# The Improvement of DS Evidence Theory and Its Application in IR/MMW Target Recognition

Authors: Yibing Li; Jie Chen; Fang Ye; Dandan Liu
Journal: Journal of Sensors
 (2016)

---

## Abstract

ATR system has a broad application prospect in the military field, especially in the field of modern defense technology. When paradoxes are in existence in ATR system due to adverse battlefield environment, integration cannot be effectively and reliably carried out only by traditional DS evidence theory. In this paper, a modified DS evidence theory is presented and applied in IR/MMW target recognition system. The improvement of DS evidence theory is realized by three parts: the introduction of sensor priority and evidence credibility to realize the discount processing of evidences, the modification of DS combination rule to enhance the accuracy of synthesis results, and the compound decision-making rule. The application of the modified algorithm in IR/MMW system is designed to deal with paradoxes, improve the target recognition rate, and ensure the reliability of target recognition system. Experiments are given to illustrate that the introduction of the modified DS evidence theory in IR/MMW system is better able to realize satisfactory target recognition performance through multisensor information fusion than any single-mode system.


---

## Body

## 1. Introduction

Automatic target recognition system, namely, the ATR system, has become an important part of the present and the future weapon system. For ATR systems in the modern warfare, with the growing complexity of battlefield environment, information acquired from sensors is often incomplete and inaccurate and has some degree of uncertainty and fuzziness, possibly even contradictory. ATR systems with single sensor have many limitations like weak anti-interference ability and low recognition reliability. Single-mode systems cannot suit the demand of future battlefield because of their dependence on the observation environment. The complementarity and synergistic interaction of sensors can improve the ATR ability in dynamic scene. In consideration of defects and limitations of single-mode systems, IR/MMW fusion, as an inevitable trend of compound target recognition algorithm, can adopt others’ strong points while overcoming its weak point and make the system adapt to continued deterioration of battlefield environment and dynamical changes of objectives.Uncertainty reasoning is the foundation of IR/MMW system, which can deal with incomplete, uncertain, and unclear information that exists in IR/MMW system effectively. DS evidence theory, also known as Dempster-Shafer theory, is common and widely used uncertainty reasoning [1, 2]. It has been widely used in many fields like expert system [3, 4], artificial intelligence [5, 6], fault diagnosis [7, 8], target recognition [9, 10], target tracking [11], decision-making [12], information fusion [13], and so forth. However, some researches found that traditional DS evidence theory cannot produce reasonable synthesis results in the case of paradoxes (because of the complexity of practical environment and probable conflicts of evidence sources) [14]. Therefore, DS evidence theory is improved primarily before its application in IR/MMW system.Domestic and foreign researchers have done a lot of researches to solve paradoxes, which are mainly divided into two categories: the improved DS combination rules and the modified conflict evidences methods.Some researchers think that unreasonable results are mainly caused by the normalization step of DS combination rule. Thus, it optimizes the DS combination rule by giving evidence conflicts to certain subset with certain proportion, which is called the improved DS combination rule. In Smets’s opinion,Θ is regarded as an incomplete set and conflicts are given to an unknown proposition [15]. This method solves paradoxes theoretically, but it actually increases the uncertainty of synthesis results by introducing an unknown proposition. Yager allotted conflicts directly to Θ [16]. However, it can only settle paradoxes efficiently with two evidence sources in system, and it is too conservative to admit the useful information that exists in conflict evidences. On this basis, an improved method is proposed by dividing evidences into support evidences and conflict evidences, which solves the problem of unequal information quantity among evidences [17]. In addition, another algorithm proportionally allocates the conflicts to every focal element of conflict evidences through the introduction of a weight factor that is proportional to the conflicts [18]. In order to distinguish the local conflicts and global conflicts effectively and make the system robust simultaneously, a new DS combination rule named absorptive method is put forward by allocating local conflicts directly to local propositions [19].Other scholars consider that paradoxes are mainly caused by unreliable evidences. Therefore, this kind of method modifies evidences before evidences combine instead of changing the DS combination rule, which is called the modified conflict evidences method. Among them, an improved method is put forward by considering the average mean of evidences as a new evidence before evidences combination [20]. But it is obvious that the idea only averages the evidences without considering their differences. Thus, Horiuchi came up with a method through calculating a new evidence by the weight sum of evidences instead of simple average [21]. In order to efficiently combine high conflict evidences, a novel method is proposed through the introduction of the distance function introduced by Lefevre, which can improve the reliability of system [21]. To manage conflict evidences, global conflicts are allocated in detail to several local conflicts, and a new method is presented based on number, reliability, and relevance of evidence sources [22].These two mainstream improved methods solve paradoxes just on the sight of single angle without fully considering the differences among sensors and uncertainty of observation environment. In this paper, the novel algorithm firstly takes the consistency and reliability of evidence sources into consideration concurrently and thus introduces sensor priority and evidence credibility to realize discount processing of evidences. Then, it cancels the normalization step of DS combination rule to decrease the interference and unpredictability of observation environment and further enhance the reliability and rationality of synthesis results. Finally, it puts the synthesis results into decision-making rule to get target recognition results. Experiment results demonstrate that the application of the novel method in IR/MMW system can improve the target recognition rate, ensure reliable operation of system, and enhance the battlefield adaptability, antijamming immunity, antistealth performance, and precision of target identification.This paper is organized as follows. The foundation and discussions of traditional DS evidence theory are summarized in Section2. Then, as the cores of this paper, Section 3 highlights the modified DS evidence theory, and Section 4 provides the specific implementation steps of its application in IR/MMW target recognition system. In Section 5, experiment results and analyses are shown to manifest the validity of the novel algorithm from the points of theory and application. And conclusions are presented in Section 6 at the end.


## 2. DS Evidence Theory

Researches were done for ATR system using DS evidence theory back in the 90s [23, 24]. The feasibility of applying DS evidence theory in ATR system has been testified in [25]. And in [26], it is shown that, compared with Sugero’s theory and possibility theory, the best performance in ATR system is achieved by DS evidence theory. Thus, DS evidence theory makes the system not only more powerful but also more robust [27].


### 2.1. Preliminaries

DS evidence theory firstly supposes the definition of a finite nonempty set of hypotheses as the frame of discernment (calledF
o
D for short), which consists of N mutually exclusive and exhaustive hypotheses. The F
o
D is defined as follows:(1)
Θ
=
H
1
,
H
2
,
…
,
H
N
,where N is the number of hypotheses in system and H is a hypothesis of the F
o
D.For example, in pattern recognition system, there are a set ofN mutually disjoint classes in the pattern space. Thus, the F
o
D should be defined as Θ
=
{
w
1
,
w
2
,
…
,
w
N
}, where w is a class in the pattern space.Then, let us denote2
Θ as the power set, which is composed of 2
N propositions of Θ:(2)
2
Θ
=
Φ
,
H
1
,
H
2
,
…
,
H
N
,
H
1
∪
H
2
,
H
1
∪
H
3
,
…
,
H
1
∪
H
2
∪
⋯
H
N
,where Φ is the empty set and any proposition in 2
Θ is a subset of Θ.The basic probability assignment (calledB
P
A for short) on 2
Θ, also called the basic belief assignment (called B
B
A for short), is a function m
:
2
Θ
→
[
0,1
], which should satisfy the following conditions:(3)
m
Φ
=
0
∑
A
⊆
Θ
m
A
=
1
,where A is a proposition in 2
Θ which contains one or more hypotheses and m
(
A
) represents the initial support degree for proposition A.Due to the lack of further information, theB
P
A of proposition A cannot be subdivided into its proper subsets. Any proposition A satisfying m
(
A
)
>
0
(
A
⊆
Θ
) is called a focal element, and the set of all focal elements is called the core of B
P
A.The belief function (calledB
e
l for short) and plausibility function (called P
l for short) in 2
Θ are defined, respectively, as(4)
Bel
⁡
A
=
∑
B
⊆
A
m
B
Pl
⁡
A
=
∑
B
∩
A
≠
Φ
m
B
A
,
B
⊆
Θ
,where A and B are both the propositions in 2
Θ.It can be seen from formula (4) that B
e
l of proposition A is interpreted as the minimum uncertainty value of A which constitutes a lower limit function on the probability of A, while its P
l is interpreted as the maximum uncertainty value of A which constitutes an upper limit function. And the relationship between Bel
⁡
(
A
) and Pl
⁡
(
A
) is defined as follows, which is shown in Figure 1:(5)
Bel
⁡
A
≤
Pl
⁡
A
Pl
⁡
A
=
1
-
Bel
⁡
A
¯
A
⊆
Θ
,where A
¯ is the complement set of A.Figure 1
The relationship diagram ofBel
⁡
(
A
) and Pl
⁡
(
A
).The interval[
Bel
⁡
A
,
Pl
⁡
(
A
)
] is named the belief interval or uncertainty interval, which represents the uncertainty and imprecision of DS evidence theory. Suppose that m
1
,
m
2
,
…
,
m
N are N mutually independent B
P
A
s from N different sensors in the same F
o
D based on information detection. The DS combination rule, denoted by m
=
m
1
⊕
m
2
⊕
⋯
⊕
m
N, is also called the orthogonal sum of evidences. Thus, the combination of m
i
,
m
j
(
i
,
j
=
1,2
,
…
,
N
) can be defined as(6)
m
i
j
A
=
1
1
-
k
∑
A
i
∩
A
j
=
A
m
i
A
i
·
m
j
A
j
m
Φ
=
0
A
⊆
Θ
,
A
≠
Φ
,where k is called the total conflict factor and it represents the total conflicts between evidences m
i and m
j:(7)
k
=
∑
A
i
∩
A
j
=
Φ
m
i
A
i
·
m
j
A
j
.k demonstrates the degree of conflicts between evidences, and 1
/
1
-
k is the normalization factor which ensures that the sum of B
P
A
s can be unit, and the B
P
A for null set is none. According to similar principle, we can calculate corresponding Bel
⁡
(
A
) and Pl
⁡
(
A
).Obviously, the DS combination rule satisfies both commutative law and associate law, which are shown separately as(8)
m
1
⊕
m
2
=
m
2
⊕
m
1
m
1
⊕
m
2
⊕
m
3
=
m
1
⊕
m
2
⊕
m
3
.


### 2.2. Common Paradoxes

Uncertainty of system can be categorized into four areas shown in [28]. Firstly, knowledge representation can be defined as confirmed, probable, possible, doubtful, and improbable to support the propositions, but the system cannot guarantee its objectivity. Secondly, algorithms decide how the uncertainty model performs operations with information, which greatly affect uncertainty reduction. Thirdly, evidence representation usually depends on human experience, which has large invalidity. Lastly, the evidence sources are sensors that exist in environment with a variety of noises.In this paper, due to the scarce knowledge of observation environment, the system may import a lot of interference and clutter. And the limitation of sensor precision will ulteriorly cause uncertainty and imprecision in DS reasoning. Thus, the paradox research is the premise to analyze the defects of DS evidence theory and realize its improvement.The common paradoxes are divided into four classes: complete conflict paradox, 0 trust paradox (one ballot veto), 1 trust paradox, and high conflict paradox.


#### 2.2.1. Complete Conflict Paradox

Assuming that theF
o
D of system is Θ
=
{
A
,
B
,
C
} and there are two sources of evidence, the B
P
A
s are presented, respectively, as(9)
m
1
A
=
1
,
m
1
B
=
0
,
m
1
C
=
0
m
2
A
=
0
,
m
2
B
=
1
,
m
2
C
=
0
.According to formula (7), the total conflict factor k can be calculated as k
=
1. It is apparent that evidences m
1 and m
2 conflict completely, which causes the denominator of formula (6) to become zero. Under such circumstances, DS combination rule is unable to synthesize. If there are two more sources of evidence, whose B
P
A
s are shown as(10)
m
3
A
=
0.8
,
m
3
B
=
0.1
,
m
3
C
=
0.1
m
4
A
=
0.8
,
m
4
B
=
0.1
,
m
4
C
=
0.1
,from intuitive judgment, the accurate synthesis results should drastically support propositionA because evidences m
1, m
3, and m
4 all support proposition A with large B
P
A
s. But DS combination rule cannot normally be used when k
=
1. This kind of illogical situation is called complete conflict paradox.


