The paper is devoted to ltering algorithms of satellite images. Inadvisability of applying the simplest mathematical models of random elds with non-uniform ltering material is shown. We consider the comparative analysis of e ectiveness of the ltering and calculate the gain of the proposed algorithm. In addition, we have su cient by adequate enough satellite image restore when applying doubly stochastic models. Restoration algorithm that can easily be implemented from different positions of the image is described. Dispersion values for recovery errors were found under using di erent models. We also have received the gain in image restoration by providing adequate description of satellite images unlike in application of autoregression (AR) models.
In many cases, transfer of multidimensional data with errors shadowing images or badly damaged them by noise arises the problem of recovering the missing fragments of images [1{3, 8] or ltering [4{6].
The white noise ltering is possible in the case of using the well-proven Kalman lter, which allows one of the reasonably accurate estimation without requiring signi cant computing expenditures.
One of the methods of restoration, essentially is in an image replacement by some model in the damaged area. However, in real-world images the damaged area can contain any objects, description of which is possible using inhomogeneous models. Therefore, to use this method, we must nd adequate model. Most of the existing models [2, 7] are unable to provide adequate replacement of damaged areas due to some reasons. However, we can use the combination-mixed models of the images.
Quite a common option of such models is doubly stochastic ones [3, 5, 9] or models, which vary its parameters from pixel to pixel.
Another important feature of the ltering and restoration results is the need for their use in solving problems of signal detection in images [10, 11].
Thus, the purpose of this work is to improve e ectiveness of the image ltering and restoration by applying models of images with varying parameters. Note that a comparison will be made on the criterion of minimum error dispersion. 2
Although signal detection is very important, e ectiveness of the work of all algorithms signi cantly depends on the source material, and usually images are distorted versions of the raw data. So, received images may have di erent shifts, shading, as well as, be quite noisy nuisance. Moreover, strong interference leads to almost total loss of information at the site of exposure. Therefore, the most important stage of preprocessing is ltration.
We consider the following doubly stochastic model of random elds:
xi;j = xi;j xi 1;j + yi;j xi;j 1
xi;j yi;j xi 1;j 1 + i;j ;
(
The random elds that describe changes in the correlation coe cients are
described as follows:
~xi;j = r1x ~xi 1;j + r2x ~xi;j 1
~yi;j = r1y ~yi 1;j + r2y ~yi;j 1
r1xr2x ~xi 1;j 1 + ξj ;
r1yr2y ~yi 1;j 1 + &yi;j ;
(
Thus, in order to solve the problem of parameter estimation, it is necessary
to estimate random elds xi;j and yi;j in model (
Suppose that the input signal of a monitoring system at the entrance is the
sum of the useful signal (
We shall use the vector nonlinear Kalman lter to make the ltering process
of the at image. Therefore, we must obtain a vector of the image line items
that can be written as follows:
(
In this case, we write a generalized expression model for the at image in accordance in the following form: xi = diag ( xi ) xi 1 + # ( xi ; yi ) i; xi = r1x xi 1 + # x xi; yi = r1y yi 1 + # y yi; where diag ( xi) is the diagonal matrix with elements xi1 ; xi2 ; : : : ; xiN .
Finally, expression for the process of line-by-line estimation is written as follows: x^epi) :
It should be noted that application of the nonlinear vector Kalman lter is possible if the signal model is known. Thus, to make signal model known, it is necessary to have information about coe cients r1x; r2x; r1y, and r2y and, in addition, the statistical characteristics of the model such as m x; m y, and 2x; 2y; x2. If the receiving part has no a priori information tagged with parameters, we must perform a preliminary assessment, for which it is proposed to use the algorithm for pseudogradient search [3, 7].
In addition to lter (
1) Vector Kalman lter for the AR model with xi;j =const and yi;j =const. 2) Wiener lter for AR model with the covariance function
B (l; k) =
x2mjlxjmjkyj: 3) Vector Kalman lter for the AR model with reverse swing (interpolation). 4) Vector Kalman lter for the doubly stochastic model with reverse swing (interpolation), for which 0 x^i;j 1
(xe; xe; ye; Vn) :
Figure 1 presents ltering error dispersion dependencies on noise dispersion. We can see that if the image is close to identical and similar only in certain segments, we have e ective ltration by only the fourth algorithm.
As for the gain (Figure 2), note that the maximum gain is achieved at low values of the noise dispersion, and then we get the stabilization of the gains. We note that for a single noise dispersion, algorithm of the Kalman lter with vector interpolation works almost 2 times more precisely than the Kalman and Wiener lters with interpolation con gured for the AR model. Furthermore, the gain compared to Kalman without interpolation is much larger.
Obviously, the heterogeneous image ltering with variations in brightness (Figure 3) can not be used without analysis of correlation parameters. So, here, there is the fourth algorithm that is also applied more accurately. However, in terms of gains (Figure 4), such algorithm provides smaller indicators since the main image has the heterogeneity of the sharp variations in brightness.
