Design of fuzzy set-based deep fusion algorithm for multi-sensor data

With the development of electronic information technology and wireless communication technology, sensor-based information acquisition mode is more and more common. Sensor is a kind of information detection device, can sense the external temperature, humidity, pressure, strength and other information, and output in the form of electrical signals to meet the requirements of information control, transmission and storage [1-2]. At this stage, the application range of sensors is very wide, in the wireless sensor network formation, industrial control, special detection and military fields have more common applications [3-4].

However, in the actual application of a single category and a single number of sensors to obtain the amount of information is limited, more often than not to the number of sensors and multiple categories of information collected by the fusion process, in order to more accurately describe the real-time state of the object being monitored [5-6]. In the actual measurement or monitoring, as the customer’s requirements for the measurement accuracy of the target continue to improve, can only be increased by increasing the number of sensors, types of ways to improve the reliability of the measurement and monitoring results [7-8]. Distributed sensor information fusion is to integrate the information collected by all sensors and form a big data set, and then do comprehensive analysis and processing of the big data set to comprehensively assess the current state of the target based on different perspectives. Selecting multiple sensors to acquire information synchronously will significantly improve the monitoring range of the monitoring system, and different categories of sensors can achieve cross-coverage of the monitoring range and complementary information, even if individual sensors failures will not affect the accurate judgment of the state of the monitoring object [9-12].

Multi-sensor data fusion is the key to accurately obtain the monitoring object status information and make accurate judgments, the current single sensor by its own device limitations and environmental noise and other factors, the measurement of the data reliability is poor. The information obtained by the multi-sensor system is richer, and all the measured data can be fused to obtain the results with smaller error and higher reliability [13-14]. Multi-source information fusion is a kind of information processing, through the rational use of the information obtained from sensors, all the information for the unification of time and space and comprehensive judgment of the validity and accuracy of the information [15-16]. The concept of information fusion was first applied to the military field in the last century, and is now widely used in automatic target recognition, battlefield surveillance, automatic vehicle navigation, robotics, and other fields due to the higher accuracy and reliability of the data obtained [17-18].

The traditional single-sensor detection system can no longer meet the growing demand for complex environment detection due to the limitation of its information acquisition capability. Grouping multiple sensors into a network for collaborative detection can obtain a wider range, more types, and higher dimensions of information compared to a single node, and obtain better detection results through information fusion [19-20]. Distributed multi-sensor multi-target tracking is one of the important ways to realize multi-sensor collaborative detection, which utilizes multiple sensors distributed over a wide area, and is able to realize joint estimation of the number of multi-targets and spatial states in the context of the presence of noise and incomplete and uncertain observations [21-22].

The processing of data in a complete multi-sensor system is hierarchical, and different processing levels have different abstraction levels of the original data. Therefore, the accuracy of the data fusion method makes it a key technology. This paper takes multi-sensor data fusion as the core, and focuses on the fusion algorithm in the process of data acquisition and processing. Aiming at the problems existing in the current multi-sensor data fusion method, this paper is based on the fuzzy weighted D-S evidence theory fusion algorithm, and summarizes its limitations and shortcomings, and finally puts forward the data fusion method based on the improved fuzzy weighting. Finally, the effect of the data fusion method in this paper is verified through simulation experiments, and its application effect is compared with typical data fusion methods, so as to verify the rationality and applicability of the method proposed in this paper.

2

Based on fuzzy sets in fusion algorithms

2.1

Multi-sensor based data fusion

Multi-sensor data fusion, also known as data knockdown, refers to the comprehensive processing of data obtained from different data sources and multiple sensors, eliminating possible contradictions between multi-sensor data, utilizing data complementarity, and reducing uncertainty in order to form a relatively complete and consistent understanding of the system environment, thus improving the scientific nature of intelligent system decision-making and planning, the rapidity and correctness of response, and thus reducing the decision-making risk of the Process.

Multi-sensor data fusion is actually a functional simulation of the human brain’s integrated processing of complex problems. In a multi-sensor system, the data provided by various sensors may have different characteristics: time-varying or non-time-varying, real-time or non-real-time, fast-varying or slow-varying, fuzzy or deterministic, precise or incomplete, reliable or non-reliable, mutually supportive or complementary, and may also be contradictory or conflicting.

The multi-sensor data fusion process is shown in Figure 1. The data obtained from multiple sensors are first calibrated. Because different sensors work independently and asynchronously, there will be differences in time and space. Therefore, the sensor data needs to be unified into the same reference time and space. Also, the interfering data needs to be removed, leaving only the real and useful data. Next, data correlation is performed. The calibrated data will be processed to determine whether they belong to the same target, thus helping to detect, classify, and judge the object being tested. Parameter estimation and target identification are carried out again. The new observation results are merged with the original results to estimate the target parameters based on the observed values. Based on the multi-sensor observations, a feature vector is formed, and the measured features are compared with those of known types to determine the target category. Finally, action estimation. The dataset of all targets is compared with the patterns of behaviors previously identified as possible postures to make decisions.

2.2

Fuzzy set theory

A fuzzy set is distinct from an ordinary set, which is composed of elements with identical explicit properties, and the nature of the elements described by its concepts is clearly and well-defined. Not only are the concepts of the elements in a set clear and well-defined, but also the affiliation between ordinary sets is relatively clear and definite, either one or the other. Unlike fuzzy sets, fuzzy sets describe non-deterministic concepts. The concept is the sum of the stated characteristics of something with a fuzzy definition, as the concept is fuzzy and undefined, making the relationship between fuzzy sets unclear and unambiguous.

The classical set theory, also known as Cantor’s set theory, is the simplest type of set theory in the field of mathematics.

