Download Quantum Mechanical Approach to Modelling Reliability of Sensor

arXiv:1705.01013v1 [cs.OH] 17 Apr 2017
Combining Evidences Based on Quantum Mechanical
Approach
Zichang Hea , Wen Jianga,∗
a
School of Electronics and Information, Northwestern Polytechnical University, Xi’an,
Shaanxi, 710072, China
Abstract
Dempster-Shafer evidence theory is wildly applied in multi-sensor data fusion. However, lots of uncertainty and interference exist in practical situation,
especially in the battle field. It is still an open issue to model the reliability of
sensor reports. Many methods are proposed based on the relationship among
collected data. In this letter, we proposed a quantum mechanical approach
to evaluate the reliability of sensor reports, which is based on the properties
of a sensor itself. The proposed method is used to modify the combining of
evidences.
Keywords: Dempster-Shafer evidence theory, multi-sensor data fusion,
quantum mechanical approach, Sensor report reliability.
Corresponding author at: School of Electronics and Information, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China. Tel: +8613363912605. E-mail address: [email protected], [email protected]
∗
Preprint submitted to Elsevier
May 3, 2017
1. Introduction
Data fusion has been widely studied in the last decades, especially its military
applications. Multi-sensor data fusion (MSDF) technology plays a more and
more significant role for the fighting demand. How to fuse the sensor data
is still an open issue[1–4]. Due to the powerful ability of handling uncertain
information, DS evidence theory is widely used in MSDF[5–8]. However,
Lots of interference exist in the complex practical situation. The information
provided by a sensor report is likely to be disturbed and incorrect. In this
case, strong conflict may exist among evidences and lead to a wrong fusion
result. Handling conflict is crucial in data fusion[9–12]. To address it, many
approaches have been proposed[13–15].
To deal with conflictive information, most previous methods handle evidences
based on the relationship among the data collected by sensors[16–19]. In this
letter, however, an method which bases on the properties of a sensor itself
is proposed. To evaluate the reliability of sensor reports, a confidence coefficient curve is determined based on a quantum mechanical approach. Interest in quantum approach to classical fuzzy logic has increased over the last
decades[20–23]. In classical mechanics, a particle is located in an exact place.
If a particle is known to be in M, then it can never in any other places, like in
N. In quantum mechanics, however, a particle can never be exactly located
due to the well-known Heisenberg’s uncertainty relation. Only the probability of finding the particle in a given area like M or N can be determined
(shown as Figure ??). This interesting property of quantum mechanics is
used to describe the reliability degree of a sensor report as it is hard to assert
that one sensor report is totally reliable or unreliable. Then we use the curve
to calculate the credibility of evidences. The fusion results of the modified
evidences show the effectiveness of our method.
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2. Preliminaries
Dempster-Shafer evidence theory was proposed by Dempster in 1967[24] and
modified by Shafer in 1978[25]. In evidence theory, the basic set Θ, called the
frame of distribution, consists of a set of N mutually exclusive and exhaustive
hypotheses, symbolized by Θ= {X1 , X2 , . . . , XN } Let P (Θ) denote the power
set composed of 2N elements of Θ.
P (Θ) = {∅, {X1 } , {X2 } , . . . , {XN } , . . . , {X1 ∪ X2 } , {X1 ∪ X3 } , . . . , Θ}
Basic probability assignment (BPA) is a mapping from P (Θ) to [0, 1], defined
by:
m : P (Θ) → [0, 1]
satisfying the following conditions:
X
m (A) = 1
(1)
(2)
A∈2N
m (∅) = 0
(3)
The mass function m represents a supporting degree to A. The elements
of P (Θ) that have a non-zero mass are called focal elements. A body of
evidence (BOE) is the set of all the focal elements[? ]:
o
n
(R, m) = [A, m (A)] ; A ∈ P (Θ) and m (A) > 0
R is a subset of P (Θ), and each of A ∈ P (Θ) has a fixed value. The classical
Dempster’s combining rule of two BOE m1 and m2 is defined as following:
P
m1 (B) m2 (C)
m (A) = B∩C=A
(4)
1−K
where K is called conflict coefficient:
X
K=
m1 (B) m2 (C)
B∩C=∅
3
(5)
3. Quantum mechanical modelling of the sensor reliability in data
fusion
Radar plays an important role in the modern battlefield. Usually, to obtain
the overall information, data from several radars need to be fused. Aiming
to do a more reasonable fusion, we propose an method based on quantum
mechanics to determine the confidence coefficient curve of radar sensor reports. We assume that the reliability of sensor reports relates to the distance
between object and sensor in some degrees. For each distance x, the sensor
has an according confidence coefficient whose maximum value is 1. Hence,
confidence coefficient curve µ (x) is defined as a function to describe this
relationship.
The signal of the object is received by k radars. The transmit power of
the object is Pt , the antenna gain of the object is Gt , the antenna gain of
the reconnaissance radar is Gr , the distance between object and a radar is
denoted as x. The signal power received by radar is:
Pr =
Pt Gt Gr σλ2
(4πx)2
(6)
where λ is the wavelength and σ is Radar Cross-Section which is the product
of geometric cross-section, reflection coefficient and direction coefficient.
If the sensitivity of a radar is Pr min, the maximal reconnaissance distance xr
is calculated as follows. If the object is far beyond this distance, it will not
be effectively reconnoitred.
Pt Gt Gr σλ2
xr =
(4π)2 Pr min
1
2
(7)
According to the quantum-mechanical rules of quantification, we should write
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an operator which corresponds to the received signal power:
H = −c2
∂2
− V (x)
∂x2
(8)
where c is a scale factor. V (x) is a quasi-potential function to model the
received power.
