Method for detecting small targets in sea clutter based on fractional integral and fractals

Abstract

The application discloses a sea clutter small target detection method based on fractional integral and fractals, and aims at the problem that the detection effect of the existing method for estimating the Hurst index by using the fractal characteristics of an object is poor under the condition of high sea state and low signal-to-clutter ratio. The method comprises the following steps: S1, obtaining radar echo data in the form of time-distance units, removing the interference of land factors in each distance unit, retaining sea clutter and target information, and performing extension processing on the data; S2, selecting a suitable Grunwald-Letnikov fractional integral order according to the data characteristics by using a particle swarm algorithm, and performing fractional integral operation on the data; S3, performing layer-by-layer decomposition on the data by using a wavelet transform method, and calculating the fractal Hurst index of each distance unit; and S4, putting the Hurst index as the fractal characteristics into a random forest for training and testing on a test set, and outputting a target detection rate and a false alarm rate.

Description

Technical Field

[0001] This invention belongs to the field of ship target detection technology in marine clutter background. Specifically, it is a method for detecting small targets in sea clutter based on fractional integral and fractal. Technical Background

[0002] With the rapid development of marine radar detection technology, the importance of sea surface target identification in military defense and marine monitoring is becoming increasingly prominent. However, due to the complex electromagnetic scattering characteristics of the sea surface, strong sea clutter mixed in radar echoes is often highly coupled with target signals, leading to a significant decline in the performance of traditional target detection methods based on amplitude and Doppler frequency. Therefore, researchers have gradually turned their attention to the fractal characteristic analysis of signals. The essential differences between sea clutter and targets in terms of geometry and scattering mechanisms can be reflected in the inconsistency of fractal characteristics. The Hurst exponent, as a correlation parameter of fractal dimension, can effectively quantify the long-range dependence of time series, providing a new dimension for distinguishing targets from clutter.

[0003] Researchers often use wavelet transform to estimate the Hurst exponent. However, in real-world scenarios, sea clutter is influenced by dynamic factors such as wind speed and wave height, exhibiting non-stationarity, nonlinearity, and non-Gaussianity, which limits the accuracy of traditional wavelet transform-based Hurst exponent estimation methods. Furthermore, the fractal characteristics of target signals in complex clutter backgrounds are easily obscured by noise, further reducing discriminative power. Therefore, an improved method is urgently needed to enhance the accuracy of target identification in sea clutter backgrounds. Summary of the Invention

[0004] The purpose of this invention is to propose a method for detecting small targets in sea clutter based on fractional integrals and fractals.

[0005] The technical solution to achieve the purpose of this invention is: a method for detecting small targets in sea clutter based on fractional integrals and fractals, comprising the following steps:

[0006] S1: Acquire radar echo data. The data is organized as a two-dimensional matrix of time-range units. Land factor interference in each range unit is removed, sea clutter and target information are retained, and the retained data is subjected to interval sampling and segmented expansion processing.

[0007] S2: Search for the optimal order d of Grünwald-Letnikov fractional integral using the particle swarm optimization algorithm, where the objective function of the particle swarm is the difference between the detection probability and the false alarm rate obtained by subsequent steps S3 and S4 at this order, and perform Grünwald-Letnikov fractional integral operation on the data of each distance cell according to the optimal order.

[0008] S3: Perform multi-level discrete wavelet transform on the distance cell data output by S2 to obtain the detail coefficients of each level. Introduce intermediate variables related to the energy of each level detail coefficient, perform linear fitting between the intermediate variables and the decomposition level, and calculate the Hurst exponent of each distance cell from the fitting slope.

[0009] S4: Using the Hurst exponent as a fractal feature, the training set and test set are divided proportionally, and the random forest classifier is input for training. The detection probability Pd and false alarm rate FAR are output on the test set.

[0010] Compared with the traditional Hurst exponent estimation method based on wavelet decomposition, this invention has significant advantages: (1) It improves the method of calculating the Hurst exponent by wavelet transform, introduces Grünwald-Letnikov fractional integral to process the data, adjusts the long correlation of the data, and makes the fractal features of the target and clutter more distinguishable; (2) It uses the PSO particle swarm algorithm to find the optimal Grünwald-Letnikov fractional integral order to maximize the distinguishability between the target and clutter; (3) It combines machine learning algorithms, puts the fractal feature Hurst exponent as data into a random forest, automatically learns the classification strategy, and can achieve good classification results. Attached Figure Description

[0011] Figure 1 This is a flowchart of the method of the present invention.

[0012] Figure 2 This is a schematic diagram illustrating the calculation of the Hurst exponent using the wavelet decomposition method.