Definition I. Assuming that X is a pre-given thesis domain, then a fuzzy subset A over X is defined as: (1) $A = {〈 x, μ_{A} (x) 〉 | x \in X}$ $$A = \left\{ {\left\langle {x,{\mu _A}(x)} \right\rangle |x \in X} \right\}$$

where $μ_{A} (x) : X \to [0, 1]$ $${\mu _A}(x):X \to \left[ {0,1} \right]$$ is the degree of affiliation function of A, and the interval of definition of μ_A(x) is $[0, 1]$ $$\left[ {0,1} \right]$$. Often fuzzy subsets are loosely referred to as sets of modes in a way that does not cause confusion. Where μ_A(x) denotes the magnitude of the degree of affiliation of the element x in the set A, and the probability of not belonging to A is denoted by 1 − μ_A(x). If the value of the degree of affiliation is approximately equal to 1, it means that it is more likely that element x belongs to the set A, and by the same token, the value of the degree of affiliation is approximately equal to 0, it means that it is less likely that element x belongs to the set A.

Fuzzy sets are usually represented by ordinal pairs, such as fuzzy set A is usually represented by equation (1), or summation or integration in equation (2): (2) $A = {\begin{array}{l} \int_{x} \frac{μ_{A} (x)}{x} & X - c o n t i n u o u s \\ \sum_{i = 1}^{n} \frac{μ_{A} (x_{i})}{x_{i}} & X D i s c r e t e \end{array}$ $$A = \left\{ {\begin{array}{l} {\int_x {\frac{{{\mu _A}\left( x \right)}}{x}} }&{X {\text -} continuous} \\ {\sum\limits_{i = 1}^n {\frac{{{\mu _A}\left( {{x_i}} \right)}}{{{x_i}}}} }&{X{\text{ }}Discrete} \end{array}} \right.$$

Note that the above equation is a representation and does not represent a true summation or integration.

Fuzzy set theory extends the classical $[0, 1]$ $$\left[ {0,1} \right]$$ set theory, so that the representation of the relation of the set is no longer just a binary representation of 0 or 1, but can be expressed by any value in the $[0, 1]$ $$\left[ {0,1} \right]$$ set, and the set relation is also extended from a simple “belongs” or “does not belong” to the expression of “can or the other”.

2.3

Typical application of fuzzy sets in fusion algorithms

In the process of multivariate information fusion, fuzzy logic inference can represent the uncertainty of multiple sensors in its inference process, and its consistent fuzzy inference can be generated by transforming the uncertainty of sensors in systematic modeling inference process. The basic process is to fuzzy the observations and then measure the relationship between the observations, where the degree to which the observations are supported by other observations is the focus of the algorithm.

2.3.1

Fuzzy closeness based data fusion algorithm

1)

Measurement value fuzzification

In order to express the fuzzy amount of sensor data measurement value, let the observed mean value x_i and variance σ_i be obtained after the mnd measurement for the ist sensor, then the fuzzy amount of observation value is expressed as follows: (3) $A_{i} = (a_{i 1}, a_{i 2}, a_{i 3}) = (x_{i} - 2 σ_{i}, x_{i}, x_{i} + 2 σ_{i})$ $${A_i} = \left( {{a_{i1}},{a_{i2}},{a_{i3}}} \right) = \left( {{x_i} - 2{\sigma _i},{x_i},{x_i} + 2{\sigma _i}} \right)$$

A target estimate of x₀ and an estimated variance of σ₀ are available: (4) $x_{0} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ $${x_0} = \frac{1}{n}\sum\limits_{i = 1}^n {{x_i}}$$ (5) $σ_{0}^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i} - x_{0})}^{2}$ $$\sigma _0^2 = \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left( {{x_i} - {x_0}} \right)}^2}}$$

And the fuzzy amount of the estimate is: (6) $\tilde{A_{0}} = (a_{01}, a_{02}, a_{03}) = (x_{0} - 2 σ_{0}, x_{0}, x_{0} + 2 σ_{0})$ $$\widetilde {{A_0}} = \left( {{a_{01}},{a_{02}},{a_{03}}} \right) = \left( {{x_0} - 2{\sigma _0},{x_0},{x_0} + 2{\sigma _0}} \right)$$ 2)

Definition and calculation of fuzzy closeness

Let A_i and A_j, obtained for the ist sensor and the jnd sensor, be the fuzzy quantities of the two observations, respectively, then the closeness of A_i and A_j is defined as follows: (1)

0 ≤ S ≤ 1

(2)

For $\tilde{A_{i}} = \tilde{A_{j}}$ $$\widetilde {{A_i}} = \widetilde {{A_j}}$$, S = 1.

(3)

$S ({\tilde{A}}_{i}, {\tilde{A}}_{j}) = S ({\tilde{A}}_{j}, {\tilde{A}}_{i})$ $$S\left( {{{\tilde A}_i},{{\tilde A}_j}} \right) = S\left( {{{\tilde A}_j},{{\tilde A}_i}} \right)$$

(4)

There is $S ({\tilde{A}}_{i}, {\tilde{A}}_{j}) = 0$ $$S\left( {{{\tilde A}_i},{{\tilde A}_j}} \right) = 0$$ when and only when ${\tilde{A}}_{i} \cap {\tilde{A}}_{j} = Θ$ $${\tilde A_i} \cap {\tilde A_j} = \Theta$$.

(5)

When $\tilde{A_{i}} \subset \tilde{A_{j}} \subset \tilde{A_{s}}$ $$\widetilde {{A_i}} \subset \widetilde {{A_j}} \subset \widetilde {{A_s}}$$, there is $S (\tilde{A_{i}}, \tilde{A_{j}}) \geq S (\tilde{A_{j}}, \tilde{A_{i}})$ $$S\left( {\widetilde {{A_i}},\widetilde {{A_j}}} \right) \geq S\left( {\widetilde {{A_j}},\widetilde {{A_i}}} \right)$$.

