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SN Applied Sciences

, 2:79 | Cite as

Evaluation method for HRRP imaging quality based on intuitionistic interval fuzzy set

  • Xinghai LiuEmail author
  • Jian Yang
  • Jian Lu
  • Zhicheng Yao
Review Paper
  • 133 Downloads
Part of the following topical collections:
  1. Engineering: Advances in Robotics, Control and Automation (ARCA)

Abstract

Aimed at the performance evaluation of radar target recognition system based on high resolution range profile (HRRP), this paper proposes an evaluation method of HRRP quality based on intuitionistic interval fuzzy set under the background that HRRP has the characteristics of translation, attitude and velocity sensitivity. Compared with the general evaluation method, this method has two improvements. One is to extend the index value from single value to interval value by interval estimation method in mathematical statistics, which introduces sample variance into calculation to improve the utilization rate of sample information. The other is to construct intuitionistic fuzzy set by using correct recognition rate derivative sequence (DSCRR) and false recognition rate derivative sequence (DSFRR), which can make full use of experimental data and improve the reliability of experimental results. The distance calculation method of intuitionistic interval fuzzy set is used to get the distance between each scheme and the ideal scheme. Finally, the score of schemes which represent different HRRP imaging quality are obtained, and the performance evaluation is completed. The evaluation results obtained from simulation validation experiments are in accordance with objective facts, which proves the validity of this evaluation method.

Keywords

Radar seeker Performance evaluation High resolution range profile Intuitionistic interval fuzzy set Correct recognition rate 

1 Introduction

Target recognition is a process of detecting target signals, extracting target features and judging target types and attributes using optical, infrared and radar sensor etc. platforms. The key of the recognition process is to obtain information sources that fully reflect the characteristics of the target and to extract and classify the features reasonably. This is the research focus in the field of target recognition. Intelligent target recognition algorithm based on neural network and infrared image is introduced in the literature [1]. An effective nonlinear data processing method is proposed in the literature [2, 3] and this method is useful in target perception algorithm based on recurrent neural network. Robust radar target recognition and classification method based on SAR image and fusion decision and automatic target recognition method based on image registration is introduced in the literature [4]. The application effect in the field of automatic target recognition of two classification methods of neural network and support vector machine are compared in the literature [5]. These processing methods have achieved good recognition results.

With the development of high-speed data acquisition and processing technology, new system radars with wide band and full polarization measurement have been applied, which greatly promotes radar target detection and recognition technology. Radar system has become an important part of target detection and recognition system. In the current application of radar target recognition, feature extraction and recognition based on high resolution range profile (HRRP) is an important part. Thermal effect under G-N theory and uniqueness results for a boundary value problem in dipolar thermo elasticity are discussed in the literature [6, 7]. Literature [8, 9] uses dynamic system to model the target HRRP, which overcomes the impact of HRRP attitude sensitivity on target detection to a certain extent. Besides a detector for detecting range-diffusion targets based on HRRP in noisy environment is designed, which achieves good detection results. The target detection algorithm using waveform entropy under the background of less prior knowledge is studied. HOBW method and Haar wavelets scheme for solving the nonlinear and unsteady process is introduced in the literature [10, 11]. These methods can make statistical model for compound HRRP and the application method of HRRP phase and amplitude information in target detection and recognition. At present there are few studies on HRRP imaging quality assessment. In similar studies, the performance evaluation method of adaptive target detection system for echo fluctuation radar is studied in the literature [12]. The performance of target recognition for Spectrum Sharing MIMO radar is studied in the literature [13]. The influence of sea clutter on radar target detection performance is analysed in literature [14]. These existing research results provide useful support for radar target recognition performance and HRRP imaging quality evaluation in this paper.

The remainder of this paper is organized as follows. Section 2 introduces HRRP basic characteristics and takes DSCRR and DSFRR into the HRRP imaging quality evaluation. Section 3 proposes a TOPSIS evaluation method based on intuitionistic interval fuzzy set (IIFS). Section 4 completes a HRRP evaluation method effectiveness validation process with simulation experiment. Section 5 is a simple summarization of this paper. Algorithm main structure in this paper is shown in Fig. 1.
Fig. 1

Evaluation algorithm schematic configuration

2 HRRP basic characteristics and DSCRR

2.1 Radar target scattering centre and HRRP

The classical electromagnetic scattering theory show that target scattering characteristics vary greatly when the size relationship between \(L\) and \(\lambda\) are different, where \(L\) is the size of different targets in the direction perpendicular to the electromagnetic wave front and \(\lambda\) is the incident electromagnetic wavelength. When \(L \gg \lambda\), the interaction between scatters is weak and radar echo information is mainly from stationary points and integral endpoints. Scatters can be equivalent to a set of discrete points. These discrete points are called target scattering centre. Here the corresponding electromagnetic frequency band is in the Optical region, also known as the high frequency region. In the literature [15, 16], Keller and Akira develop the mathematical description of the scattering centre at the tip and edge of the target according to the theory of geometric diffraction. In the literature [17, 18], Bechtel uses scattering centre model to classify local scattering sources. With the development and application of broadband radar, target scattering characteristics measured in high frequency region can be well described by target scattering centre model, which verifies the effectiveness of target scattering centre model [19].

