=Paper=
{{Paper
|id=Vol-2391/paper5
|storemode=property
|title=Double stochastic wave models of multidimensional random fields
|pdfUrl=https://ceur-ws.org/Vol-2391/paper5.pdf
|volume=Vol-2391
|authors=Victor Krasheninnikov,Aleksey Subbotin
}}
==Double stochastic wave models of multidimensional random fields ==
https://ceur-ws.org/Vol-2391/paper5.pdf
Double stochastic wave models of multidimensional random
fields
V R Krasheninnikov1, A U Subbotin1
1
Department of Applied Mathematics and Informatics, Ulyanovsk State Technical University,
Ulyanovsk, Russia
e-mail: kvrulstu@mail.ru
Abstract. The paper deals with the development of mathematical models of random fields to
describe and simulate images. In the wave model, a random field is the result of the influence
of perturbations (waves) that occur at random times in random places and have random shapes.
This model allows representing and simulate isotropic and anisotropic images (and their
temporal sequences) defined on arbitrary areas of multidimensional space, as well as on any
surfaces. The problems of correlation analysis and synthesis can be relatively easily solved.
However, this model allows representing only homogeneous fields. In this paper, we consider
«double stochastic» wave models, when the first wave random field (control field) sets the
parameters of the second (controlled field). As a result, the controlled field becomes non-
uniform, since its parameters vary randomly. We also consider options when two fields
mutually influence each other. These models allow us to represent and simulate
multidimensional inhomogeneous images (and their temporal sequences), as well as systems of
such images with mutual correlations.
1. Introduction
A rigorous mathematical formulation of image processing tasks is required for their effective solution.
This formulation primarily includes the model of the image as an object of study. The representation
of images by random fields (RF) is generally accepted. To date, there is an extensive literature on
image models (or RF models), for example, [1-6]. Much attention is paid to the modeling and
processing of medical images for example, [7-11]. However, the vast majority of works consider two-
dimensional flat RF, less often their sequence. Therefore, it is rather difficult to solve the problems of
image imitation, descriptions of inhomogeneous images, correlation analysis and synthesis. A wave
model of a random field was proposed in [12-14], which makes it possible to describe homogeneous
images and their sequences defined on regions and surfaces of any dimension with small
computational costs for simulation. In this model, the RF is the result of the influence of perturbations
(waves) that occur at random times in random places and have random shapes. The generated fields
are isotropic if waves are spherical. Anisotropic RF can be obtained, for example, with ellipsoidal
waves. The main influence on the form of the correlation function of RF is exerted by only one
parameter, namely the probability distribution of the wave scale factor, therefore, the problems of
correlation analysis and synthesis are relatively easy to solve.
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019)
Image Processing and Earth Remote Sensing
V R Krasheninnikov, A U Subbotin
However, this model describes only homogeneous RF. In [15, 16], a double stochastic
autoregressive model of RF was proposed for describing inhomogeneous images. In this model, the
first wave RF (control field) sets the parameters of the second RF (controlled field), which turns out to
be non-uniform, since its parameters randomly vary in space. So the structure of the resulting image is
significantly different in different places. This idea is used in the present paper to the wave model,
which made it possible to obtain models of a wide class of inhomogeneous images defined on regions
and surfaces of any dimensions.
2. Basic wave model of a random field
Let us consider the RF wave model [12] that generalizes a number of other models and helps to solve
the tasks of analysis and synthesis effectively. This model is simple enough and can serve as a basis
for simulating images and their sequences with given covariance function (CF) without increasing the
number of model parameters.
In the wave model, the RF is determined by the stochastic equation
xtj f ( j , t ),(uk , k ),k (1)
{k : k t }
where an (n+1)-dimensional domain { ( j, t ) } may be discrete or continuous, {(uk , τ k )} is a discrete
field of random points (FRP) in an (n+1)-dimensional continuous space, t and τ k are interpreted as
time, k is a random vector of function f parameters. This field can be represented as the effect of
random disturbances or waves f ( j , t ),(uk , k ),k appearing in random places u k at random time k
and changing according to a given law in time and space. Selection of function f, the FRP parameters
and ω allow us to obtain a vast class of fields.
Let us consider a particular case of a wave model, for which correlation tasks of analysis and
synthesis can easily be solved:
f ( j , t ),(uk , k ),k g( k / Rk )exp( / t k ) k (2)
where the FRP is a Poisson one with constant density , k | j uk | is a distance between j and u k ,
{ Rk } is a set of independent non-negative equally distributed random variables density w() , { k } is
a system of independent equally distributed random variables with zero mean. Waves are motionless,
independent, have spherical sections in space and exponentially attenuate over time. System { k }
determines a wave intensity and { Rk } is their spatial scale. If g ( y ) c exp( 2 y 2 ) then the generated
field X is stationary, homogeneous, has zero mean and its CF
c 2 n /2
exp t n exp 2 / 2 w( )d
V ( ,t ) n 1
(3)
2 0
is spatially isotropic with variance
c 2 n / 2
n2 n1 M [ R n ] . (4)
2
_
Simulation of a discrete field on an n-dimensional grid S { j} with time quantization t can be
implemented by the following algorithm. At the initial time t0 = 0, the field values in all nodes are
equal to zero. At each subsequent moment tm = mt, a Poisson FRP with density t is formed over
continuous space or grid, which somehow overlaps S. At each generated point u k , random values k
_
and Rk are formed. After that the following transformation of all field values on grid { j } is carried
out:
xtjm xtjm1 exp( t ) c exp( k / Rk )k , (5)
k
which forms a sequence of frames that are n-dimensional images defined on S.
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 42
Image Processing and Earth Remote Sensing
V R Krasheninnikov, A U Subbotin
Note that the field values are calculated independently of each other. This makes it possible to
simulate images on the desired area S, for example, on a certain surface.