3)

Definition of triangular similarity

There are many ways to calculate the closeness of S is A_i to A_j. For convenience, the degree of similarity is fixed as: (7) $S ({\tilde{A}}_{i}, {\tilde{A}}_{j}) = \frac{1}{1 + d (\tilde{A_{j}}, \tilde{A_{i}})}$ $$S\left( {{{\widetilde A}_i},{{\widetilde A}_j}} \right) = \frac{1}{{1 + d\left( {\widetilde {{A_j}},\widetilde {{A_i}}} \right)}}$$ (8) $d (\tilde{A_{i}}, \tilde{A_{j}}) = | \frac{a_{i 1} + 4 a_{i 2} + a_{i 3} - a_{j 1} - 4 a_{j 2} - a_{j 3}}{6} |$ $$d\left( {\widetilde {{A_i}},\widetilde {{A_j}}} \right) = \left| {\frac{{{a_{i1}} + 4{a_{i2}} + {a_{i3}} - {a_{j1}} - 4{a_{j2}} - {a_{j3}}}}{6}} \right|$$

If the value of the closeness is closer to 1, it is considered that the compatibility between sensor S_i and sensor S_j is better, and it is said that the closeness of observation data A_i and A_j is higher; if the value of the closeness is closer to 0, it means that the compatibility between observation data A_i and A_j is worse.

2.3.2

Data fusion algorithm based on fuzzy composite function

1)

Fuzzy composite function definition set: (9) $U = {1, 2, \dots, j, \dots, \dots, N}$ $$U = \left\{ {1,2, \cdots ,j, \cdots , \cdots ,N} \right\}$$ (10) $S = {1, 2, \dots, i, \dots, \dots, M}$ $$S = \left\{ {1,2, \cdots ,i, \cdots , \cdots ,M} \right\}$$

Let the data fusion system be a collection of sequences consisting of N target attribute of and M sensors m_ij denotes the degree of support of the ith sensor for class j of the target attribute and 0 ≤ m_i,j ≤ 1, o_i denotes that the target attribute is of class j and the distribution of the target attribute is: (11) $\prod_{i \in U} m_{i, j} / o_{j}, \forall i \in s$ $$\mathop \prod \limits_{i \in U} {m_{i,j}}/{o_j},\forall i \in s$$

A property is selected that corresponds to it: (12) $m_{i, j} = {\begin{array}{l} 1 & W h e n j = j_{j} \\ 0 & W h e n j \neq j_{l} \end{array}$ $${m_{i,j}} = \left\{ {\begin{array}{l} 1&{When\;j = {j_j}} \\ 0&{When\;j \ne {j_l}} \end{array}} \right.$$

When the target attribute is a set, then there is: (13) $m_{i, j} = {\begin{array}{l} 1 & W h e n j \in U \\ 1 - f_{i} & W h e n j \notin U_{1} \end{array}$ $${m_{i,j}} = \left\{ {\begin{array}{l} 1&{When\;j \in U} \\ {1 - {f_i}}&{When\;j \notin {U_1}} \end{array}} \right.$$ 2)

Define the trust function: (14) $f_{i} = μ_{i} \cdot c_{i}$ $${f_i} = {\mu _i} \cdot {c_i}$$

Where: μ_i is the reliability of the observations of the ind sensor, c_i is the support of the ith sensor by the other sensors. 3)

Define the affiliation function: (15) $μ (z) = {\begin{array}{l} 1 - \frac{z - μ}{2 σ} & | z - μ | \geq 2 σ \\ 0 & | z - μ | \leq 2 σ \end{array}$ $$\mu (z) = \left\{ {\begin{array}{l} {1 - \frac{{z - \mu }}{{2\sigma }}}&{\left| {z - \mu } \right| \geq 2\sigma } \\ 0&{\left| {z - \mu } \right| \leq 2\sigma } \end{array}} \right.$$

4)

Define the fuzzy synthesis function: (16) $m_{j} = S_{M} [m_{1, j}, m_{2, j}, \dots, m_{M, j}]$ $${m_j} = {S_M}\left[ {{m_{1,j}},{m_{2,j}}, \cdots ,{m_{M,j}}} \right]$$ (17) $S_{M} [m_{1, j}, m_{2, j}, \dots, m_{M, j}] = {[\prod_{i = 1}^{M} m_{i, j}]}^{1 / M}$ $${S_M}\left[ {{m_{1,j}},{m_{2,j}}, \cdots ,{m_{M,j}}} \right] = {\left[ {\prod\limits_{i = 1}^M {{m_{i,j}}} } \right]^{1/M}}$$

5)

Data fusion results: (18) $Π = \sum_{j \in U} m_{j} / o_{j}$ $$\Pi = {{\sum\limits_{j \in U} {{m_j}} } \mathord{\left/ {\vphantom {{\sum\limits_{j \in U} {{m_j}} } {{o_j}}}} \right. } {{o_j}}}$$

2.3.3

Data fusion algorithm based on fuzzy confidence distance consistency

1)

