=Paper=
{{Paper
|id=Vol-2391/paper10
|storemode=property
|title=Optimal filtering of multidimensional random fields generated by autoregressions with multiple roots of characteristic equations
|pdfUrl=https://ceur-ws.org/Vol-2391/paper10.pdf
|volume=Vol-2391
|authors=Nikita Andriyanov,Konstantin Vasiliev
}}
==Optimal filtering of multidimensional random fields generated by autoregressions with multiple roots of characteristic equations ==
https://ceur-ws.org/Vol-2391/paper10.pdf
Optimal filtering of multidimensional random fields generated
by autoregressions with multiple roots of characteristic
equations
N A Andriyanov1,2, K K Vasiliev1
1
Ulyanovsk State Technical University, Severny Venets, 32, Ulyanovsk, Russia, 432027
2
Ulyanovsk Civil Aviation Institute, Mozhaiskogo, 8/8, Ulyanovsk, Russia, 432071
e-mail: nikita-and-nov@mail.ru
Abstract. The use of mathematical models allows to compare the theoretical expressions and
simulation results. Autoregressive random fields can be used for description of the images,
however, such models have pronounced anisotropy, and the simulated images are too sharp.
The elimination of this drawback is possible through the use of models with multiple roots of
characteristic equations. The analysis shows that using models with multiple roots in filtering
images with smoothly varying brightness provides smaller errors than the use of autoregressive
random fields. However, studies of the dependences of filtering efficiency on various model
parameters and signal-to-noise ratios for multidimensional autoregressive random fields were
almost not carried out. The article discusses the solution of the problem of optimal filtering of
images based on models with multiple roots of characteristic equations. Theoretical
dependences of the relative variance of the filtering error on the dimension of random fields are
obtained. Furthermore, it was presented some results of filtering real images by such model in
comparison with autoregressive model.
1. Introduction
Currently there are many different mathematical models of random fields (RF) using for describing
images [1-5]. The popularity of this approach is due to a number of advantages that mathematical
models provide. First of all, it is the generation of sufficiently large volumes of material for research,
and also mathematical models act as a tool for developing and testing various algorithms. The simplest
autoregressive models generate RF with pronounced anisotropy and such models are suitable for
describing only a narrow class of real multidimensional images. Doubly stochastic models [6,7]
provide a change in the probabilistic properties of the generated RF at each point, but on average the
properties of such a model depend on the model chosen for the main RF simulation. Therefore, to
obtain RFs that are close to isotropic fragments of multispectral images, it is necessary to use
autoregression with multiple roots of characteristic equations [8–10].
However, one of the main tasks of signal processing is the noise reducing or filtering. It is often
considered that the observed signal is an additive mixture of the information (useful) signal and white
noise. In this paper we analyze the efficiency of spatial Wiener filtering of multidimensional
autoregressive RFs with multiple roots of characteristic equations against additive white Gaussian
noise background. At the same time the investigation is aimed at such models of different
multiplicities, which provide equivalent correlation properties. The developed filtering algorithms can
become very useful tool in solving various applied problems of image processing, among which an
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019)
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important place is occupied by the detection and localization of various objects in the image [11,12].
Furthermore filtering and segmentation tasks are of interesting [13,14].
2. Model of a multidimensional random field and its linear filter
The following equations are commonly used to describe a multidimensional autoregressive Gaussian
RF
xi = ∑ α j xi − j + σ x β 0ξ i , i ∈ Ω , (1)
j ∈D
where =X { x i , i ∈ Ω} is simulated RF defined on N-dimensional grid
Ω
= { i= ( i , i , ... i ) : i= 1 M , k= 1 N } ;
1 2 N k k
{β 0 , α j , j ∈ D} are coefficients of the model;
{ξ i , i ∈ Ω} is RF of random values with Gaussian distribution having zero mathematical expectation,
and its variance is equal to one; σ x2 is variance of RF
x i ; D ⊂ Ω is causal region of local states.
