Journal of Computational Information Systems 9: 16 (2013) 6417–6424 Available at http://www.Jofcis.com

Multisensor Information Fusion Based on Dempster-shafer Theory and Power Average Operator ⋆ Xinyang DENG 1 , Yong DENG 1,2,∗ 1 School

of Computer and Information Science, Southwest University, Chongqing 400715, China 2 School

of Engineering, Vanderbilt University, Nashville, NT 37235, USA

Abstract In multisensor information fusion, the key problems are the representation of sensor report and the combination methodology of sensor information. In this paper, we propose a novel method for the fusion of multisensor information. Within the proposed method, the sensor report has been represented by using Dempster-shafer theory. Then an evidence-driven method is proposed to obtain the relative credibility of each sensor based on the power average operator. At last, a weighted balance evidence theory is employed to combine the sensor reports. The proposed method is efficient for the representation of uncertain information and fusion of conflicting sensor reports. A numerical example is given to demonstrate the effectiveness of the proposed method. Keywords: Multisensor Information Fusion; Dempster-shafer Theory; Power Average Operator; Conflict Management

1

Introduction

Multisensor system is ubiquitous in the many applications [1, 2]. Usually, the report or output of each sensor can be seen as an opinion to the environment. The information fusion of Multisensor system has emerged as an important research field. As a result, an open issue is how to fuse these multiple sensors. Generally speaking, the key problems in the multisensor fusion are the representation of sensor report and the combination methodology of sensor information. Due to the limitation of sensor observation, an uncertain information will be obtained for each sensor. In previous study, many methods, for example fuzzy set theory, possibility theory, are developed to express the uncertainty of environment. By comparing with other methods, a method is the Dempster-shafer theory [3, 4], ⋆ The work is partially supported by National Natural Science Foundation of China (Grant No.61174022), Chongqing Natural Science Foundation for Distinguished Young Scientists (Grant No.CSCT, 2010BA2003), National High Technology Research and Development Program of China (863 Program) (No.2013AA013801), Doctor Funding of Southwest University (Grant No.SWU110021). ∗ Corresponding author. Email address: [email protected] (Yong DENG).

1553–9105 / Copyright © 2013 Binary Information Press DOI: 10.12733/jcis7841 August 15, 2013

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which is widely applied many fields [5-11]. On the one hand, the mass function of Dempster-shafer theory is flexible and clearly understandable, and more effective to express uncertain information. On the other hand, the generation of mass function is easy and convenient. The following problem is thus to combine these evidences. Dempster-shafer theory provides a classical combination rule to fuse multiple evidences. However, when the evidential system is highly conflicting, a counterintuitive conclusion will be obtained [12]. To this problem, many methods have been proposed to handle highly conflicting evidences, for instance Murphy’s simple average combination method [13], Deng et al’s weighted average combination method [14], and Schubert’s the degree of falsity-based method [15]. In this paper, we propose use a novel method for the conflict management. The proposed method uses the power average operator [16] to calculate the credibility of sensor and then combines the evidences by using a weighted balance evidence theory (WBET) [17]. A numerical example is given to demonstrate the effectiveness of the proposed method. The rest of this paper is organized as follows. An introduction to the Dempster-shafer theory and power average operator is given in Section 2. The proposed method is presented in Section 3. An example is presented in Section 4. Section 5 concludes this paper.

2 2.1

Preliminaries Dempster-shafer theory

The Dempster-Shafer theory [3, 4] supposes the definition of a set of hypotheses Θ called the frame of discernment. Definition 1 A mass function, called basic probability assignment(BPA), is a mapping m: 2Θ → [0, 1], which satisfies: m(∅) = 0

and

∑

m(A) = 1

(1)

A⊆Θ

Definition 2 Dempster’s rule of combination, denoted by (m1 ⊕ m2 )(also called the orthogonal sum of m1 and m2 ), is defined as follows: m(A) =

∑ 1 m1 (B)m2 (C) 1 − K B∩C=A

K=

∑

m1 (B)m2 (C)

(2)

(3)

B∩C=∅

Note that K is called the normalization constant of the orthogonal sum (m1 ⊕ m2 ), which measures the degree of the conflict between m1 and m2 . In the literature [18], a concept of Jousselme distance between two evidence is presented, which is an effective principled distance between two BPAs.

