Space cone target translation compensation and micro-motion feature extraction method

Abstract

The application discloses a method for translational compensation and micro-motion feature extraction of a space cone target, which comprises the following steps: m-D curve separation; construction of a snapshot sequence required by DMD decomposition; DMD decomposition; curve reconstruction; determination of an optimal value of a shift stack number; mode selection; translational compensation and m-D curve frequency extraction. Under a low signal-to-noise ratio condition, the translational compensation still achieves good effects, and the micro-motion frequency features of the target can be accurately extracted. The translational compensation and micro-motion feature extraction of the space cone target can be realized, and the overall working performance of a radar is improved.

Description

Technical Field

[0001] This invention relates to signal and information processing technology, specifically to a method for translational compensation and micro-motion feature extraction of a spatial cone target. Background Technology

[0002] Space cone targets move at high speeds and exhibit micro-motions (or micro-dynamics) such as spin or precession. The micro-motion characteristics of space cone targets differ significantly from those of debris and wreckage, providing an effective method for space cone target identification. However, the translational motion of space cone targets causes changes in their micro-Doppler (mD) curves, complicating subsequent feature extraction. Professor Luo Ying of the Air Force Engineering University used the trend term obtained from Empirical Mode Decomposition (EMD) to compensate for the target's translational motion and extracted micro-motion features such as spin frequency and cone spin frequency. However, this method suffers from mode aliasing during micro-motion feature extraction. Dr. Gu Fufei of the China Satellite Maritime Tracking and Control Department used delayed conjugate multiplication to estimate the target's translational acceleration and estimated micro-motion parameters such as spin frequency by searching for peak values ​​in the compensated signal spectrum. This method is only applicable to micro-motion signals with sinusoidal or cosine forms. Li Jingqing, a master's student at the Air Force Engineering University, developed a translational compensation method for rotationally symmetric targets based on the micro-Doppler symmetry cancellation effect, but this method is not applicable to translational compensation for asymmetric targets. Feng Cunqian, a professor at the Air Force Engineering University, completed the estimation of translational parameters for multiple targets using a method combining high-order fuzzy functions, delay conjugate multiplication, and time-frequency analysis; however, this method is significantly affected by noise.

[0003] In recent years, Schmid proposed a method for predicting and analyzing the stability of flow fields using eigenvalues ​​of dynamic systems, namely the Dynamic Mode Decomposition (DMD) algorithm. The essence of this algorithm is to seek a reduced-order model of the flow field dynamics. By performing eigenvalue decomposition on a snapshot of the flow field, it extracts low-order modes that characterize the flow field structure and their corresponding eigenvalue frequencies. These frequencies are singular, and the flow field can be sorted based on frequency, allowing observation of the influence and contribution of flow structures at different frequencies on the flow field. Currently, the DMD algorithm is widely used in fluid mechanics, electromagnetic signal analysis, and neuroscience. Rowley et al. discussed the connection between the DMD algorithm and the Koopman operator, pointing out that DMD can approximate Koopman modes and eigenvalues, possessing the ability to describe the time evolution of observables (such as pressure and velocity) in nonlinear flows. In other words, the DMD decomposition process is a linear estimation process, achieving nonlinear estimation through linear assumptions. Considering that mD signals are non-stationary and nonlinear in the time-frequency domain or slow time-range domain, and that the micro-motion echoes of spatial cone targets usually contain multiple frequency components such as spin and cone spin, and that the frequency of the mode obtained by DMD decomposition is unique, it is very suitable to apply the DMD algorithm to perform feature decomposition on mD signals and extract the micro-motion frequency components. Summary of the Invention

[0004] The present invention will now be described in detail with reference to the accompanying drawings.

[0005] To address the shortcomings of existing technologies, this invention proposes a method for translational compensation and micro-motion feature extraction of a spatial cone target, comprising the following steps:

[0006] Step 1: mD curve separation

[0007] The slow-time-range image sequence is preprocessed by first performing Gaussian smoothing and binarization on the spatial cone target echo, followed by skeleton extraction. The mD curve corresponding to each scattering point is then searched and separated. At this point, the vector composed of the data of each mD curve still contains the micro-motion characteristics of each scattering point. That is, the mD curve exhibits non-stationary and nonlinear periodic changes with slow time. The DMD algorithm is used to describe the evolution of this nonlinear change over time. Based on this, the DMD decomposition method is used for translational compensation and extraction of micro-motion characteristics.

[0008] Step 2: Construct the snapshot sequence required for DMD decomposition

[0009] By performing curve separation on the slow-time-range image sequence of the target in the image domain, data for each scattering point of the target at N slow times are obtained. These N slow-time data are then arranged in chronological order of echo time to form a data vector [x1, x2, ..., x]. N Let x be the data for the i-th slow time. i Let Δt be the time interval between any two data points, where Δt is the pulse repetition interval PRI. Define two snapshot sequences as follows:

[0010]

[0011]

[0012] In the formula, This indicates taking the data vector [x1, x2, ..., x...] N The first data x1 to the (N-1)th data x in the [] N-1 The arranged vector; This indicates taking the data vector [x1, x2, ..., x...] N The second data x2 to the Nth data x in ] N The arranged vector;

[0013] Construct the first augmented data matrix using equation (1):

[0014]

[0015] In the formula, h represents the number of shift stacking operations;

[0016] The second augmented data matrix is ​​formed by shifting and stacking equation (2):

[0017]

[0018] Assuming that the micro-motion echo data of the target scattering point satisfies linear correlation within a short time period, i.e., that a linear operator A exists, then:

[0019] X′ aug =AX aug (5)

[0020] Given a linear operator A, which is a high-dimensional matrix of size h × h, find a low-dimensional similarity matrix of rank r. To approximate a high-dimensional linear operator A, where C represents the set of complex numbers, r << h; and a low-dimensional matrix. It is a similarity matrix of a high-dimensional linear operator A.