#### 2.2.2.
0 Trust Paradox

Assume thatF
o
D of the system is Θ
=
{
A
,
B
,
C
}, and four B
P
A
s of evidences are(11)
m
1
A
=
0.5
,
m
1
B
=
0.2
,
m
1
C
=
0.3
m
2
A
=
0
,
m
2
B
=
0.9
,
m
2
C
=
0.1
m
3
A
=
0.5
,
m
3
B
=
0.2
,
m
3
C
=
0.3
m
4
A
=
0.5
,
m
4
B
=
0.2
,
m
4
C
=
0.3
.The total conflict factor can be calculated ask
=
0.99 in the same way as shown before. Applying formula (7), the synthesis results are(12)
m
A
=
0
,
m
B
=
0.727
,
m
C
=
0.273
.It can be checked that because evidencem
2 totally denies proposition A, the B
P
A for proposition A in the synthesis results will always be zero no matter how strongly evidences m
1, m
3, and m
4 support proposition A. That is, DS combination rule has the disadvantage of one ballot veto.


#### 2.2.3.
1 Trust Paradox

Assume that theF
o
D of system is Θ
=
{
A
,
B
,
C
}, and there are four evidences. The B
P
A
s are(13)
m
1
A
=
0.9
,
m
1
B
=
0.1
,
m
1
C
=
0
m
2
A
=
0
,
m
2
B
=
0.1
,
m
2
C
=
0.9
m
3
A
=
0.1
,
m
3
B
=
0.15
,
m
3
C
=
0.75
m
4
A
=
0.1
,
m
4
B
=
0.15
,
m
4
C
=
0.75
.The total conflict factor can be calculated ask
=
0.9998 and the synthesis results are(14)
m
A
=
0
,
m
B
=
1
,
m
C
=
0
.Although all sources of evidence give propositionB small B
P
A
s, the synthesis results completely believe proposition B is the correct proposition, which is perverse in practical application.


#### 2.2.4. High Conflict Paradox

Assume that theF
o
D of system is Θ
=
{
A
,
B
,
C
,
D
,
E
}, and there are five evidences. The B
P
A
s and synthesis results are, respectively, presented in formula (15) and formula (16).The total conflict factor can be calculated ask
=
0.9999. It can be proved in a similar way that precise synthesis results should support proposition A as evidences m
1, m
3, m
4, and m
5 all give proposition A large B
P
A
s. But high conflicts among evidences actually lead to error reasoning shown in formula (16). Hence,(15)
m
1
A
=
0.7
,
m
1
B
=
0.1
,
m
1
C
=
0.1
,
m
1
D
=
0
,
m
1
E
=
0.1
m
2
A
=
0
,
m
2
B
=
0.5
,
m
2
C
=
0.2
,
m
2
D
=
0.1
,
m
2
E
=
0.2
m
3
A
=
0.6
,
m
3
B
=
0.1
,
m
3
C
=
0.15
,
m
3
D
=
0
,
m
3
E
=
0.15
m
4
A
=
0.55
,
m
4
B
=
0.1
,
m
4
C
=
0.1
,
m
4
D
=
0.15
,
m
4
E
=
0.1
m
5
A
=
0.6
,
m
5
B
=
0.1
,
m
5
C
=
0.2
,
m
5
D
=
0
,
m
5
E
=
0.1
(16)
m
A
=
0
,
m
B
=
0.3571
,
m
C
=
0.4286
,
m
D
=
0
,
m
E
=
0.2143
.


### 2.3. Relationship among Bayesian Theory, DS Evidence Theory, and DSmT Reasoning

Relative to probability theory and Bayesian theory, DS evidence theory can settle imprecise information in the absence of prior knowledge and can be viewed as a generalization of probability theory. It can be seen in the previous section that DS evidence theory cannot handle the information with paradoxes. In order to overcome the problem, Jean Dezert and Florentin Smarandache put forward the DSmT reasoning which can combine uncertain, imprecise, and contradictory information [29]. DSmT reasoning is an extension of traditional DS evidence theory, and it is widely applied in edge detection in color images [30], aircraft recognition [31], remote sensing image classification [32], and so forth.In 2003, Jean Dezert presents the hyperpower set notation asD
Θ for DSmT. It retains the contradictory focal elements, while greatly increasing the total number of focal elements. Examples are given as follows to show the increasing calculational complexity of DSmT compared with DS evidence theory:(1)
Θ
=
{
}(empty): 2
Θ
=
{
Φ
}, D
Θ
=
{
Φ
}.
(2)
Θ
=
{
H
1
}: 2
Θ
=
{
Φ
,
H
1
}, D
Θ
=
{
Φ
,
H
1
}.
(3)
Θ
=
{
H
1
,
H
2
}: 2
Θ
=
{
Φ
,
H
1
,
H
2
,
H
1
∪
H
2
}, D
Θ
=
{
Φ
,
H
1
,
H
2
,
H
1
∪
H
2
,
H
1
∩
H
2
}.
(4)
Θ
=
{
H
1
,
H
2
,
H
3
}: 2
Θ
=
{
Φ
,
H
1
,
H
2
,
H
3
,
H
1
∪
H
2
,
H
1
∪
H
3
,
H
2
∪
H
3
,
H
1
∪
H
2
∪
H
3
}, D
Θ
=
{
Φ
,
H
1
,
H
2
,
H
3
,
H
1
∪
H
2
,
H
1
∪
H
3
,
H
2
∪
H
3
,
H
1
∩
H
2
,
H
1
∩
H
3
,
H
2
∩
H
3
,
H
1
∪
H
2
∪
H
3, H
1
∩
H
2
∩
H
3
,
(
H
1
∪
H
2
)
∩
H
3
,
(
H
1
∪
H
3
)
∩
H
2
,
(
H
2
∪
H
3
)
∩
H
1
,
(
H
1
∩
H
2
)
∪
H
3, (
H
1
∩
H
3
)
∪
H
2
,
(
H
2
∩
H
3
)
∪
H
1
,
(
H
1
∪
H
2
)
∩
(
H
1
∩
H
3
)
∩
(
H
2
∪
H
3
)
}.From these discussions, it is obvious that DS evidence theory is an extension of Bayesian theory, and DSmT reasoning is viewed as a general flexible approach for managing uncertainty and conflicts for a wide class of fusion problems where the information to combine is modeled as a finite set of belief functions provided by different independent sources [33, 34]. And these three algorithms are used in similar applications. It is easy to check that, with the increasing number of F
o
D, the gap on the calculational amount between DS evidence theory and DSmT reasoning becomes larger. Thus, considering DSmT’s large computational burden, this paper uses DS evidence theory as the uncertain reasoning method in the IR/MMW target recognition system, which can not only ensure the identification speed of system, but also build the basis for DSmT’s application implementation.


### 2.4. Comparison between Two Mainstream Improved Methods and PCR5

In the DSmT reasoning’sF
o
D, proportional conflict redistribution 5 (called P
C
R
5 for short) is used generally to combine the B
P
A
s [33]. P
C
R
5 transfers the conflict mass only to the elements involved in the conflict and proportionally to their individual masses, which is similar to the way of assigning local conflicts directly to local propositions [19]. As P
C
R
5 can be used in DS evidence theory as well, we take two mainstream improved methods to give a simple comparison.TheP
C
R
5 rule in DS evidence theory is defined as follows (assuming that there are two evidences):(17)
m
A
=
∑
A
1
∩
A
2
=
A
m
1
A
1
·
m
2
A
2
+
Δ
A
,
2
Θ
,where Δ
(
A
,
2
Θ
)
=
∑
A
3
∩
A
=
Φ
[
m
1
(
A
)
2
·
m
2
(
A
3
)
/
m
1
(
A
)
+
m
2
(
A
3
)
+
m
2
(
A
)
2
·
m
1
(
A
3
)
/
m
2
(
A
)
+
m
1
(
A
3
)
] and A
1
,
A
2
,
A
3
⊆
2
Θ.The general description of the improved DS combination rules presented in Section1 can be shown as(18)
m
A
=
∑
A
1
∩
A
2
=
A
m
1
A
1
·
m
2
A
2
+
Δ
A
,
Θ
,where Δ
(
A
,
Θ
) depends on the specific algorithm and A
1
,
A
2
⊆
Θ. In Yager [16] and Sun et al. [17], Δ
Y
a
g
e
r
(
A
,
Θ
)
=
k, if A
=
Θ; otherwise, it becomes zero, and Δ
S
u
n
(
A
,
Θ
)
=
k
·
e
-
k
·
(
m
1
(
A
)
+
m
2
(
A
)
/
2
) when system has two evidence sources.According to formulae (17) and (18), it is easy to see that the improved part changes from Δ
(
A
,
Θ
) in DS evidence theory to Δ
(
A
,
2
Θ
) in P
C
R
5, which demonstrates that P
C
R
5 rule can get higher combination precision with extra and larger computational complexity.In addition, the general description of the modified conflict evidences methods expressed in Section1 is shown as (assuming that there are n evidences)(19)
m
A
=
∑
A
i
⊆
Θ
m
i
A
i
·
w
i
A
i
,where w
i
(
A
i
) denotes the weighted factor assigned to m
i
(
A
i
). In Murphy [20] and Deng et al. [35], w
i
-Murphy
(
A
i
)
=
1
/
n, where n is the evidence number, and w
i
-Deng
(
A
i
) represents evidence credibility gotten from the introduction of Euclidean distance.Through comparison betweenP
C
R
5 and the modified conflict evidences method in formulae (17) and (19), it is easy to verify that, different from P
C
R
5 rule, this kind of approach takes the ambient noises into consideration and handles conflicts from the perspective of evidence credibility, which is more proximate to actual applications.From what was mentioned above,P
C
R
5 rule in DS evidence theory can be classified into the improved DS combination rules with better validity and larger computational complexity, and the modified conflict evidences methods take factors like environmental noises and sensors precision into account, which have better ability of practical applications relative to P
C
R
5 rule. Thus, in order to get accurate, effective, and fast target recognition performance in IR/MMW system, this paper chooses to combine two mainstream improved methods and proposes the modified DS evidence theory presented in the next section.


## 2.1. Preliminaries

DS evidence theory firstly supposes the definition of a finite nonempty set of hypotheses as the frame of discernment (calledF
o
D for short), which consists of N mutually exclusive and exhaustive hypotheses. The F
o
D is defined as follows:(1)
Θ
=
H
1
,
H
2
,
…
,
H
N
,where N is the number of hypotheses in system and H is a hypothesis of the F
o
D.For example, in pattern recognition system, there are a set ofN mutually disjoint classes in the pattern space. Thus, the F
o
D should be defined as Θ
=
{
w
1
,
w
2
,
…
,
w
N
}, where w is a class in the pattern space.Then, let us denote2
Θ as the power set, which is composed of 2
N propositions of Θ:(2)
2
Θ
=
Φ
,
H
1
,
H
2
,
…
,
H
N
,
H
1
∪
H
2
,
H
1
∪
H
3
,
…
,
H
1
∪
H
2
∪
⋯
H
N
,where Φ is the empty set and any proposition in 2
Θ is a subset of Θ.The basic probability assignment (calledB
P
A for short) on 2
Θ, also called the basic belief assignment (called B
B
A for short), is a function m
:
2
Θ
→
[
0,1
], which should satisfy the following conditions:(3)
m
Φ
=
0
∑
A
⊆
Θ
m
A
=
1
,where A is a proposition in 2
Θ which contains one or more hypotheses and m
(
A
) represents the initial support degree for proposition A.Due to the lack of further information, theB
P
A of proposition A cannot be subdivided into its proper subsets. Any proposition A satisfying m
(
A
)
>
0
(
A
⊆
Θ
) is called a focal element, and the set of all focal elements is called the core of B
P
A.The belief function (calledB
e
l for short) and plausibility function (called P
l for short) in 2
Θ are defined, respectively, as(4)
Bel
⁡
A
=
∑
B
⊆
A
m
B
Pl
⁡
A
=
∑
B
∩
A
≠
Φ
m
B
A
,
B
⊆
Θ
,where A and B are both the propositions in 2
Θ.It can be seen from formula (4) that B
e
l of proposition A is interpreted as the minimum uncertainty value of A which constitutes a lower limit function on the probability of A, while its P
l is interpreted as the maximum uncertainty value of A which constitutes an upper limit function. And the relationship between Bel
⁡
(
A
) and Pl
⁡
(
A
) is defined as follows, which is shown in Figure 1:(5)
Bel
⁡
A
≤
Pl
⁡
A
Pl
⁡
A
=
1
-
Bel
⁡
A
¯
A
⊆
Θ
,where A
¯ is the complement set of A.Figure 1
The relationship diagram ofBel
⁡
(
A
) and Pl
⁡
(
A
).The interval[
Bel
⁡
A
,
Pl
⁡
(
A
)
] is named the belief interval or uncertainty interval, which represents the uncertainty and imprecision of DS evidence theory. Suppose that m
1
,
m
2
,
…
,
m
N are N mutually independent B
P
A
s from N different sensors in the same F
o
D based on information detection. The DS combination rule, denoted by m
=
m
1
⊕
m
2
⊕
⋯
⊕
m
N, is also called the orthogonal sum of evidences. Thus, the combination of m
i
,
m
j
(
i
,
j
=
1,2
,
…
,
N
) can be defined as(6)
m
i
j
A
=
1
1
-
k
∑
A
i
∩
A
j
=
A
m
i
A
i
·
m
j
A
j
m
Φ
=
0
A
⊆
Θ
,
A
≠
Φ
,where k is called the total conflict factor and it represents the total conflicts between evidences m
i and m
j:(7)
k
=
∑
A
i
∩
A
j
=
Φ
m
i
A
i
·
m
j
A
j
.k demonstrates the degree of conflicts between evidences, and 1
/
1
-
k is the normalization factor which ensures that the sum of B
P
A
s can be unit, and the B
P
A for null set is none. According to similar principle, we can calculate corresponding Bel
⁡
(
A
) and Pl
⁡
(
A
).Obviously, the DS combination rule satisfies both commutative law and associate law, which are shown separately as(8)
m
1
⊕
m
2
=
m
2
⊕
m
1
m
1
⊕
m
2
⊕
m
3
=
m
1
⊕
m
2
⊕
m
3
.