[0013] Figure 3 The original distance-time image of the data used.

[0014] Figure 4 The Hurst exponent image for each distance cell of the data used. Detailed Implementation

[0015] The present invention will be further described below with reference to the accompanying drawings and embodiments, but this is not intended to limit the scope of the invention.

[0016] This invention discloses a method for small target detection in sea clutter based on fractional integrals and fractals. By processing the data using Grünwald-Letnikov fractional integrals, the long-term correlation between target and clutter units is adjusted to varying degrees, making their Hurst exponents more distinguishable. The method first preprocesses the radar echo data and expands the dataset using interval sampling. Then, a particle swarm optimization algorithm is used to select an appropriate Grünwald-Letnikov fractional integral order to maximize the Hurst exponent distinguishability between targets and clutter. Next, wavelet transform is used to decompose the data layer by layer, introducing intermediate variables related to the detail coefficients of each layer. The linear relationship between the decomposition level and the intermediate variables is fitted to calculate the fractal Hurst exponent for each range unit. Finally, the Hurst exponent is used as a fractal feature and trained in a random forest to output the target detection rate and false alarm rate.

[0017] Specifically, it includes the following steps, such as Figure 1 As shown:

[0018] Example:

[0019] S1: Obtain radar echo data in a two-dimensional matrix format with time-range cells. Remove land-related interference from each range cell, retain sea clutter and target information, and perform data extension processing.

[0020] In this embodiment, the selected data is IPIX dataset #40 obtained by McMaster University in Canada on the Dartmouth coast in 1993. This dataset contains 14 range cells, each with 131,072 data points, a wave height of 0.9 meters, a range resolution of 30 meters, and a sampling depth of 15 meters. After preprocessing, land interference factors in the data were removed. The main technical parameters of the IPIX radar are shown in Table 1.

[0021] Table 1 Radar Parameters

[0022] Radar parameters Parameter value band X-band Operating frequency 9.39GHz Peak power 8KW Antenna gain 45.7dB polarization mode HH Antenna beamwidth 0.9° Pulse repetition frequency 1000Hz Pulse width 200ns glancing angle 1°

[0023] S1: In this embodiment, distance units are used as the unit, and the data expansion method is interval sampling and segmentation, with a segmentation interval of 64. After interval sampling and segmentation processing, the original data can be expanded into 64 smaller data points {x}. n}1, {x n}2……{x n} 64 The calculation formula is:

[0024]

[0025] Where M=[N / 64]; other distance cells are expanded using the same operation.

[0026] S2: Using the particle swarm optimization algorithm, select an appropriate Grünwald-Letnikov fractional integral order based on the data characteristics:

[0027] In this embodiment, the particle swarm optimization (PSO) algorithm parameters are: number of particles: 10, maximum number of iterations: 10, inertia weight: 0.9, cognitive coefficient: 1.5, social coefficient: 1.5, order search range: 0.01~1, and speed limit: 0.3. The algorithm objective function is the difference between the detection rate and false alarm rate output by the random forest when the integral order is d. It aims to maximize the difference between the detection probability (Pd) and the false alarm rate (FAR). The mathematical expression is: J(d)=Pd(d)−FAR(d).

[0028] The optimal order d = 0.5978 was obtained using the particle swarm optimization algorithm on dataset #40. For data {x} n}, after the Grünwald-Letnikov fractional integral, {y n The expression is:

[0029]

[0030] S3: Perform multi-level discrete wavelet transform on the integrated data to obtain the detail coefficients of each level. Introduce intermediate variables related to the energy of each level's detail coefficients, perform linear fitting between these intermediate variables and the decomposition levels, and calculate the Hurst exponent of each distance cell based on the fitting slope.

[0031] S3.1: Perform wavelet transform. Assume the data has already undergone Grünwald-Letnikov fractional integral processing, and the processed data sequence is denoted as {y}. n The wavelet decomposition process is as follows: Figure 2 As shown. This embodiment uses the Haar wavelet as the wavelet basis, with 8 layers in the wavelet transform. The wavelet function Ψ and scaling function φ at different scales and positions are obtained by the following formulas:

[0032]

[0033] Where j is the scaling factor and k is the translation factor; by decomposing the data with different versions of wavelet functions, low-frequency components with different resolutions at different locations are obtained, and by decomposing the data with different versions of scaling functions, the corresponding high-frequency components are obtained.