[0021] Step 3: DMD Decomposition

[0022] (1) For the first augmented data matrix X augPerform singular value decomposition:

[0023] X aug =U∑V * (6)

[0024] In the formula, the left unitary matrix U∈C h×r Right unitary matrix V∈C (N-h)×r And satisfying U*U=I, V*V=I, where I is the identity matrix, * denotes the complex conjugate transpose, and the first diagonal matrix ∑∈C r×r It has r non-zero singular values ​​(σ1, σ2, ..., σ) on its diagonal. r ), where σ1, σ2,…,σ r These are the 1st, 2nd, ..., rth non-zero singular values, respectively.

[0025] (2) Solving for the similarity matrix

[0026] Substituting equation (6) into equation (5), we get:

[0027] A = X' ang V∑ -1 U * (7)

[0028] Then similar matrix Represented as:

[0029]

[0030] (3) For similar matrices Perform eigenvalue decomposition:

[0031]

[0032] In the formula, the columns of the first matrix W are eigenvectors, and the second diagonal matrix Λ contains the corresponding eigenvalues ​​λ. i If the eigenvalue lies inside the unit circle, the mode is stable; otherwise, it is unstable. Calculate the logarithmic form of the eigenvalue:

[0033] ω i =lnλ i / Δt (10)

[0034] In the formula, the logarithmic form of the eigenvalues ​​is ω i The real part represents the growth / decrease rate of the corresponding DMD mode, while the imaginary part determines the frequency of the mode.

[0035]

[0036] In the formula, imag(·) represents taking ω i The imaginary part;

[0037] (4) Reconstruct the eigenvalues ​​of the linear operator A using the first matrix W and the second diagonal matrix Λ. λ in the second diagonal matrix Λ i Let Φ be the eigenvalues ​​of A, and Φ be the column of the second matrix Φ. i Let A be the eigenvector of the linear operator A, i.e., the DMD mode:

[0038] Φ=X′ aug V∑ -1 W (12)

[0039] Step 4: Curve Reconstruction

[0040] Reconstruction using the first r-order modes:

[0041]

[0042] In the formula, X DMD (t) represents the reconstructed data, t represents time, and the third diagonal matrix Ω = diag(ω) contains elements ω. i b i It is the initial amplitude of each mode, and b is the value determined by b. i The vector formed;

[0043] If we take the data at the initial time, i.e., X aug The first column X aug1 Substituting into equation (13), we get X. aug1 =Φb, at this time the modal amplitude b is:

[0044] b = Φ + X aug1 (14)

[0045] In the formula, Φ + It is the pseudo-inverse matrix of matrix Φ; the corresponding modes are reordered according to the modal amplitude b from large to small, that is, each two-order mode is regarded as a single mode, so as to obtain the sorted DMD modes. According to equation (11), each mode corresponds to a frequency component. The main frequency components of the mD curve can be obtained through the first few main modes, and according to equation (13), the mD curve can be reconstructed further through the first few main modes.

[0046] Step 5: Determine the optimal number of shift stacking operations.

[0047] Define a loss function to select the optimal number of stacking operations:

[0048]

[0049] In the formula, ||·|| F It is the Frobenius norm;

[0050] Step 6: Modal Selection

[0051] Based on the optimal value of the number of shift stacking, the number of modes is selected by combining equations (13) and (15). First, the mode order r is substituted into equation (13), and then equation (13) is substituted into equation (15). It is determined whether the loss function tends to a constant value. If the loss function does not tend to a constant value, the mode order r is increased until the loss function tends to a constant value.

[0052] Step 7: Translational Compensation and mD Curve Frequency Extraction

[0053] Following the third step of performing DMD decomposition on the mD curve and sorting it according to the amplitude b defined in equation (14), the amplitudes of the first 5 modes are the largest compared to the amplitudes of other modes. The frequencies of these 5 modes include 0Hz, Ω s +Ω c Ω s Ω c and |Ω s -Ω c These five main frequency components include the 0Hz frequency component, which corresponds to the translational component, Ω. s Ω is the spin frequency. c The frequency of the cone rotation, Ω s +Ω c The sum of the spin frequency and the cone spin frequency, |Ω s -Ω c | represents the absolute value of the difference between the spin frequency and the conic spin frequency; DMD decomposition of the mD curve can extract micro-motion features such as spin frequency and conic spin frequency, as well as zero-frequency modes that require translational compensation.

[0054] Each column of the slow-time range profile is a single range profile of the target. By using the zero-frequency mode and employing a cyclic shifting method, translational compensation can be achieved by shifting the range profile.

[0055] In one embodiment of the present invention, in the fourth step, the modal amplitude sorting method is used to arrange the modes according to their contribution and influence on the mD curve.

[0056] In one specific embodiment of the present invention, the number of primary modes is 10.

[0057] In another specific embodiment of the present invention, the optimal number of stacking times is 700.