## 2.2. Common Paradoxes

Uncertainty of system can be categorized into four areas shown in [28]. Firstly, knowledge representation can be defined as confirmed, probable, possible, doubtful, and improbable to support the propositions, but the system cannot guarantee its objectivity. Secondly, algorithms decide how the uncertainty model performs operations with information, which greatly affect uncertainty reduction. Thirdly, evidence representation usually depends on human experience, which has large invalidity. Lastly, the evidence sources are sensors that exist in environment with a variety of noises.In this paper, due to the scarce knowledge of observation environment, the system may import a lot of interference and clutter. And the limitation of sensor precision will ulteriorly cause uncertainty and imprecision in DS reasoning. Thus, the paradox research is the premise to analyze the defects of DS evidence theory and realize its improvement.The common paradoxes are divided into four classes: complete conflict paradox, 0 trust paradox (one ballot veto), 1 trust paradox, and high conflict paradox.


### 2.2.1. Complete Conflict Paradox

Assuming that theF
o
D of system is Θ
=
{
A
,
B
,
C
} and there are two sources of evidence, the B
P
A
s are presented, respectively, as(9)
m
1
A
=
1
,
m
1
B
=
0
,
m
1
C
=
0
m
2
A
=
0
,
m
2
B
=
1
,
m
2
C
=
0
.According to formula (7), the total conflict factor k can be calculated as k
=
1. It is apparent that evidences m
1 and m
2 conflict completely, which causes the denominator of formula (6) to become zero. Under such circumstances, DS combination rule is unable to synthesize. If there are two more sources of evidence, whose B
P
A
s are shown as(10)
m
3
A
=
0.8
,
m
3
B
=
0.1
,
m
3
C
=
0.1
m
4
A
=
0.8
,
m
4
B
=
0.1
,
m
4
C
=
0.1
,from intuitive judgment, the accurate synthesis results should drastically support propositionA because evidences m
1, m
3, and m
4 all support proposition A with large B
P
A
s. But DS combination rule cannot normally be used when k
=
1. This kind of illogical situation is called complete conflict paradox.


### 2.2.2.
0 Trust Paradox

Assume thatF
o
D of the system is Θ
=
{
A
,
B
,
C
}, and four B
P
A
s of evidences are(11)
m
1
A
=
0.5
,
m
1
B
=
0.2
,
m
1
C
=
0.3
m
2
A
=
0
,
m
2
B
=
0.9
,
m
2
C
=
0.1
m
3
A
=
0.5
,
m
3
B
=
0.2
,
m
3
C
=
0.3
m
4
A
=
0.5
,
m
4
B
=
0.2
,
m
4
C
=
0.3
.The total conflict factor can be calculated ask
=
0.99 in the same way as shown before. Applying formula (7), the synthesis results are(12)
m
A
=
0
,
m
B
=
0.727
,
m
C
=
0.273
.It can be checked that because evidencem
2 totally denies proposition A, the B
P
A for proposition A in the synthesis results will always be zero no matter how strongly evidences m
1, m
3, and m
4 support proposition A. That is, DS combination rule has the disadvantage of one ballot veto.


### 2.2.3.
1 Trust Paradox

Assume that theF
o
D of system is Θ
=
{
A
,
B
,
C
}, and there are four evidences. The B
P
A
s are(13)
m
1
A
=
0.9
,
m
1
B
=
0.1
,
m
1
C
=
0
m
2
A
=
0
,
m
2
B
=
0.1
,
m
2
C
=
0.9
m
3
A
=
0.1
,
m
3
B
=
0.15
,
m
3
C
=
0.75
m
4
A
=
0.1
,
m
4
B
=
0.15
,
m
4
C
=
0.75
.The total conflict factor can be calculated ask
=
0.9998 and the synthesis results are(14)
m
A
=
0
,
m
B
=
1
,
m
C
=
0
.Although all sources of evidence give propositionB small B
P
A
s, the synthesis results completely believe proposition B is the correct proposition, which is perverse in practical application.


### 2.2.4. High Conflict Paradox

Assume that theF
o
D of system is Θ
=
{
A
,
B
,
C
,
D
,
E
}, and there are five evidences. The B
P
A
s and synthesis results are, respectively, presented in formula (15) and formula (16).The total conflict factor can be calculated ask
=
0.9999. It can be proved in a similar way that precise synthesis results should support proposition A as evidences m
1, m
3, m
4, and m
5 all give proposition A large B
P
A
s. But high conflicts among evidences actually lead to error reasoning shown in formula (16). Hence,(15)
m
1
A
=
0.7
,
m
1
B
=
0.1
,
m
1
C
=
0.1
,
m
1
D
=
0
,
m
1
E
=
0.1
m
2
A
=
0
,
m
2
B
=
0.5
,
m
2
C
=
0.2
,
m
2
D
=
0.1
,
m
2
E
=
0.2
m
3
A
=
0.6
,
m
3
B
=
0.1
,
m
3
C
=
0.15
,
m
3
D
=
0
,
m
3
E
=
0.15
m
4
A
=
0.55
,
m
4
B
=
0.1
,
m
4
C
=
0.1
,
m
4
D
=
0.15
,
m
4
E
=
0.1
m
5
A
=
0.6
,
m
5
B
=
0.1
,
m
5
C
=
0.2
,
m
5
D
=
0
,
m
5
E
=
0.1
(16)
m
A
=
0
,
m
B
=
0.3571
,
m
C
=
0.4286
,
m
D
=
0
,
m
E
=
0.2143
.


## 2.2.1. Complete Conflict Paradox

Assuming that theF
o
D of system is Θ
=
{
A
,
B
,
C
} and there are two sources of evidence, the B
P
A
s are presented, respectively, as(9)
m
1
A
=
1
,
m
1
B
=
0
,
m
1
C
=
0
m
2
A
=
0
,
m
2
B
=
1
,
m
2
C
=
0
.According to formula (7), the total conflict factor k can be calculated as k
=
1. It is apparent that evidences m
1 and m
2 conflict completely, which causes the denominator of formula (6) to become zero. Under such circumstances, DS combination rule is unable to synthesize. If there are two more sources of evidence, whose B
P
A
s are shown as(10)
m
3
A
=
0.8
,
m
3
B
=
0.1
,
m
3
C
=
0.1
m
4
A
=
0.8
,
m
4
B
=
0.1
,
m
4
C
=
0.1
,from intuitive judgment, the accurate synthesis results should drastically support propositionA because evidences m
1, m
3, and m
4 all support proposition A with large B
P
A
s. But DS combination rule cannot normally be used when k
=
1. This kind of illogical situation is called complete conflict paradox.


## 2.2.2.
0 Trust Paradox

Assume thatF
o
D of the system is Θ
=
{
A
,
B
,
C
}, and four B
P
A
s of evidences are(11)
m
1
A
=
0.5
,
m
1
B
=
0.2
,
m
1
C
=
0.3
m
2
A
=
0
,
m
2
B
=
0.9
,
m
2
C
=
0.1
m
3
A
=
0.5
,
m
3
B
=
0.2
,
m
3
C
=
0.3
m
4
A
=
0.5
,
m
4
B
=
0.2
,
m
4
C
=
0.3
.The total conflict factor can be calculated ask
=
0.99 in the same way as shown before. Applying formula (7), the synthesis results are(12)
m
A
=
0
,
m
B
=
0.727
,
m
C
=
0.273
.It can be checked that because evidencem
2 totally denies proposition A, the B
P
A for proposition A in the synthesis results will always be zero no matter how strongly evidences m
1, m
3, and m
4 support proposition A. That is, DS combination rule has the disadvantage of one ballot veto.


## 2.2.3.
1 Trust Paradox

Assume that theF
o
D of system is Θ
=
{
A
,
B
,
C
}, and there are four evidences. The B
P
A
s are(13)
m
1
A
=
0.9
,
m
1
B
=
0.1
,
m
1
C
=
0
m
2
A
=
0
,
m
2
B
=
0.1
,
m
2
C
=
0.9
m
3
A
=
0.1
,
m
3
B
=
0.15
,
m
3
C
=
0.75
m
4
A
=
0.1
,
m
4
B
=
0.15
,
m
4
C
=
0.75
.The total conflict factor can be calculated ask
=
0.9998 and the synthesis results are(14)
m
A
=
0
,
m
B
=
1
,
m
C
=
0
.Although all sources of evidence give propositionB small B
P
A
s, the synthesis results completely believe proposition B is the correct proposition, which is perverse in practical application.


## 2.2.4. High Conflict Paradox

Assume that theF
o
D of system is Θ
=
{
A
,
B
,
C
,
D
,
E
}, and there are five evidences. The B
P
A
s and synthesis results are, respectively, presented in formula (15) and formula (16).The total conflict factor can be calculated ask
=
0.9999. It can be proved in a similar way that precise synthesis results should support proposition A as evidences m
1, m
3, m
4, and m
5 all give proposition A large B
P
A
s. But high conflicts among evidences actually lead to error reasoning shown in formula (16). Hence,(15)
m
1
A
=
0.7
,
m
1
B
=
0.1
,
m
1
C
=
0.1
,
m
1
D
=
0
,
m
1
E
=
0.1
m
2
A
=
0
,
m
2
B
=
0.5
,
m
2
C
=
0.2
,
m
2
D
=
0.1
,
m
2
E
=
0.2
m
3
A
=
0.6
,
m
3
B
=
0.1
,
m
3
C
=
0.15
,
m
3
D
=
0
,
m
3
E
=
0.15
m
4
A
=
0.55
,
m
4
B
=
0.1
,
m
4
C
=
0.1
,
m
4
D
=
0.15
,
m
4
E
=
0.1
m
5
A
=
0.6
,
m
5
B
=
0.1
,
m
5
C
=
0.2
,
m
5
D
=
0
,
m
5
E
=
0.1
(16)
m
A
=
0
,
m
B
=
0.3571
,
m
C
=
0.4286
,
m
D
=
0
,
m
E
=
0.2143
.