[0034] The data is decomposed at level j to obtain the low-frequency component SA. j With high-frequency components SD j The calculation process is as follows:

[0035]

[0036] When performing the (j+1)th level decomposition, based on the j-th level, SA is... j Perform another decomposition, following the same steps as above; after L-level wavelet decomposition, the relationship between the decomposed data and the low-frequency and high-frequency components of each decomposition level is as follows:

[0037]

[0038] S3.2: Introducing an intermediate variable; Based on the wavelet decomposition results, a variable Γ related to the high-frequency components of each decomposition level is introduced, and its calculation method is as follows:

[0039]

[0040] in It is the number of wavelet coefficients at the j-th layer.

[0041] S3.3: Calculate the Hurst exponent using linear fitting; establish the relationship between Γ(j) and the number of decomposition layers j:

[0042]

[0043] Where c0 is a constant bias that does not affect the slope estimation. The slope k is estimated using the least squares method, and then the following calculations are performed.

[0044]

[0045] The Hurst exponent H can then be estimated.

[0046] The distance-time graph and Hurst exponent plot of the #40 data before and after the Grünwald-Letnikov integration are shown below. Figure 3 , Figure 4 As shown.

[0047] S4: Use the Hurst exponent as a fractal feature to train a random forest, outputting the target detection rate and false alarm rate. The specific steps are as follows:

[0048] S4.1: Data Labeling and Segmentation

[0049] All distance cell samples are labeled with ground truth values: pure sea clutter samples are labeled 0, and samples containing targets are labeled 1. The training and test sets are randomly divided in a 7:3 ratio.

[0050] S4.2: Hyperparameter Grid Search

[0051] The number of decision trees n in a random forest tree With the minimum number of leaf node samples n leafPerform a grid search: n tree The preferred integer is in the range of 5 to 25, n leaf The optimal range is integers from 1 to 10; classification accuracy is used as the evaluation metric, with the initial optimal accuracy set at a preset threshold of 0.6; all combinations are iterated, and if the accuracy of the current combination is higher than the historical best, it is updated, ultimately yielding the optimal parameter n. tree * with n leaf *

[0052] S4.3: Training and Testing

[0053] A random forest classifier is constructed using the optimal hyperparameters obtained in S4.2, trained on the training set, and used for prediction on the test set. The detection probability and false alarm rate are calculated using the following formula:

[0054]

[0055] TP, FN, FP, and TN represent the number of true positives, false negatives, false positives, and true negatives, respectively.

[0056] In this embodiment, the optimal number of decision trees was determined to be 15, the minimum number of leaf nodes was 9, and the final detection accuracy was 93.08%, with a false alarm rate of 3.61%. The detection results show that the target detection accuracy of the method of this invention is good.

Claims

1. A method for detecting small targets in sea clutter based on fractional integrals and fractals, characterized in that... This includes the following steps: S1: Acquire radar echo data. The data is organized as a two-dimensional matrix of time-range units. Land factor interference in each range unit is removed, sea clutter and target information are retained, and the retained data is subjected to interval sampling and segmented expansion processing. S2: Search for the optimal order d of Grünwald-Letnikov fractional integral using the particle swarm optimization algorithm, where the objective function of the particle swarm is the difference between the detection probability and the false alarm rate obtained by subsequent steps S3 and S4 at this order, and perform Grünwald-Letnikov fractional integral operation on the data of each distance cell according to the optimal order. S3: Perform multi-level discrete wavelet transform on the distance cell data output by S2 to obtain the detail coefficients of each level. Introduce intermediate variables related to the energy of each level detail coefficient, perform linear fitting between the intermediate variables and the decomposition level, and calculate the Hurst exponent of each distance cell from the fitting slope. S4: Using the Hurst exponent as a fractal feature, the training set and test set are divided proportionally, and the random forest classifier is input for training. The detection probability Pd and false alarm rate FAR are output on the test set.

2. The method according to claim 1, characterized in that: In S1, the specific method of the interval sampling segmented expansion processing is as follows: Let the original time series length of each distance unit be N, the expansion factor be k, and the extraction be performed with an interval step size of k. The specific calculation formula is as follows: Where M = [N / k] is the total number of samples, {x n } i This indicates that the data {x} n The i-th sample obtained from sampling.

3. The method for detecting small targets in sea clutter based on fractional integrals and fractals according to claim 2, characterized in that, In S2, the data is subjected to a Grünwald-Letnikov fractional integral, and the data before integration {x} is recorded. n }, after integration, the data {y n },but: Where Γ is the gamma function, d is the order of integration, and m is the summation index.