[0058] The advantages of this invention are as follows:

[0059] 1. Compared with traditional algorithms such as EEMD and VMD, the DMD algorithm can better achieve translational compensation for spatial cone targets;

[0060] 2. In describing the evolution of periodic observations (such as mD curves) over time, DMD decomposition of mD curves can not only accurately extract the main frequency components in the target echo, but also the frequency corresponding to each mode has a single characteristic. Compared with EMD-type methods (including EMD, EEMD, etc.), there is no mode aliasing phenomenon, which is beneficial for accurately extracting the micro-motion frequency components of the target.

[0061] 3. Compared to the EEMD and VMD algorithms, the DMD-based method for extracting micro-motion features from spatial cone targets falls between the two in terms of computation time. With improvements in hardware and faster algorithms, it is expected that the computational load can be further reduced and the real-time performance of the algorithm improved. Attached Figure Description

[0062] Figure 1 This invention illustrates the flowchart of translational compensation and micro-motion feature extraction for a spatial cone target.

[0063] Figure 2 This diagram illustrates the relationship between the spatial cone-shaped target and the radar position.

[0064] Figure 3 The image shows the preprocessed results of the slow time-range image sequence of the space cone target, where Figure 3 (a) shows a slow time-range image sequence of a space cone target. Figure 3 (b) shows the skeleton extraction results. Figure 3 (c) shows the curve separation results;

[0065] Figure 4 This shows the relationship between the number of stacking operations and the loss function;

[0066] Figure 5 The distribution of DMD modal eigenvalues ​​is shown, where Figure 5 (a) shows the modal eigenvalues ​​of DMD. Figure 5 (b) shows a local magnification of the DMD modal eigenvalues;

[0067] Figure 6 This shows the relationship between the number of DMD modes and the loss function;

[0068] Figure 7 The relationship between DMD mode frequency and amplitude is shown, where Figure 7 (a) shows the frequency-amplitude relationship for all DMD modes. Figure 7 (b) A partially enlarged view showing the relationship between DMD modal frequencies and amplitudes;

[0069] Figure 8 The first 10 DMD modes are shown, where Figure 8 (a) shows the first and second order DMD modes. Figure 8 (b) shows the DMD modes of orders 3-10;

[0070] Figure 9 The results of the first 10 DMD mode reconstructions are shown.

[0071] Figure 10 The translational compensation results are shown.

[0072] Figure 11 The distribution of modal eigenvalues ​​under low signal-to-noise ratio conditions is shown, where Figure 11 (a) shows the modal eigenvalues. Figure 11 (b) Shows a magnified view of the modal eigenvalues;

[0073] Figure 12 The relationship between mode frequency and amplitude under low signal-to-noise ratio conditions is shown, where Figure 12 (a) shows the frequency-amplitude relationship for all modes. Figure 12 (b) A magnified view of the relationship between modal frequency and amplitude is shown;

[0074] Figure 13 The first 10 modes after DMD decomposition under low signal-to-noise ratio conditions are shown, among which Figure 13 (a) shows the first and second order modes. Figure 13 (b) shows the 3rd to 10th modes;

[0075] Figure 14 This demonstrates the translational compensation effect under low signal-to-noise ratio conditions;

[0076] Figure 15 The results of EEMD decomposition and translational compensation are shown, in which Figure 15 (a) shows the EEMD decomposition results of the mD curve at point A. Figure 15 (b) shows the EEMD decomposition results of the mD curve at point B. Figure 15 (c) shows the translational compensation results of EEMD;

[0077] Figure 16 The VMD decomposition and translational compensation results are shown, where Figure 16 (a) shows the VMD decomposition results of the mD curve at point A. Figure 16 (b) shows the VMD decomposition results of the mD curve at point B. Figure 16 (c) shows the VMD decomposition and translational compensation results. Detailed Implementation

[0078] The present invention will be further described below with reference to the embodiments and accompanying drawings.

[0079] This invention proposes a method for translational compensation and micro-motion feature extraction of a spatial cone target, comprising the following steps:

[0080] Step 1: mD curve separation

[0081] Under broadband radar, the echoes from the scattering points of a spatial cone target exhibit periodicity in the image domain (i.e., the two-dimensional plane composed of slow time and range). That is, the slow time-range image sequence of the target scattering point, whether it is the cone apex scattering point or the tail fin scattering point, shows a periodic change in the peak position of the range image at each moment with slow time. Preprocessing of the slow-time-range image sequence is required. First, Gaussian smoothing and binarization are performed on the echo of the spatial cone target. Then, skeleton extraction is performed, and the mD curve corresponding to each scattering point can be searched and separated (ZHANG Q, LUO Y, and CHEN YA. Micro-Doppler Characteristics of Radar Targets[M]. Amsterdam: Elsevier, 2017.). At this time, the vector composed of the data of each mD curve still contains the micro-motion characteristics of each scattering point. That is, the mD curve exhibits non-stationary and nonlinear periodic changes with slow time. The DMD algorithm (KUTZ JN, BRUNTON SL, BRUNTON W, et al. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems[M]. Philadelphia: Society for Industrial and Applied Mathematics, 2016.) can describe the evolution of this nonlinear change over time. Based on this, the DMD decomposition method can be used for translational compensation and extraction of micro-motion characteristics.