## 2.3. Relationship among Bayesian Theory, DS Evidence Theory, and DSmT Reasoning

Relative to probability theory and Bayesian theory, DS evidence theory can settle imprecise information in the absence of prior knowledge and can be viewed as a generalization of probability theory. It can be seen in the previous section that DS evidence theory cannot handle the information with paradoxes. In order to overcome the problem, Jean Dezert and Florentin Smarandache put forward the DSmT reasoning which can combine uncertain, imprecise, and contradictory information [29]. DSmT reasoning is an extension of traditional DS evidence theory, and it is widely applied in edge detection in color images [30], aircraft recognition [31], remote sensing image classification [32], and so forth.In 2003, Jean Dezert presents the hyperpower set notation asD
Θ for DSmT. It retains the contradictory focal elements, while greatly increasing the total number of focal elements. Examples are given as follows to show the increasing calculational complexity of DSmT compared with DS evidence theory:(1)
Θ
=
{
}(empty): 2
Θ
=
{
Φ
}, D
Θ
=
{
Φ
}.
(2)
Θ
=
{
H
1
}: 2
Θ
=
{
Φ
,
H
1
}, D
Θ
=
{
Φ
,
H
1
}.
(3)
Θ
=
{
H
1
,
H
2
}: 2
Θ
=
{
Φ
,
H
1
,
H
2
,
H
1
∪
H
2
}, D
Θ
=
{
Φ
,
H
1
,
H
2
,
H
1
∪
H
2
,
H
1
∩
H
2
}.
(4)
Θ
=
{
H
1
,
H
2
,
H
3
}: 2
Θ
=
{
Φ
,
H
1
,
H
2
,
H
3
,
H
1
∪
H
2
,
H
1
∪
H
3
,
H
2
∪
H
3
,
H
1
∪
H
2
∪
H
3
}, D
Θ
=
{
Φ
,
H
1
,
H
2
,
H
3
,
H
1
∪
H
2
,
H
1
∪
H
3
,
H
2
∪
H
3
,
H
1
∩
H
2
,
H
1
∩
H
3
,
H
2
∩
H
3
,
H
1
∪
H
2
∪
H
3, H
1
∩
H
2
∩
H
3
,
(
H
1
∪
H
2
)
∩
H
3
,
(
H
1
∪
H
3
)
∩
H
2
,
(
H
2
∪
H
3
)
∩
H
1
,
(
H
1
∩
H
2
)
∪
H
3, (
H
1
∩
H
3
)
∪
H
2
,
(
H
2
∩
H
3
)
∪
H
1
,
(
H
1
∪
H
2
)
∩
(
H
1
∩
H
3
)
∩
(
H
2
∪
H
3
)
}.From these discussions, it is obvious that DS evidence theory is an extension of Bayesian theory, and DSmT reasoning is viewed as a general flexible approach for managing uncertainty and conflicts for a wide class of fusion problems where the information to combine is modeled as a finite set of belief functions provided by different independent sources [33, 34]. And these three algorithms are used in similar applications. It is easy to check that, with the increasing number of F
o
D, the gap on the calculational amount between DS evidence theory and DSmT reasoning becomes larger. Thus, considering DSmT’s large computational burden, this paper uses DS evidence theory as the uncertain reasoning method in the IR/MMW target recognition system, which can not only ensure the identification speed of system, but also build the basis for DSmT’s application implementation.


## 2.4. Comparison between Two Mainstream Improved Methods and PCR5

In the DSmT reasoning’sF
o
D, proportional conflict redistribution 5 (called P
C
R
5 for short) is used generally to combine the B
P
A
s [33]. P
C
R
5 transfers the conflict mass only to the elements involved in the conflict and proportionally to their individual masses, which is similar to the way of assigning local conflicts directly to local propositions [19]. As P
C
R
5 can be used in DS evidence theory as well, we take two mainstream improved methods to give a simple comparison.TheP
C
R
5 rule in DS evidence theory is defined as follows (assuming that there are two evidences):(17)
m
A
=
∑
A
1
∩
A
2
=
A
m
1
A
1
·
m
2
A
2
+
Δ
A
,
2
Θ
,where Δ
(
A
,
2
Θ
)
=
∑
A
3
∩
A
=
Φ
[
m
1
(
A
)
2
·
m
2
(
A
3
)
/
m
1
(
A
)
+
m
2
(
A
3
)
+
m
2
(
A
)
2
·
m
1
(
A
3
)
/
m
2
(
A
)
+
m
1
(
A
3
)
] and A
1
,
A
2
,
A
3
⊆
2
Θ.The general description of the improved DS combination rules presented in Section1 can be shown as(18)
m
A
=
∑
A
1
∩
A
2
=
A
m
1
A
1
·
m
2
A
2
+
Δ
A
,
Θ
,where Δ
(
A
,
Θ
) depends on the specific algorithm and A
1
,
A
2
⊆
Θ. In Yager [16] and Sun et al. [17], Δ
Y
a
g
e
r
(
A
,
Θ
)
=
k, if A
=
Θ; otherwise, it becomes zero, and Δ
S
u
n
(
A
,
Θ
)
=
k
·
e
-
k
·
(
m
1
(
A
)
+
m
2
(
A
)
/
2
) when system has two evidence sources.According to formulae (17) and (18), it is easy to see that the improved part changes from Δ
(
A
,
Θ
) in DS evidence theory to Δ
(
A
,
2
Θ
) in P
C
R
5, which demonstrates that P
C
R
5 rule can get higher combination precision with extra and larger computational complexity.In addition, the general description of the modified conflict evidences methods expressed in Section1 is shown as (assuming that there are n evidences)(19)
m
A
=
∑
A
i
⊆
Θ
m
i
A
i
·
w
i
A
i
,where w
i
(
A
i
) denotes the weighted factor assigned to m
i
(
A
i
). In Murphy [20] and Deng et al. [35], w
i
-Murphy
(
A
i
)
=
1
/
n, where n is the evidence number, and w
i
-Deng
(
A
i
) represents evidence credibility gotten from the introduction of Euclidean distance.Through comparison betweenP
C
R
5 and the modified conflict evidences method in formulae (17) and (19), it is easy to verify that, different from P
C
R
5 rule, this kind of approach takes the ambient noises into consideration and handles conflicts from the perspective of evidence credibility, which is more proximate to actual applications.From what was mentioned above,P
C
R
5 rule in DS evidence theory can be classified into the improved DS combination rules with better validity and larger computational complexity, and the modified conflict evidences methods take factors like environmental noises and sensors precision into account, which have better ability of practical applications relative to P
C
R
5 rule. Thus, in order to get accurate, effective, and fast target recognition performance in IR/MMW system, this paper chooses to combine two mainstream improved methods and proposes the modified DS evidence theory presented in the next section.


## 3. The Modified DS Evidence Theory

The modified DS evidence theory proposed in this paper is designed to deal with the paradoxes caused by environmental uncertainty and imperfect knowledge and make the novel algorithm more suitable for IR/MMW target recognition system. It imitates the human brain’s information reception processing and integration processing to realize information fusion in IR/MMW system. It firstly accurately assigns sensor priority through classifying evidences from different sources by their type and precision and calculates evidence credibility of each evidence through introduction of Minkowski distance, then modifies DS combination rule to avoid probable paradoxes caused by the normalization step, and, at last, uses decision-making rule to produce reliable synthesis results.


### 3.1. Discount Processing of Evidences

The discount processing of evidences is composed of two parts. One is the definition of source priority for distinguishing the differences of data obtained from different sensors according to their types and precision, and the other is the introduction of evidence credibility for analyses of different reliability of data from different sensors in consideration of weather and observation environment. The discount processing of evidences can decrease the conflicts caused by environmental uncertainty and differences among sensors, which is the precondition of DS combination rule.


#### 3.1.1. Sensor Priority

Suppose that there areM sensors with N different types in IR/MMW system; we regard the data received from different sensors as the sources of evidence, whose number is also M and type is N. Because the system has N types of sensors, sensor priorities are also quantized into N levels. According to the differences of sensor types and precision, the sensor priority can be defined to represent the dominance and importance of different sensors under certain identification mission. And for a particular environment of target recognition, sensor priorities of different sensors differ and satisfy pr
⁡
(
m
i
)
∈
[
0,1
]
(
i
=
1,2
,
…
,
N
).This paper only assigns sensor priority by considering sensor types and precision to meet real-time demand of system; readers can take more factors into account by analogy. As sensor priorities are deemed to be the prior knowledge, assignment of evidence priorities depends on practical application. An example will make it clear: If there are five sensors in IR/MMW system with three same precision activeI
R
s and two same precision passive M
M
W
s, then these three active I
R
s must have the same sensor priorities, and two passive M
M
W
s also have the same sensor priorities. The concrete value of sensor priority should be set by artificial operators based on system requirements. If system requires high spatial resolution and strong penetrating power, M
M
W
s have obvious superiority and their sensor priorities should be higher than that of I
R
s, such as assignment as pr
⁡
(
m
I
R
s
)
=
0.9, pr
⁡
(
m
M
M
R
s
)
=
0.4. On the contrary, if system demands great night work ability, sensor priorities of I
R
s will be much higher than that of M
M
W
s.


#### 3.1.2. Evidence Credibility

Sensor priority takes inequality and inconsistence caused by different types and precision of sensors into account, but uncertainty caused by factors like weather, observation environment, and missile-target distance is still the obstacle of obtaining accurate synthesis results. In order to relieve uncertainty and imperfection among evidences, evidence credibility is put forward to evaluate the reliability of different sensors before evidence combines.Primarily, Minkowski distance is introduced as a distance function between two evidences. Assume that there areN evidence sources, and m
i and m
j
(
i
,
j
=
1,2
,
…
,
N
) are two B
P
A
s of mutually exclusive and exhaustive F
o
D
:
Θ
=
{
A
1
,
A
2
,
…
,
A
M
}, which are, respectively, shown as(20)
m
i
A
1
=
p
1
,
m
i
A
2
=
p
2
,
…
,
m
i
A
M
=
p
M
m
j
A
1
=
q
1
,
m
j
A
2
=
q
2
,
…
,
m
j
A
M
=
q
M
.Minkowski distance is a distance measurement in European space, and it is an extension of Euclidean distance and Manhattan distance. According to the introduction of Minkowski distance, the distance function ofm
i and m
j can be calculated as(21)
d
i
j
=
d
m
i
,
m
j
=
∑
l
=
1
M
p
l
-
q
l
m
1
/
m
,where m is a variable parameter. By definition, Minkowski distance will degrade into Manhattan distance when m
=
1, Euclidean distance when m
=
2, and Chebyshev distance when m
→
∞. In this paper, m is unified to be 2 for the experimental convenience in Section 5.Thus, the distance matrix is defined as(22)
DM
=
0
d
12
⋯
d
1
N
d
21
0
⋯
d
2
N
⋮
⋮
⋱
⋮
d
N
1
d
N
2
⋯
0
.Then, the similarity function between two evidences is introduced as(23)
s
i
j
=
s
m
i
,
m
j
=
1
-
d
i
j
.It is obvious that the similarity function between two evidences will increase as the distance function decreases.The similarity matrix of evidences is defined in the same way as shown before:(24)
SM
=
1
s
12
⋯
s
1
N
s
21
1
⋯
s
2
N
⋮
⋮
⋱
⋮
s
N
1
s
N
2
⋯
1
.Further, the support measurement and credibility of evidencem
i are calculated separately as(25)
sup
⁡
m
i
=
∑
j
=
1
,
j
≠
i
N
s
m
i
,
m
j
(26)
crd
⁡
m
i
=
sup
⁡
m
i
∑
i
=
1
N
sup
⁡
m
i
.It can be seen from formula (26) that the credibility of evidences satisfies crd
⁡
(
m
i
)
∈
[
0,1
] and ∑
i
=
1
N
‍
crd
⁡
(
m
i
)
=
1. The support measurement represents the similarity degree between certain evidence and other evidences, and credibility reflects the normalized support degree of certain evidence. It is clear that the support measurement and credibility of evidences will both increase when similarity function between certain evidence and other evidences increases.


#### 3.1.3. Discount Processing

Assume sensor priority and evidence credibility for discount processing before evidences combine can enhance the consistency of sensors and reduce conflicts among evidences. The discount processing is defined as(27)
m
′
A
=
∑
i
=
1
N
m
i
A
·
pr
⁡
m
i
·
crd
⁡
m
i
m
′
Φ
=
0
∀
A
⊆
Θ
,
A
≠
Φ
.Because ofpr
⁡
(
m
i
)
∈
[
0,1
] and crd
⁡
(
m
i
)
∈
[
0,1
], the discounted evidence satisfies ∑
A
⊆
Θ
m
′
(
A
)
≤
1. To ensure the normalization of synthesis results, the discounted evidence should be normalized before evidences combine, which is defined as follows:(28)
m
′
′
A
=
m
′
A
∑
A
⊆
Θ
m
′
A
A
⊆
Θ
.