The relationship between radar range resolution \(\Delta R\) and radar bandwidth \(B\) is as shown below [20]:
$$\Delta R = c/2B$$
(1)
where \(c\) is the propagation rate of electromagnetic wave.

For narrow-band radar, the range resolution is small, and the equivalent scattering centres along the radar line of sight cannot be distinguished. In this case, the target is a “point target” whose electromagnetic characteristics are expressed by Radar Cross Section (RCS). With the increase of radar bandwidth \(B\), radar range resolution \(\Delta R\) increases. When the radar signal bandwidth is large enough to satisfy the range resolution far less than the target size, the equivalent scattering centres of the target can be distinguished in the radar line of sight. The wide-band radar echoes show amplitude fluctuations on multiple range units. These amplitude fluctuations are consistent with the scattering characteristics of the target along the radar line of sight direction, which is called High Resolution Range Profile (HRRP). HRRP can be regarded as the projection of target scattering centre in the direction of radar line of sight, reflecting the radial position relationship between target scattering centres and the structural characteristics of part of the target, and it is an important information for radar target recognition. The HRRP mathematical form of linear frequency modulated waveform emission signal are described in the literature [21].

2.2 HRRP sensitivity characteristics

The feature of HRRP varies with the interception position of range window, which is called the translation sensitivity of HRRP. In the target recognition process based on HRRP features, considering its translation sensitivity, the initial range profile should be processed. The processing methods include translation alignment, translation registration and compensation, translation invariant feature extraction and so on [22].

Amplitude sensitivity of HRRP refers to the amplitude fluctuation caused by different detection radar performance and detection environment for the same target. In the target recognition process based on HRRP feature, considering its amplitude sensitivity, the commonly used solution is to normalize the range profile amplitude. This method discards the absolute range information of HRRP and retains the relative range information between scattering centres. Therefore, the target relative shape information can be obtained for next target feature extraction and recognition process.

when the relative sight line angle between target and radar changes, the distribution characteristics and scattering characteristics of target scattering centres in each range unit will change, and the synthesis target echoes will also change too. This feature is called HRRP attitude sensitivity. Therefore, in the target recognition process based on HRRP, it is of great significance to extract features that are not sensitive to attitude [23].

For a moving target whose radar sight line direction velocity is \(v\), Doppler effect will change time delay from echo signal to transmitted signal and HRRP which is synthesize by echo signal. This characteristic is known as HRRP velocity sensitivity. Condition sensitivity will affect HRRP imaging quality and bring many difficulties in target recognition process.

2.3 DSCRR and DSFRR

At present, the main application area of HRRP is target recognition. The HRRP quality directly affects the target recognition effect. For the target recognition process based on HRRP, when recognition algorithm is same, the target recognition effect can directly reflect the HRRP image quality. From this point of view, we can evaluate HRRP image quality using the target recognition effect. In this paper, we take the Derivative Sequence of Recognition Rate (DSRR) as HRRP evaluation indexes, which is initially used to evaluate target recognition effect referring to the literature [24].

In the target recognition process, the total recognition results are divided into groups on average. Dividing the number of correct recognition times by total test numbers in each group, a sample of Derivative Sequence of Recognition Rate (DSCRR) can be obtained. Similar processing can result in Derivative Sequence of False Recognition Rate (DSFRR) and Derivative Sequence of Unrecognized Rate (DSUR).

In order to get DSCRR samples, it is necessary to test the target recognition system \(m\,*\,n\) times, \(m\) is the number of test groups, and \(n\) is the test numbers in each group. Single test results are independent with each other. For a group test, When the test target is correctly recognized for the \(i\)th time, the test results are recorded as \(r_{i} = 1\), otherwise \(r_{i} = 0\). When the test target is falsely recognized for the \(i\)th time, the test results are recorded as \(f_{i} = 1\), otherwise \(f_{i} = 0\). When the test target cannot be recognized for the \(i\)th time, the test results are recorded as \(u_{i} = 1\), otherwise \(u_{i} = 0\). Then, a DSCRR, a DSFRR and a DSUR sample can be obtained, their value are \(R = ( {\sum\nolimits_{i = 1}^{n} {r_{i} } } )/n\), \(F = ( {\sum\nolimits_{i = 1}^{n} {r_{i} } } )/n\), \(U = ( {\sum\nolimits_{i = 1}^{n} {u_{i} } } )/n\) respectively. After completing the \(m\) groups test, the corresponding \(m\) samples can be obtained.