V (x) =
where γ ∝
Pt Gt Gr σλ2
(4π)2
(
γ
x2
0 < x ≤ xr
∞ x ≤ 0, x > xr
(9)
corresponds to the parameters in Eq. (6). The quasi-
potential function V (x) is roughly illustrated as Figure 1.
Figure 1: The quasi-potential function V (x)
Based on quantum-mechanical rules, a quasi time-independent Schrödinger
equation can be obtained.
Hψ (x) = Lψ (x)
(10)
where L relates to the level of the radar sensitivity Pr min .
The solution of Eq. (10) is a quasi-amplitude distribution ψ (x). When the
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object is within the maximal reconnaissance distance xr , we can obtain:
"
√ !#
√ !
√
L
L
ψ (x) ∝ x Jα
+ Yα
(11)
c
c
where Jα and Yα are the Bessel function of the first kind and the second kind
respectively. α is their order:
1
α=
2
r
c2 − 4γ
c2
(12)
Then let us consider the other situation, when the object is beyond xr , the
value of V (x) is infinite. According to quantum mechanics, it is impossible
for a particle to penetrate the well wall if it is within a infinite well potential.
Hence, we can conclude that ψ (x) = 0 in this case.
Then we can obtain the probability distribution P (x), which is illustrated
graphically in Figure 2.
"
P (x) = |ψ (x)|2 ∝ x Jα
√ !
L
+ Yα
c
√ !#2
L
c
(13)
By amplifying Eq. (13), we can obtain the confidence coefficient curve µ (x).
Seen from Figure 3, the curve rises rapidly when x is smaller than x0 and
comes to its maximum when x equals to x0 . Then it declines slowly until x
comes to xr , which is reasonable. In practical situation, due to precision and
some other intricate issues, a radar do not work well when it is too close to
the object. There exists an optimal distance x0 for a radar to work. Then the
performance of a radar becomes poorer as it is located further. When the distance is further than the maximal reconnaissance distance, the radar can not
reconnoitre the object effectively. With the basis of this curve, we can evaluate the reliability of radar reports effectively. For different types of radars, we
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Figure 2: The probability distribution P (x)
Figure 3: The confidence coefficient curve µ (x)
can obtain their according confidence coefficient curves as Figure 4. The parameters of these curves are in Table 1. In the following, the curves are used
in combining evidences. Assume we have k pieces of BOEs: m1 , m2 , . . . , mk ,
collected from k radar sensors. By using confidence coefficient curves, each
BOE corresponds to one confidence coefficient: µ1 , µ2 , . . . , µk . The credibility
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Figure 4: The confidence coefficient curves of different radars
Table 1: Curves of different radars
c
L
r
Radar a
10 0.7
14
Radar b
10 0.8
12
Radar c
10 1.0
10
Radar d
10 1.1
13
Radar e
10 1.3
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degree Crdi of BOE mi is defined as:
Crdi =
µi
k
P
(14)
µi
i=1
It is easy to find that
Pk
i=i
Crdi . Hence, the credibility degree reveals the
relatively importance of the collected evidence. After determining the credibility of each BOE, we do a modified average for all k pieces of BOEs to
obtain a new evidence m′ .
m′ =
k
X
i=1
Crdi × mi
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(15)
Then we can combine m′ with itself for k − 1 times by using classical combining rule (Eq. (4)), which is same as Murphy’s approach[18]. Obviously,
if a BOE is collected from a sensor with high reliability, it will have more
effect on the final combination results. On the contrary, if a BOE is collected
from a sensor with relatively low reliability, it will matter little in the final
combination results.
4. Numerical example
In this section, a numerical example is illustrated to show the effectiveness of
our method. In a target recognition system, five radar sensors have collected
five pieces of BOEs shown as follows:
(R1 , m1 ) = ([{A} , 0.6] , [{B} , 0.15] , [{A, C} , 0.25])
(R2 , m2 ) = ([{A} , 0.5] , [{B} , 0.3] , [{C} , 0.2])
(R3 , m3 ) = ([{B} , 0.95] , [{C} , 0.05])
(R4 , m4 ) = ([{A} , 0.55] , [{B} , 0.25] , [{A, C} , 0.2])
(R5 , m5 ) = ([{A} , 0.6] , [{B} , 0.3] , [{B, C} , 0.1])
The reliability of these sensor reports is 0.55, 0.6, 0.25, 0.45 and 0.5 respectively, which is obtained based on their confidence coefficient curves. Then
fusion results and comparison are shown in Table 2. Four evidences prefer to
Table 2: Fusion results and comparison
m (A)
Classical rule
0
m (B)
m (C)
0.9057 0.0943
Murphy’s approach
0.7971 0.2011 0.0018
Our method
0.9373 0.0609 0.0018
recognizing the target as A. Hence, data from the third sensor is probable to
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be interfered and incorrect. As can be seen from Table 2, in this situation,
our method works better than Murphy’s while the classical combining rule
does not work. The target can be effectively recognized with our method.
5. Conclusion
In summary, we propose a new method to model the reliability of sensor
reports. Unlike previous methods, we focus on the properties of a sensor
itself. The confidence coefficient curve of a radar sensor is obtained by solving
a a quasi time-independent Schrödinger equation. The method is used in
combining of evidences. The result shows the efficiency of our method.
6. Acknowledgement
The work is partially supported by National Natural Science Foundation
of China (Grant No. 61671384), Natural Science Basic Research Plan in
Shaanxi Province of China (Program No. 2016JM6018), Aviation Science
Foundation (Program No. 20165553036), the Fund of SAST (Program No.
SAST2016083)
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