=Paper=
{{Paper
|id=Vol-2005/paper-17
|storemode=property
|title=Synthesis and analysis of doubly stochastic models of images
|pdfUrl=https://ceur-ws.org/Vol-2005/paper-17.pdf
|volume=Vol-2005
|authors=Konstantin K. Vasiliev,Nikita A. Andriyanov
}}
==Synthesis and analysis of doubly stochastic models of images==
https://ceur-ws.org/Vol-2005/paper-17.pdf
Synthesis and analysis of doubly stochastic
models of images
Konstantin K. Vasiliev1 and Nikita A. Andriyanov 1
1
Ulyanovsk State Technical University, Severny Venets, 32, 432027, Russia
nikita-and-nov@mail.ru,
WWW home page: http://tk.ulstu.ru/
Abstract. The problem of describing inhomogeneous images by the
models with varying parameters is considered in the paper. Synthesis
of doubly stochastic images based on autoregressive random fields was
performed. In addition, the possibility of simulating various images is
shown under taking into account the choice of parameters and methods
of their transformation. We investigate the characteristics of inhomoge-
neous images and get the “image-correlation” dependences. Particular
attention is paid to the problems of estimating the doubly stochastic
models’ parameters. We describe a technique that allows one to form a
doubly stochastic model with its implementation for real images. We also
consider algorithms for detecting point anomalies against a background
of doubly stochastic signals and images. An optimal detection algorithm
has been developed. The algorithm provides a gain in comparison with
a detector based on the autoregressive models. The results of practical
application of the elaborated algorithms to real images are presented.
Keywords: Doubly stochastic models, inhomogeneous images, statisti-
cal analysis of images, model parameters estimation, anomalies detection,
random processes and fields, image processing
1 Introduction
Classical regression with constant correlation coefficients is widely used in scien-
tific studies [1–7]. However, the constant coefficients do not allow describing real
satellite data, which are characterized by spatial heterogeneity. Indeed, to simu-
late the image to be similar to the real one, its formation on a multidimensional
grid can not be performed at each point by the same model. Therefore, the prob-
lem of regression models “dynamization” is so urgent. One of the approaches to
ensuring such “dynamization” is a recurrent change in the parameters of the
model.
2 Synthesis of doubly stochastic model
Consider the synthesis of a doubly stochastic model both on the basis of statis-
tical modeling and on the basis of real data.
146
2.1 The idea of doubly stochastic model
Let the modeling of the image take place in accordance with the three-stage
simulation algorithm [8]. The first stage involves generation of a component of
a homogeneous random field (RF) % (basic RF). In the second stage, it is needed
to perform transformation of the obtained RF % to make the RF with correlation
parameters {ρj , j ∈ (j1 , j2 , ..., jM )}, where M is a parameter describing the
dimensions of the simulated image. These parameters characterize the correlation
between the generated pixel of the image and its neighbors, similarly to the usual
autoregression (AR) model. Finally, using RF with the parameters ρj , we can
get the main image.
To form the basic RF, we may use different models of RF [14]. Let us take,
for example, the presentation of a two-dimension RF X={xi , i ∈ Ω}, that we
can get by presentation of first order AR model. If we use this model, we need to
get the basic RF, and then transform its values into a row correlation {ρxij , i =
1, 2, ..., M1 , j = 1, 2, ..., M2 } and column correlation {ρyij , i = 1, 2, ..., M1 , j =
1, 2, ..., M2 }
ρ̃xi,j = r1x ρ̃xi−1,j + r2x ρ̃xi,j−1 − r1x r2x ρ̃xi−1,j−1 + ςxi,j ,
(1)
ρ̃yi,j = r1y ρ̃yi−1,j + r2y ρ̃yi,j−1 − r1y r2y ρ̃yi−1,j−1 + ςyi,j ,
where ςxi,j and ςyi,j are the two-dimensional RF of independent Gaussian ran-
dom values (RV) with zero means and variances
2 2
M {ςxi,j } = σςx ,
2 2 2
2
σςx = σρx 1 − r1x 1 − r2x ,
2 2
M {ςyi,j } = σςy 2
= σρ2y 1 − r1y 2
1 − r2y ;
2 2
σρx and σρy define the variances of the basic random fields of correlation
parameters for the row and column, respectively.
Parameters ρ̃xi,j and ρ̃yi,j allow us to adjust the geometric characteristics of
objects imitated on the image while we are simulating the basic RF. The closer
their values to unity, the larger the area in the image is occupied by such an
object.