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Definition 3 Let m1 and m2 be two BPAs on the frame of discernment U, containing N mutually exclusive and exhaustive hypotheses. The distance between m1 and m2 is: √ 1 − →−− →) →−− →)T D(− dBP A (m1 , m2 ) = (m m m m (4) 1 2 1 2 2 → and − → are the vector representation of m and m , D is an 2N × 2N matrix whose where − m m 1

2

elements are D(A, B) =

2.2

1

|A∩B| , |A∪B|

2

A, B ⊆ U .

Power average operator

The power average operator [16] first proposed by Yager is an aggregation operator that allows argument values to support each other in the aggregation process. Definition 4 To a set of data (a1 , a2 , · · · , an ), the power average operator is n ∑

P − A(a1 , a2 , · · · , an ) =

(1 + T (ai ))ai

i=1 n ∑

(5) (1 + T (ai ))

i=1

where T (ai ) =

n ∑

Sup(ai , aj )

(6)

j=1 j̸=i

Here, Sup(a, b) is the support for a from b, and it satisfies the following three properties: (1) Sup(a, b) ∈ [0, 1]; (2) Sup(a, b) = Sup(b, a); (3) Sup(a, b) ≥ Sup(x, y), if |a − b| ≤ |x − y|. Usually, ∑n 1+T (ai ) is regarded as the weight of data ai . (1+T (ai )) i=1

3

Proposed Method

In the fusion of multiple sensors, two key problems, namely the representation of sensor report and fusion method of multiple sensors, should be considered. Here, a novel method is proposed to fuse multiple sensors based on the Dempster-Shafer theory and power average operator. Within the proposed method, mass function is used to represent the sensor report. Then a concept of relative credibility is defined to measure the credibility of each sensor related with other sensors based on the power average operator. Finally, the sensors are fused based on WBET method [17].

3.1

Collection of evidence from the sensor

In a complicated measurement system with multiple sensors, each sensor can obtain some information which are partial due to the limitation of observation. Moreover, these information might be fuzzy, imprecise and vague, etc. Thus various uncertainty is included in the report of sensor. Many methods, such as fuzzy set theory and rough set theory, can represent the uncertainty. Here, the Dempster-shafer theory is adopted to represent the report of sensor. There are

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mainly two reasons. At first, the mass function is flexible and clearly understandable to express uncertain information. As can be seen from the definition of mass function, it is very simple and understandable to indicate the report of sensor. At second, the generation of mass function is relatively easy and convenient. So, the report of sensor can be collected and represented as evidence or BPA.

3.2

Calculation of relative credibility of sensor

Once the report of sensor has been represented as evidence, the credibility or reliability of each sensor can be considered. The sensor with higher reliability should be assigned higher credibility. In the paper, an evidence-driven method is proposed to obtain the relative credibility of each sensor. At first, the support degree of a sensor by another sensor is defined as follows. Let < S1 , m1 >, < S2 , m2 > be two sensors S1 , S2 , and m1 , m2 are two evidences derived from S1 , S2 , respectively. The degree of sensor S1 supports sensor S2 is Sup(S1 , S2 ) = 1 − dBP A (m1 , m2 )

(7)

where dBP A (m1 , m2 ) is the Jousselme distance of evidences between m1 and m2 . It is noted that Sup(S2 , S1 ) = Sup(S1 , S2 ) due to dBP A (m2 , m1 ) = dBP A (m1 , m2 ). Then, assume there are n sensors, the total support degree of a sensor Si by other sensors can be calculated by n ∑ Tsup (Si ) = Sup(Si , Sj ) (8) j=1 j̸=i

Finally, the relative credibility of each sensor related with other sensors is calculated according to power average operator 1 + Tsup (Si ) ReCrd(Si ) = ∑ (9) n (1 + Tsup (Si )) i=1

It can be found that the derived credibility of each sensor is relative with other sensors. The sum of credibility of all sensors is equal to 1. In the process, only the evidences from sensors have been used, hence this is an evidence-driven method which makes full use of the sensor report.