Definition of confidence distance measure

There are n homogeneous sensors composed of sensor sequence $S = {S_{1}, S_{2}, \dots, S_{n}}$ $$S = \left\{ {{S_1},{S_2}, \ldots ,{S_n}} \right\}$$, using direct observation method, the same characteristics of the parameter X in different directions for independent observation, the ith sensor and the jth sensor to obtain the observation data for X_i and X_j, then define the two sensors of the confidence distance measure is defined as follows: (19) $d_{i j} = 2 \int_{x_{i}}^{x_{j}} p_{i} (x | x_{i}) d x = 2 A$ $${d_{ij}} = 2\int_{{x_i}}^{{x_j}} {{p_i}} \left( {x|{x_i}} \right)dx = 2A$$ (20) $d_{i, j} = 2 \int_{x_{j}}^{x_{j}} p_{i} (x | x_{j}) d x = 2 B$ $${d_{i,j}} = 2\int_{{x_j}}^{{x_j}} {{p_i}} \left( {x|{x_j}} \right)dx = 2B$$

In the above equation: (21) $p_{i} (x | x_{i}) = \frac{1}{\sqrt{2 π} σ_{i}} \exp {- \frac{1}{2} {(\frac{x - x_{i}}{σ_{i}})}^{2}}$ $${p_i}\left( {x|{x_i}} \right) = \frac{1}{{\sqrt {2\pi } {\sigma _i}}}\exp \left\{ { - \frac{1}{2}{{\left( {\frac{{x - {x_i}}}{{{\sigma _i}}}} \right)}^2}} \right\}$$ (22) $p_{j} (x | x_{j}) = \frac{1}{\sqrt{2 π} σ_{j}} \exp {- \frac{1}{2} {(\frac{x - x_{j}}{σ_{j}})}^{2}}$ $${p_j}\left( {x|{x_j}} \right) = \frac{1}{{\sqrt {2\pi } {\sigma _j}}}\exp \left\{ { - \frac{1}{2}{{\left( {\frac{{x - {x_j}}}{{{\sigma _j}}}} \right)}^2}} \right\}$$ 2)

Calculation of the error function

The smaller the difference in value between the ist sensor and the jnd sensor, the smaller the error value is said to be for sensor i and sensor j, and conversely, the larger the error value is said to be for sensor i and sensor j. In order to represent the proximity between sensor observations, the error function is defined as: (23) $e r f (θ) = \frac{2}{π} \int_{0}^{θ} e^{- μ^{2} d u}$ $$erf\left( \theta \right) = \frac{2}{\pi }\int\limits_0^\theta {{e^{ - {\mu ^2}du}}}$$

And there is: (24) $d_{i, j} = e r f (\frac{x - x_{i}}{\sqrt{2} σ_{i}})$ $${d_{i,j}} = erf\left( {\frac{{x - {x_i}}}{{\sqrt 2 {\sigma _i}}}} \right)$$ (25) $d_{j, i} = e r f (\frac{x - x_{j}}{\sqrt{2} σ_{j}})$ $${d_{j,i}} = erf\left( {\frac{{x - {x_j}}}{{\sqrt 2 {\sigma _j}}}} \right)$$ 3)

Calculation of confidence distance matrix

Let there is M homogeneous sensor in the multi-sensor data fusion system to measure the same target parameter, then there is a confidence distance matrix is calculated as follows: (26) $D_{m} = [\begin{matrix} d_{11} & d_{12} & \dots & d_{1 m} \\ d_{21} & d_{21} & \dots & d_{2 m} \\ ⋮ \\ d_{m 1} & d_{m 2} & \dots & d_{m m} \end{matrix}]$ $${D_m} = \left[ {\begin{array}{*{20}{c}} {{d_{11}}}&{{d_{12}}}& \cdots &{{d_{1m}}} \\ {{d_{21}}}&{{d_{21}}}& \cdots &{{d_{2m}}} \\ {}&{}& \vdots &{} \\ {{d_{m1}}}&{{d_{m2}}}& \cdots &{{d_{mm}}} \end{array}} \right]$$

4)

Relationship matrix definition

Define a threshold, when higher than the threshold its sensor confidence is 1, when less than or equal to the threshold period confidence is 0, then there is a threshold function for: (27) $r_{i, j} = {\begin{array}{l} 1 & d_{i, j} \leq β_{i, j} \\ 0 & d_{i, j} > β_{i, j} \end{array}$ $${r_{i,j}} = \left\{ {\begin{array}{l} 1&{{d_{i,j}} \leq {\beta _{i,j}}} \\ 0&{{d_{i,j}} > {\beta _{i,j}}} \end{array}} \right.$$

The confidence distance matrix processed by the threshold function is transformed into a relation matrix: (28) $R_{m} = [\begin{matrix} r_{11} & r_{12} & \dots & r_{1 m} \\ r_{21} & r_{21} & \dots & r_{2 m} \\ ⋮ \\ r_{m 1} & r_{m 2} & \dots & r_{m m} \end{matrix}]$ $${R_m} = \left[ {\begin{array}{*{20}{c}} {{r_{11}}}&{{r_{12}}}& \cdots &{{r_{1m}}} \\ {{r_{21}}}&{{r_{21}}}& \cdots &{{r_{2m}}} \\ {}&{}& \vdots &{} \\ {{r_{m1}}}&{{r_{m2}}}& \cdots &{{r_{mm}}} \end{array}} \right]$$

3

Data fusion algorithm based on improved fuzzy weighted support degree

The effectiveness of existing data fusion methods is mainly characterized by the magnitude of accuracy of the target fusion result and the magnitude of fusion time in the case of the same data source.