Meanwhile, choosing the method of the basic RF transformation into a set of
correlation coefficients, it is possible to provide acceptable covariance functions
(CF) of the images. Usually, the selected conversion method may contain some
desired correlation structure of the original image. Thus, the proposed algorithm
can be used to simulate images that are close (in probability) to real images from
satellites.
The averages of RF ςxi,j and ςyi,j are equal to zero; so, the average for ρ̃xi,j
and ρ̃yi,j (1) is also equal to zero. In view of the foregoing, we restrict ourselves
to such a choice of the transformation of the values of the basic RF, at which
the value of the RF ρ̃xi,j and ρ̃yi,j will be increased by a constant at every point,
i.e. mathematical expectations mρx and mρy will be the following:
ρxi,j = ρ̃xi,jbase + mρx ,
(2)
ρyi,j = ρ̃yi,jbase + mρy ,
147
where index “base” characterizes the value of the basic RF.
Figure 1 shows the way to imitate an image whose correlation parameters
change according to the transformation of another simulated image based on the
first-order AR model, so called the Habibi model.
Fig. 1. Algorithm for simulating a doubly stochastic image
Fig. 2. Images generated by a doubly stochastic model based on the first-order AR
Figure 2 shows an example of using the algorithm according to the scheme of
Fig. 1 for generating doubly stochastic RF, whose parameters vary in accordance
with the equations of the AR. Here, (a, d, h) are presentations of the basic RF
148
{ρxi,j }; (b, e, i) are presentations of the basic RF {ρyi,j }; (c, f, i) are presentations
of the doubly stochastic model based on AR model.
Analysis of the results (Fig. 2) shows that application of a doubly stochastic
model of a RF allows imitating images with given correlation properties. Calcu-
lating the variances of the obtained RF, we can conclude that they are close to
the given ones, and the implementation of the doubly stochastic RF is station-
ary. The additional advantage of this approach is that the formation of small
images does not require large computational costs.
Figure 3 shows examples of images generated by the doubly stochastic models
with different average values of correlation coefficients.
Fig. 3. Doubly stochastic images
Investigation of the correlation properties of the basic image % ={ρi , i ∈ Ω}
has shown that on its basis it is possible to obtain objects of different geometric
shapes against the background of the main image. At the same time, by increas-
ing the dimension of N -dimensional rectangular grid Ω, we can make a model
for RF of higher orders. This provides a description of the images, which are
inherent in a fairly complex terrain.
Obviously, changing the parameters allows us to form very different images.
In this case, different algorithms for parameter transformation are possible. Thus,
the simplest models of images were obtained.
2.2 Synthesis using parameters estimation
The problem of estimating parameters is reduced to estimating the fields ρxi,j
and ρyi,j in model (1) [9, 10]. In the considered case, in order to identify the
parameters of such a model, it is necessary to evaluate the statistical parameters
149
mρx , mρy , σρ2x and σρ2y . To do this, we shall use the sliding window method.
Using some simple mathematical transformations, we can obtain the following
estimates of the mathematical expectation, variance, and correlation by row and
column at each point, proceeding from the necessity of presentation of an AR of
the second order Pi+N −1 Pj+N −1
mz(i+ N −1 )(j+ N −1 ) = N12 ∗ l=i k=j Xlk ,
2 2
2 1
Pi+N −1 Pj+N −1
σz(i+ N −1 )(j+ N −1 ) = N 2 −1 ∗ l=i k=j (Xlk − mzlk )2 ,
2 2 v Pi+N −2 Pj+N −1
(Xlk −mzlk )∗(X(l+1)k −mz(l+1)k )
u
1−t1−( l=i k=j
)2
u
2
(N −1)∗(N )∗σ
z(i+ N −1 )(j+ N −1 )
2 2
ρxz(i+ N −1 )(j+ N −1 ) = Pi+N −2 Pj+N −1
(Xlk −mzlk )∗(X(l+1)k −mz(l+1)k )
,
2 2 l=i k=j
(N −1)∗(N )∗σ 2
z(i+ N −1 )(j+ N −1 )
v 2 2
Pi+N −1 Pj+N −2
(Xlk −mzlk )∗(Xl(k+1) −mzl(k+1) )
u
l=i k=j
)2
u
1−t1−(
(N −1)∗(N )∗σ 2
z(i+ N −1 )(j+ N −1 )
2 2
ρyz(i+ N −1 )(j+ N −1 ) = Pi+N −1 Pj+N −2
(Xlk −mzlk )∗(Xl(k+1) −mzl(k+1) )
. where
2 2 l=i k=j
(N −1)∗(N )∗σ 2
z(i+ N −1 )(j+ N −1 )
2 2
z-index is introduced to descript the observed data.
It should be noted that the relations found for the mathematical expectation
and variance remain valid for other versions of the doubly stochastic model.