For each test group, a binomial distribution is formed by \(r_{i}\), which is correctly identifying sample sequence at a single time. For \(\xi_{n} = \sum\nolimits_{i = 1}^{n} {r_{i} }\), according to the central limit theorem of independent identical distribution, there are:
$$\lim P\left\{ {\frac{{\zeta_{n} - np}}{{\sqrt {np(1 - p)} }} \le x\left| {n \to \infty } \right.} \right\} = \int\limits_{ - \infty }^{x} {\frac{1}{{\sqrt {2\pi } }}} e^{{ - t^{2} /2}} dt$$
(2)

Here \(p\) is mathematical expectation of sample distribution and \(P\) is probability distribution function.

It shows that when the number of samples is large enough (generally considered as \(n \ge 50\)), \(\xi_{n}\) and \(\xi_{n} /n\) are normal distribution respectively. That is to say, the sample value of DSCRR \(\left\{ {R_{i} \left| {i = 1,2, \ldots ,m} \right.} \right\}\) is normal distribution for \(m\) group test. The same way can show that DSFRR and DSUR are also normal distribution. Such a conclusion can also be verified by the measured data in Fig. 2.
Fig. 2

100 sets DSCRR test values (left) and their statistical distribution (right) of a target recognition system

2.4 DSCRR sample size calculation

To obtain accurate DSCRR characteristics, it is necessary to meet the sample size requirements. We use hypothesis test method to determine the sample size range. Efficiency function synthesizes two kinds of risks in a unified form through rejection domain, which can help to obtain appropriate test sample size. The minimum sample size under given risk constraints can be determined using target recognition results mean as test statistics. Make hypothesis tests:
$$H_{0} :p = p_{0} ;\quad H_{1} :p \ne p_{0}$$
where \(p_{0}\) is the true value of system recognition rate, \(p\) is the mean value of system recognition results,\(p = \bar{x} = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {x_{i} }\), \(\sigma^{2} = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {x_{i} - \bar{x}} \right)^{2} }\) is the sample variance. Taking \(\sqrt n \,*\,\frac{{p - p_{0} }}{\sigma }\) as test statistics, the rejection domain of the hypothesis test is:
$$W_{1} = \left\{ {(x_{1} ,x_{2} , \ldots x_{n} ):\left| {\sqrt n \,*\,\frac{{\bar{x} - p_{0} }}{\sigma }} \right| > u_{1 - \alpha /2} } \right\}.$$
(3)
The efficiency function of the hypothesis test is:
$$\beta (\bar{x}) = 1 - P\left\{ {\left| {\sqrt n \,*\,\frac{{\bar{x} - p_{0} }}{\sigma }} \right| > u_{1 - \alpha /2} } \right\}.$$
(4)
The class II risk \(\beta (\bar{x})\) of the hypothesis test is:
$$\begin{aligned} 1 - \beta (\bar{x}) & = \varphi \left( {u_{1 - \alpha /2} - \sqrt n \,*\,\frac{{p - p_{0} }}{\sigma }} \right) \\ & \quad - \varphi \left( { - u_{1 - \alpha /2} - \sqrt n \,*\,\frac{{p - p_{0} }}{\sigma }} \right). \\ \end{aligned}$$
(5)

Set significance level \(\alpha = 0.1\), when \(\left| {p - p_{0} } \right|\) is not less than 0.016, the class II risk \(\beta (\bar{x})\) is less than 0.1, The minimum sample size \(n\) obtained by numerical calculation is shown in the table below.

Table 1 shows the relationship between sample size and recognition rate, where R represents recognition rate true value and S represents minimum sample size [24]. It shows that given the significance level \(\alpha = 0.1\) and class II risk \(\beta (\bar{x}) < 0.1\), when the true value of recognition rate is more than 0.5, the minimum sample size is less than 95. When the sample size is greater than 95, our estimation of the statistical characteristics of DSCRR is credible. This calculation results provide a reference for the simulation experiments design.
Table 1

Relationship between sample size and recognition rate

R

0.5

0.6

0.7

0.76

0.8

0.86

0.9

S

95

93

87

81

76

66

57

2.5 DSCRR statistical property interval estimation

HRRP imaging quality assessment is a systematic project. To get an objective evaluation conclusion, it is necessary to establish a complete evaluation index set and use different indexes to describe all aspects of HRRP characteristics. In following part, the relationship between the statistical characteristics of DSCRR and the performance indicators is analysed. Then, the interval estimation of the statistical parameters of DSCRR is carried out using the mathematical statistics method. Finally, HRRP imaging quality is evaluated based on the intuitionistic interval fuzzy set and TOPSIS evaluation method.
  1. 1.

    Robustness Index and Variance Interval Estimation of DSCRR

     

DSCRR variance is used to measure the deviation from the samples mean at different sample points, which reflects the DSCRR fluctuation characteristics. This feature can be used to describe the robustness of HRRP imaging quality evaluation.