[0082] Step 2: Construct the snapshot sequence required for DMD decomposition

[0083] By performing curve separation on the slow-time-range image sequence of the target in the image domain, data for each scattering point of the target at N slow times can be obtained. Furthermore, the data from these N slow times are arranged in chronological order of echo time to form a data vector [x1, x2, ..., x]. N Let x be the data for the i-th slow time. i Let Δt be the time interval between any two data points (Δt is the pulse repetition interval PRI), and define two snapshot sequences as follows:

[0084]

[0085]

[0086] In the formula, This indicates taking the data vector [x1, x2, ..., x...] N The first data x1 to the (N-1)th data x in the [] N-1The arranged vector; This indicates taking the data vector [x1, x2, ..., x...] N The second data x2 to the Nth data x in ] N The vector after arrangement.

[0087] When applying the DMD algorithm to radar target mD signal processing, the vector composed of data from each mD curve is a row vector with fewer rows than columns. If the data vector is directly decomposed using DMD, only a single real eigenvalue can be obtained, rather than conjugate pairs of complex eigenvalues. Therefore, the periodically oscillating mD curve cannot be completely captured. To address this, the present invention shifts and stacks the data vector into an augmented data matrix. By increasing the number of rows in the matrix, the frequency components in the periodically oscillating mD curve can be completely captured. Specifically, a first augmented data matrix can be constructed from equation (1):

[0088]

[0089] In the formula, h represents the number of shift stacking operations.

[0090] Similarly, the second augmented data matrix can be formed by shifting and stacking equation (2):

[0091]

[0092] Assuming that the micro-motion echo data of the target scattering point satisfies linear correlation within a short period of time, i.e., that a linear operator A exists, then:

[0093] X′ aug =AX aug (5)

[0094] As can be seen, this process is a linear estimation process, that is, achieving nonlinear estimation through linear assumptions. Furthermore, DMD decomposition is performed on the augmented data matrix. Since the mode, frequency, and growth / decay rate information obtained by the DMD algorithm are closely related to the eigenvalues ​​of the linear operator A, but the linear operator A is usually a high-dimensional matrix of h×h, it is difficult to solve. Therefore, in practical processing, it is necessary to perform dimensionality reduction to find a low-dimensional similarity matrix with rank r (r << h). Let C be used to approximate a high-dimensional linear operator A, where C represents the set of complex numbers. Low-dimensional matrix. It is a similarity matrix of a high-dimensional linear operator A, i.e. (This expression is already in equation (8), and U is obtained through equation (6).)

[0095] Step 3: DMD Decomposition

[0096] (1) For the first augmented data matrix X aug Perform singular value decomposition:

[0097] X aug =U∑V * (6)

[0098] In the formula, the left unitary matrix U∈C h×r Right unitary matrix V∈C (N-h)×r And satisfying U*U=I, V*V=I, where I is the identity matrix, * denotes the complex conjugate transpose, and the first diagonal matrix ∑∈C r×r It has r non-zero singular values ​​(σ1, σ2, ..., σ) on its diagonal. r ), where σ1, σ2,…,σ r These are the 1st, 2nd, ..., rth non-zero singular values, respectively.

[0099] (2) Solving for the similarity matrix

[0100] Substituting equation (6) into equation (5), we get:

[0101] A = X' ang V∑ -1 U * (7)

[0102] Then similar matrix It can be represented as:

[0103]

[0104] (3) For similar matrices Perform eigenvalue decomposition:

[0105]

[0106] In the formula, the columns of the first matrix W are eigenvectors, and the second diagonal matrix Λ contains the corresponding eigenvalues ​​λ. i If the eigenvalues ​​lie inside the unit circle, the mode is stable; otherwise, it is unstable. The logarithmic form of the eigenvalues ​​can be calculated.

[0107] ω i =lnλ i / Δt (10)

[0108] In the formula, the logarithmic form of the eigenvalues ​​is ω i The real part represents the growth / decrease rate of the corresponding DMD mode, while the imaginary part determines the frequency of the mode:

[0109]

[0110] In the formula, imag(·) represents taking ω i The imaginary part.

[0111] (4) Reconstruct the eigenvalues ​​of the linear operator A using the first matrix W and the second diagonal matrix Λ. λ in the second diagonal matrix Λ i Let Φ be the eigenvalues ​​of A, and Φ be the column of the second matrix Φ. i For the eigenvectors (i.e., DMD modes) of the linear operator A:

[0112] Φ=X′ aug V∑ -1 W (12)

[0113] Step 4: Curve Reconstruction

[0114] Reconstruction can be achieved through the first r-order modes:

[0115]

[0116] In the formula, X DMD (t) represents the reconstructed data, t represents time, and the third diagonal matrix Ω = diag(ω) contains elements ω. i b i It is the initial amplitude of each mode, and b is the value determined by b. i The vector formed;

[0117] If we take the data at the initial time, i.e., X aug The first column X aug1 Substituting into equation (13), we can obtain X. aug1 =Φb, then the modal amplitude b is:

[0118] b = Φ + X aug1 (14)

[0119] In the formula, Φ + It is the pseudo-inverse matrix of matrix Φ. The corresponding modes (i.e., every two-order modes can be regarded as a single mode) are reordered according to the modal amplitude b from large to small, so as to obtain the sorted DMD modes. The modal amplitude sorting method can be arranged according to the contribution and influence of each mode on the mD curve. According to equation (11), each mode corresponds to a frequency component (this technique is well known to those skilled in the art). The main frequency components of the mD curve can be obtained through the first few main modes (in a specific embodiment of the present invention, the number of main modes is 10 (i.e., 5), but this step cannot be obtained in advance. This number can only be solved through the sixth step). According to equation (13), the mD curve can be reconstructed further through the first few main modes. MR, SCHMID PJ, and NICHOLS JW. Sparsity-Promoting Dynamic Mode Decomposition[J] Physics of Fluids, 2014, 26(2): 024103.).