### 3.2. Improvement of DS Combination Rule

In the DS combination rule, paradoxes are mainly caused by incompleteF
o
D and the normalization step under the hypothesis of reliable evidences. In this section, paradoxes are only caused by the normalization step through assuming a complete F
o
D. Evidence conflicts have been greatly reduced by the discount processing with the introduction of sensor priority and evidence credibility. Under this circumstance, traditional DS combination rule can simply deal with common conflicts. However, conflicts caused by imperfect knowledge are still an urgent problem. In order to handle four common paradoxes described in Section 2, DS combination rule is improved correspondingly. The improvement of DS combination rule proportionally assigns global conflicts to propositions based on the discounted evidence instead of blindly negating information that hides in conflict evidences, which enhances the reliability and rationality of synthesis results.The improved DS combination rule is defined as(29)
m
A
=
∑
A
1
∩
A
2
=
A
m
′
′
A
1
·
m
′
′
A
2
+
k
·
m
′
′
A
m
Φ
=
0
∀
A
⊆
Θ
,
A
≠
Φ
.Here, we need to further explain one point. The reason why it adopts assigning global conflicts to all propositions instead of assigning local conflicts directly to local propositions is that, without discount processing, local assignment method is more rational than global assignment, but in the current situation with discount processing, evidence conflicts have been drastically decreased. Global assignment method will lead to better convergence and less computation.


### 3.3. Decision-Making Rule

In DS evidence theory, two common decision-making rules are maximum belief function and maximum plausibility function method. But it is obvious that both of them are too simple to obtain accurate and satisfactory recognition results in IR/MMW system. Therefore, a compound decision-making rule is applied. Assume that theF
o
D is Θ
=
{
A
1
,
A
2
,
…
,
A
M
}; that is, M targets are existent in target recognition system. If the synthesis results satisfy(30)
m
A
1
=
max
⁡
m
A
i
,
A
i
⊆
Θ
m
A
2
=
max
⁡
m
A
j
,
A
j
⊆
Θ
,
A
j
≠
A
1
m
A
1
-
m
A
2
≥
ε
1
m
Θ
≥
ε
2
m
A
1
≥
m
Θ
, the recognition result will be target A
1, where ε
1 and ε
2 are preset threshold values.


## 3.1. Discount Processing of Evidences

The discount processing of evidences is composed of two parts. One is the definition of source priority for distinguishing the differences of data obtained from different sensors according to their types and precision, and the other is the introduction of evidence credibility for analyses of different reliability of data from different sensors in consideration of weather and observation environment. The discount processing of evidences can decrease the conflicts caused by environmental uncertainty and differences among sensors, which is the precondition of DS combination rule.


### 3.1.1. Sensor Priority

Suppose that there areM sensors with N different types in IR/MMW system; we regard the data received from different sensors as the sources of evidence, whose number is also M and type is N. Because the system has N types of sensors, sensor priorities are also quantized into N levels. According to the differences of sensor types and precision, the sensor priority can be defined to represent the dominance and importance of different sensors under certain identification mission. And for a particular environment of target recognition, sensor priorities of different sensors differ and satisfy pr
⁡
(
m
i
)
∈
[
0,1
]
(
i
=
1,2
,
…
,
N
).This paper only assigns sensor priority by considering sensor types and precision to meet real-time demand of system; readers can take more factors into account by analogy. As sensor priorities are deemed to be the prior knowledge, assignment of evidence priorities depends on practical application. An example will make it clear: If there are five sensors in IR/MMW system with three same precision activeI
R
s and two same precision passive M
M
W
s, then these three active I
R
s must have the same sensor priorities, and two passive M
M
W
s also have the same sensor priorities. The concrete value of sensor priority should be set by artificial operators based on system requirements. If system requires high spatial resolution and strong penetrating power, M
M
W
s have obvious superiority and their sensor priorities should be higher than that of I
R
s, such as assignment as pr
⁡
(
m
I
R
s
)
=
0.9, pr
⁡
(
m
M
M
R
s
)
=
0.4. On the contrary, if system demands great night work ability, sensor priorities of I
R
s will be much higher than that of M
M
W
s.


### 3.1.2. Evidence Credibility

Sensor priority takes inequality and inconsistence caused by different types and precision of sensors into account, but uncertainty caused by factors like weather, observation environment, and missile-target distance is still the obstacle of obtaining accurate synthesis results. In order to relieve uncertainty and imperfection among evidences, evidence credibility is put forward to evaluate the reliability of different sensors before evidence combines.Primarily, Minkowski distance is introduced as a distance function between two evidences. Assume that there areN evidence sources, and m
i and m
j
(
i
,
j
=
1,2
,
…
,
N
) are two B
P
A
s of mutually exclusive and exhaustive F
o
D
:
Θ
=
{
A
1
,
A
2
,
…
,
A
M
}, which are, respectively, shown as(20)
m
i
A
1
=
p
1
,
m
i
A
2
=
p
2
,
…
,
m
i
A
M
=
p
M
m
j
A
1
=
q
1
,
m
j
A
2
=
q
2
,
…
,
m
j
A
M
=
q
M
.Minkowski distance is a distance measurement in European space, and it is an extension of Euclidean distance and Manhattan distance. According to the introduction of Minkowski distance, the distance function ofm
i and m
j can be calculated as(21)
d
i
j
=
d
m
i
,
m
j
=
∑
l
=
1
M
p
l
-
q
l
m
1
/
m
,where m is a variable parameter. By definition, Minkowski distance will degrade into Manhattan distance when m
=
1, Euclidean distance when m
=
2, and Chebyshev distance when m
→
∞. In this paper, m is unified to be 2 for the experimental convenience in Section 5.Thus, the distance matrix is defined as(22)
DM
=
0
d
12
⋯
d
1
N
d
21
0
⋯
d
2
N
⋮
⋮
⋱
⋮
d
N
1
d
N
2
⋯
0
.Then, the similarity function between two evidences is introduced as(23)
s
i
j
=
s
m
i
,
m
j
=
1
-
d
i
j
.It is obvious that the similarity function between two evidences will increase as the distance function decreases.The similarity matrix of evidences is defined in the same way as shown before:(24)
SM
=
1
s
12
⋯
s
1
N
s
21
1
⋯
s
2
N
⋮
⋮
⋱
⋮
s
N
1
s
N
2
⋯
1
.Further, the support measurement and credibility of evidencem
i are calculated separately as(25)
sup
⁡
m
i
=
∑
j
=
1
,
j
≠
i
N
s
m
i
,
m
j
(26)
crd
⁡
m
i
=
sup
⁡
m
i
∑
i
=
1
N
sup
⁡
m
i
.It can be seen from formula (26) that the credibility of evidences satisfies crd
⁡
(
m
i
)
∈
[
0,1
] and ∑
i
=
1
N
‍
crd
⁡
(
m
i
)
=
1. The support measurement represents the similarity degree between certain evidence and other evidences, and credibility reflects the normalized support degree of certain evidence. It is clear that the support measurement and credibility of evidences will both increase when similarity function between certain evidence and other evidences increases.


### 3.1.3. Discount Processing

Assume sensor priority and evidence credibility for discount processing before evidences combine can enhance the consistency of sensors and reduce conflicts among evidences. The discount processing is defined as(27)
m
′
A
=
∑
i
=
1
N
m
i
A
·
pr
⁡
m
i
·
crd
⁡
m
i
m
′
Φ
=
0
∀
A
⊆
Θ
,
A
≠
Φ
.Because ofpr
⁡
(
m
i
)
∈
[
0,1
] and crd
⁡
(
m
i
)
∈
[
0,1
], the discounted evidence satisfies ∑
A
⊆
Θ
m
′
(
A
)
≤
1. To ensure the normalization of synthesis results, the discounted evidence should be normalized before evidences combine, which is defined as follows:(28)
m
′
′
A
=
m
′
A
∑
A
⊆
Θ
m
′
A
A
⊆
Θ
.


## 3.1.1. Sensor Priority

Suppose that there areM sensors with N different types in IR/MMW system; we regard the data received from different sensors as the sources of evidence, whose number is also M and type is N. Because the system has N types of sensors, sensor priorities are also quantized into N levels. According to the differences of sensor types and precision, the sensor priority can be defined to represent the dominance and importance of different sensors under certain identification mission. And for a particular environment of target recognition, sensor priorities of different sensors differ and satisfy pr
⁡
(
m
i
)
∈
[
0,1
]
(
i
=
1,2
,
…
,
N
).This paper only assigns sensor priority by considering sensor types and precision to meet real-time demand of system; readers can take more factors into account by analogy. As sensor priorities are deemed to be the prior knowledge, assignment of evidence priorities depends on practical application. An example will make it clear: If there are five sensors in IR/MMW system with three same precision activeI
R
s and two same precision passive M
M
W
s, then these three active I
R
s must have the same sensor priorities, and two passive M
M
W
s also have the same sensor priorities. The concrete value of sensor priority should be set by artificial operators based on system requirements. If system requires high spatial resolution and strong penetrating power, M
M
W
s have obvious superiority and their sensor priorities should be higher than that of I
R
s, such as assignment as pr
⁡
(
m
I
R
s
)
=
0.9, pr
⁡
(
m
M
M
R
s
)
=
0.4. On the contrary, if system demands great night work ability, sensor priorities of I
R
s will be much higher than that of M
M
W
s.


## 3.1.2. Evidence Credibility

Sensor priority takes inequality and inconsistence caused by different types and precision of sensors into account, but uncertainty caused by factors like weather, observation environment, and missile-target distance is still the obstacle of obtaining accurate synthesis results. In order to relieve uncertainty and imperfection among evidences, evidence credibility is put forward to evaluate the reliability of different sensors before evidence combines.Primarily, Minkowski distance is introduced as a distance function between two evidences. Assume that there areN evidence sources, and m
i and m
j
(
i
,
j
=
1,2
,
…
,
N
) are two B
P
A
s of mutually exclusive and exhaustive F
o
D
:
Θ
=
{
A
1
,
A
2
,
…
,
A
M
}, which are, respectively, shown as(20)
m
i
A
1
=
p
1
,
m
i
A
2
=
p
2
,
…
,
m
i
A
M
=
p
M
m
j
A
1
=
q
1
,
m
j
A
2
=
q
2
,
…
,
m
j
A
M
=
q
M
.Minkowski distance is a distance measurement in European space, and it is an extension of Euclidean distance and Manhattan distance. According to the introduction of Minkowski distance, the distance function ofm
i and m
j can be calculated as(21)
d
i
j
=
d
m
i
,
m
j
=
∑
l
=
1
M
p
l
-
q
l
m
1
/
m
,where m is a variable parameter. By definition, Minkowski distance will degrade into Manhattan distance when m
=
1, Euclidean distance when m
=
2, and Chebyshev distance when m
→
∞. In this paper, m is unified to be 2 for the experimental convenience in Section 5.Thus, the distance matrix is defined as(22)
DM
=
0
d
12
⋯
d
1
N
d
21
0
⋯
d
2
N
⋮
⋮
⋱
⋮
d
N
1
d
N
2
⋯
0
.Then, the similarity function between two evidences is introduced as(23)
s
i
j
=
s
m
i
,
m
j
=
1
-
d
i
j
.It is obvious that the similarity function between two evidences will increase as the distance function decreases.The similarity matrix of evidences is defined in the same way as shown before:(24)
SM
=
1
s
12
⋯
s
1
N
s
21
1
⋯
s
2
N
⋮
⋮
⋱
⋮
s
N
1
s
N
2
⋯
1
.Further, the support measurement and credibility of evidencem
i are calculated separately as(25)
sup
⁡
m
i
=
∑
j
=
1
,
j
≠
i
N
s
m
i
,
m
j
(26)
crd
⁡
m
i
=
sup
⁡
m
i
∑
i
=
1
N
sup
⁡
m
i
.It can be seen from formula (26) that the credibility of evidences satisfies crd
⁡
(
m
i
)
∈
[
0,1
] and ∑
i
=
1
N
‍
crd
⁡
(
m
i
)
=
1. The support measurement represents the similarity degree between certain evidence and other evidences, and credibility reflects the normalized support degree of certain evidence. It is clear that the support measurement and credibility of evidences will both increase when similarity function between certain evidence and other evidences increases.