The effectiveness of existing data fusion methods is mainly reflected in the magnitude of accuracy of the target fusion results and the magnitude of fusion time in the case of the same data sources. K as the conflict coefficient between data sources, in the case of K = 1, will lead to fusion result conflict and serious deviation of fusion results. In addition, the data extracted from many data sources in the process of data fusion are not single and isolated, most of them have an inseparable connection before, and the above method does not make good use of this relationship to improve the accuracy of the fusion result in the process of fusion of multiple data sources, and simply acquires these data as irrelevant data, and then utilizes its own fusion method to simply sequentially and get the final fusion result, which leads to the problem of bias in the fusion result. This paper proposes a fuzzy weighted D-S evidence theory method based on the following formula: (29) $m (C) = \sum_{A_{i} \cap B_{j} \cap \dots = C} g_{i} \cdot m_{1} (A_{i}) m_{2} (B_{j}) + \dots + k * q (C)$ $$m\left( C \right) = \sum\limits_{{A_i} \cap {B_j} \cap \ldots = C} {{g_i}} \cdot {m_1}\left( {{A_i}} \right){m_2}\left( {{B_j}} \right) + \cdots + k^*q\left( C \right)$$

Among them: (30) $k = \sum_{A_{i} \cap B_{j} \cap \dots = C} m_{1} (A_{i}) m_{2} (B_{j})$ $$k = \sum\limits_{{A_i} \cap {B_j} \cap \cdots = C} {{m_1}} \left( {{A_i}} \right){m_2}\left( {{B_j}} \right)$$ (31) $q (C) = \frac{1}{n} \sum_{i = 1}^{n} m_{i} (C), is the number of data sources$ $$q\left( C \right) = \frac{1}{n}\sum\limits_{i = 1}^n {{m_i}} \left( C \right){\text{, is the number of data sources}}$$

The method improves the fusion accuracy of the fusion calculation formula by calculating the average trust degree and utilizing the fuzzy affiliation degree as the weights, and it can effectively avoid the complete conflict problem of the fusion result between data by adopting the concept of the average trust degree when the conflict coefficient is K = 1, so as to ultimately obtain a more accurate fusion result of the multi-sensor data.

The fuzzy weighting based D-S evidence theory method is described as follows:

Input: target A to be detected, trust function of the sensor $m_{i} (K)$ $${m_i}\left( K \right)$$.

Output: detection result of target A.

A weighted average algorithm for fusion of multi-source information based on fuzzy similarity matrix, which fully considers the degree to which each evidence is supported by other evidence, and the fused results are more accurate. In the same year, Peng Huiping, Cao Xiaojun proposed a conflict handling method of D-S evidence theory based on decision distance measurement, which can overcome the conflict problem of D-S evidence theory and ensure the accuracy of information fusion results. In which the decision distance is measured as follows: (32) $D_{i j} = 2 | \int_{x_{i}}^{x_{j}} p_{i} (x | x_{i}) p (x_{i}) d x |$ $${D_{ij}} = 2\left| {\int_{{x_i}}^{{x_j}} {{p_i}} \left( {x|{x_i}} \right)p\left( {{x_i}} \right)dx} \right|$$ (33) $D_{j i} = 2 | \int_{x_{j}}^{x_{i}} p_{j} (x | x_{j}) p (x_{j}) d x |$ $${D_{ji}} = 2\left| {\int_{{x_j}}^{{x_i}} {{p_j}} \left( {x|{x_j}} \right)p\left( {{x_j}} \right)dx} \right|$$

where x_i, x_j are information points and $p (x_{i}) p (x_{j})$ $$p\left( {{x_i}} \right)p\left( {{x_j}} \right)$$ is the probability distance matrix for the occurrence of the point: (34) $D = [\begin{array}{l} d_{11} & d_{12} & d_{13} & \dots \\ d_{21} & d_{22} & d_{23} & \dots \\ d_{31} & d_{32} & d_{33} & \dots \\ d_{41} & d_{42} & d_{43} & \dots \\ d_{51} & d_{52} & d_{53} & \dots \\ \dots & \dots & \dots & \dots \end{array}]$ $$D = \left[ {\begin{array}{l} {{d_{11}}}&{{d_{12}}}&{{d_{13}}}& \ldots \\ {{d_{21}}}&{{d_{22}}}&{{d_{23}}}& \ldots \\ {{d_{31}}}&{{d_{32}}}&{{d_{33}}}& \ldots \\ {{d_{41}}}&{{d_{42}}}&{{d_{43}}}& \ldots \\ {{d_{51}}}&{{d_{52}}}&{{d_{53}}}& \ldots \\ \ldots & \ldots & \ldots & \ldots \end{array}} \right]$$

where D denotes the distance matrix between all points, where each element denotes the decision distance between any two points Define the relation matrix R on the basis of D as: (35) $R = [\begin{matrix} r_{11} & r_{12} & r_{13} & \dots \\ r_{21} & r_{22} & r_{23} & \dots \\ r_{31} & r_{32} & r_{33} & \dots \\ r_{41} & r_{42} & r_{43} & \dots \\ r_{51} & r_{52} & r_{53} & \dots \\ \dots & \dots & \dots & \dots \end{matrix}]$ $$R = \left[ {\begin{array}{*{20}{c}} {{r_{11}}}&{{r_{12}}}&{{r_{13}}}& \ldots \\ {{r_{21}}}&{{r_{22}}}&{{r_{23}}}& \ldots \\ {{r_{31}}}&{{r_{32}}}&{{r_{33}}}& \ldots \\ {{r_{41}}}&{{r_{42}}}&{{r_{43}}}& \ldots \\ {{r_{51}}}&{{r_{52}}}&{{r_{53}}}& \ldots \\ \ldots & \ldots & \ldots & \ldots \end{array}} \right]$$

Where R is the relationship matrix built on the basis of the decision distance r_ij the value of which is determined by the threshold value (determined empirically) e, $r_{i j} = {\begin{array}{l} 1, W h i c h d_{i j} \leq e \\ 0, W h i c h d_{i j} \geq e \end{array}$ $${r_{ij}} = \left\{ {\begin{array}{l} {1,\;Which{\text{ }}{d_{ij}} \leq e} \\ {0,\;Which{\text{ }}{d_{ij}} \geq e} \end{array}} \right.$$, r_ij = r_ji = 0 denotes mutual independence, r_ij = 0, r_ji = 1 denotes weak support, r_ij = r_ji = 1 denotes strong support. The method is complex in calculating the decision distance as well as the support calculation, which is prone to unavoidable errors and leads to anomalies in the fusion results.

