forming the distribution of background radiation, taking
into account the influence of the average ambient temperature; ηw (x, y)–is the
forming function of the background irradiation, ηe (x, y) - the receiver noises OES
RMPC.
Based on the resulting function using Matlab and a library IPT (Image Pro-
cessing Toolbox) [20] a program of RM image synthesis in OES RMPC was
written. Modeling results are illustrated in Fig. 3.
5 Performance analysis of restoration methods of
distorted RM images synthesized by the computer
model
Modeling of the original RM position at 20 px smear length with RM image
coordinates (319.15 px, 239.92 px) has demonstrated that the error of RM image
coordinates definition for restored images is 0.0118 px for Wiener method, 0.0136
px for Tikhonov method, 0.0148 px for Lucy-Richardson method, and 0.0149 px
for blind deconvolution method.
Results of applying restoration methods (3) – (6) of RM images are illustrated
in Fig. 4.
6 Pham N.T et al.
Fig. 3. RM image synthesized by a computer model in Matlab: a) an ideal RM image
without noise; b) RM image distorted by noise and background; c) an image exposed
to a smear caused by relative motion of OES RMPC and RM.
Fig. 4. Results of RM image coordinates definition with Matlab: a) coordinates of the
smeared RM image energy center is (320.05 px, 240.57 px); RM image energy center
coordinates in the image restored by b) inverse filtration are not defined; c) Wiener
parametric filtration are (319.17 px; 239.93 px); d) Tikhonov method are (319.15 px;
239.93 px); e) Lucy-Richardson method are (319.13 px; 240.52 px); f) blind deconvo-
lution are (319.12 px; 240.52 px).
Title Suppressed Due to Excessive Length 7
Thus, application of different methods for RM image restoration has demon-
strated that:
- inverse filtration (Fig. 3b) demonstrates significant distortions and doesn’t
allow to define energy center coordinates after image restoration;
- applying of Wiener parametric filtration (Fig. 3c) and the method of smooth-
ing functional minimization (Tikhonov regularization) (Fig 3d) for image restora-
tion has demonstrated the best results and advisability of their use in OES
RMPC. Besides this, the developed computer model can be used for estimation
of energy center shift evaluation, with irradiance in the receiver plane consid-
ered, as well as for optimization of OES RMPC parameters in accordance to
operational conditions.
6 Conclusion
In the paper it is proven that the shift transfer function (7) can set a RM image
position with the precision up to a hundredth of a pixel. Mathematical descrip-
tion of RM distorted image, with RM motion relative to the system considered,
is developed.
It is discovered that applying of Wiener parametric filtration and the method
of smoothing functional minimization (Tikhonov regularization) for image restora-
tion demonstrate the best restoration results and can be recommended for use
in the information processing system in OES RMPC.
Efficiency of applying the developed computer model to estimation of dis-
torted RM image restoration methods is proven.
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