## 3.1.3. Discount Processing

Assume sensor priority and evidence credibility for discount processing before evidences combine can enhance the consistency of sensors and reduce conflicts among evidences. The discount processing is defined as(27)
m
′
A
=
∑
i
=
1
N
m
i
A
·
pr
⁡
m
i
·
crd
⁡
m
i
m
′
Φ
=
0
∀
A
⊆
Θ
,
A
≠
Φ
.Because ofpr
⁡
(
m
i
)
∈
[
0,1
] and crd
⁡
(
m
i
)
∈
[
0,1
], the discounted evidence satisfies ∑
A
⊆
Θ
m
′
(
A
)
≤
1. To ensure the normalization of synthesis results, the discounted evidence should be normalized before evidences combine, which is defined as follows:(28)
m
′
′
A
=
m
′
A
∑
A
⊆
Θ
m
′
A
A
⊆
Θ
.


## 3.2. Improvement of DS Combination Rule

In the DS combination rule, paradoxes are mainly caused by incompleteF
o
D and the normalization step under the hypothesis of reliable evidences. In this section, paradoxes are only caused by the normalization step through assuming a complete F
o
D. Evidence conflicts have been greatly reduced by the discount processing with the introduction of sensor priority and evidence credibility. Under this circumstance, traditional DS combination rule can simply deal with common conflicts. However, conflicts caused by imperfect knowledge are still an urgent problem. In order to handle four common paradoxes described in Section 2, DS combination rule is improved correspondingly. The improvement of DS combination rule proportionally assigns global conflicts to propositions based on the discounted evidence instead of blindly negating information that hides in conflict evidences, which enhances the reliability and rationality of synthesis results.The improved DS combination rule is defined as(29)
m
A
=
∑
A
1
∩
A
2
=
A
m
′
′
A
1
·
m
′
′
A
2
+
k
·
m
′
′
A
m
Φ
=
0
∀
A
⊆
Θ
,
A
≠
Φ
.Here, we need to further explain one point. The reason why it adopts assigning global conflicts to all propositions instead of assigning local conflicts directly to local propositions is that, without discount processing, local assignment method is more rational than global assignment, but in the current situation with discount processing, evidence conflicts have been drastically decreased. Global assignment method will lead to better convergence and less computation.


## 3.3. Decision-Making Rule

In DS evidence theory, two common decision-making rules are maximum belief function and maximum plausibility function method. But it is obvious that both of them are too simple to obtain accurate and satisfactory recognition results in IR/MMW system. Therefore, a compound decision-making rule is applied. Assume that theF
o
D is Θ
=
{
A
1
,
A
2
,
…
,
A
M
}; that is, M targets are existent in target recognition system. If the synthesis results satisfy(30)
m
A
1
=
max
⁡
m
A
i
,
A
i
⊆
Θ
m
A
2
=
max
⁡
m
A
j
,
A
j
⊆
Θ
,
A
j
≠
A
1
m
A
1
-
m
A
2
≥
ε
1
m
Θ
≥
ε
2
m
A
1
≥
m
Θ
, the recognition result will be target A
1, where ε
1 and ε
2 are preset threshold values.


## 4. Application in IR/MMW Target Recognition

As Section3 gives specific evolution thread of the novel algorithm, this section will present the implementation steps of its application in IR/MMW system. Combining different features of I
R
s and M
M
W
s in target recognition system, the modified DS evidence theory is applied as a decision-making fusion method. Figure 2 shows the flow diagram of IR/MMW system (take an example of 2 I
R
s and 2 M
M
W
s corresponding to experiments in Section 5).Figure 2
The flow diagram of IR/MMW system.Specific implementation steps of the application in IR/MMW system for target recognition are summarized as follows.Step 1.
Collect information ofI
R
s and M
M
W
s as evidences and assume that all sensors have the same F
o
D; obtain B
P
A
s from all sensors through data acquisition and preprocessing. This step establishes a platform for information fusion.Step 2.
Utilize the modified DS evidence theory to combine evidences and make synthesis decision.Step 3.
Output the decision results as the recognition results.


## 5. Experiment Results and Analyses

In this section, we give two experiments. One is the comparison of performances between the modified algorithm and several existent methods, which proves its validity and reliability theoretically. The other is its application in IR/MMW system, which demonstrates its feasibility in practice.


### 5.1. Experiment  1: Theoretical Comparison between the Modified Algorithm and Existent Methods

Take four paradoxes described in Section2 as examples to discuss the rationality and validity of the modified algorithm. The B
P
A
s of four common paradoxes presented in Section 2.2 are shown in Table 1 and Figure 3.Table 1
BPAs of four common paradoxes.
Paradoxes
Evidences
Propositions
A
B
C
D
E
Complete conflict paradox
m
1
1
0
0
∖
∖
m
2
0
1
0
∖
∖
m
3
0.8
0.1
0.1
∖
∖
m
4
0.8
0.1
0.1
∖
∖
0 trust paradox
m
1
0.5
0.2
0.3
∖
∖
m
2
0.5
0.2
0.3
∖
∖
m
3
0
0.9
0.1
∖
∖
m
4
0.5
0.2
0.3
∖
∖
1 trust paradox
m
1
0.9
0.1
0
∖
∖
m
2
0
0.1
0.9
∖
∖
m
3
0.1
0.15
0.75
∖
∖
m
4
0.1
0.15
0.75
∖
∖
High conflict paradox
m
1
0.7
0.1
0.1
0
0.1
m
2
0
0.5
0.2
0.1
0.2
m
3
0.6
0.1
0.15
0
0.15
m
4
0.55
0.1
0.1
0.15
0.1
m
5
0.6
0.1
0.2
0
0.1Figure 3
B
P
A
s of four common paradoxes.
(a)
B
P
A of complete conflict paradox
(b)
B
P
A of 0 trust paradox
(c)
B
P
A of 1 trust paradox
(d)
B
P
A of high conflict paradoxIt can be seen from Table1 and Figure 3 that evidences in IR/MMW system can be divided into consistent evidences and conflict evidences. Apparently, the relatively consistent evidences are m
1, m
3, and m
4 in complete conflict paradox, m
1, m
2, and m
4 in 0 trust paradox, m
2, m
3, and m
4 in 1 trust paradox, and m
1, m
3, m
4, and m
5 in high conflict paradox. Thus, accurate synthesis results should agree with these consistent evidences while being away from conflict evidences.It is evident that traditional DS combination rule is unable to manage all four paradoxes, so we choose four existent improved methods, respectively, proposed by Yager [16], Sun et al. [17], Murphy [20], and Deng et al. [35] (called Yager, Sun, Murphy, and Deng for short) for comprehensive analyses with the modified method (called Modified for short). And the synthesis results are presented in Table 2 and Figure 4.Table 2
Comparison of the synthesis results.
Paradoxes
Methods
Propositions
A
B
C
D
E
Θ
Complete conflict paradox (k = 1)
Yager
0
0
0
∖
∖
1
Sun
0.0917
0.0423
0.0071
∖
∖
0.8589
Murphy
0.8204
0.1748
0.0048
∖
∖
0
Deng
0.8166
0.1164
0.0670
∖
∖
0
Modified
0.9242
0.0502
0.0256
∖
∖
0
0 trust paradox (k = 0.99)
Yager
0
0.7273
0.2727
∖
∖
0
Sun
0.0525
0.0597
0.0377
∖
∖
0.8501
Murphy
0.4091
0.4091
0.1818
∖
∖
0
Deng
0.4318
0.2955
0.2727
∖
∖
0
Modified
0.4679
0.2800
0.2521
∖
∖
0
1 trust paradox (k = 0.9998)
Yager
0
1
0
∖
∖
0
Sun
0.0388
0.0179
0.0846
∖
∖
0.8587
Murphy
0.1676
0.0346
0.7978
∖
∖
0
Deng
0.1388
0.1318
0.7294
∖
∖
0
Modified
0.0791
0.0743
0.8466
∖
∖
0
High conflict paradox (k = 0.9999)
Yager
0
0.3571
0.4286
0
0.2143
0
Sun
0.0443
0.0163
0.0136
0.0045
0.0118
0.9094
Murphy
0.7637
0.1031
0.0716
0.0080
0.0538
0
Deng
0.5324
0.1521
0.1462
0.0451
0.1241
0
Modified
0.6320
0.1227
0.1171
0.0316
0.0967
0Figure 4
Comparison of the synthesis results.
(a)
Synthesis result of complete conflict paradox
(b)
Synthesis result of 0 trust paradox
(c)
Synthesis result of 1 trust paradox
(d)
Synthesis result of high conflict paradoxAccording to Table2 and Figure 4, we make the following discussion.


#### 5.1.1. In Complete Conflict Paradox (k
=
1)

m
1
,
m
3, and m
4 are the relatively consistent evidences. Yager gives Θ the whole belief as m
(
Θ
)
=
1, which, on the contrary, increases propositions’ uncertainty. Sun only solves part of the conflicts as the B
P
As of propositions A
,
B, and C match the BPAs’ proportion of these consistent evidences, but the B
P
A for Θ is still high as m
(
Θ
)
=
0.8589, which still has high uncertainty. At the same time, Murphy, Deng, and Modified can all get relatively rational results in complete conflict paradox.


#### 5.1.2. In 0 Trust Paradox

m
1
,
m
2, and m
4 are the relatively consistent evidences. It is easy to check that the consistent evidences are the same, so the most valid algorithm should have the minimum difference between the synthesis results and the consistent evidences. As can be seen intuitively, Yager presents totally wrong results, and Modified is the most effective algorithm in this kind of paradox.


#### 5.1.3. In 1 Trust Paradox

m
2, m
3, and m
4 are the relatively consistent evidences. Yager and Sun cannot handle this paradox practically which can be proved in similar way as shown in the discussion of complete conflict paradox. And Murphy, Deng, and Modified can manage 1 trust paradox in different degrees.


#### 5.1.4. In High Conflict Paradox

m
1, m
3, m
4, and m
5 are the relatively consistent evidences. Yager and Sun are logical theoretically but cannot be put into practice because of their increasing uncertainty. And Murphy, Deng, and Modified all produce relatively reasonable results in this kind of paradox.It is verified that Yager always produces wrong synthesis results under the condition of paradoxes, and it is unable to handle any kind of paradox. Sun allots most of the conflicts directly toΘ, which just solves paradoxes theoretically. It is not suitable for practical application because of the increasing uncertainty of synthesis results. Murphy averages all evidences without separating consistent evidences and conflict evidences. And it is clear that it cannot solve paradoxes fundamentally because evidences’ contributions to the synthesis results are totally different, in spite of its low computation. Therefore, only Deng and Modified can generate relatively reasonable synthesis results for all these four common paradoxes.Then, the comparison of Deng and Modified is discussed here in detail. The evaluation criterion is composed of two parts: the weight distance measurement (w
d) between the synthesis result and original evidences and the decrement (∇
w
d) between Deng and Modified, which are defined as(31)
wd
=
∑
i
=
1
N
d
m
i
,
m
·
crd
⁡
m
i
∇
wd
=
wd
2
-
w
d
1
w
d
1
·
100
%
,where m
i
(
i
=
1,2
,
…
,
N
) and m, respectively, represent N original evidences and the synthesis result and w
d
1 and w
d
2 separately represent the weight distance measurement of Deng and Modified.In formula (31), we can see that when the synthesis result is consistent with original evidences, the weight distance measurement becomes small. This means that the smaller w
d is, the more efficient this combination method is. The comparison is shown in Table 3.Table 3
Comparison betweenDeng and Modified.
Paradoxes
Methods
wd
∇
wd
Complete conflict paradox
Deng
0.1064
11.69%
Modified
0.0940
0 trust paradox
Deng
0.1472
78.69%
Modified
0.0314
1 trust paradox
Deng
0.1275
20.37%
Modified
0.1015
Complete conflict paradox
Deng
0.1407
43.68%
Modified
0.0793It can be seen from Table3 that w
d of Modified is far less than that of Deng. It demonstrates that Modified can solve all the four paradoxes better, the evidences processed by Modified have better consistency and reliability, and the synthesis results produced by Modified are more rational and valid. Therefore, experiments here successfully certify theoretical rationality and feasibility of the modified DS evidence theory before its practical application.There are two details needed to be illustrated. On the one hand, experiments here just discuss theoretical feasibility of the modified algorithm without considering its application; thus sensor priorities are all set to be 1. This experiment represents the notion that every sensor is equally suitable for the current demands of system, while, in practical application, experiment results will be more effective and reliable with the introduction of sensor priority. On the other hand, in theoretical experiments, there is no need to prove the validity of decision-making rule, so it only outputs the synthesis results without decision-making. The advantages of sensor priority and decision-making rule are demonstrated in the next experiment.