Under the prior conclusion that the DSCRR sample satisfies the normal distribution, the interval estimation method of DSCRR variance is as follows. Assuming that \(y_{1} ,y_{2} , \ldots ,y_{m}\) is the sample observation value from the population \(Y\), the mean and variance of the sample can be calculated separately:
$$\bar{y} = \frac{1}{m}\sum\limits_{i = 1}^{m} {y_{i} } \quad s^{2} = \frac{1}{m}\sum\limits_{i = 1}^{m} {\left( {y_{i} - \bar{y}} \right)^{2} } .$$
(6)
A rigorous variance test is used, assuming that:
$$H_{0} :\sigma^{2} = \sigma_{0}^{2} ;\quad H_{1} :\sigma^{2} \ne \sigma_{0}^{2}$$
Referring to literature [25], the test statistics are as follows:
$$\frac{{\left( {m - 1} \right)s^{2} }}{{\sigma_{0}^{2} }} \sim \chi^{2} \left( {m - 1} \right).$$
(7)
At a given saliency level \(\alpha\), there are:
$$P\left\{ {\chi_{1 - \alpha /2}^{2} \left( {m - 1} \right) < \frac{{\left( {m - 1} \right)s^{2} }}{{\sigma_{0}^{2} }} < \chi_{\alpha /2}^{2} \left( {m - 1} \right)} \right\} = 1 - \alpha .$$
(8)
The confidence interval of \(\sigma_{ 0}^{ 2}\) can be obtained as follows:
$$\left[ \begin{aligned} \left( {m - 1} \right) \,*\,s^{2} /\chi_{\alpha /2}^{2} \left( {m - 1} \right), \, \hfill \\ \left( {m - 1} \right) \,*\,s^{2} /\chi_{{1{ - }\alpha /2}}^{2} \left( {m - 1} \right) \hfill \\ \end{aligned} \right]$$
The calculation result is the variance estimation interval of DSCRR distribution.
  1. 2.

    Correctness Index and Mean Interval Estimation of DSCRR

     
The DSCRR sample mean is more reasonable and accurate to reflect the recognition ability than single DSCRR sample value, and the larger the sample size, the more accurate the evaluation conclusion is. For populations that conform to normal distribution, the mean \(\mu_{ 0}\) of population distribution can be estimated by hypothesis test when the mean \(\bar{y}\) and variance \(s^{2}\) of observation samples are known. Using a rigorous mean test method:
$$H_{0} :\mu_{ 0} = \bar{y};\quad H_{ 1} :\mu_{ 0} \ne \bar{y}$$
The test statistics are as follows:
$$\sqrt m \frac{{\bar{y} - \mu_{0} }}{s} \sim t(m - 1).$$
(9)
At a given saliency level \(\alpha\), the confidence interval of \(\mu_{ 0}\) can be obtained as follows:
$$\left[ {\bar{y} - t_{\alpha /2} (m - 1)\frac{{\sigma_{0} }}{\sqrt m }, \, \bar{y} + t_{\alpha /2} (m - 1)\frac{{\sigma_{0} }}{\sqrt m }} \right]$$
The calculation result is the mean estimation interval of DSCRR distribution.
  1. 3.

    Conditional independence index

     
Conditional independence index \(I_{ind}\) is suitable for measuring the independence between HRRP quality and external conditions. \(I_{ind}\) is measured by the change degree of target recognition effect when external conditions change in a certain range. The control variable method can be used to calculate the independence. That is, to control the change of a certain condition to be measured while the other conditions remain unchanged. By calculating the change value \(\Delta \mu\) of DSCRR mean and variance, the ratio \(I_{ind}\) between change value and the threshold \(\eta\) is obtained.
$$I_{ind} = \frac{\Delta \mu }{\eta }.$$
(10)

The threshold \(\eta\) is related to application requirement. When \(\Delta \mu\) is interval value, the conditional independence index \(I_{ind}\) calculated is also interval value.

3 HRRP quality fuzzy evaluation method

3.1 Fuzzy set

Fuzziness refers to the membership ambiguity in the process of transition from one side to the other, such as the state uncertainty in the process of transition from “stability” to “instability”. Fuzziness reflects the ambiguity of set boundaries. Since 1965, Zadah defined fuzzy sets, considerable progress has been made in fuzzy mathematics and related research based on it [26, 27]. Atanassov is one of the important scholars who promote development of fuzzy theory. He proposed and perfected intuitionistic fuzzy theory and interval intuition theory, which expanded application scope of classical fuzzy theory [28, 29, 30]. Fuzzy evaluation introduces fuzzy mathematics into classical evaluation methods and effectively converts ambiguous subjective evaluation into definite and quantitative evaluation results.

Definition 1

For set \(A\) with unclear boundary, object \(x_{i}\) no longer belong to or do not belong to \(A\) explicitly, but have a certain membership degree represented by \(\mu_{i}\).Corresponding set \(A = \{ < x_{i} ,\mu_{i} > |i = 1,2, \ldots n\}\) is a Fuzzy Set. \(< x_{i} ,\mu_{i} >\) is a fuzzy element with membership degree.

