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.
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 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]. is xi = ∑α j xi − j +σ xβ 0ξ i , i ∈ Ω , j∈D simulated RF defined on
N-dimensional where
X
={x i , i ∈ Ω} Ω ={ i =( i1 , i2 , ... iN ) : ik =1 M k , k =1 N}; {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; is variance of
RF σ 2 x xi ; 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
H ( z ) = σ β
x 0 1 − ∑α j z − j j∈D , (1) grid (2) (4) (5)
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 ) = ∏ Sk (zk ) 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 k zk−1 ) mk xi = σ xβ 0ξ i , i ∈ Ω , k =1 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 xβ 0 ∏ (1 − ρ k zk−1 ) mk ,
k =1
N mk −1 where β 0 = ∏ β k ; β k (mk ) = (1 − ρ k2 )2mk −1 / ∑ (C mlk −1ρ kl )2 , С ij = j! (i!( j − i)!) .
k =1 l=0
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 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 j =x j + n j , j =( j1 j2 ... jN )T ∈ Ω, which are the sum of informational (useful) RF and additive white Gausian noise with a variance σ 2 = M {n2j } it is necessary to make the best (in the sense of the minimum error variance) linear estimate xˆ = ∑ h z 0 j∈Ω j j of element x 0 in informational RF. This estimation will use coefficients h j which will determine the optimal filtering. The search of minimum error variance equations
2 σ ε2 = M {(xˆ0 − x0 )2 }= M ∑j∈Ω hj z j − x0 can be written as a system of linear hq σ 2 + ∑ h j B( r − j ) =B(r ),
r ∈ Ω , 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 where q = σ 2 σ 2 is signal-to-noise ratio, N is the dimension of RF, mk is the model’s multiplicity x for k-th dimension, ρ k is correlation parameter in k-th dimension. (6) (7) a) b)
Figure 1. Relative variances of errors of multidimensional RF.
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.
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.
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 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.
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.