[0120] Step 5: Determine the optimal number of shift stacking operations.

[0121] To determine the optimal value of the number of shift stacking operations, h (in one specific embodiment of the invention, the optimal number of stacking operations is 700), while avoiding excessive shift stacking operations that waste computational resources, it is necessary to ensure that the DMD is completely decomposed and accurately captures the periodically oscillating mD curve, extracting its micro-motion features such as spin and conical frequencies. A loss function can be defined to select the optimal number of stacking operations:

[0122]

[0123] In the formula, ||·|| F It is the Frobenius norm.

[0124] If the loss function tends to a constant value as the number of stackings increases, it means that the periodic oscillations have been captured, and continuing to stack will not increase the accuracy of DMD decomposition in principle.

[0125] Step 6: Modal Selection

[0126] Based on determining the optimal number of shift stacking times, the number of modes is selected by combining equations (13) and (15). Specifically, the mode order r can be substituted into equation (13) first, and then equation (13) can be substituted into equation (15) to determine whether the loss function tends to a constant value. If the loss function does not tend to a constant value, the mode order r is increased (in a specific embodiment of the present invention, the number of modes is 10, i.e. 5), until the loss function tends to a constant value.

[0127] Step 7: Translational Compensation and mD Curve Frequency Extraction

[0128] As can be seen from equation (11), each mode obtained by DMD decomposition has the characteristic of a single frequency, which can reflect the frequency components contained in the micro-motion signal. The mD curve corresponding to the tail fin scattering point B mainly includes the Ω generated by micro-motion modulation. s +Ω c Ω s Ω c and |Ω s -Ω c These four frequency components are the frequency components of the translation term, among which Ω s Ω is the spin frequency. cThe frequency of the cone rotation, Ω s +Ω c The sum of the spin frequency and the cone spin frequency, |Ω s -Ω c | represents the absolute value of the difference between the spin frequency and the conic spin frequency, but the frequency component of the translational term is zero. Following the third step, the mD curve is decomposed into DMD components, and after sorting by the amplitude b defined in equation (14), the amplitudes of the first five modes are the largest compared to the other modes. The frequencies of these five modes include 0Hz, Ω... s +Ω c Ω s Ω c and |Ω s -Ω c These five main frequency components include the 0Hz frequency component, which corresponds to the translational component. DMD decomposition of the mD curve can extract micro-motion features such as spin frequency and conospin frequency, as well as the zero-frequency mode that needs translational compensation. The other modes with smaller amplitudes are caused by quantization errors introduced in the distance dimension sampling, resulting in the separated mD curves not being ideally smooth.

[0129] Whether for the cone apex scattering point or the tail fin scattering point, the translational terms are the same, and the zero-frequency modes obtained by DMD decomposition of the mD curve are also the same. Therefore, for the mD curve of any scattering point, translational compensation can be completed using the zero-frequency modes obtained by DMD decomposition. Specifically, each column of the slow-time range profile is a primary range profile of the target. Using the zero-frequency modes and a cyclic shifting method, translational compensation can be completed by shifting the range profile (this technique is well known to those skilled in the art).

[0130] Example: Algorithm Feasibility Analysis and Simulation Verification

[0131] The simulation parameters for the broadband linear frequency modulated radar and the space cone target used in the example are shown in Table 1. A schematic diagram of the positional relationship between the space cone target and the radar is shown below. Figure 2 As shown.

[0132] Table 1 Parameters of Radar and Space Cone Targets

[0133]

[0134] First, without considering noise and clutter, the preprocessing results of the slow time-range image sequence of the space cone target are as follows: Figure 3 As shown, where Figure 3 (a) is a slow time-range image sequence of a spatial cone target. Figure 3 (b) shows the skeleton extraction result. Figure 3(c) shows the curve separation results. It can be seen that the mD curve of the target scattering point in the slow time-range plane (image domain) exhibits a sinusoidal or approximately sinusoidal variation law. Furthermore, due to the existence of the translational term, all mD curves show an overall downward trend, which makes subsequent micro-motion feature extraction difficult.

[0135] Taking scattering point B as an example, the data vector composed of the separated mD curves is shifted and stacked to form an augmented data matrix. To determine the optimal stacking number, the relationship between the loss function and the stacking number is analyzed in conjunction with equation (15). Figure 4 As can be seen, for the micro-motion data of the separated target scattering points, the loss function is 0.88% when stacked 550 times. When stacked 600 times, the loss function drops rapidly to 0.16%. As the number of stacks increases to about 700 times, the loss function tends to a constant value of about 0.06%. This indicates that the periodic oscillation of the mD curve has been completely captured at this point. Continuing to increase the number of stacks will not increase the accuracy and precision of DMD decomposition, but will instead lead to a decrease in computational efficiency.