3.3

Fusion of multiple sensors

In this paper, the report of sensor is indicated by mass function of Dempster-Shafer theory. Consequently, Dempster’s rule of combination can be used to fuse the evidences from multiple sensors. However, as Zadeh [12] pointed out, the classical Dempster’s rule of combination is unable to fuse conflicting evidences. To solve the problem, some methods, such as Murphy’s simple average combination method [13] and Deng et al.’s weighted average combination method [14], have been developed. In this paper, we use a method called weighted balance evidence theory (WBET) proposed by Guo and Zhang [17] to fuse the evidences derived from sensors. The

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WBET method combines the evidences in pairs until all mass functions have been combined. It is shown as follows. For two evidences indicated by m1 and m2 , their weights are w1 and w2 , where w1 + w2 = 1. ′ In WEBT method, to calculate the combination of m1 and m2 , denoted as m1 ⊕ m2 , firstly, the weighted average evidence m ¯ is obtained by m ¯ =

2 ∑

w i mi

(10)

i=1

Secondly, use m ¯ to balance the original evidences, the derived balanced evidences are indicated ′ ′ by m1 and m2 { ′ m1 = m1 , if w1 > w2 (11) ′ m 2 = 2m ¯ − m1 { ′ m1 = m1 , if w1 = w2 (12) ′ m2 = m2 { ′ m 1 = 2m ¯ − m2 , if w1 < w2 (13) ′ m2 = m2 ′

′

At last, combine m1 and m2 using classical Dempster’s rule of combination. This is the process of combining two evidences using WBET method. Supposing there are n sensors and their reports are indicated by < S1 , m1 , ReCrd1 >, < S2 , m2 , ReCrd2 >, · · · , < Si , mi , ReCrdi >, · · · , < Sn , mn , ReCrdn >, where ReCrdi is the relative credibility of sensor Si . The combination of these sources can be divided into multiple pairwise combinations of evidences as mentioned above. It should be noted that the evidences should be ordered according to their relative credibility. The evidence with higher relative credibility should be fused first. The weights of evidences is calculated by { wi = ReCrdi /(ReCrdi + ReCrdj ) (14) wj = ReCrdj /(ReCrdi + ReCrdj ) After fusing the two evidences, the relative credibility of combing evidence is set as (ReCrdi + ReCrdj ) to sequentially combine with other evidence. By this means, multiple sensors have been fused finally.

4

Numerical Example

In this section, a numerical example coming from literature [14] is given to demonstrate the effectiveness of the proposed method. A target is detected by a multisensor-based automatic target recognition system. Assume the real target is A. From five different sensors, the system has collected five bodies of evidence shown as follows • < S1 , m1 >= ([{A}, 0.5], [{B}, 0.2], [{C}, 0.3])

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• < S2 , m2 >= ([{A}, 0.0], [{B}, 0.9], [{C}, 0.1]) • < S3 , m3 >= ([{A}, 0.55], [{B}, 0.1], [{A, C}, 0.35]) • < S4 , m4 >= ([{A}, 0.55], [{B}, 0.1], [{A, C}, 0.35]) • < S5 , m5 >= ([{A}, 0.6], [{B}, 0.1], [{A, C}, 0.3]) Table 1: Total support degree and relative credibility of each sensor Sensor

S1

S2

S3

S4

S5

Total support degree

2.5865

1.0149

2.9174

2.9174

2.8741

Relative credibility

0.2072

0.1164

0.2263

0.2263

0.2238

Table 2: Fusion results of multiple sensors using different methods Combination method

m1 , m 2

m 1 , m 2 , m3

m1 , m 2 , m 3 , m 4

m1 , m2 , m3 , m4 , m5

Dempster’s

m(A) = 0

m(A) = 0

m(A) = 0

m(A) = 0

rule of

m(B) = 0.8571

m(B) = 0.6316

m(B) = 0.3288

m(B) = 0.1404

combination

m(C) = 0.1429

m(C) = 0.3684

m(C) = 0.6712

m(C) = 0.8596

Murphy’s average

m(A) = 0.1543

m(A) = 0.5568

m(A) = 0.8653

m(A) = 0.9688

combination rule [13]