Definition 2

[31, 32]: When the membership degree is difficult to be expressed by a single numerical value, and it can be expressed by a series of numerical values, which is set as \(\{\mu_{li} |l = 1,2, \ldots m\}\). Corresponding set \(A = \{ < x_{i} ,\mu_{li} > |i = 1,2, \ldots n,l = 1,2, \ldots m\}\) is a hesitant fuzzy set.

Definition 3

Let \(\mu_{A} (x)\) be the membership degree, and \(v_{A} (x)\) be the none-membership degree which is used to measure the degree of object \(x_{i}\) not belong to \(A\). Then \(A = \{ (x,\mu_{A} (x),v_{A} (x))|x \in X\}\) is an intuitionistic fuzzy set. Here \(0 \le \mu_{A} (x) \le 1\), \(0 \le v_{A} (x) \le 1\), \(0 \le \mu_{A} (x) + v_{A} (x) \le 1\), \(\pi_{A} (x) = 1 - \mu_{A} (x) - v_{A} (x)\) is the hesitant degree of object \(x_{i}\) belong to \(A\). \(a = [\mu_{a} ,v_{a} ]\) is named intuitionistic fuzzy element.

When it is difficult to give \(\mu_{A} (x)\) and \(v_{A} (x)\) with one or more real numbers, the interval number can be used to express them instead. Then we can extend the fuzzy sets, hesitant fuzzy sets and intuitionistic fuzzy sets to the corresponding fuzzy interval sets, hesitant interval fuzzy sets and intuitionistic interval fuzzy sets using interval number. The advantage of using interval numbers to describe membership degree of fuzzy sets is to give full play to the evaluation opinions. For example, when the membership degree given by the evaluation subject is between 0.5 and 0.7 according to his own judgment, the classical fuzzy set can take \(\mu { = 0} . 6\), the hesitant fuzzy sets can take \(\mu = \left\{ {0.5,0.6,0.7} \right\}\), the interval fuzzy sets can take \(\mu = \left[ {0.5,0.7} \right]\). By comparison, the interval fuzzy set can make full use of the assessor’s judgment information. For intuitionistic fuzzy sets given in Definition 3, when \(\mu_{A} (x)\) and \(v_{A} (x)\) is replaced by \(\tilde{\mu }_{A} (x)\) and \(\tilde{v}_{A} (x)\) which are interval numbers, \(< x,\tilde{\mu }_{A} (x),{\tilde{\text{v}}}_{A} (x) >\) is an intuitionistic interval fuzzy element.

To compare the size relationship of different interval numbers, we define the score function and deviation function. Let \(\tilde{h}_{A} (x) = [h_{A}^{ - } (x),h_{A}^{ + } (x)]\) be an interval fuzzy number, \(h_{A}^{ - } (x)\) be its lower limit and \(h_{A}^{ + } (x)\) be its upper limit. Its interval length is \(\Delta h = h_{A}^{ + } (x) - h_{A}^{ - } (x)\). The score function of \(\tilde{h}_{A} (x)\) is:
$$s\left( {\tilde{h}_{A} (x)} \right) = \left( {h_{A}^{ - } (x) + h_{A}^{ + } (x)} \right)/2.$$
(11)

When comparing the size relationship of two interval numbers, we can compare their score function. The larger the score function, the larger the corresponding interval number. When the score functions of two interval numbers are equal and the interval lengths are not equal, it shows that they are the same in the relation of numerical magnitude. The interval numbers with larger interval lengths contain broader information, such as \([0.4,0.6]\) and \([0.3,0.7]\).

Let \(\tilde{h}_{A} (x) = \left\{ {\tilde{h}_{A}^{i} (x)|i = 1,2, \ldots ,m} \right\}\) be an interval fuzzy numbers including \(m\) interval numbers, the calculation method of its score function and deviation function is as follows:
$$s\left( {\tilde{h}_{A} (x)} \right) = \frac{1}{m}\sum\limits_{i = 1}^{m} {s\left( {\tilde{h}_{A}^{i} (x)} \right)}$$
(12)
$$e\left( {\tilde{h}_{A} (x)} \right) = \frac{1}{m}\sum\limits_{i = 1}^{m} {\left( {s\left( {\tilde{h}_{A}^{i} (x)} \right) - s\left( {\tilde{h}_{A} (x)} \right)} \right)^{2} } .$$
(13)

When comparing the size relationship between two interval element arrays, we give priority to comparing their scoring function \(s\left( {\tilde{h}_{A} (x)} \right)\). The larger the scoring function, the larger the corresponding interval set. When the score function is equal, the smaller the deviation function \(e\left( {\tilde{h}_{A} (x)} \right)\) is, the larger the corresponding interval set is. When the score function and deviation function of two interval sets are both equal, it can be considered that they are equivalent in the relation of numerical size.

