The article describes probabilistic properties of autoregressive models with multiple roots of characteristic equations as well as the results of such properties investigation. Particular attention is paid to the study of covariance functions. However, we investigate only such models of different orders that provides equal intervals of correlation in row and column. So the multiplicity of such models by row and column is the same. Asymptotic graphs of covariance functions cross-sections are constructed. It is shown that the cross-sections tend to ellipses when model orders are improving. We also describe in detail the problem of filtering images generated by autoregressions with multiple roots. Dependences of variances of filtering errors on the correlation parameter are obtained. The efficiency of filtering at various signal noise ratios and various orders of models is investigated. The effectiveness and expediency of applying autoregressions with multiple roots are shown in comparison with other autoregressions of high orders.
A lot of real images is characterized by a smooth change in brightness. It means that real images have rather strong correlation links in some neighborhood. The known autoregressive (AR) image models [1-3] either do not adequately describe the nature of such images even at high values of the correlation coefficient in case of first order model.
However, the use of higher order models requires complex operations to calculate the set of correlation parameters. Meanwhile, there are models of AR random field (RF), that generated by the characteristic equations in image processing and representation literature [4-6,12]. Such models are called AR with multiple roots. In this case, the application of algorithms based on such models, when processing real signals and in various applied problems, can contribute to improving the efficiency of solving such problems. In recent years, particular interest is caused by the processing of satellite images [13-15].
A lot of tasks connecting with preliminary processing are known, but the important task is to suppress noise or filter images [7,8]. The advantage of models with multiple roots is the possibility of applying recurrent filtration procedures [9]. The article deals with the problem of image modeling on the basis of such models as well as problems of analysis of probability properties and optimal filtering of random fields generated by such models. 2. Autoregressions with multiple roots of characteristic equations The widely known class of image models is the AR model of RF [1-3]. One of the main reasons for the spread of AR models is the efficient mathematical apparatus developed for modeling random sequences. The AR class of RF models can be obtained on the basis of linear stochastic difference equations of the following form: xi j xi j i i ,
jD modeled RF defined where
X {xi , i }
is {i (i1, i2 ,...,iN ) : {ik 1...M k }, k 1...N}; { j , , j D} are coefficients of the model; { , i } is a random variable with Gaussian distribution; D is a causal region of local states. i
The choice of a normally distributed RF with independent components is convenient and simple enough to describe some brightness properties of the image by the model. In this case, the RF X will also obey the Gaussian distribution. As an example, let us consider the formation of a two-dimensional RF using of the first order AR model known also as Habibie model: on the
N-size xi, j x xi1, j y xi, j1 x y xi1, j1 i, j ,i 1...M1; j 1...M 2 , where x and y are coefficients of correlation of neighboring elements in row and column, respectively; { i, j } is a two-dimensional field of independent Gaussian random variables with zero mean M{ i, j } 0 and variance 2 M{ i2,j } (1 x2 )(1 y2 ) x2 ; x2 M{xi2, j }; M M 2 is 1 the size of the image that we imitate.
Due to the specifics of this way to generate the image, the process of estimating the parameters of the AR model can be performed without special difficulties. Nevertheless, such models are not good enough for describing real images, since they provide only small regions of local states. And this is an important factor in modeling multi-zone images. Therefore, if it is necessary to use the AR equation, we must increase the order of the model for the expansion of local states. So this fact entails an inevitable growth of computational costs. Sometimes the image can be so difficult that it is required to perform imitation in a class other than AR. Otherwise, to achieve adequate replacement of the image is unlikely to succeed.
In addition, the weakness of such models is that they are not suitable for an adequate description of
isotropic RFs, for example, with a covariation function (CF) of the form
R(k) R(k k12 k22 ... kN2 ) . To overcome this drawback, one can use the characteristic equations
[10]. In work [6] models based on the generalization of one-dimensional ARs to the multidimensional
case were proposed. For example, if we take an AR with multiple roots of characteristic equations of
the second order
xi 2xi1 2xi2 i ,
(
It can be noted that the model (
Figure 1(a) shows an image formed from the values of RF (
(а) multiplicity (2;2) (b) multiplicity (3;3)
Figure 1. Implementation of models having different multiplicities.
Thus, using a model with multiple roots of characteristic equations when imitating images, it is possible to simply obtain such implementations of RF that will be close to real images. In this case, an important property of the generated RF will be their quasi-isotropy property. A general formula for models of different multiplicities can be written in the form:
N1 N2 xi, j i, j i1, j1 xii1, j j1 i10 j10 the corresponding coefficients of one-dimensional AR along the axes x and y :
i1, j1 xi1y j1 .
The coefficients of one-dimensional AP (
1 x y j1 ( y , N2 ) (1) j11CNj12 xj1 n! where Сnm m!(n m)! is number of combinations of n by m ; x , y are parameters of the model.
Finally, the two-dimensional model coefficient is the normalized product of the corresponding coefficients of one-dimensional ARs along the axes x and y :
x x y . (
x N11C i i 2
N11 x , y 1 2 2N2 1
y N21C i i 2
N21 y
.
i0 i0
So we can use expressions (
Thus, it is possible to construct models of arbitrary multiplicity, obtaining images with slow varying brightness properties. It should be noted that speed of properties change will depend on parameters and orders of models. 3. Covariance functions of autoregressive random fields with multiple roots In order to construct AR models of images with multiple roots of characteristic equations having given statistical characteristics, we perform an analysis of the data of the characteristics of the considered models. An important task arising in the course of statistical analysis of the mathematical model of a n RF is to determine the type of its CF.