m(B) = 0.7469

m(B) = 0.3562

m(B) = 0.0891

m(B) = 0.0156

m(C) = 0.0988

m(C) = 0.0782

m(C) = 0.0382

m(C) = 0.0127

m(A, C) = 0.0088

m(A, C) = 0.0074

m(A, C) = 0.0029

Deng et al.’s

m(A) = 0.1543

m(A) = 0.6500

m(A) = 0.9305

m(A) = 0.9846

weighted average

m(B) = 0.7469

m(B) = 0.2547

m(B) = 0.0274

m(B) = 0.0024

combination rule [14]

m(C) = 0.0988

m(C) = 0.0858

m(C) = 0.0339

m(C) = 0.0098

m(A, C) = 0.0095

m(A, C) = 0.0082

m(A, C) = 0.0032

Proposed

m(A) = 0

m(A) = 0.8571

m(A) = 0.9927

m(A) = 0.9996

method

m(B) = 0.8571

m(B) = 0.0617

m(B) = 0.0012

m(B) = 0

m(C) = 0.1429

m(C) = 0.0812

m(C) = 0.0060

m(C) = 0.0004

m(A, C) = 0.0001

m(A, C) = 0

According to our proposed method, firstly, the relative credibility of each sensors can be calculated. Take S1 as an example. The distance between the BPA of sensor S1 and BPAs of other sensors are dBP A (m1 , m2 ) = 0.6245, dBP A (m1 , m3 ) = 0.2622, dBP A (m1 , m4 ) = 0.2622, dBP A (m1 , m5 ) = 0.2646. So the support degree of S1 by other sensors are Sup(S1 , S2 ) = 0.3755,

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Sup(S1 , S3 ) = 0.7378, Sup(S1 , S4 ) = 0.7378, Sup(S1 , S5 ) = 0.7354. According to power average operator, Tsup (S1 ) = 2.5865. In a similar way, we can obtain Tsup (S2 ) = 1.0149, Tsup (S3 ) = 2.9174, Tsup (S4 ) = 2.9174, Tsup (S5 ) = 2.8741. Hence, the relative credibility of each sensor is derived, as shown in TABLE 1. Then, these sensors can be fused in pairs by using the WBET method. Due to the ranking of relative credibility of sensors is ReCrd3 ≻ ReCrd4 ≻ ReCrd5 ≻ ReCrd1 ≻ ReCrd2 , so the fusion ′ ′ ′ ′ ′ of these sensors is implemented by [[[[m3 ⊕ m4 ] ⊕ m5 ] ⊕ m1 ] ⊕ m2 ], where ⊕ represents the combination rule of WBET method. The combining result of these five sensors is m(A) = 0.9996, m(B) = 0, m(C) = 0.0004, m(A, C) = 0, as shown in TABLE 2. TABLE 2 also shows each step fusion result of evidence cumulation, and the fusion results of different methods. As can be seen from the TABLE 2, the real target A has been recognized, which has the largest value of support. With the cumulation of evidence, the support to proposition “the target is A” increase while the values of other propositions decrease, which means that the systems takes advantage of multi sensor data fusion technology. By comparing with other methods, the proposed method shows more effective, not only the correctness, but also the convergence speed. Obviously, the BPA m2 from sensor S2 is an abnormal evidence, which supports target B. It is conflicting with other sensors that support target A. Using Dempster’s rule of combination, a nonsense and counterintuitive conclusion is obtained that target C is with highest support. Murphy’s combination rule and Deng et al.’s method both can derive the correct conclusion. Comparing with these two methods, our proposed method shows higher speed in convergence. When m1 , m2 , m3 are considered, the support of target A reaches 0.8571. It is greatly faster than Murphy’s and Deng et al.’s method. When all sensor are fused, the support of target A is 0.9996, which is very close to 1. Therefore, the proposed method is more effective in the fusion of multiple sensors.

5

Conclusion

In this paper, two key problems in multisensor information fusion, the representation of sensor report and combination of multisensor information, have been addressed. A new method is proposed to solve the problems. Within the proposed method, mass function is adopted to express the sensor report, and the power average operator is used to obtain the relative credibility of each sensor. At last, the evidences coming from different sensors have been fused by using WBET method. A numerical example has demonstrated the effectiveness of the proposed method.

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