One of the most important problems of image processing is the problem of mathematical description of objects appearing on image's background. There are many algorithms for current imitation of images, but unfortunately they have some drawbacks. Often, such models are not able to describe images that are inhomogeneous in space, or they can describe only images with slowly changing properties. Therefore, a model is proposed for describing sharp change in the properties of an image. However, due to the use of autoregression with multiple roots, the discrete model will also contain objects with slowly changing brightness characteristics. The paper is devoted to the development of images' models with variable parameters, which can take a limited number of values. We propose methods for discretizing the obtained random parameter elds by transforming the initial elds with a continuous distribution. Furthermore, the probabilistic properties of the proposed model are analyzed. We also describe in details methods for simulating images containing a given number of structures and investigate segmentation algorithms for the proposed model of images.
Solutions to a number of important problems in various elds of science and technology can be achieved by using the results of processing multidimensional data sets. Among the image processing tasks, a special place is occupied by problems of segmentation, detecting anomalies, reconstruction, and prediction. One of the most important sources of such data arrays are various multidimensional images. Examples of such images are various information signals, digital photographs and video sequences, remote sensing data of the Earth (RS), etc.
The particular urgency of solving these problems is associated with wide dissemination of methods for recording images of various objects including multispectral (up to 10 spectral ranges) and hyperspectral (up to 300 ranges) of recording the Earth areas. As a result of such registration, multidimensional arrays of information are obtained. Thus, there is an increase in the volume of information received, and although signi cant amounts of processed information potentially improve the quality of satellite material processing, new methods are necessary for qualitative and quantitative analysis of aerospace observations as a single multidimensional aggregate.
In addition, the problem of describing images can not be regarded as solved. Despite the variety of random elds (RF) and random processes models [1{11], most of them do not give a satisfactory description of real images and signals due to a number of reasons. The main problem is presence on an image of several objects having di erent nature, description of which is di cult for one model. Digital image processing algorithms [11] can be synthesized for some particular cases of mixed image models [7, 8]. In such models, auxiliary RFs are used to implement the basic RF, which are also a set of model parameters.
In this article, we will consider modi cation of a doubly stochastic model [1, 3, 4, 7{9], which allows one to obtain a smooth change in the brightness properties of an image, as well as the possibility of using a given number of correlation levels. Finally, we show that the use of a countable number of correlation levels allows one to perform e ective segmentation of such images. 2
Doubly stochastic model of images based on Habibi
and autoregression with multiple roots of characteristic
equations models
The main drawbacks of the doubly stochastic model based on the Habibi model
or the usual rst-order autoregressive (AR) model are its anisotropy, as well as
the use of only three neighboring elements to form a new element of the RF.
Indeed, it is obvious that the Habibi model is a special case of an AR model
with multiple roots of the characteristic equations [2, 6, 10]. The multiplicity
of such a model is (
The process of such model synthesis is analogous to the process considered in [1], but it has its own peculiarities and is implemented in three stages. First, the basic RFs are created. Secondly, the values of the obtained RFs are converted into a set of correlation parameters f j ; j 2 (j1; j2; :::; jM )g, where M is the dimension of the image being formed. These parameters characterize the connection between the current pixel of the simulated image and the neighboring image elements. Finally, the image is formed as a model of a RF with varying correlation parameters j .
For simplicity, the basic RFs are simulated using the rst order AR model as well as the doubly stochastic RF model based on the Habibi model. After the formation of two basic RFs, the brightness values of one of them should be transformed into a set of correlation parameters f xij ; i = 1; 2; :::; M1; j = 1; 2; :::; M2g, and the second RF should be transformed into correlation parameters f yij ; i = 1; 2; :::; M1; j = 1; 2; :::; M2g.
We can write AR with multiple roots of characteristic equations of the second
order in the one-dimensional case as follows:
(
2xi 2 + i:
On the basis of formula (
4 x yxi 1;j 1 2xxi 2;j
2yxi;j 2 + 2 2x yxi 2;j 1 +2 2y xxi 1;j 2
2x 2yxi 2;j 2 + b i;j ; where b is the coe cient ensuring the equality of the variance of the simulated RF to a given value.