3.2 Distance measure

In the decision-making process, C.L. Hwang and K. Yoon firstly proposed TOPSIS (Technology for Order Preference by Similarity to an Ideal Solution) method to do scheme ranking, which is a method of sequencing schemes using distance measure among schemes [33]. TOPSIS method ranks a limited number of evaluation objects according to their proximity to idealized objectives. TOPSIS method is to rank evaluation objects by measuring their comprehensive distance to the idealized objectives including optimal and worst solutions [34, 35]. The ranking principle is that the closer the evaluation object is to the optimal solution and the further the evaluation object is to the worst solution, the higher the ranking order is. For benefit-oriented indexes, the optimal solution, which is also called positive ideal solution, is obtained by taking each index maximum value among all evaluation objects. The worst solution is obtained by taking each index minimum value among all evaluation objects, which is also called negative ideal solution. TOPSIS requires the utility function to be monotonic [36, 37].

Distance measure is a key part in TOPSIS. Generally, the distance measure between elements satisfies the following conditions:
  1. 1.
    $$d\left( {A,B} \right) = d\left( {B,A} \right)$$
     
  2. 2.

    \(d\left( {A,B} \right) \ge 0\), If and only \(A = B\), then \(d\left( {A,B} \right) = 0\)

     
  3. 3.
    $$d\left( {A,B} \right) \le d\left( {A,C} \right) + d\left( {B,C} \right)$$
     
Commonly used distance measures include Hamming distance, Euclidean distance and Hausdorff distance. In this paper, the distance calculation methods of fuzzy sets are introduced based on Euclidean distance.
  1. (1)

    For fuzzy set, the calculation method for Euclidean distance measure \(d(\mu_{i} ,\mu_{j} )\) between two fuzzy elements \(\mu_{i}\) and \(\mu_{j}\) is as follows:

    $$d(\mu_{i} ,\mu_{j} ) = \left| {\mu_{i} { - }\mu_{j} } \right|$$
    (14)
     
  2. (2)

    For hesitant fuzzy set, the calculation method for Euclidean distance measure \(d(\mu_{i} (m),\mu_{j} (m))\) between two hesitate fuzzy elements \(\mu_{i} (m)\) and \(\mu_{j} (m)\) is as follows:

    $$d(\mu_{i} (m),\mu_{j} (m)) = \sqrt {\frac{1}{m}\sum\limits_{l = 1}^{m} {\left| {\mu_{li} { - }\mu_{lj} } \right|^{2} } }$$
    (15)
     

Here \(\mu_{li}\) and \(\mu_{lj}\) are the \(l\)th fuzzy element in hesitate fuzzy elements \(\mu_{i} (m)\) and \(\mu_{j} (m)\) respectively.

Obviously, formula (13) is only applicable to the calculation of hesitant fuzzy elements with the same length. When two groups of hesitant fuzzy elements \(\mu_{i} (m)\) and \(\mu_{j} (m^{'} )\) have different lengths (\(m \ne m^{'}\)), it is unable to calculate their distances directly by formula (13). Therefore, it is necessary to align the lengths [38]. Assuming \(m < m^{'}\), the specific method is to take \(\overline{{\mu_{i} }} (m)\) as supplementary elements to align the length of hesitant fuzzy elements, where \(\overline{{\mu_{i} }} (m)\) is the mean of \(\mu_{i} (m)\).
  1. (3)

    For interval fuzzy set, The calculation method for Euclidean distance measure \(d\left( {\tilde{h}_{A} (x),\tilde{h}_{B} (x)} \right)\) between two interval fuzzy elements \(\tilde{h}_{A} (x)\) and \(\tilde{h}_{B} (x)\) is as follows:

    $$\begin{aligned} & d\left( {\tilde{h}_{A} (x),\tilde{h}_{B} (x)} \right) \\ & \quad = \sqrt {\left( {\left| {\tilde{h}_{A}^{ + } (x) - \tilde{h}_{B}^{ + } (x)} \right|^{2} + \left| {\tilde{h}_{A}^{ - } (x) - \tilde{h}_{B}^{ - } (x)} \right|^{2} } \right)\big/2} . \\ \end{aligned}$$
    (16)
     
  2. (4)

    For hesitant interval fuzzy set, the calculation method for Euclidean distance measure \(d\left( {\tilde{H}_{A} (x),\tilde{H}_{B} (x)} \right)\) between two hesitant interval fuzzy element \(\tilde{H}_{A} (x) = \left\{ {\tilde{h}_{A}^{i} (x)|i = 1,2, \ldots ,m} \right\}\) and \(\tilde{H}_{B} (x) = \left\{ {\tilde{h}_{B}^{i} (x)|i = 1,2, \ldots ,m} \right\}\) is as follows:

    $$d\left( {\tilde{H}_{A} (x),\tilde{H}_{B} (x)} \right) = \frac{1}{m}\sum\limits_{i = 1}^{m} {d\left( {\tilde{h}_{A}^{i} (x),\tilde{h}_{B}^{i} (x)} \right)}$$
    (17)
     

Here, \(\tilde{h}_{A}^{i} (x)\) and \(\tilde{h}_{B}^{i} (x)\) are the \(i\)th fuzzy element respectively in interval fuzzy element array \(\tilde{H}_{A} (x)\) and \(\tilde{H}_{B} (x)\) which ranking their elements from small to large relationships. When calculating the distance between two interval hesitant sets, we will also encounter the problem of different array length. Similarly, we can use the complement method to align the array length firstly, then calculate its distance.