An important property of models with multiple roots is the factorizability of CF. For example, the
RF generated by the model (
x generalization of the CF of a one-dimensional first order AR to a two-dimensional case. It can be shown [4] that it is described by the following expression: distances between the elements of the RF X along the axes x and y.
The use of the model (
In order to obtain the CF of models with arbitrary orders, one can use expressions for onedimensional CF of AR with multiple roots of the characteristic equations
m1 2(ml1)
Bx (k) x2 g(m, l, k) ,
l0 (1 2 )2kl1
(
l0
Correspondingly, if the AR multiplicity is (m1,m2) then the expression for the CF can be written as
the product of the RF variance and expressions of the form (
l0 l0
(
The obtained relations completely determine the CF and the parameters of the RFs generated by the AR with multiple roots of the characteristic equations of multiplicity (m1,m2).
For analysis, an interesting case is when the correlation coefficients of the model provide the same correlation intervals on row and column for models of different orders, i.e.
Bm1( x1, y2 , k0 , k0 ) Bm2 ( x2 , y2 , k0 , k0 ) ... Bmn ( xn , yn , k0 , k0 ) ... x . (15) e
For simplicity, we will assume that the multiplicity of the AR for each of the axes is the same, and the parameter is also the same for both axes. Then we can reduce condition (15) to the following form
Bmn ( , k0 ) x .
e
This simplification makes it possible to obtain a set of parameters that ensure the same correlation intervals, analogous to the one-dimensional case [5]. Table 1 presents the dependences between the correlation parameter and correlation interval k0 .
Analysis of the data presented in Table 1 shows that in order to ensure equal correlation interval, it is necessary to decrease the value of the parameter . Furthermore, it is possible to single out the (16) dependence of expression m 2(1 m )k0 . The parameter m tends to 1 if m 1 . Figure 2 shows dependence of parameter m on correlation interval k0 .
Тable 1. Results of numerical calculations of parameters for different correlation intervals. m1 k0 m2 m3 m4
1
Analysis of the curves in Figure 2 shows that the parameter m tends to a constant value for large k0 . At the same time, the greater the multiplicity of the model, then the greater the parameter m .
Figure 3 shows CF of different order models providing a correlation interval k0 10 . Figure 3(a)
shows the model of multiplicity (
Figure 3. CF of the AR RF with multiple roots of the characteristic equations.
Figure 4 shows the CF cross-sections for the models of the 1st order (solid line) and the 2nd order (dashed line) with the correlation interval k0 15 . Figure 5 shows CF cross-sections for models of different orders with the same value of the parameter 0.8615 .
As can be seen from Figure 4, the second order CF model has a "bell-shaped" vertex, and the correlation links between the RF elements generated by such models are stronger within the correlation interval.
Fig. 5 shows that the CF cross-sections tend to ellipsoids with increasing multiplicities AR. However, with a significant increase in the orders, the CF decreases much more slowly than when using the first and second order ARs.
Thus, the correlation properties of AR with multiple roots of the characteristic equations ensuring equal correlation intervals are investigated. Dependences of the correlation parameter of such ARs on the correlation interval are obtained. 4. Covariance functions of autoregressive random fields with multiple roots Let us consider the case when against a background of white noise, it is necessary to perform a filtration of an RF simulated by the AR with multiple roots of the characteristic equations zij xij nij , (17) where white noise has zero mathematical expectation and variance 2 . n
We will use the following simple algorithm for image filtering. At the first stage, we perform line by line filtering of all the elements in each row. Then the same procedure applies to each column. Finally, we get the total estimation of each element as the average between the estimates for the row and the column. So, to filter a string, you can use the following algorithm [5]. We introduce the extended state vector:
xi (xi xi1...xim1)T .
xi xi1 i ,i 1, 2,...,
Then the observation model will be written as:
zi Cxi ni , i 1, 2,..., where C (1 0...0) .
The equation of state of a string is also can be written in the vector-matrix form: (18) (19) (20) where the error) P11i M xi xˆi 2 where Pэi Pi1T V ; xˆэi xˆi1 . m At the every i-th estimation step we build the optimal forecast xˆэi 1 j xˆi j based on previous j1 estimates xˆi j , j 1,2,...,m and we find the best estimate (in the sense of the minimum variance of 1 xˆi xˆэi P11i 2 (zi xˆэi ) n
, where P11i Pэ11i 1 Pэ11i n2 . The remaining components of the vector xˆi are calculated on the basis of interpolation of previous estimates taking into account the next observation zi and the mutual covariance of estimation errors.
Figure 6 shows the variances of filtering errors obtained for models of different multiplicity and for different ratios of signal to noise q x2 2 . It is worth noting that the variance estimate of the filter n error was averaged over five processed images of size 150×150. (21) (21)
Thus, RF models based on AR with multiple roots of characteristic equations are presented. The equality of correlation interval is the interesting property of investigated models. For such models, dependencies between correlation properties and multiplicities are found. An effective quasi-optimal filtering algorithm for such RFs is proposed. It is shown that the use of high-order models makes it possible to obtain gains in the case of filtering in the case of identical correlation intervals. Acknowledgments The study was supported by RFBR, project № 17-01-00179.