### 5.2. Experiment  2: Practical Experiment on Application of IR/MMW System

To illustrate practical application of the modified method, three groups of data are set here for target recognition, including two ordinary pieces of data (data 1 and data 2) and one adverse piece of data (data 3). According to the data acquisition and preprocessing method described in [36], we extract the features of information from IRs and MMWs and then use grey correlation classifier to get B
P
A
s as the input of the modified DS evidence fusion.Experiment conditions are as follows: in IR/MMW system, assume that theF
o
D is Θ
=
{
A
1
,
A
2
,
A
3
}, where A
1, A
2, and A
3 separately represent reconnaissance aircraft, bomber aircraft, and fighters. I
R
s and M
M
W
s provide information about targets as evidences, and there are four evidence sources including I
R
1, I
R
2, M
M
W
1, and M
M
W
2. The sensor priorities are expressed as pr
⁡
m
1, pr
⁡
m
2, pr
⁡
m
3, and pr
⁡
(
m
4
), and the preset threshold values in decision-making rule are ε
1
=
0.2, ε
2
=
0.25.Data  1 (Target Recognition for
A
1
). Assume that there is a lot of smoke, fog, and cloud in observation environment (actual battlefield). And system requires high spatial resolution and strong penetrating power. Under these conditions, M
M
W
s have obvious advantage over I
R
s. As prior knowledge, sensor priorities are set as pr
⁡
(
m
1
)
=
pr
⁡
(
m
2
)
=
0.53, pr
⁡
(
m
3
)
=
pr
⁡
(
m
4
)
=
0.85.Data  2 (Target Recognition for
A
2
). Assume that target recognition mission is marched in the night and system requires great night work ability and strong concealment. It is obvious that I
R
s have obvious advantage over M
M
W
s in this situation. As prior knowledge, sensor priorities are set as pr
⁡
(
m
1
)
=
pr
⁡
(
m
2
)
=
0.86, pr
⁡
(
m
3
)
=
pr
⁡
(
m
4
)
=
0.45.Data  3 (Target Recognition for
A
3
). Assume that observation environment is abominable due to environmental clutter and artificial interference. Single-mode system will not produce correct recognition results. In order to ensure the reliability of target recognition, the fusion of IRs and MMWs is an inevitable trend. If system requires wide range of search and interception, M
M
W
s have obvious advantage over I
R
s. As prior knowledge, sensor priorities are set as pr
⁡
(
m
1
)
=
0.40, pr
⁡
(
m
2
)
=
0.35, pr
⁡
(
m
3
)
=
0.60, and pr
⁡
(
m
4
)
=
0.65.According to the above data, the recognition results of IR/MMW system are presented in Table4 and Figure 5.Table 4
Recognition results of IR/MMW system.
Targets
Sensors
m
i
(
A
j
)
,
i
=
1,2
,
3,4
,
A
j
⊆
Θ
Recognition results
Sensor type
Sensor priority
A
1
A
2
A
3
Θ
A
1
IR1
0.53
0.25
0.15
0.20
0.40
Θ
IR2
0.53
0.20
0.35
0.20
0.25
A
2
MMW1
0.85
0.60
0.15
0.10
0.15
A
1
MMW2
0.85
0.65
0.10
0.15
0.10
A
1
Discounted evidence
0.4759
0.1715
0.1526
0.2000
A
1
IR/MMW fusion
0.5885
0.1599
0.1394
0.1122
A
1
A
2
IR1
0.86
0.10
0.60
0.10
0.20
A
2
IR2
0.86
0.10
0.70
0.10
0.10
A
2
MMW1
0.45
0.20
0.30
0.10
0.40
Θ
MMW2
0.45
0.20
0.25
0.35
0.20
A
3
Discounted evidence
0.1330
0.5251
0.1407
0.2012
A
2
IR/MMW fusion
0.1144
0.6576
0.1221
0.1059
A
2
A
3
IR1
0.40
0.12
0.21
0.32
0.35
Θ
IR2
0.35
0.35
0.06
0.29
0.30
A
1
MMW1
0.60
0.29
0.03
0.28
0.40
Θ
MMW2
0.65
0.05
0.20
0.43
0.32
A
3
Discounted evidence
0.1894
0.1261
0.3376
0.3469
Θ
IR/MMW fusion
0.2167
0.1363
0.4362
0.2108
A
3Figure 5
Recognition results of IR/MMW system.
(a)
Target recognition forA
1
(b)
Target recognition forA
2
(c)
Target recognition forA
3It can be seen from Table4 and Figure 5 that, in target recognition of A
1, system is led mainly by M
M
W
s with complement by I
R
s according to their sensor priorities. In the single-mode systems, M
M
W
s can identify objective correctly, while I
R
s cannot. And IR/MMW fusion can not only identify real objective, but also enhance the accuracy of recognition results by greatly increasing the B
P
A of real target. In similar principle when identifying A
2, the syncretic recognition results verify rationality and validity of the modified DS evidence theory in IR/MMW system. Experiment results of data 3 show that all sensor priorities are low because of poor observation environment, and only M
M
W
2 in single-mode systems can identify real objective but with large B
P
A for Θ. This demonstrates that large uncertainty exists in this single-mode’s recognition results. However, under that experimental condition, recognition results in fusion system significantly decrease uncertainty of system to facilitate decision-making. In addition, the modified algorithm in IR/MMW system can still accurately identify objective, even under the condition that two or three of the sensors are unable to identify real objective, and greatly increase reliability of IR/MMW system.Through the above analyses, application of the modified DS evidence theory in IR/MMW system can efficiently fuse multisensor information, accurately identify objective, and improve recognition rate and system reliability. It directly proves the validity and rationality of the modified algorithm and adaptability and flexibility of the system.


## 5.1. Experiment  1: Theoretical Comparison between the Modified Algorithm and Existent Methods

Take four paradoxes described in Section2 as examples to discuss the rationality and validity of the modified algorithm. The B
P
A
s of four common paradoxes presented in Section 2.2 are shown in Table 1 and Figure 3.Table 1
BPAs of four common paradoxes.
Paradoxes
Evidences
Propositions
A
B
C
D
E
Complete conflict paradox
m
1
1
0
0
∖
∖
m
2
0
1
0
∖
∖
m
3
0.8
0.1
0.1
∖
∖
m
4
0.8
0.1
0.1
∖
∖
0 trust paradox
m
1
0.5
0.2
0.3
∖
∖
m
2
0.5
0.2
0.3
∖
∖
m
3
0
0.9
0.1
∖
∖
m
4
0.5
0.2
0.3
∖
∖
1 trust paradox
m
1
0.9
0.1
0
∖
∖
m
2
0
0.1
0.9
∖
∖
m
3
0.1
0.15
0.75
∖
∖
m
4
0.1
0.15
0.75
∖
∖
High conflict paradox
m
1
0.7
0.1
0.1
0
0.1
m
2
0
0.5
0.2
0.1
0.2
m
3
0.6
0.1
0.15
0
0.15
m
4
0.55
0.1
0.1
0.15
0.1
m
5
0.6
0.1
0.2
0
0.1Figure 3
B
P
A
s of four common paradoxes.
(a)
B
P
A of complete conflict paradox
(b)
B
P
A of 0 trust paradox
(c)
B
P
A of 1 trust paradox
(d)
B
P
A of high conflict paradoxIt can be seen from Table1 and Figure 3 that evidences in IR/MMW system can be divided into consistent evidences and conflict evidences. Apparently, the relatively consistent evidences are m
1, m
3, and m
4 in complete conflict paradox, m
1, m
2, and m
4 in 0 trust paradox, m
2, m
3, and m
4 in 1 trust paradox, and m
1, m
3, m
4, and m
5 in high conflict paradox. Thus, accurate synthesis results should agree with these consistent evidences while being away from conflict evidences.It is evident that traditional DS combination rule is unable to manage all four paradoxes, so we choose four existent improved methods, respectively, proposed by Yager [16], Sun et al. [17], Murphy [20], and Deng et al. [35] (called Yager, Sun, Murphy, and Deng for short) for comprehensive analyses with the modified method (called Modified for short). And the synthesis results are presented in Table 2 and Figure 4.Table 2
Comparison of the synthesis results.
Paradoxes
Methods
Propositions
A
B
C
D
E
Θ
Complete conflict paradox (k = 1)
Yager
0
0
0
∖
∖
1
Sun
0.0917
0.0423
0.0071
∖
∖
0.8589
Murphy
0.8204
0.1748
0.0048
∖
∖
0
Deng
0.8166
0.1164
0.0670
∖
∖
0
Modified
0.9242
0.0502
0.0256
∖
∖
0
0 trust paradox (k = 0.99)
Yager
0
0.7273
0.2727
∖
∖
0
Sun
0.0525
0.0597
0.0377
∖
∖
0.8501
Murphy
0.4091
0.4091
0.1818
∖
∖
0
Deng
0.4318
0.2955
0.2727
∖
∖
0
Modified
0.4679
0.2800
0.2521
∖
∖
0
1 trust paradox (k = 0.9998)
Yager
0
1
0
∖
∖
0
Sun
0.0388
0.0179
0.0846
∖
∖
0.8587
Murphy
0.1676
0.0346
0.7978
∖
∖
0
Deng
0.1388
0.1318
0.7294
∖
∖
0
Modified
0.0791
0.0743
0.8466
∖
∖
0
High conflict paradox (k = 0.9999)
Yager
0
0.3571
0.4286
0
0.2143
0
Sun
0.0443
0.0163
0.0136
0.0045
0.0118
0.9094
Murphy
0.7637
0.1031
0.0716
0.0080
0.0538
0
Deng
0.5324
0.1521
0.1462
0.0451
0.1241
0
Modified
0.6320
0.1227
0.1171
0.0316
0.0967
0Figure 4
Comparison of the synthesis results.
(a)
Synthesis result of complete conflict paradox
(b)
Synthesis result of 0 trust paradox
(c)
Synthesis result of 1 trust paradox
(d)
Synthesis result of high conflict paradoxAccording to Table2 and Figure 4, we make the following discussion.


### 5.1.1. In Complete Conflict Paradox (k
=
1)

m
1
,
m
3, and m
4 are the relatively consistent evidences. Yager gives Θ the whole belief as m
(
Θ
)
=
1, which, on the contrary, increases propositions’ uncertainty. Sun only solves part of the conflicts as the B
P
As of propositions A
,
B, and C match the BPAs’ proportion of these consistent evidences, but the B
P
A for Θ is still high as m
(
Θ
)
=
0.8589, which still has high uncertainty. At the same time, Murphy, Deng, and Modified can all get relatively rational results in complete conflict paradox.


### 5.1.2. In 0 Trust Paradox

m
1
,
m
2, and m
4 are the relatively consistent evidences. It is easy to check that the consistent evidences are the same, so the most valid algorithm should have the minimum difference between the synthesis results and the consistent evidences. As can be seen intuitively, Yager presents totally wrong results, and Modified is the most effective algorithm in this kind of paradox.


### 5.1.3. In 1 Trust Paradox

m
2, m
3, and m
4 are the relatively consistent evidences. Yager and Sun cannot handle this paradox practically which can be proved in similar way as shown in the discussion of complete conflict paradox. And Murphy, Deng, and Modified can manage 1 trust paradox in different degrees.