For such a model it is easy to find the transfer function of a linear filter. Using Z-transformation for
model (1) it is possible to get a spatial linear filter, which is described by the transfer function of the
following form
σ x β0
H (z) = , (2)
1 − ∑α j z − j
j∈D
−j − j1 − j2 − jN
where z = z z ...z . 1 2 N
It should be noted that the transfer function (2) also depends on the parameters of the signal model,
as does the energy spectrum of such a RF. The relationship of the transfer function (2) and the energy
spectrum of the RF X is determined by the expression
Sx ( z ) = H ( z ) H z ( ). −1
(3)
The analysis of probabilistic properties of the RF isN simplified if the transfer function of a
multidimensional filter can be factorized: H (z ) = ∏ H k (z k ) . Then the energy spectrum
k =1
N N
S x ( z ) = ∏ S k ( z k ) and correlation function (CF) B(r ) = ∏ Bk (rk ) are also can be factorized. Simple
k =1 k =1
and very useful for applications multidimensional splittable RF xi can be represented using spatial
autoregression
N
∏ (1 − ρ z ) x = σ β ξ , i ∈ Ω ,
k =1
k
−1 mk
k i x 0 i
(4)
with multiple roots ρ k of characteristic equations having multiplicities mk , k = 1, 2,..., N .
The transfer function of such a RF will be factorizable and will be written as
N
H (z) = σ x β0 ∏ (1 − ρ z )
k =1
k
−1 mk
k , (5)
mk −1
∑ (C
N
where β 0 = ∏ β k ; β k ( mk )= (1 − ρ k2 ) 2 mk −1 / l
mk −1
ρ kl ) 2 , С j = j! (i!( j − i )!) .
i
k =1 l =0
3. Filtering efficiency of multidimensional random fields with multiple roots of characteristic
equations
One of the difficult tasks of filtering image sequences on multidimensional grids is the analysis of the
effectiveness of such filtering. In this case, the necessary criterion for analysis is the dependence of the
variance of the filtering error on various model parameters and noise. Formally, spatial covariance
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matrices of estimation errors can be calculated using the recurrence relations for the Kalman filter
[6,7]. However, if it is necessary to compare the algorithms for different values of the parameters of
the stochastic equations and noise levels, the determination of even steady-state values of the elements
of the covariance matrices becomes a very laborious task.
Consider a relatively simple way to determine the effectiveness of estimating homogeneous fields
on infinite grids based on the basic principles of Wiener's filter theory [8]. Using the observations
z= xj + nj , =
j ( j1 j2 ... j N ) ∈ Ω, which are the sum of informational (useful) RF and additive
T
j
white Gausian noise with a variance σ = M n j
2
{ } it is necessary to make the best (in the sense of the
2
minimum error variance) linear estimate xˆ 0 = ∑ h j z j of element x 0 in informational RF. This
j∈Ω
estimation will use coefficients h j which will determine the optimal filtering. The search of minimum
σ ε = M {(xˆ 0 − x0 ) }
2
= M ∑ h j z j − x0 can be written as a system of linear
2 2
error variance
j∈Ω
equations
hq σ +
2
∑ h B=
( r − j ) B ( r ),
j
r ∈Ω, (6)
j ∈Ω
which can be considered as a spatial analogue of the Wiener-Hopf equations.
Using multidimensional z − transformation it is possible to find equations system solution and
expression for the relative error variance [8]:
π π
σ ε2 1 β 02
σ ч2 (2π ) N −∫π −∫π N
= ... dλ . (7)
∏ (1 + ρ − 2 ρ cos λ ) + qβ
2 mk 2
k k k 0
k =1
where q = σ x σ
2 2
is signal-to-noise ratio, N is the dimension of RF, mk is the model’s multiplicity
for k-th dimension, ρ k is correlation parameter in k-th dimension.
a) b)
Figure 1. Relative variances of errors of multidimensional RF.
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 74
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Figure 1 shows the dependence of the relative error of the filter variance on the correlation interval
k0 for models of different dimensions and orders with q=0.1. Correlation interval is less than 100 in
figure 1a and less than 500 in figure 1b. This interval determines the equivalence of models of
different multiplicities. On the graphs, the dimension is indicated as N, the multiplicity as m.
The analysis of the obtained dependences shows that with sufficiently small correlation intervals
(k0 < 10) the variances of filtering errors of autoregressive RF of the 1st and 2nd orders are rather
close. An increase in the dimensions and a further increase in the correlation interval leads to the
distancing of the curves. At the same time the smallest values of relative variances of filtering errors
are obtained for the cases N=3, m=(2,2,2), N=4, m=(2,2,2,2). This is because when m=1
autoregression along the axes are quite prickly and their filtering is a more difficult task.