In this paper, the proposed fuzzy weighted D-S evidence theory based method is improved and the fuzzy weighted support based D-S evidence theory data fusion method is proposed, which utilizes the distance between the evidence sources to calculate a distance matrix of all the evidence sources, and then calculates the overall trust of the target in all the information sources based on this distance matrix, and then averages it to find the average support of the target in the The average support degree W_i in the whole system, and combined with the calculated fuzzy affiliation degree g_i as the weights for fusion calculation, so that the overall weights can be known that is D_i = W_i · g_i, and then use the existing formula to calculate the final fusion results. 1)

Determination and calculation method of affiliation function

The fuzzy statistical method uses fuzzy statistical experiments to determine the value of the affiliation degree of an element, and its basic requirement is to make a judgment on whether the element u fixed in the domain U belongs to the changeable set A_x, where A_x is the elastic frontier of the fuzzy set A. In each experiment, u is fixed and A_x is changeable, and after conducting n experiments, the affiliation degree of u is the ratio of its number of times in A to n. That is, U is an argument domain, μ is a function that maps any A ∈ U to some value on [0, 1], and can be set $μ (A)$ $$\mu \left( A \right)$$ to denote the affiliation of u within U.

The fuzzy statistical method is calculated to define that if multiple sensors are used to detect multiple objects, if the number of sensors supporting an object can be determined to be n, and the total number of sensors is N, the degree of affiliation μ(Ai) can be calculated using the affiliation function μ(Ai) = n/N, Ui for the ith object. Where, N is the total number of sensors and n is the number of sensors supporting the measured object (sensors whose trust is not 0) A larger affiliation degree indicates a greater correlation with the U thesis. An affiliation matrix is obtained: (36) $G = [g_{1}, g_{2}, g_{3}, g_{4}, g_{5} \dots \dots]$ $$G = \left[ {{g_1},{g_2},{g_3},{g_4},{g_5} \cdots \cdots } \right]$$ (37) $g_{i} = μ (A_{i}), i = 1, 2, 3, 4, 5 \dots \dots$ $${g_i} = \mu \left( {{A_i}} \right),i = 1,2,3,4,5 \cdots \cdots$$

2)

Operations on fuzzy sets

Some of the properties of fuzzy sets can be expressed in terms of the affiliation function. This includes fuzzy set inclusion, equivalence relations, and their operations of concatenation, intersection and complement. Let U be the domain, A and B are two fuzzy sets on it, μ_A and μ_B are their fuzzy affiliation functions respectively, then there are: (1)

If for every element u in U, there is μ_A(u) ≥ μ_B(u), then A contains B;

(2)

If for any element u in U, there is μ_A(u) = μ_B(u), then A and B are said to be equal;

(3)

The complement of A (38) $\bar{A} μ_{\bar{A}} (u) = 1 - μ_{A} (u)$ $$\bar A{\mu _{\bar A}}(u) = 1 - {\mu _A}(u)$$

(4)

Fuzzy subsets of the concatenation (39) $μ A \cup B (u) = μ_{A} (u) \lor μ_{B} (u) = \max [μ_{A} (u), μ_{B} (u)]$ $$\mu A \cup B(u) = {\mu _A}(u) \vee {\mu _B}(u) = \max \left[ {{\mu _A}(u),{\mu _B}(u)} \right]$$

(5)

Intersection of fuzzy subsets (40) $μ A \cap B (u) = μ_{A} (u) \land μ_{B} (u) = \min [μ_{A} (u), μ_{B} (u)]$ $$\mu A \cap B(u) = {\mu _A}(u) \wedge {\mu _B}(u) = \min \left[ {{\mu _A}(u),{\mu _B}(u)} \right]$$

Here max and min represent the operation of taking big and small respectively. According to the relationship operation between the above fuzzy sets, the relationship between the affiliation degree of each information source for the information can be directly judged, which is conducive to the judgment of the reliability of the information and the accuracy of the information, in order to facilitate effective state detection of the target and the identification of the nature of the target, and finally arrive at the most accurate fusion information. 3)

Calculation of support degree

Multi-sensor information fusion in each information source on the information trust is different, and each pair of information sources is also the existence of mutual trust, so the size of the trust value of the information and all the information sources are closely related to each other’s degree of trust, m_i, m_j for any two sources of information corresponding to the information of the trust function, each pair of information sources of the trust function of the distance between the information sources on behalf of the trust between these pairs of sources, the trust can not be directly solved to derive. The degree of trust can not be derived directly from the solution, you need to calculate the distance between each pair of information sources d, which can be expressed as $d (m_{i}, m_{j})$ $$d\left( {{m_i},{m_j}} \right)$$, using the distance between each pair of information sources to derive the distance matrix corresponding to all the information sources, the corresponding degree of trust that is the inverse of the distance between the information sources. The specific definitions are as follows:

Definition: m₁ and m₂ are two trust degree functions on the identification framework Θ, then the distance between m₁ and m₂ can be expressed as: (41) $d (m_{1}, m_{2}) = {\sqrt{{(m_{1} - m_{2})}^{2}}}^{2}$ $$d\left( {{m_1},{m_2}} \right) = {\sqrt {{{\left( {{m_1} - {m_2}} \right)}^2}} ^2}$$