### 5.1.4. In High Conflict Paradox

m
1, m
3, m
4, and m
5 are the relatively consistent evidences. Yager and Sun are logical theoretically but cannot be put into practice because of their increasing uncertainty. And Murphy, Deng, and Modified all produce relatively reasonable results in this kind of paradox.It is verified that Yager always produces wrong synthesis results under the condition of paradoxes, and it is unable to handle any kind of paradox. Sun allots most of the conflicts directly toΘ, which just solves paradoxes theoretically. It is not suitable for practical application because of the increasing uncertainty of synthesis results. Murphy averages all evidences without separating consistent evidences and conflict evidences. And it is clear that it cannot solve paradoxes fundamentally because evidences’ contributions to the synthesis results are totally different, in spite of its low computation. Therefore, only Deng and Modified can generate relatively reasonable synthesis results for all these four common paradoxes.Then, the comparison of Deng and Modified is discussed here in detail. The evaluation criterion is composed of two parts: the weight distance measurement (w
d) between the synthesis result and original evidences and the decrement (∇
w
d) between Deng and Modified, which are defined as(31)
wd
=
∑
i
=
1
N
d
m
i
,
m
·
crd
⁡
m
i
∇
wd
=
wd
2
-
w
d
1
w
d
1
·
100
%
,where m
i
(
i
=
1,2
,
…
,
N
) and m, respectively, represent N original evidences and the synthesis result and w
d
1 and w
d
2 separately represent the weight distance measurement of Deng and Modified.In formula (31), we can see that when the synthesis result is consistent with original evidences, the weight distance measurement becomes small. This means that the smaller w
d is, the more efficient this combination method is. The comparison is shown in Table 3.Table 3
Comparison betweenDeng and Modified.
Paradoxes
Methods
wd
∇
wd
Complete conflict paradox
Deng
0.1064
11.69%
Modified
0.0940
0 trust paradox
Deng
0.1472
78.69%
Modified
0.0314
1 trust paradox
Deng
0.1275
20.37%
Modified
0.1015
Complete conflict paradox
Deng
0.1407
43.68%
Modified
0.0793It can be seen from Table3 that w
d of Modified is far less than that of Deng. It demonstrates that Modified can solve all the four paradoxes better, the evidences processed by Modified have better consistency and reliability, and the synthesis results produced by Modified are more rational and valid. Therefore, experiments here successfully certify theoretical rationality and feasibility of the modified DS evidence theory before its practical application.There are two details needed to be illustrated. On the one hand, experiments here just discuss theoretical feasibility of the modified algorithm without considering its application; thus sensor priorities are all set to be 1. This experiment represents the notion that every sensor is equally suitable for the current demands of system, while, in practical application, experiment results will be more effective and reliable with the introduction of sensor priority. On the other hand, in theoretical experiments, there is no need to prove the validity of decision-making rule, so it only outputs the synthesis results without decision-making. The advantages of sensor priority and decision-making rule are demonstrated in the next experiment.


## 5.1.1. In Complete Conflict Paradox (k
=
1)

m
1
,
m
3, and m
4 are the relatively consistent evidences. Yager gives Θ the whole belief as m
(
Θ
)
=
1, which, on the contrary, increases propositions’ uncertainty. Sun only solves part of the conflicts as the B
P
As of propositions A
,
B, and C match the BPAs’ proportion of these consistent evidences, but the B
P
A for Θ is still high as m
(
Θ
)
=
0.8589, which still has high uncertainty. At the same time, Murphy, Deng, and Modified can all get relatively rational results in complete conflict paradox.


## 5.1.2. In 0 Trust Paradox

m
1
,
m
2, and m
4 are the relatively consistent evidences. It is easy to check that the consistent evidences are the same, so the most valid algorithm should have the minimum difference between the synthesis results and the consistent evidences. As can be seen intuitively, Yager presents totally wrong results, and Modified is the most effective algorithm in this kind of paradox.


## 5.1.3. In 1 Trust Paradox

m
2, m
3, and m
4 are the relatively consistent evidences. Yager and Sun cannot handle this paradox practically which can be proved in similar way as shown in the discussion of complete conflict paradox. And Murphy, Deng, and Modified can manage 1 trust paradox in different degrees.


## 5.1.4. In High Conflict Paradox

m
1, m
3, m
4, and m
5 are the relatively consistent evidences. Yager and Sun are logical theoretically but cannot be put into practice because of their increasing uncertainty. And Murphy, Deng, and Modified all produce relatively reasonable results in this kind of paradox.It is verified that Yager always produces wrong synthesis results under the condition of paradoxes, and it is unable to handle any kind of paradox. Sun allots most of the conflicts directly toΘ, which just solves paradoxes theoretically. It is not suitable for practical application because of the increasing uncertainty of synthesis results. Murphy averages all evidences without separating consistent evidences and conflict evidences. And it is clear that it cannot solve paradoxes fundamentally because evidences’ contributions to the synthesis results are totally different, in spite of its low computation. Therefore, only Deng and Modified can generate relatively reasonable synthesis results for all these four common paradoxes.Then, the comparison of Deng and Modified is discussed here in detail. The evaluation criterion is composed of two parts: the weight distance measurement (w
d) between the synthesis result and original evidences and the decrement (∇
w
d) between Deng and Modified, which are defined as(31)
wd
=
∑
i
=
1
N
d
m
i
,
m
·
crd
⁡
m
i
∇
wd
=
wd
2
-
w
d
1
w
d
1
·
100
%
,where m
i
(
i
=
1,2
,
…
,
N
) and m, respectively, represent N original evidences and the synthesis result and w
d
1 and w
d
2 separately represent the weight distance measurement of Deng and Modified.In formula (31), we can see that when the synthesis result is consistent with original evidences, the weight distance measurement becomes small. This means that the smaller w
d is, the more efficient this combination method is. The comparison is shown in Table 3.Table 3
Comparison betweenDeng and Modified.
Paradoxes
Methods
wd
∇
wd
Complete conflict paradox
Deng
0.1064
11.69%
Modified
0.0940
0 trust paradox
Deng
0.1472
78.69%
Modified
0.0314
1 trust paradox
Deng
0.1275
20.37%
Modified
0.1015
Complete conflict paradox
Deng
0.1407
43.68%
Modified
0.0793It can be seen from Table3 that w
d of Modified is far less than that of Deng. It demonstrates that Modified can solve all the four paradoxes better, the evidences processed by Modified have better consistency and reliability, and the synthesis results produced by Modified are more rational and valid. Therefore, experiments here successfully certify theoretical rationality and feasibility of the modified DS evidence theory before its practical application.There are two details needed to be illustrated. On the one hand, experiments here just discuss theoretical feasibility of the modified algorithm without considering its application; thus sensor priorities are all set to be 1. This experiment represents the notion that every sensor is equally suitable for the current demands of system, while, in practical application, experiment results will be more effective and reliable with the introduction of sensor priority. On the other hand, in theoretical experiments, there is no need to prove the validity of decision-making rule, so it only outputs the synthesis results without decision-making. The advantages of sensor priority and decision-making rule are demonstrated in the next experiment.


## 5.2. Experiment  2: Practical Experiment on Application of IR/MMW System

To illustrate practical application of the modified method, three groups of data are set here for target recognition, including two ordinary pieces of data (data 1 and data 2) and one adverse piece of data (data 3). According to the data acquisition and preprocessing method described in [36], we extract the features of information from IRs and MMWs and then use grey correlation classifier to get B
P
A
s as the input of the modified DS evidence fusion.Experiment conditions are as follows: in IR/MMW system, assume that theF
o
D is Θ
=
{
A
1
,
A
2
,
A
3
}, where A
1, A
2, and A
3 separately represent reconnaissance aircraft, bomber aircraft, and fighters. I
R
s and M
M
W
s provide information about targets as evidences, and there are four evidence sources including I
R
1, I
R
2, M
M
W
1, and M
M
W
2. The sensor priorities are expressed as pr
⁡
m
1, pr
⁡
m
2, pr
⁡
m
3, and pr
⁡
(
m
4
), and the preset threshold values in decision-making rule are ε
1
=
0.2, ε
2
=
0.25.Data  1 (Target Recognition for
A
1
). Assume that there is a lot of smoke, fog, and cloud in observation environment (actual battlefield). And system requires high spatial resolution and strong penetrating power. Under these conditions, M
M
W
s have obvious advantage over I
R
s. As prior knowledge, sensor priorities are set as pr
⁡
(
m
1
)
=
pr
⁡
(
m
2
)
=
0.53, pr
⁡
(
m
3
)
=
pr
⁡
(
m
4
)
=
0.85.Data  2 (Target Recognition for
A
2
). Assume that target recognition mission is marched in the night and system requires great night work ability and strong concealment. It is obvious that I
R
s have obvious advantage over M
M
W
s in this situation. As prior knowledge, sensor priorities are set as pr
⁡
(
m
1
)
=
pr
⁡
(
m
2
)
=
0.86, pr
⁡
(
m
3
)
=
pr
⁡
(
m
4
)
=
0.45.Data  3 (Target Recognition for
A
3
). Assume that observation environment is abominable due to environmental clutter and artificial interference. Single-mode system will not produce correct recognition results. In order to ensure the reliability of target recognition, the fusion of IRs and MMWs is an inevitable trend. If system requires wide range of search and interception, M
M
W
s have obvious advantage over I
R
s. As prior knowledge, sensor priorities are set as pr
⁡
(
m
1
)
=
0.40, pr
⁡
(
m
2
)
=
0.35, pr
⁡
(
m
3
)
=
0.60, and pr
⁡
(
m
4
)
=
0.65.According to the above data, the recognition results of IR/MMW system are presented in Table4 and Figure 5.Table 4
Recognition results of IR/MMW system.
Targets
Sensors
m
i
(
A
j
)
,
i
=
1,2
,
3,4
,
A
j
⊆
Θ
Recognition results
Sensor type
Sensor priority
A
1
A
2
A
3
Θ
A
1
IR1
0.53
0.25
0.15
0.20
0.40
Θ
IR2
0.53
0.20
0.35
0.20
0.25
A
2
MMW1
0.85
0.60
0.15
0.10
0.15
A
1
MMW2
0.85
0.65
0.10
0.15
0.10
A
1
Discounted evidence
0.4759
0.1715
0.1526
0.2000
A
1
IR/MMW fusion
0.5885
0.1599
0.1394
0.1122
A
1
A
2
IR1
0.86
0.10
0.60
0.10
0.20
A
2
IR2
0.86
0.10
0.70
0.10
0.10
A
2
MMW1
0.45
0.20
0.30
0.10
0.40
Θ
MMW2
0.45
0.20
0.25
0.35
0.20
A
3
Discounted evidence
0.1330
0.5251
0.1407
0.2012
A
2
IR/MMW fusion
0.1144
0.6576
0.1221
0.1059
A
2
A
3
IR1
0.40
0.12
0.21
0.32
0.35
Θ
IR2
0.35
0.35
0.06
0.29
0.30
A
1
MMW1
0.60
0.29
0.03
0.28
0.40
Θ
MMW2
0.65
0.05
0.20
0.43
0.32
A
3
Discounted evidence
0.1894
0.1261
0.3376
0.3469
Θ
IR/MMW fusion
0.2167
0.1363
0.4362
0.2108
A
3Figure 5
Recognition results of IR/MMW system.
(a)
Target recognition forA
1
(b)
Target recognition forA
2
(c)
Target recognition forA
3It can be seen from Table4 and Figure 5 that, in target recognition of A
1, system is led mainly by M
M
W
s with complement by I
R
s according to their sensor priorities. In the single-mode systems, M
M
W
s can identify objective correctly, while I
R
s cannot. And IR/MMW fusion can not only identify real objective, but also enhance the accuracy of recognition results by greatly increasing the B
P
A of real target. In similar principle when identifying A
2, the syncretic recognition results verify rationality and validity of the modified DS evidence theory in IR/MMW system. Experiment results of data 3 show that all sensor priorities are low because of poor observation environment, and only M
M
W
2 in single-mode systems can identify real objective but with large B
P
A for Θ. This demonstrates that large uncertainty exists in this single-mode’s recognition results. However, under that experimental condition, recognition results in fusion system significantly decrease uncertainty of system to facilitate decision-making. In addition, the modified algorithm in IR/MMW system can still accurately identify objective, even under the condition that two or three of the sensors are unable to identify real objective, and greatly increase reliability of IR/MMW system.Through the above analyses, application of the modified DS evidence theory in IR/MMW system can efficiently fuse multisensor information, accurately identify objective, and improve recognition rate and system reliability. It directly proves the validity and rationality of the modified algorithm and adaptability and flexibility of the system.


## 6. Conclusion

This paper proposes a modified DS evidence theory method for target recognition in IR/MMW system. With the discount processing and modified DS combination rule, the novel algorithm theoretically solves paradoxes caused by inconsistency and inequality of evidences, reduces the negative effects caused by complex environmental factors, and improves effectiveness and accuracy of the synthesis results. And the introduction of the compound decision-making rule further ensures producing satisfactory recognition results. Experiment results and analyses demonstrate that application of the modified algorithm as an information fusion method in IR/MMW system can enhance recognition rate of system and improve the accuracy and antijamming immunity of guidance system. Thus, it greatly enhances the operational performance of weapon system under various environmental conditions and realizes the all-weather operations. That is, it has great engineering application value.In the further study, there are two technical researches that need to be studied. With the increasing number of sensors, there will be a huge computation burden for system along with great recognition performance. And the application of DSmT in target recognition system should be studied to realize the uncertainty reasoning better. Therefore, this method should be simplified and optimized.



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*Source: 1903792-2015-12-28.xml*