It should be noted that model (
Making coe cients of model (
Thus, using the doubly stochastic AR models, one can generate images and their sequences adequate to real multi-zone images at relatively low computational costs. This allows us to use such models for statistical analysis of the e ciency of image processing algorithms. In this case, it is possible to achieve a smooth change in the brightness properties of the image due to models with multiple roots of the characteristic equations.
In model (
M f&y2i;j g = &2y = 2y 1 r12y 1 r22y , 2x = M f 2xij g and m x and m y de ne average correlation in row and column. 2y = M f 2yij g,
The correlation coe cients obtained from expression (
xij L
maxf xij g
minf xij g
);
(
Then it is necessary to convert the obtained discrete RF (
Thus, by varying the number of discretization levels L, it is possible even for
identical model parameters to obtain its various implementations. We also can see that on
the graphs of Figure 3 the covariance relationships of discrete models decrease more
slowly than those in continuous ones.
Let us nd the number of elements of the vector obtained after the transformation (
Lmax1 + Lmax2 ; 2 where Lmax1 and Lmax2 are the brightness values of the rst and second maximum of the histogram.
Let us transform the equalized image f~xijeqg with 256 gradations of brightness into an image fRxij g with 2 gradations of brightness. So the image fRxij g is binary image, which is obtained by the formula
Rxij = 0; if f~xijeqg < Lthr; Rxij = 255; else:
Expression (
Figure 4 shows histogram of an image simulated by doubly stochastic model with possible parameter values x1 = y1 = 0:8 and x2 = y2 = 0:95. 4
Segmentation of images with varying correlation properties Now consider the segmentation algorithm for the generated images in the case where the number of correlation levels is two. To select homogeneous areas, you need to build a histogram of the image. To do this, all the values of the RF f xij g are written in the vector of brightness values. It should be noted that all the values have before passed the equalization procedure, and we also taking into account the average additive. So, the result vector is written as follows:
L(k) = ~xijeq; i = 1; 2; :::; M1; j = 1; 2; :::; M2; 1; 2; :::; M1xM2:
(
It is clear that the best segmentation will be provided by combinational processing of correlation parameters by row and column.
Figure 5 shows the result of the segmentation algorithm using the histogram: Figure 5a corresponds to basic RF; Figure 5b corresponds to nonhomogeneous image with homogeneous regions; Figure 5c corresponds to binarization of the basic RF; Figure 5d corresponds to segmentation of the nonhomogeneous image.
The percentage of the correspondence between Figure 5c and Figure 5d is quite large, namely, 85.7%. Analysis of Figure 5 shows that it is possible to increase segmentation e ciency if small objects are included in a larger neighboring area.
Nevertheless, the result obtained for simulated images allows us to reasonably hope that in the case of applying more complex segmentation procedures in processing the correlation parameter eld, the segmentation algorithm found can be applied to real images. Indeed, Figure 6 shows the results of 2-level segmentation (binarization) of a satellite image by applying a combination of the proposed algorithm and the ISODATA algorithm to the eld of correlation parameters. In this case, Figure 6a is the original image; Figure 6b shows the correlation parameter elds for the original image; Figure 6c shows the results of segmentation by correlation parameters jointly ISODATA algorithm; and Figure 6d shows the results of applying the ISODATA algorithm to the original image.
The gain for the presented case on segmentation accuracy was about 6%. However, it is worth to note that the task of assessing the accuracy of segmentation when processing real images is signi cantly complicated, since there is no predetermined correct version, and most often, we have to use expert estimates. 5
Conclusion Doubly stochastic models of images based on the AR with multiple roots of characteristic equations are considered. The transition to the discrete case is carried out. Probabilistic properties of the proposed models are analyzed. An algorithm for image segmentation by correlation parameters is described. Gains are obtained when processing images consisting of two structures. The gains are about of 6 {10% in comparison with segmentation by the ISODATA algorithm.
Acknowledgements. The study was supported by RFBR, project 17-01-00179 A.