[0136] Furthermore, it can be seen that, after sorting the decomposed modes according to the modal amplitude defined in equation (14), the eigenvalue distribution of the modes is as follows: Figure 5 As shown in (a), the eigenvalues ​​are all distributed in pairs. Because their eigenvalues ​​are conjugate complex pairs, each pair of modes (i.e., every two modes) can be considered as a single mode. In the figure, the larger the mode amplitude, the darker the circle corresponding to the eigenvalue. Figure 5 (b) The magnified view of the modal eigenvalue distribution shows that the amplitude of the first 10 modes after mode sorting is the largest (the dark circle in the middle is the overlap of the 1st and 2nd modes), and the eigenvalues ​​are basically located on the unit circle, indicating that each mode is close to a stable state.

[0137] The number of modes is selected again using equation (15) to ensure that the zero-frequency mode that needs translational compensation and other main modes containing micro-motion frequencies are obtained. Figure 6 The relationship between the number of DMD modes and the loss function shows that the loss function of the first 10 modes is constant at around 0.03%, which is sufficient to reflect the main frequency composition of the mD curve. Therefore, the mD curve can be reconstructed well through the first 10 modes.

[0138] The frequencies of each mode will be further analyzed below. Figure 7 The graph shows the frequency-amplitude relationship of DMD modes. Although the amplitudes of the first and second modes are the largest, their frequencies are 0. Figure 8 As can be seen in (a), the first and second order modes correspond to the translational terms of the target, and... Figure 3The baseline trend of the mD curve is consistent; the frequency of the 3rd-4th order modes is 3.5Hz, corresponding to the target Ω. s +Ω c Frequency components; the frequency of the 5th-6th modes is 3Hz, corresponding to the target's spin frequency component, i.e., Ω. s The frequency of the 7th-8th modes is approximately 0.5 Hz, corresponding to the conic-rotation frequency component of the target, i.e., Ω. c Components; the 9th-10th order modes have the smallest amplitudes and a frequency of approximately 2.5 Hz, corresponding to the target's |Ω s -Ω c | Quantity. Figure 8 In (b), the 3rd to 10th modes exhibit essentially periodic simple harmonic motion, corresponding to... Figure 7 The frequencies of the corresponding modes are shown. It can be seen that each mode has a single frequency, corresponding to a specific frequency component in the target echo; this is an advantage of DMD decomposition. Therefore, DMD decomposition can accurately extract the micro-motion characteristics such as the spin frequency and cone rotation frequency of a space cone target.

[0139] Modal energy is also one of the indicators reflecting the importance of modes. Table 2 shows the proportion of the sum of the first i (i = 2, 4, 6, 8, 10) modal energies in the total modal energy. It can be seen that the sum of the first two modal energies is close to 100% of the total modal energy. The first two modes correspond to the translational term of the space cone target, reflecting the Doppler echo of the target body. The modal energies of the 3rd to 10th modes are relatively small, reflecting the mD echo signal of the micro-motion scattering point. Table 3 shows the proportion of the sum of the first i (i = 2, 4, 6, 8) modes (i.e., the original 3rd to 10th modes) in the total remaining modal energy after removing the first two modes. It can be seen that the sum of these 8 modal energies is close to 100%, and their contribution to the mD curve cannot be ignored. These 8 modes represent the Ω of the space cone target during precession. s +Ω c Ω s Ω c and |Ω s -Ω c These are the four frequency components.

[0140] Table 2 shows the modal energies and proportions of the first i (i = 2, 4, 6, 8, 10) modes.

[0141]

[0142] Table 3 shows the energy and proportion of the first i (i = 2, 4, 6, 8) modes.

[0143]

[0144]

[0145] Further reconstruction of the micro-motion echoes from the target scattering point in the slow time-range domain. For example... Figure 9 As shown, the mD curve of the target scattering point can be reconstructed well using the first 10 modes, which is very close to the mD curve before DMD decomposition. It can be seen that the first 10 main modes basically contain the echo information of the target micro-motion scattering point and can reflect the frequency components of the mD curve.

[0146] The three mD curves at scattering points A, B, and C are then subjected to DMD decomposition to obtain the first and second order modal superposition results for each curve, i.e., the translational term of each mD curve. After averaging, translational compensation is performed, and the compensation result is as follows: Figure 10 As shown, the downward tilt of the curve caused by translation is suppressed, achieving a good compensation effect, which facilitates the accurate extraction of subsequent micro-motion features.

[0147] Experimental verification

[0148] To verify the robustness of the method under low signal-to-noise ratio conditions, white noise with a signal-to-noise ratio of -15dB was added to the target echo. The simulation parameter settings for the broadband linear frequency modulated radar and the spatial cone target are shown in Table 1, and the schematic diagram of the positional relationship between the spatial cone and the radar is shown below. Figure 2 As shown.

[0149] from Figure 11 As can be seen from the eigenvalue distribution in (a), the eigenvalues ​​obtained after decomposition with added noise are still distributed in pairs; from Figure 11 (b) A magnified view of the distribution of eigenvalues ​​shows that although some eigenvalues ​​are affected by noise and deviate from the unit circle, i.e. they increase or decrease, the eigenvalues ​​of the first 10 modes are still located on the unit circle. This indicates that the first 10 modes basically contain the main frequency characteristics of the target mD curve.

[0150] from Figure 12 As can be seen from the modal frequency versus amplitude graph in (a), mode 1-2 still has the largest amplitude and a frequency of 0Hz, which also corresponds to the translational term of the space cone target; from Figure 12 (b) The magnified view of the modal frequency versus amplitude relationship shows that modes 3-8 still correspond to the Ω of the target echo. s +Ω c Ω s Ω c and |Ω s -Ω c The frequency positions of these four frequency components remain relatively accurate. Figure 13The first 10 modes after DMD decomposition show that, at a signal-to-noise ratio of -15dB, the first and second modes still reflect the target's translational motion well. Similarly, the third to tenth modes still exhibit periodic motion, containing the spin and conoscillation frequency components of the target's scattering point. This demonstrates that even under low signal-to-noise ratio conditions, DMD decomposition can still accurately capture the micro-motion characteristics contained in the target's mD curve.