Using m₁, m₂ as vectors, i.e: (42) $\begin{array}{rcl} m_{l} & = & [m_{l} (a), m_{l} (b), m_{l} (c), m_{l} (d), \dots \dots] \\ m_{2} & = & [m_{2} (a), m_{2} (b), m_{2} (c), m_{2} (d), \dots \dots] \\ {(m_{l} - m_{2})}^{2} = m_{l}^{2} + m_{2}^{2} + 2 m_{2} m_{l} \end{array}$ $$\begin{array}{*{20}{rcl}} {{m_l}}& = &{\left[ {{m_l}(a),{m_l}(b),{m_l}(c),{m_l}(d), \ldots \ldots } \right]} \\ {{m_2}}& = &{\left[ {{m_2}(a),{m_2}(b),{m_2}(c),{m_2}(d), \ldots \ldots } \right]} \\ {}&{}&{{{\left( {{m_l} - {m_2}} \right)}^2} = m_l^2 + m_2^2 + 2{m_2}{m_l}} \end{array}$$

The distance matrix $D = [\begin{matrix} 0 & d_{21} & d_{31} & \dots \\ d_{12} & 0 & d_{32} & \dots \\ d_{13} & d_{23} & 0 & \dots \end{matrix}]$ $$D = \left[ {\begin{array}{*{20}{c}} 0&{{d_{21}}}&{{d_{31}}}& \cdots \\ {{d_{12}}}&0&{{d_{32}}}& \cdots \\ {{d_{13}}}&{{d_{23}}}&0& \cdots \end{array}} \right]$$ is derived from the distance before each sensor, and the average value of each column is sought as the average distance of m_i(i = 1……n). The larger the average distance indicates that its system support is smaller, so the inverse of the average distance is used to express the degree of trust (support), i.e., $W_{i} = \frac{1}{d_{i}}$ $${W_i} = \frac{1}{{{d_i}}}$$.

Through the fuzzy affiliation degree of the information source and the support degree between the information sources sought above, these two values about the accuracy of the information are multiplied to get the effective fusion of the two, and the target fusion results calculated according to this method are closer to the actual target results. The calculation formula is as follows: (43) $m (C) = \sum_{A_{i} \cap B_{j} \cap \dots = C} W_{i} \cdot g_{i} \cdot m_{1} (A_{i}) m_{2} (B_{j}) + \dots + k * q (C)$ $$m\left( C \right) = \sum\limits_{{A_i} \cap {B_j} \cap \ldots = C} {{W_i}} \cdot {g_i} \cdot {m_1}\left( {{A_i}} \right){m_2}\left( {{B_j}} \right) + \cdots + k^*q\left( C \right)$$

Where, g_i = μ(Ai) = n/N is the overall affiliation of the information and $W i = \frac{1}{d i}$ $$Wi = \frac{1}{{di}}$$ is the overall trustworthiness of each information in the whole source, i.e., it is the value of the accuracy data obtained by the information in the whole source.

4

Validation of data fusion method based on improved fuzzy support degree

4.1

Experimental data sources and processing

The data for this simulation experiment comes from the humidity data (corrected relative humidity, range 0~100%) obtained from the real sensor network deployed by the Berkeley-Intel research group. In this simulation experiment, 44 humidity data obtained from the monitoring of nodes 4~6 are taken as experimental data, and the data collection interval is 11 minutes/time, with a total collection time of 7.9 hours. The humidity data collected by the three sensor nodes are relatively similar, and the trend of data sequence changes over time is more apparent and diverse. Sensor nodes 4, 5 and 6 are located in the central hall of the laboratory, and the accurate and reasonable fusion of the humidity data from the three sensors is an important basis for judging the indoor environmental conditions in the area.

The time series data constituted by the original sample data has the problems of large fluctuation and poor smoothness, which will have a great impact on the accuracy of the subsequent data fusion, so the polynomial least squares filtering method is firstly used to process the sample data. After many experiments, the optimal parameters are obtained, i.e., the fitting order is 2, and the length of the data window is 5. The filtering effect is shown in Fig. 2, and (a) ~ (c) are the results of the noise reduction processing of the humidity sample data collected from three sensor nodes using polynomial least squares filtering. The noise-reduced humidity data has better stability and smoothness than the original sample data, and the difference is only between 0-0.5% and 0.5%. The difference is only between 0-0.5% RH, which basically retains the shape of the original humidity curve, and can provide effective and reliable data for the establishment of the subsequent multi-sensor data fusion model. Therefore, polynomial least squares filtering can play its advantage in sensor data preprocessing to achieve the purpose of smoothing time series data.

4.2

Analysis of the effect of data fusion

In this section, the dynamic time regularization algorithm is used to calculate the dynamic bending distance between two by two of the time series data consisting of the data collected by the three sensor nodes throughout the time period, as a substitute for the absolute distance in the traditional fuzzy support function. Among them, adjusting the value of K can change the degree of fuzziness of the support function. In the whole monitoring time period, three humidity sensor nodes measured data composed of time series data between the dynamic bending distance calculation and weighted value is shown in Table 1. It can be seen that the dynamic bending distance is distributed between 5 and 12, so the appropriate K value is selected so that the fuzzy support corresponding to the dynamic bending distance in this range has a greater degree of differentiation, and the K value of the improved fuzzy support function is selected to be 0.3. At the same time, the variance of the time series data for each time period is computed as a part of the weighting, and the final data fusion is performed, and the final fusion weighting values of nodes 1 to 3 are respectively 0.213, 0.291 and 0.496.

Table 1.