Thus, ltering algorithm based on the doubly stochastic model provides the best results for real images. 3
Consider restoring the square area on the image using a model with variable parameters. Let the brightness values of the image representing the random eld be fZi;j ;i = 1; 2,: : : , M1;j = 1; 2,: : : , M2g. There is the damaged area starting at the point (i0; j0) of the c c-dimension. Denote this area as D. We introduce the following restoration model [12] where 1i;j ; 2i;j are the estimations of the correlation coe cients for row and column at the point (i; j); Xi;j is the estimation of the average value at the point (i; j); i;j is the Gaussian random eld with average M f i;j g = 0 and dispersion 2 = i;j q 1 21i;j )(1 22i;j , where i2;j is the estimation of the dispersion at the point (i; j).
It is advisable to assess parameters in the sliding window excluding the points
that lay in the damaged area. Model (
u= N2 P N2 i2;j = N12 P N2 q= N (Zi+u;j+q u= N2 P N2
2 1i;j = i2;j1N2 PuN2= N2 PqN2= N2 (Zi+u;j+q i2;j1N2 PuN2= N2 PqN2= N2 (Zi+u;j+q
Xi;j )2
Xi;j ) (Zi+u 1;j+q Xi;j ) (Zi+u;j+q 1
Xi;j ) Xi;j )
To restore the damaged area, we shall make an assessment in neighbourhood
of the area D. In addition, to ensure greater heterogeneity, we divide area D
into sub-areas, onto each of which we shall expand the model with the estimated
parameters (
Evaluation system (
To assess the performance of the proposed algorithm, we shall implement
restoration method on the di erent images. When we do this we compare
restoration (
So, the restoration algorithm with the Habibie model (
Figures 5-7 show the di erent damaged images and the result of their recovery: a) damaged image, b) restore from the upper left corner, c) restore from a right corner, d) restore from the left bottom corner, e) restore from a right corner, f) restore based on the Habibie model from the upper-left corner.
Dispersions of the restoration error (Figure 5) are the following: b: 0.046, c: 0.060, d: 0.045, e: 0.043, f: 0.335.
Dispersions of the restoration error (Figure 6) are the following: b: 0.031, c: 0.040, d: 0.035, e: 0.034, f: 0.043.
Analysis of the errors for restoration in Figure 6 shows that in the case of a homogeneous area of the image, results in the implementation of algorithms are close enough regardless the initial point. However, slight variation of the dispersion values of the error may be due to the fact that the implementation of a model uses a random eld. Consequently, the value of the restored pixel brightness is accidental. It also should be noted that the image selected in Figure 6 consists the inhomogeneities. Such structure also a ected the calculation of variance of the restoration error. However, the restoration results in Figure 6b | e are signi cantly better than restoration results in Figure 6f. Firstly, it is due to the fact that the model with variable parameters is better suited for the description of the original image. Secondly, implementation of the Habibie model leads to using the constant correlation coe cients, although the connection between the real image pixel does not correspond to this description. Finally, we consider the restoration of the image area when the neighborhood is di erent from di erent sides, i:e:, there are either diverse objects at the corners of the damaged section or there is a di erence in the brightness values. Dispersions of the restoration error (Figure 7) are the following: b: 0.005, c: 0.003, d: 0.006, e: 0.008, f: 0.015.
An error dispersion investigation for the Figure 7 revealed that when the damaged area is bounded on di erent sides of the pixels with di erent brightness, it is a very important factor what side is basic for restoration. Indeed, restoration from the right upper corner of the neighborhood is based on the dark area closest to the brightness of the damaged portion. Further, the order of similarity of the neighborhood brightness coincides exactly with the order of increasing error variance. Therefore, when the brightness of the vicinity parts is closer to the brightness of the damaged area, we have the better restoration results. Thus, for Figure 7, restoration algorithm using the Habibie model is considerably inferior to the algorithm based on the use of complex (doubly stochastic) models. This is primarily due to the heterogeneity of the original image.
The size of the damage is c = 40 for all images. The images sizes are the following: Figure 5 290 290, Figure 6 330 330, Figure 7 440 440. Dispersions of the error were calculated from relations for dispersion of the images.
The analysis shows that restoring (
It should be noted that application of the doubly stochastic model allows one to have information on the undamaged neighborhoods in the form of parameter elds that provides better restoration. Analysis of the results shows that the e ciency of restoring depends on the starting position of the model implementation that makes it possible to increase the e ciency of the restoration of the damaged area due to splitting into the smaller subareas. For these subareas, we choose the best neighborhood, in which estimation is made.
Thus, considered restoring algorithm based on the use of models with varying parameters unlike the Habibie model restoration is able to restore the heterogeneity areas and is generally superior to the latter. 4
Comparative analysis of four algorithms of ltering was described in details. Researches were performed for the di erent images.
It was found that the gain of the vector Kalman lter stabilizes with noise dispersion increasing.
The vector Kalman lter for the doubly stochastic models with reverse swing provides the signi cant gain (40 | 50%) for the actual images that cannot be adequately described by the AR models of random elds.
The algorithm of restoring damaged areas on images based on mathematical modeling was suggested. Analysis of the results obtained shows that to restore satellite images, it is appropriate to use models with varying parameters.
Improvement of e ciency of the algorithm in future can be obtained by aggregation of the restoration results obtained for di erent directions. Acknowledgements. This work was partly supported by the Russian Education Ministry, project Goszadanie no. 2014/232.