Fig. 4. Real and simulated images
Figure 4 shows how basing on estimates obtained by using the sliding window,
we can represent the original image (a) as an implementation of the model of
images with varying parameters (b).
3 Anomalies detection
We solve the problem of detecting anomalies for simulated and real images.
3.1 Point anomalies detection on the simulated data
We shall use the Neyman-Pearson approach, according to which we choose the
detection rule, that ensures the maximum probability of correct detection. This
150
is provided by the fact that the probability of a false alarm does not exceed a
predetermined value PF 0 . Thus, the optimal detection rule (in the sense of the
Neyman-Pearson test) maximizes the probability
R R
PD = 1 − PM = G1 ... ω(z/H1 )dz
with an additional restriction
R R
G1
... ω(z/H0 )dz = PF 0 .
We shall calculate the characteristics of the detection of a point signal with
level S, which can appear at a discrete point in time i. We write down the
decisive rule [11–13] as follows:
If λ = S T (PE + Vn )−1 (zS − x̂E ) > λ0 , then there is the signal S,
Else if λ = S T (PE + Vn )−1 (zS − x̂E ) ≤ λ0 , then there is not the signal S.
Then it can be reduced to the form
If λ = S(ziσ−x̂
2
Ei )
> λ0 , then there is the signal S,
n
Else if λ = S(ziσ−x̂
2
Ei )
≤ λ0 , then there is not the signal S, where σn2 is the
n
noise variance.
On the next step, we find the mathematical expectations and variances of
the left part of the detection rule under conditions of presence and absence of a
signal
2
M {λ/H1 } = σS2 ,
n
M {λ/H0 } = 0,
4
σλ2 = D{λ/H1 } = D{λ/H0 } = σS4 (σn2 + σe2 ), where σe2 is parameter that
n
defines variance of main RF.
Then the probabilities of false alarm and missed target can be written as
follows:
PF = 0.5 − Φ0 ( σλλ0 ) is the false alarm probability,
PM = 0.5 − Φ0 ( h−λσλ ) is the missed target probability, where Φ0 is Laplace
0
2 2
function, h = S /σn .
So, to calculate the probability of correct detection it is necessary to subtract
from the unit the probability of the missed target. We find the threshold value
λ0 considering that the probability of the false alarms is PF 0 = 0.001.
Figures 5a and 5b show graphs of the correct detection probability depending
on the signal-to-noise ratio q in the forecast using one and two observations,
respectively. It is seen that the detection on the basis of a doubly stochastic
RF model is more effective than detection based on the classic AR model. In
Fig. 5 the solid line is the theoretical probability for the algorithm based on the
doubly stochastic RF model, the dashed line is the experimental probability for
the algorithm based on the doubly stochastic RF model, the dot-and-dashed line
is the probability for the algorithm based on the AR model.
151
Fig. 5. Probability of correct detection with a prediction based on one (a) and on two
(b) observartions
Analysis of the graphs in Fig. 5 shows that the detection efficiency increases
in the case of using the doubly stochastic RF model and the larger volume of
information for forecasting. The gain on 0.5 correct detection probability level
is about 5-6% if we use two observations for prediction.
3.2 Extensive anomalies detection on real data
Now let us compare the work of two detectors of anomalies constructed on the
basis of the doubly stochastic model (Algorithm 1) and on the basis of the
autoregressive model (Algorithm 2). In this case, the detection will be performed
on the real images obtained from the LandSat-8 satellite. Studies are conducted
for three images. At the same time, in each image, 4 regions are selected where an
anomaly can be located. It is worth to note that the areas are selected based on
the structure of the images being examined with taking into account the greater
and less heterogeneity and the fact that the detection procedures are performed
not for the entire image, but only for these areas.
Figure 6 shows an example of one of the processed images with signals located
in different parts of the images. The picture also reflects the probabilities of
correct detection obtained using two algorithms. The dimensions of all images
are 250x250. The images are distorted by the white Gaussian noise with a single
dispersion. The size of the square is 4x4, the radius of the circle is 2. The signal-
to-noise ratio is 1. The statistics are taken out 500 times. We also provide the
probability of correct detection. The probabilities for Algorithm 1 are at the left
, the probabilities for Algorithm 2 are at the top.
152
Fig. 6. Noisy (left) and source images (right) with probabilities of correct detection of
a square signal
Figure 7 shows the gain of Algorithm 1 with respect to Algorithm 2 for the
magnitude of the threshold signal at the probability of correct detection of 0.5
and the probability of false alarm 0.001, which corresponds to the threshold
L = 3.1σz2 .
