An impact of evaluation error of reference mark image energy 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 analyzed. 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 ltration and Tikhonov regularization are the most e ective.
When a railway track is built or repaired with modern high-performance track
machines, de nition of the actual railway track position and estimation of the
result with technical means are important aspects for improvement of an operation
of railway track placing in a required position [
Impact of evaluation error of RM image energy center
on OES RMPC accuracy
In the category of systems we are considering [
XEC = 4 2 M N
X X (xi;j Qi;j )5 i=1 j=1 3,2 M N 3 4X X Qi;j 5
i=1 j=1 YEC = 4 2 M N
X X (yi;j Qi;j )5 i=1 j=1 3,2 M N 3 4X X Qi;j 5 i=1 j=1 1 :G(u; v) (1) (2) (3) (4) 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 in uence generally consists in using
specialized 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 [
Restoration methods of the distorted image
For an e ective energy center coordinates de nition, the following known
methods for distorted RM image restoration can be used [
N (u; v) H(u; v) optimal Wiener ltration
0 _ 1 jH(u; v)j2
F (u; v) = @ H(u; v) jH(u; v)j2 + SSnf((uu;;vv)) A :G(u; v) smoothing functional method (Tikhonov method) _ F (u; v) =
H (u; v) jH(u; v)j2 + jP (u; v)j2 Lucy-Richardson method _ _ f k+1(x; y) = f k(x; y) h( x; y) g(x; y)
_ h(x; y) f k(x; y) ! (6) _ 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 de ned 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 [
Restoration of distorted images with the methods considered doesn't prevent emergence of edge e ects in restored images (e.g., false waves e ect, Gibbs e ect), 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 e ect, which is necessary for estimation of RM image restoration algorithms, when image coordinates are de ned. 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 de ned as:
Hd (u; v) = e i2 (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
function of the optical system, in turn, can be found through FT of function [
1 2 ab
x2 y2 e a2 e b2 e i2 (ux+uv)dxdy (8)
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 de ned as time integral of instant exposure T. In this case, the image smear model transfer function can be found through FT as: Hl (u; v) =
T (ua + vb) sin ( (ua + vb)) e i (ua+vb) (9) where a, b - are RM image shifts during exposure time T. Besides the functions shown above, RM image formation model also considers the following parameters: 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 optical radiation.
Based on proposed elements of generalized OES RMPC imitation model a
synthesis function of RM digital image is obtained [
F (x; y) = nf (x; y)M [Hop(u; v):Hc(u; v):Hd(u; v)] 1
+ b(x; y)o : : w(x; y) + e(x; y) (10) 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; 1 { Inverse Fourier transform; b(x; y) { the function of forming the distribution of background radiation, taking into account the in uence of the average ambient temperature; w(x; y){is the forming function of the background irradiation, e(x; y) - the receiver noises OES RMPC.
Based on the resulting function using Matlab and a library IPT (Image
Processing Toolbox) [
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 de nition 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.
Thus, application of di erent methods for RM image restoration has demonstrated that:
- inverse ltration (Fig. 3b) demonstrates signi cant distortions and doesn't allow to de ne energy center coordinates after image restoration; - applying of Wiener parametric ltration (Fig. 3c) and the method of smoothing functional minimization (Tikhonov regularization) (Fig 3d) for image restoration 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 considered, 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 description of RM distorted image, with RM motion relative to the system considered, is developed.
It is discovered that applying of Wiener parametric ltration and the method of smoothing functional minimization (Tikhonov regularization) for image restoration demonstrate the best restoration results and can be recommended for use in the information processing system in OES RMPC.
E ciency of applying the developed computer model to estimation of distorted RM image restoration methods is proven.