Dynamic bending distance calculation and weighted value

Dynamic bending distance	Node 1	Node 2	Node 3	Support value	The value of its own reliability	Final fusion weighted value
Node 1	0	11.23	5.70	0.220	0.322	0.213
Node 2	11.23	0	6.40	0.299	0.333	0.291
Node 2	5.70	6.40	0	0.481	0.345	0.496

The fusion results are compared with the original data as shown in Fig. 3. After filtering the time series data of the three humidity nodes and performing the fusion operation, the fusion results have eliminated the interference of anomalous data, and the fusion curves are smoother, which are in line with the normal trend of the indoor humidity changes. The fusion method accurately fuses the three humidity sensor node data, reasonably reflects the indoor environment humidity changes covered by these three humidity sensor nodes, and verifies the effect of fuzzy set-based multi-sensor data fusion in this paper.

4.3

Comparative analysis of data fusion applications

To further validate the performance and practicality of the data fusion algorithm in this paper, target recognition performed by a UAV during an agricultural inspection operation is used as a background, with reference values A, B and C for normal, insect-infested and diseased rice, respectively. The target identification sensor system includes infrared, radar, gimbal cameras, and electronic support facilities. Targets were identified using six methods, including the methods of this paper, Yager, Dempster, Murphy, Zhang, and Ghiasi. A comparison of the recognition results of fusing the four evidences is shown in Table 2.From the recognition results, it can be seen that the method of this paper can still correctly recognize the target as a normal crop after fusing the four evidences and has a high support rate (0.9996) for A, thus proving that the method of this paper also has a good recognition ability in the case of fusing the four evidences.The methods of Dempster and the method of this paper are both able to obtain accurate recognition Yager’s method fails to give correct identification results when there are conflicts between the evidence and the evidence becomes more positive for A. Ghiasi’s method also lacks high support for A. The reason may be that Ghiasi’s method is more suitable for the case of a large sample size. In the event that the evidence becomes too much, the other four approaches can precisely identify the target, except for Yager’s and Ghiasi’s methods.

Table 2.

Comparison of the four evidence of the fusion

Method	m(A)	m(B)	m(C)	m(Φ)
Yager	0.3880	0	0.000034	0.6120
Dempster	0.9991	0	0.000086	0
Murphy	0.9994	0.000022	0.000648	0
Zhang	0.9994	0.000022	0.000322	0
Ghiasi	0.5146	0	0.000495	0
Ours	0.9996	0.00000009	0.000006	0

Comparison of target recognition success rate when fusing four evidences is shown in Fig. 4, for example, the method in this paper fully describes the correlation between evidences, and when conflicting data evidences are generated, the conflict is not assigned to unknown item m(Φ), but combines the characteristics of correlation between evidences in accordance with the reliability of conflicting evidences by implementing the scientific and appropriate weights, efficiently utilizing correlation between evidences, and properly handling the conflict between evidences, which reduces the interference of evidence conflict with the data fusion results and has a strong anti-interference ability. It reduces the interference of evidence conflicts on the data fusion results and has strong anti-interference capability. From the figure, it can be intuitively seen that when fusing four pieces of evidence, compared with the other five methods, the optimized method in this paper can obtain the highest recognition success rate, with a recognition success rate of 82.6% with better feasibility. This further validates the effectiveness of the optimized method proposed in this paper.

5

Conclusion

This paper focuses on the relevant theoretical methods, concepts, results of fusion, as well as advantages and disadvantages of fuzzy set approach and evidence theory approach in data fusion algorithms, proposes data fusion algorithms based on fuzzy weighted support, and carries out relevant research, the main work is as follows: 1)

The basic theoretical knowledge of multi-sensor data fusion is studied. It mainly includes the process of data fusion, and the methods of data fusion and the efficiency and scope of application of various methods, in which the fusion method based on the evidence theory is studied in depth, which provides a theoretical basis for the subsequent research of this paper.

2)

This paper proposes a data fusion algorithm based on the improved fuzzy weighted support degree, through the dynamic time regularization algorithm, calculates the dynamic bending distance between two and two of the time series data composed of the data collected by the three sensor nodes in the whole time period, so as to replace the absolute distance in the traditional fuzzy support degree function. The K value of the improved fuzzy support function is selected to be 0.3, and the final fusion weighted values of nodes 1~3 are 0.213, 0.291 and 0.496, respectively.After filtering the time-series data of the three humidity nodes and performing the fusion operation, the fusion curves are smoother and in line with the normal trend of change. The data fusion method in this paper can accurately fuse the data of multiple sensor nodes, which reasonably reflects the data changes covered by the sensor nodes, and verifies the effect of multi-sensor data fusion based on fuzzy sets in this paper.

3)

In the data fusion application comparison, this paper’s method still correctly identifies the target as a normal crop after fusing four evidences from the sensors, with the highest support rate of 0.9996. The optimized method in this paper is applied to the actual target detection, the recognition success rate is as high as 82.6%, compared with other typical data fusion methods can be able to obtain a better recognition success rate, which illustrates the data fusion in this paper has a higher adaptability and application value.

Language:: English

Publication timeframe:: 1 times per year
Journal Subjects:: Life Sciences, Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics, Physics, other

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Design of fuzzy set-based deep fusion algorithm for multi-sensor data

Weibing Li

Shenggang Wu

Published Online: Mar 21, 2025

Received: Oct 18, 2024

Accepted: Feb 01, 2025

DOI: https://doi.org/10.2478/amns-2025-0676

KeywordsMulti-sensor, Fuzzy set, Data fusion, Weighted support degree

© 2025 Weibing Li et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Multi-sensor, Fuzzy set, Data fusion, Weighted support degree