Using expression (7), one can obtain the following equations for models of various dimensions,
presented in Table 1.
Table 1. Filtering efficiency of RFs of different dimensions.
σ ε2
N σ x2
1 π β 02
1 ∫ dλ1
2π −π (1 + ρ12 − 2 ρ1 cosλ1 ) m + qβ 02 1
1 π π β 2
∫ ∫ 0
dλ1dλ2
2 ( 2π ) π π (1 + ρ − 2 ρ cosλ ) (1 + ρ − 2 ρ cosλ )
2 − −
1
2
1 1
m1 2
2 2 2
m2
+ qβ 02
1 π π π β 02
3 3 ∫−π ∫−π ∫−π
dλ1dλ2 dλ3
( 2π ) (1 + ρ1 − 2 ρ1 cosλ1 ) (1 + ρ 2 − 2 ρ 2 cosλ2 ) m (1 + ρ 32 − 2 ρ 3 cosλ3 ) m + qβ 02
2 m 12 2 3
1 π π π π β 2
4 4 ∫−π ∫−π ∫−π ∫−π
0
dλ1 dλ2 dλ3 dλ4
(2π ) (1 + ρ1 − 2 ρ1 cosλ1 ) (1 + ρ 2 − 2 ρ 2 cosλ2 ) (1 + ρ 32 − 2 ρ 3 cosλ3 ) m (1 + ρ 42 − 2 ρ 4 cosλ4 ) m + qβ 02
2 1m 2 m2 3 4
Figure 2. Relative variances of filtering errors for three-dimensional RF.
Figure 2 shows the dependences of the relative error of filtering variance on the signal-to-noise
ratio q for models of three-dimensional RF of different orders with k0 =50.
An analysis of the curves in the graph shows that in the case of a large dimension of the RF, for
example N = 3, the variance of the filtration error is rather small. In this connection, effective filtering
is obtained both for small signal-to-noise ratios and for large ones. At the same time, an increase in the
multiplicity of models leads to a decrease in the relative dispersion of the filtering error.
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 75
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Figure 3 shows the dependences of the relative variance of filtering errors on the dimension of the
AR for the cases k0=100, q=0.01 with multiplicities m=1 and m=2.
Figure 3. Dependence of the relative error variance on the dimension.
a) b)
c) d)
Figure 4. Filtering of satellite images.
The analysis of the curves presented in Figure 3 shows that increasing the dimension of the RF
leads to a significant increase in filtration efficiency, which is associated with a large number of
correlations in the multidimensional model. At the same time large dimensions provide variance of
filtering errors tending to 0 (~10-15) already with multiplicities m=2 along each axis. At the same time
if m=1 then the variance of the filtering error is several orders of magnitude greater.
4. Real image processing
The filtering algorithm based on a multiple-root model was tested on a multidimensional satellite
image compared to an algorithm based on autoregressive models. Figure 4 shows the filtering results
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 76
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for one of the images. Figure 4a shows the source image, figure 4b shows the noisy image, figure 4c
shows the filtering results using autoregressive model of the first order and figure 4d shows the
filtering results using autoregressions with multiple roots model.
The analysis of the presented pictures shows the model with multiple roots provides better results
in variance of filtering error, for example, the results for image on figure 4 is following: relative error
variance for figure 4c is 0.782, error variance for figure 4d is 0.358. The signal-to-noise ratio is 0.5.
5. Conclusion
Thus, in this paper, the filtration efficiency of multidimensional RF with multiple roots of
characteristic equations is investigated. At the same time, an increase in the dimensions and orders of
the models leads to a significant decrease in the relative dispersion of filtering error. Therefore, it is
advisable to use less computationally sophisticated mathematical models of RFs that provide fairly
small errors. For example, already for the dimension N=3 it is possible to achieve relative error equal
10-5 for q=0.01 and multiplicities m=(3,3,3). In addition, studies have been conducted on the
effectiveness of filtration depending on the dimension of the RF. It should be noted that in the
logarithmic axes, these dependencies are close to linear for the dimensions N=1,...,4. Such models are
also useful in processing real images having strong correlation properties.
6. References
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Acknowledgement
The study was supported by RFBR, Project №17-01-00179.
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