[0151] Figure 14 The results show the translational compensation under low signal-to-noise ratio (SNR) conditions. It can be seen that translational compensation still achieves good results under low SNR, indicating that the DMD algorithm has good noise immunity.

[0152] To further verify the accuracy of applying the DMD algorithm for translational compensation and micro-motion feature extraction, the micro-motion feature extraction algorithms such as EEMD and VMD were used to compare and simulate the target echo in the slow time-range domain under a signal-to-noise ratio of -15dB. The simulation parameters for the broadband linear frequency modulated radar and the space cone target are shown in Table 1.

[0153] Figure 15 The decomposition and translational compensation results using the EEMD algorithm are presented, from... Figure 15 (a) shows the EEMD decomposition results of the mD curve of scattering point A. Due to the quantization error introduced by range dimension sampling in the slow-time-range image sequence preprocessing step, the separated mD curve is no longer continuous and smooth, inevitably introducing high-frequency components. Furthermore, since the decomposition results of the EEMD algorithm are generally arranged from high to low frequencies, the first obtained IMF1 and IMF2 components are high-frequency components, while the IMF3 component has a frequency of 0.5 Hz, corresponding to the target's conic spiral frequency component Ω. c The IMF4 component corresponds to the translation term; Figure 15 (b) shows the EEMD decomposition result of the mD curve at scattering point B, with the IMF3 component frequency at 3.5 Hz, corresponding to the target's Ω. s +Ω c The component, and the spin frequency Ω of 3Hz. s Components and 2.5Hz |Ω s -Ω c The component is quite close to 3.5Hz, making it difficult to decompose accurately. Meanwhile, the 0.5Hz cone-rotation frequency Ω can be seen from the IMF4 components. c The components are also difficult to obtain accurately. It is evident that although the EEMD algorithm is an improved version of EMD and overcomes modal aliasing to some extent, it cannot completely eliminate it; the IMF5 component is the translational term. Figure 15(c) The translational compensation result of the trend term obtained by EEMD decomposition. Due to the existence of the endpoint effect, it can be seen from the translational compensation result that the micro-motion curves on both sides are deformed due to the influence of the endpoint effect.

[0154] Figure 16 The decomposition and translational compensation results are obtained using the VMD algorithm. Figure 16 (a) shows the VMD decomposition result of the mD curve at scattering point A, where the IMF1 component is the translational term, the IMF2 component has a frequency of 0.5 Hz, and corresponds to the conic-rotation frequency component Ω of the target. c The remaining IMF components are high-frequency components; Figure 16 (b) shows the VMD decomposition result of the mD curve of scattering point B. IMF1 is still a translational term, and the frequency of the IMF2 component is 3.5 Hz, corresponding to the target's Ω. s +Ω c The IMF3 component has a frequency of 0.5 Hz, corresponding to the target's cone-rotation frequency component Ω. c The frequency of the IMF4 component is 3Hz, corresponding to the spin frequency component Ω. s The remaining components are high-frequency components; Figure 16 (c) is the result of translational compensation using the trend term obtained by VMD decomposition. Since the VMD algorithm also has an endpoint effect, the micro-motion curves on both sides are deformed due to its influence, but the degree of influence is smaller than that of the EEMD algorithm.

[0155] The computation time of the three algorithms, EEMD, VMD, and DMD, is compared below under the same computer configuration: Intel i5-8265U processor, 8GB RAM, and Windows 10 operating system. Taking the mode decomposition of the echo from scattering point B as an example, the computation time of the three algorithms is shown in Table 4. It can be seen that the EEMD algorithm has the shortest computation time, the VMD algorithm has the longest computation time, and the DMD algorithm's computation time is in between, approximately 6.57 seconds.

[0156] Table 4 Comparison of computation time for the three algorithms

[0157]