An example of image simulation with the described wave model is represented in figure 1. It is
necessary to underline that (5) actually implements time-varying images. Therefore, this figure shows
only one frame of this process.
Each field value in (5) is a sum of random numbers of the random variables. Thus, generally
speaking, the field will not be Gaussian even with Gaussian { k }. However, when the model
parameter h M [ R n ] / grows, then the number of summands (5) with similar distributions
increase and the field is normalized.
Figure 1. Example of image simulation using a wave model.
Now consider the solution of correlation analysis and synthesis task. It follows (4) that the formed
field has an exponential time normalised CF (NCF) e t and space NCF.
n exp 2 2 d
1
r
M [ R n ] 0
(6)
Thus, solving analysis tasks, when the density ω(α) is given, the required NCF can be found
analytically or by numerical integration. Solving synthesis tasks, when the NCF r( ) is given, it is
necessary to solve integral equation (6) with respect to unknown () . As it is not always possible to
find an analytical solution of (6), we consider a method of its approximate solution. It follows from (2)
that in the case of degenerate distribution (R==const) we obtain the NCF equal to exp( ρ 2 / α 2 ) .
Let now an arbitrary non-increasing NCF r () be given. Let us approximate it with adequate accuracy
by a sum of Gaussoids with positive coefficients r h i qi exp 2 i2 , where qi 1 , when
i
r(0) = 1. Then, for discrete distribution P R i k qi , where k qi , the generated field
-1
i
n
i
n
i
will have NCF equal to h(). Thus, the generated model allows us approximately to solve a synthesis
task by changing only the density of scale R.
3. Double stochastic wave models
The wave model described above only defines homogeneous images. For the formation of
inhomogeneous images, it is necessary that the model parameters vary in the domain of S. This can be
achieved by breaking S into parts, in each of which the wave model has its own set of parameters. But,
taking into account the possible uncertainty of real images, “double stochastic” models are considered
in [6, 7], in which one or several autoregressive (control) RF set the parameters of the final
(controlled) autoregressive RF. We use this idea for wave models.
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 43
Image Processing and Earth Remote Sensing
V R Krasheninnikov, A U Subbotin
Let the control field X be given by model (2) with some values of its parameters. We use for the
controlled field Y a model of the same type but with a modified density ω(α) depending on X. For
example,
RkY rm x t (uk ) min X t Rk , (7)
where X t is the field X at the time t, u k is formed the point of FRP for Y t , x t (u k ) is field value at
this point, rm is the minimum allowable scale value RkY for Y, RkY is random scale value (when it is
distributed for the field X). Transformation (7) leads to the corresponding conversion
wX ( ) wY ( ) of the Y field scale, which should be used in (6) when calculating the CF of field Y.
Note that other doubly stochastic models are possible when other parameters of the resulting image are
also possible (c value, FRP density, and so on) are determined by one or several control images.
Figure 2 (b) shows the image Y, controlled by the top image X in figure 2 (a). It is noticeable that
the smoother areas of the image Y correspond to the brighter areas of the image X, that is, its larger
values and larger values of the wave stretch factor RkY . The sample value of the interval of absolute
correlation in a smoother region of this image is 87, and at the less smooth region is only 12.
Figure 2. An example of simulating double stochastic images: (a) is a control image, (b) is a
controlled image.
However, the boundaries between more and less smooth areas on Y are blurred, since the waves
have a considerable length and the brightness X varies smoothly. In [6], to obtain clearer boundaries
on double-stochastic autoregressive images, the quantization of the control image is applied, which
provides a stepwise change in the parameters of the controlled one. This method can also be applied to
wave models. Figure 3 (a) shows a 3-level quantized image in figure 2 (a). The image controlled by
this quantized image is shown in figure 3 (b). Here, the boundaries between areas of different
smoothness are clearer than in figure 2(b). The sample values of the correlation interval are
approximately the same as in figure 2 (b).
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 44
Image Processing and Earth Remote Sensing
V R Krasheninnikov, A U Subbotin
Figure 3. An example of simulating double-stochastic images with quantization: (a) is a control
image, (b) is a controlled image.
Figure 4. An example of simulating double stochastic images with mutual influence.
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 45
Image Processing and Earth Remote Sensing
V R Krasheninnikov, A U Subbotin
Two images are unequal in the described model: one controls the parameters of the other. In [8], a
model of autoregressive images defined on a cylinder was proposed, jointly controlling the parameters
of each other. We apply this approach to the considered wave models. Consider the sequence
X 1 , X 2 , X 3 ,... of images. Images X 1 and Y1 are formed in the same way as X and Y in the model
described above. Image X 2 aX 1 bY1 controls the formation of the image Y2 , and so on. Here
should be | a | 1 . This process is similar to autoregression. As a result, in the steady state, the frames
of the formed sequence mutually control the parameters of each other. The correlation weakens with
an increase in the time interval between frames. Figure 4 shows two consecutive images formed in the
manner described. Their correlation is visually noticeable (the sample correlation coefficient is 0.769).
This is significantly different from simple control (figure 2), where the correlation between the images
is almost zero.
4. Conclusions
This paper presents double stochastic wave models of random fields (images). In these models, the
control wave random field sets the parameters of the controlled random field. As a result, the structure
of the controlled image turns out to be significantly different in different places, which makes it
possible to describe inhomogeneous images with random inhomogeneities. Options are also
considered when two or more fields mutually influence each other. These models allow to represent
and simulate multidimensional inhomogeneous images (and their temporal sequences), as well as
systems of such images with mutual influences.
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V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 46
Image Processing and Earth Remote Sensing
V R Krasheninnikov, A U Subbotin
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V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 47