Fig. 7. Percent gains of the detection algorithm based on the doubly stochastic model
in comparison with the detection algorithm based on the AR model
Analysis of the results shows that the algorithm based on the doubly stochas-
tic model works better than the algorithm based on a simple autoregressive
model and provides reliable detection of the signal in 90-95% of cases.
Analysis of data presented in Fig. 7 shows that a similar situation is in terms
of ensuring at least the same efficiency. But in some cases, a significant gain
is retained in the case of processing other heterogeneous satellite images. The
probability of correct detection depends not only on the shape and size of the
signal itself, but, also on the brightness values in its closest neighborhood. In
153
this sense, a more universal algorithm is an algorithm based on doubly stochastic
models.
4 Conclusion
Thus, application of the doubly stochastic models provides an adequate descrip-
tion of heterogeneous images, and the detection algorithms developed for such
models lead to a gain in the detection of point signals of the order of 5-10%. We
also can improve the correct detection probability by introducing more points
for prediction or by increasing the signal level. We have investigated the model-
ing data, but the performed algorithms can become useful tool to detect various
anomalies in the real multispectral images. In addition, the processing of real
images also yielded gains in the detection of extended anomalies of different
shapes on the average of the order of 10-15% with respect to the signal-to-noise
ratio. It is worth to note that the novelty is offered by the proposed model with
varying parameters, since it, unlike autoregressive models, makes it possible to
generate images with different correlation properties. A comparison of the pro-
posed algorithms based on a doubly stochastic model was performed with known
autoregressive algorithms.
Acknowledgements. The study was supported by RFBR, project 16-41-
732027.
References
1. Kashyap, R.L., Rao, A.R.: Construction of dynamic stochastic models from exper-
imental data. Trans. from the English, ed. by V.Pugachev. Moscow. Nauka, 384
(1983)
2. Shalygin, A.S., Palagin, Yu.A.: Applied methods of statistical modeling. Mechan-
ical Engineering, 320 (1986)
3. Spektor, A.A.: Digital image processing. Novosibirsk, NSTU, 350 (2002)
4. Vasiliev, K.K. et al: Applied theory of random processes and fields. Ulyanovsk,
UlSTU, 256 (1995)
5. Vasiliev, K.K., Popov, O.V.: Autoregressive models of random fields with multiple
roots. Proceedings of the 4th Conf. “Image recognition and image analysis: new
information technologies.”, Novosibirsk, 4.1, 258–260 (1998)
6. Vasiliev, K.K., Sluzhivy, M.N.: Mathematical modeling of communication systems.
Ulyanovsk, UlSTU, 170 (2010)
7. Boxing, J., Jenkins, G.: Analysis of time series. Forecast and management. Trans.
from the English, ed. by V.P. Pisarenko. Moscow, Mir. 1, 406 (1974)
8. Vasil’ev, K. K., Dement’ev, V. E., Andriyanov, N. A.: Doubly stochastic models
of images. Pattern Recognition and Image Analysis. 25(1), 105–110 (2015)
9. Vasiliev, K.K., Dementiev, V.E., Andriyanov, N.A.: Estimation of the parameters
of doubly stochastic random fields. Radiotekhnika. 7, 103–106 (2014)
10. Vasiliev, K.K., Dementiev, V.E., Andriyanov, N.A.: Analysis of the effectiveness of
estimating the changing parameters of a doubly stochastic model. Radiotekhnika.
6, 12–15 (2015)
154
11. Andriyanov, N.A.: Investigation of the efficiency of correct detection of signals
against the background of one-dimensional doubly stochastic random processes.
Technical Sciences: Problems and Perspectives Materials of the IV International
Scientific Conference, 37-40 (2016)
12. Vasiliev, K.K., Dementiev, V.E., Andriyanov, N.A.: Detection of extended signals
on a background of doubly stochastic images. Radiotekhnika. 9, 23–27 (2016)
13. Andriyanov, N.A., Vasiliev, K.K., Dementiev, V.E.: Anomalies detection on spa-
tially inhomogeneous polyzonal images. CEUR Workshop Proceedings. Volume
1901, 2017 International Conference Information Technology and Nanotechnology.
Session Image Processing, Geoinformation Technology and Information Security,
IPGTIS-ITNT 2017; Samara; Russian Federation, 10-15 (2017)
14. Vasiliev, K.K., Dementiev, V.E., Andriyanov, N.A.: Filtration and restoration of
satellite images using doubly stochastic random fields. CEUR Workshop Proceed-
ings, “REIT 2017 - Proceedings of the 1st International Workshop on Radio Elec-
tronics and Information Technologies”, 1814, 10-20 (2017)