Claims

1. A method for translational compensation and micro-motion feature extraction of a spatial cone target, characterized in that, Includes the following steps: Step 1: mD curve separation The slow-time-range image sequence is preprocessed by first performing Gaussian smoothing and binarization on the spatial cone target echo, followed by skeleton extraction. The mD curve corresponding to each scattering point is then searched and separated. At this point, the vector composed of the data of each mD curve still contains the micro-motion characteristics of each scattering point. That is, the mD curve exhibits non-stationary and nonlinear periodic changes with slow time. The DMD algorithm is used to describe the evolution of this nonlinear change over time. Based on this, the DMD decomposition method is used for translational compensation and extraction of micro-motion characteristics. Step 2: Construct the snapshot sequence required for DMD decomposition By performing curve separation on the slow-time-range image sequence of the target in the image domain, data for each scattering point of the target at N slow times are obtained. These N slow-time data are then arranged in chronological order of echo time to form a data vector [x1, x2, ..., x]. N Let x be the data for the i-th slow time. i Let Δt be the time interval between any two data points, where Δt is the pulse repetition interval PRI. Define two snapshot sequences as follows: In the formula, This indicates taking the data vector [x1, x2, ..., x...] N The first data x1 to the (N-1)th data x in the [] N-1 The arranged vector; This indicates taking the data vector [x1, x2, ..., x...] N The second data x2 to the Nth data x in ] N The arranged vector; Construct the first augmented data matrix using equation (1): In the formula, h represents the number of shift stacking operations; The second augmented data matrix is ​​formed by shifting and stacking equation (2): Assuming that the micro-motion echo data of the target scattering point satisfies linear correlation within a short time period, i.e., that a linear operator A exists, then: X′ aug =AX aug (5) Given a linear operator A, which is a high-dimensional matrix of size h × h, find a low-dimensional similarity matrix of rank r. To approximate a high-dimensional linear operator A, where C represents the set of complex numbers, r << h; and a low-dimensional matrix. It is a similarity matrix of a high-dimensional linear operator A. Step 3: DMD Decomposition (1) For the first augmented data matrix X aug Perform singular value decomposition: X aug =U∑V * (6) In the formula, the left unitary matrix U∈C h×r Right unitary matrix V∈C (N-h)×r And satisfy U * U = I, V * V = I, where I is the identity matrix. * Let ∑∈C be the complex conjugate transpose of the first diagonal matrix. r×r It has r non-zero singular values ​​(σ1, σ2, ..., σ) on its diagonal. r ), where σ1, σ2,…,σ r These are the 1st, 2nd, ..., rth non-zero singular values, respectively. (2) Solving for the similarity matrix Substituting equation (6) into equation (5), we get: A=X′ aug V∑ -1 U * (7) Then similar matrix Represented as: (3) For similar matrices Perform eigenvalue decomposition: In the formula, the columns of the first matrix W are eigenvectors, and the second diagonal matrix Λ contains the corresponding eigenvalues ​​λ. i If the eigenvalue lies inside the unit circle, the mode is stable; otherwise, it is unstable. Calculate the logarithmic form of the eigenvalue: oh i =1nλ i / Δt (10) In the formula, the logarithmic form of the eigenvalues ​​is ω i The real part represents the growth / decrease rate of the corresponding DMD mode, while the imaginary part determines the frequency of the mode. In the formula, imag(·) represents taking ω i The imaginary part; (4) Reconstruct the eigenvalues ​​of the linear operator A using the first matrix W and the second diagonal matrix Λ. λ in the second diagonal matrix A i Let Φ be the eigenvalues ​​of A, and Φ be the column of the second matrix Φ. i Let A be the eigenvector of the linear operator A, i.e., the DMD mode: Φ=X′ aug V∑ -1 W (12) Step 4: Curve Reconstruction Reconstruction using the first r-order modes: In the formula, X DMD (t) represents the reconstructed data, t represents time, and the third diagonal matrix Ω = diag(ω) contains elements ω. i b i It is the initial amplitude of each mode, and b is the value determined by b. i The vector formed; If we take the data at the initial time, i.e., X aug The first column X aug1 Substituting into equation (13), we get X. aug1 =Φb, at this time the modal amplitude b is: b=Φ + X aug1 (14) In the formula, Φ + It is the pseudo-inverse matrix of matrix Φ; the corresponding modes are reordered according to the modal amplitude b from large to small, that is, each two-order mode is regarded as a single mode, so as to obtain the sorted DMD modes. According to equation (11), each mode corresponds to a frequency component. The main frequency components of the mD curve can be obtained through the first few main modes, and according to equation (13), the mD curve can be reconstructed further through the first few main modes. Step 5: Determine the optimal number of shift stacking operations. Define a loss function to select the optimal number of stacking operations: In the formula, ||•|| F It is the Frobenius norm; Step 6: Modal Selection Based on the optimal value of the number of shift stacking, the number of modes is selected by combining equations (13) and (15). First, the mode order r is substituted into equation (13), and then equation (13) is substituted into equation (15). It is determined whether the loss function tends to a constant value. If the loss function does not tend to a constant value, the mode order r is increased until the loss function tends to a constant value. Step 7: Translational Compensation and mD Curve Frequency Extraction Following the third step of performing DMD decomposition on the mD curve and sorting it according to the amplitude b defined in equation (14), the amplitudes of the first 5 modes are the largest compared to the amplitudes of other modes. The frequencies of these 5 modes include 0Hz, Ω s +Ω c Ω s Ω c and |Ω s -Ω c These five main frequency components include the 0Hz frequency component, which corresponds to the translational component, Ω. s Ω is the spin frequency. c The frequency of the cone rotation, Ω s +Ω c Let |Ω be the sum of the spin frequency and the cone spin frequency. s -Ω c | represents the absolute value of the difference between the spin frequency and the conic spin frequency; DMD decomposition of the mD curve can extract micro-motion features such as spin frequency and conic spin frequency, as well as zero-frequency modes that require translational compensation. Each column of the slow-time range profile is a single range profile of the target. By using the zero-frequency mode and employing a cyclic shifting method, translational compensation can be achieved by shifting the range profile.

2. The method for translational compensation and micro-motion feature extraction of a spatial cone target as described in claim 1, characterized in that, In the fourth step, the modal amplitude sorting method is used to arrange the modes according to their contribution and influence on the mD curve.

3. The method for translational compensation and micro-motion feature extraction of a spatial cone target as described in claim 1, characterized in that, The number of primary modes is 10.

4. The method for translational compensation and micro-motion feature extraction of a spatial cone target as described in claim 1, characterized in that, The optimal number of stacking operations is 700.

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