4 Simulation verification test

4.1 HRRP quality evaluation process

In this section, we propose a HRRP image quality evaluation method combining target recognition rate(DSCRR and DSFRR)and IIFS (interval intuitionistic fuzzy sets). The evaluation principle is that in target recognition experiments based on HRRP, the bigger DSCRR is, and the smaller DSFRR is, the better the HRRP image quality is. For an intuitionistic interval fuzzy set representing HRRP performance as “excellent”, its element membership can be expressed by DSCRR confidence interval and non-membership can be expressed by DSFRR confidence interval. Let scheme \(X\) to be evaluated is a set of HRRP with different image quality, \(A\) be an IIFS for performance “excellent”. The implementation steps of evaluation method are as follows:
  • Step 1 An IIFS \(A = \{ (x_{i} ,\tilde{\mu }_{A} (x_{i} ),\tilde{v}_{A} (x_{i} ))|x_{i} \in X\}\) is constructed for performance evaluation, where \(\tilde{\mu }_{A} (x_{i} )\) is the DSCRR estimation interval corresponding of scheme \(x_{i}\), and \(\tilde{v}_{A} (x_{i} )\) is the DSFRR estimation interval.

  • Step 2 The score function \(s\left( {\tilde{h}_{A} (x)} \right)\) and deviation function \(e\left( {\tilde{h}_{A} (x)} \right)\) are used to compare the interval index values of different schemes and construct the positive ideal solution \(x_{\hbox{max} }\) and negative ideal solution \(x_{\hbox{min} }\). Their corresponding IIFS are \(A^{ + }\) and \(A^{ - }\) respectively. Here,

    $$A^{ + } = \{ (x_{\hbox{max} } ,max(\tilde{\mu }_{A} (x_{i} )),min(\tilde{v}_{A} (x_{i} )))|x_{i} \in X\}$$
    $$A^{ - } = \{ (x_{\hbox{min} } ,min(\tilde{\mu }_{A} (x_{i} )),max(\tilde{v}_{A} (x_{i} )))|x_{i} \in X\}$$
  • Step 3 The TOPSIS method is used to calculate the relative distance measure \(d\left( {x_{i} ,x_{\hbox{max} } } \right)\) and \(d\left( {x_{i} ,x_{\hbox{min} } } \right)\) of each scheme to the positive and negative ideal solution respectively, then the scheme performance score \(s\left( x \right)\) is obtained. The calculating process is as follows:

$$\begin{aligned} & d\left( {x_{i} ,x_{\hbox{max} } } \right) \\ & \quad = \left( {d\left( {\tilde{\mu }_{A} (x_{i} ),max\tilde{\mu }_{A} (x_{i} )} \right) + d\left( {\tilde{v}_{A} (x_{i} ),min\tilde{v}_{A} (x_{i} )} \right)} \right)/2. \\ \end{aligned}$$
(18)
$$\begin{aligned} & d\left( {x_{i} ,x_{\hbox{min} } } \right) \\ & \quad = \left( {d\left( {\tilde{\mu }_{A} (x_{i} ),min\tilde{\mu }_{A} (x_{i} )} \right) + d\left( {\tilde{v}_{A} (x_{i} ),max\tilde{v}_{A} (x_{i} )} \right)} \right)/2. \\ \end{aligned}$$
(19)
$$s\left( {x_{i} } \right) = 1 - \frac{{d\left( {x_{i} ,x_{\hbox{max} } } \right)}}{{\sum\nolimits_{i = 1}^{m} {d\left( {x_{i} ,x_{\hbox{max} } } \right)} }} + \frac{{d\left( {x_{i} ,x_{\hbox{min} } } \right)}}{{\sum\nolimits_{i = 1}^{m} {d\left( {x_{i} ,x_{\hbox{min} } } \right)} }}.$$
(20)

According to formula (18), it can be concluded that when the DSCRR of the scheme \(x_{i}\) is larger and the DSFRR is smaller, the relative distance measure \(d\left( {x_{i} ,x_{\hbox{max} } } \right)\) to the positive ideal solution is smaller, and the relative distance measure \(d\left( {x_{i} ,x_{\hbox{min} } } \right)\) to the negative ideal solution is larger, the performance score \(s\left( {x_{i} } \right)\) is larger, and the performance of the scheme \(x_{i}\) is better.