4. The method for detecting small targets in sea clutter based on fractional integrals and fractals according to claim 3, characterized in that, In S2, the fractional integral order *d* is determined using the Particle Swarm Optimization (PSO) algorithm. The algorithm's objective function is to perform a fractional integral of order *d* on the data and estimate its Hurst exponent. The Hurst exponent is then used as input features and fed into a random forest. The difference between the detection rate and the false alarm rate output by the random forest is mathematically expressed as: The parameters and optimal range of the particle swarm optimization algorithm are as follows: number of particles 5~30, maximum number of iterations 5~50, inertia weight 0.4~0.9, cognitive coefficient 1.0~2.5, social coefficient 1.0~2.5, lower bound of the order search range 0.01~0.1, upper bound 0.5~1.0, and velocity limit 0.1~0.5; the specific steps are as follows: S2.1: Initialization Initialize the parameters of the particle swarm, and initialize the individual historical best pbest,i = di for each particle. The global historical best gbest is initialized to the position of the particle with the largest objective function value. S2.2: Evaluation of the objective function For each particle's current position di, perform steps S3 and S4 once, using di as the fractional order of integration, to obtain Pd(di) and FAR(di). Calculate the objective function value: S2.3: Individual and Global Optimal Updates For each particle i, if J(di) > J(pbest,i), then update pbest,i = di; after traversing all particles, if there exists a particle whose objective function value is greater than J(gbest), then update gbest = di. S2.4: Velocity and Position Update The velocities of each particle are updated according to the standard PSO velocity update formula; S2.5: Stopping condition judgment After each iteration, the following two stopping conditions are checked sequentially. The iteration terminates if either condition is met: (1) Convergence condition: The process terminates early when the absolute value of the difference between the global optimal objective function values ​​of two adjacent iterations is less than the convergence threshold; (2) Iteration limit condition: The iteration is forcibly terminated when the maximum number of iterations is reached; S2.6: Output the optimal order After the algorithm terminates, the global historical best position gbest is used as the optimal fractional integral order d, which is used in step S3 to perform the final Grünwald-Letnikov fractional integral operation on all data. Where Pd(d) and FAR(d) are the detection probability and false alarm rate output by the random forest on the test set after steps S3 and S4, respectively, when the fractional order of integration is d.

5. The method for detecting small targets in sea clutter based on fractional integrals and fractals according to claim 4, characterized in that, The steps for calculating the Hurst exponent based on wavelet transform in S3 are as follows: S3.1: Perform wavelet transform; assuming the data has already undergone Grünwald-Letnikov fractional integral processing, the processed data sequence is denoted as {y}. n The wavelet functions Ψ and scaling functions φ at different scales and locations are obtained by the following formulas: Where j is the scaling factor and k is the translation factor; by decomposing the data with different versions of wavelet functions, low-frequency components with different resolutions at different locations are obtained, and by decomposing the data with different versions of scaling functions, the corresponding high-frequency components are obtained. The data is decomposed at level j to obtain the low-frequency component SA. j With high-frequency components SD j The calculation process is as follows: When performing the (j+1)th level decomposition, based on the j-th level, SA is... j Perform another decomposition, following the same steps as above; after L-level wavelet decomposition, the relationship between the decomposed data and the low-frequency and high-frequency components of each decomposition level is as follows: S3.2: Introducing an intermediate variable; Based on the wavelet decomposition results, a variable Γ related to the high-frequency components of each decomposition level is introduced, and its calculation method is as follows: in It is the number of wavelet coefficients at the j-th layer; S3.3: Calculate the Hurst exponent using linear fitting; establish the relationship between Γ(j) and the number of decomposition layers j: Where c0 is a constant bias that does not affect the slope estimation; the slope k is estimated using the least squares method, and then the following calculations are performed. The Hurst exponent H can then be estimated.

6. The method according to claim 1, characterized in that, The specific steps in S4 for training a random forest using the Hurst exponent as a feature and outputting detection metrics are as follows: S4.1: Data Labeling and Segmentation All distance cell samples are labeled with ground truth values, with pure sea clutter samples labeled as 0 and samples containing targets labeled as 1; the training set and test set are randomly divided in a ratio between 6:4 and 8:

2. S4.2: Hyperparameter Grid Search The number of decision trees n in a random forest tree With the minimum number of leaf node samples n leaf Perform a grid search: n tree The preferred integer is in the range of 5 to 25, n leaf The preferred integers are those ranging from 1 to 10; the classification accuracy is used as the evaluation metric, and the initial optimal accuracy is set to a preset threshold A0. Iterate through all combinations, and update the combination if the accuracy of the current combination is higher than the historical best, finally obtaining the optimal parameter n. tree * with n leaf *; S4.3: Training and Testing A random forest classifier is constructed using the optimal hyperparameters obtained in S4.2, trained on the training set, and used for prediction on the test set. The detection probability and false alarm rate are calculated using the following formula: TP, FN, FP, and TN represent the number of true positives, false negatives, false positives, and true negatives, respectively.

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