4.2 Simulation experiments

In order to verify the evaluation method effectiveness proposed in this paper, we set up the following simulation verification experiment. In the scheme, the target recognition system is a millimetre-wave stepped radar with working frequency of 34.7–35.7 GHz, nominal range resolution of 0.15 m, stepped frequency interval of 2 MHz, polarization state of vertical launch-horizontal reception, target elevation angle of 0°, attitude range of 0°–360°. According to HRRP attitude sensitivity, we know that HRRP quality will vary with the attitude change. In the experiment, HRRP measured data comes from Boeing 707 aircraft target scale model, and the scale model size is 2 m. In the process of target recognition, the normal distribution white noise is introduced, and the target recognition rate is controlled by adjusting the noise intensity. Simulation experimental data is recorded as Table 2 follows. Where \(\mu_{DSCRR}\) is the DSCRR mean, \(\sigma_{DSCRR}^{2}\) is the DSCRR variance, \(\mu_{DSFRR}\) is the DSFRR mean, \(\sigma_{DSFRR}^{2}\) is the DSCRR variance.
Table 2

Simulation experimental data

\(\theta\)

\(\mu_{DSCRR}\)

\(\sigma_{DSCRR}^{2}\)  × 10e−3

\(\mu_{DSFRR}\)

\(\sigma_{DSFRR}^{2}\) × 10e−3

0.86

0.22

0.15

0.22

30°

0.79

0.62

0.18

0.72

60°

0.72

0.74

0.21

0.76

100°

0.61

0.35

0.33

0.48

200°

0.67

0.79

0.16

0.63

270°

0.51

0.29

0.32

0.34

350°

0.82

0.21

0.15

0.24

Set saliency level \(\alpha { = 0} . 1 0\), sample size \(m{ = 100}\), the confidence intervals of DSCRR and DSFRR are calculated by Eqs. 7 and 9. The results are as Table 3 shows. Where \(\tilde{x}\) is the estimation interval of \(x\).
Table 3

Confidence intervals of DSCRR and DSFRR

\(\theta\)

\(\tilde{\mu }_{DSCRR}\)

\(\tilde{\sigma }_{DSCRR}^{2}\)  × 10e−3

\(\tilde{\mu }_{DSFRR}\)

\(\tilde{\sigma }_{DSFRR}^{2}\) ×  10e−3

[0.63,1.09]

[0.16,0.32]

[0.05,0.25]

[0.16,0.32]

30°

[0.57,1.01]

[0.44,0.89]

[0.07,0.29]

[0.51,1.03]

60°

[0.51,0.93]

[0.53,1.06]

[0.10,0.32]

[0.54,1.09]

100°

[0.41,0.81]

[0.25,0.50]

[0.19,0.47]

[0.34,0.69]

200°

[0.47,0.87]

[0.56,1.14]

[0.06,0.26]

[0.45,0.91]

270°

[0.33,0.69]

[0.21,0.42]

[0.18,0.46]

[0.24,0.49]

350°

[0.59,1.05]

[0.15,0.30]

[0.05,0.25]

[0.17,0.34]

Based on TOPSIS raking method and IIFS, we obtain HRRP quality score vectors respectively using simulation data in Table 2 and Table 3. HRRP quality score vectors are calculated by Eq. 19. The results are as Table 4 shows, where score1 (s1) is obtained from single value in Table 2, score2 (s2) is obtained from interval value in Table 3.
Table 4

HRRP quality normalized score vector (take 0° as reference)

\(\theta\)

30°

60°

100°

200°

270°

350°

s1

0.909

0.819

0.611

0.819

0.529

0.963

s2

0.911

0.822

0.651

0.820

0.564

0.964

The score vectors show that when the angle between the target radial direction and radar sight line gets smaller, the HRRP imaging quality gets better. This conclusion is consistent with the field test result and verifies the validity of the evaluation method. Besides, the comparison between score1 and score2 shows that both results are holistic consistency and score2 is more objective and accurate based on IIFS, which can make full use of test data.

5 Conclusion

Radar target recognition method based on HRRP is an important research project. The HRRP imaging quality influence factors of broadband radar include relative position, relative attitude and relative velocity between missile and target. For the same radar target recognition system and algorithm, the HRRP image quality directly affects the target recognition effect. Relatively speaking, HRRP-based radar target recognition performance can reflect HRRP imaging quality. Based on this, this paper proposes a method to evaluate the HRRP imaging quality using DSCRR. In this evaluation process, to make full use of the test data and get more objective evaluation conclusions, we introduce IIFS and correspond confidence interval of DSCRR and DSFRR to membership and non-membership for IIFS respectively. In the progress for ranking evaluation schemes, we use TOPSIS to calculate the distance between the schemes and the ideal solution. When distance measure is synthesized into the scheme score vector, the scheme evaluation is completed. In the validation process of evaluation method, we have tested target recognition effect of same broadband radar system at different attitude angles and completed HRRP imaging quality evaluation. The evaluation results show that when the angle between the target radial direction and radar sight line gets smaller, the HRRP imaging quality gets better. This conclusion is consistent with the field test result and verifies the validity of the evaluation method.

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 61501471.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Rocket Force University of EngineeringXi’anChina
  2. 2.School of Electronic EngineeringXidian UniversityXi’anChina

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