Structural mode recognition method based on image phase in video stream

Abstract

The present invention discloses a structural modal recognition method based on the image phase in a video stream, which clarifies the corresponding relationship between principal component analysis and modal parameters; principal component analysis has the advantage of retaining high spatial resolution vibration information, and compared with the traditional method of obtaining structural frequency based on a small number of discrete points, it improves the algorithm's fault tolerance and increases the accuracy of frequency recognition; a high-resolution single-mode extraction formula is theoretically derived, which improves the vibration type recognition accuracy compared with the traditional modal vibration type extraction within the frequency range.

Description

Technical Field

[0001] The invention relates to the technical field of dynamic monitoring of civil engineering structures, and in particular to a structural mode recognition method based on image phase in a video stream. Background Art

[0002] Structural health monitoring can provide guarantee for structural safety. Among them, structural dynamic characteristics are important to ensure structural dynamic safety, and modal parameters are important indicators for evaluating dynamic characteristics.

[0003] The natural frequency and effective vibration mode of wind turbine blades provide reliable basis for avoiding wind turbine resonance and identifying and locating structural damage. Traditional modal analysis requires physically connected wired or wireless sensors, such as accelerometers mounted on the structure. Although these sensors are reliable, their installation is extremely time-consuming and labor-intensive for large wind turbine blade structures. In addition, such contact sensors can only provide sparse, discrete point measurement information. Figure 1 The sparse point measurement method in the MATLAB software is usually limited by the low spatial sensing resolution, which is not enough for modal-based local damage detection, model updating, etc. In addition, previous studies have shown that the spatial resolution of sensor measurements severely limits the effectiveness of many commonly used damage detection and location methods based on modal vibration shapes or modal vibration shape curvature.

[0004] Therefore, the modal information with rich high-resolution characteristics (see Figure 1 The dense measurement point method in the finite element method is very valuable for more accurate local damage identification, detection, correlation and updating of highly refined and detailed finite element models. Summary of the invention

[0005] In order to solve the technical problem of low spatial sensing resolution limitation caused by the existing sparse point measurement method, the present invention provides a structural mode recognition method based on image phase in video stream.

[0006] The technical solution is as follows:

[0007] A structural modal recognition method based on image phase in a video stream is performed in the following steps:

[0008] S1. High-precision modal frequency derivation is carried out according to the following steps:

[0009] S11, calculate the covariance matrix C of the video image motion information δ δδ ;

[0010] S12. Calculate the covariance matrix C δδ The r non-zero eigenvalues ​​λ of i , and sort by size;

[0011] S13, calculate the modal response q i (t) and natural frequency u r ;

[0012] S2. High-resolution modal vibration shape derivation is carried out according to the following steps:

[0013] S21, decomposing the video image into an amplitude spectrum and a phase spectrum by two-dimensional Fourier transform;

[0014] S22, by different spatial frequencies ω i The phase spectrum of the video image is processed by an adjustable pyramid filter composed of Gabor waves to obtain a reconstructed phase spectrum. The filtered response obtained by fusing the reconstructed phase spectrum with the amplitude spectrum can be reconstructed into the motion information δ` of the new video image;

[0015] S23. Using normalization processing, separate and obtain high-resolution modal vibration shapes of each order.

[0016] Compared with the prior art, the present invention has the following beneficial effects:

[0017] 1. The corresponding relationship between principal component analysis and modal parameters is clarified; principal component analysis has the advantage of retaining high spatial resolution vibration information. Compared with the traditional method of obtaining structural frequency based on a small number of discrete points, it improves the algorithm's fault tolerance and increases the accuracy of frequency identification;

[0018] 2. The high-resolution single-mode extraction formula is theoretically derived, which improves the accuracy of mode shape identification compared with the traditional mode shape extraction within the frequency range. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 Structural mode shapes for different measurement types;

[0020] Figure 2 It is the motion information of each pixel in the image structure;

[0021] Figure 3 It is the filtering process of the image by the adjustable pyramid filter;

[0022] Figure 4 This is a laboratory test diagram of the tower model;

[0023] Figure 5 is the response and power spectrum curve of the acceleration sensor;

[0024] Figure 6 It is the comparison diagram of displacement time history curve;

[0025] Figure 7 are the main components of the structural response and their spectrum;

[0026] Figure 8This is the single-order high-resolution mode shape of the tower model. DETAILED DESCRIPTION

[0027] The present invention is further described below in conjunction with embodiments and drawings.

[0028] A structural modal recognition method based on image phase in a video stream is performed in the following steps:

[0029] S1. High-precision modal frequency derivation is carried out according to the following steps:

[0030] S11, calculate the covariance matrix C of the video image motion information δ δδ , as follows:

[0031] For space R N The motion information of N pixels in the video image δ=[δ1,δ2,...,δ N ] T ∈R N×T , where R N ×T Represents an N×T dimensional real matrix. According to the principle of principal component analysis, it is assumed that the motion information δ can be obtained from the space R r The last r unrelated unknown variables η=[η1,η2,...,η r ] T ∈R r×T After the linear transformation matrix E∈R N×r Mixed, where R r×T represents an r×T-dimensional real matrix, R N×r represents an N×r dimensional real matrix, and the linear transformation matrix E satisfies the conditions:

[0032]

[0033] In formula (1), E (N×r) represents the N×r dimensional linear transformation real matrix, E T (r×N) represents the transpose of the E matrix, I (r×r) represents the r×r dimensional identity matrix;

[0034] r unrelated unknown variables η are used as the principal components of motion information δ, and the relationship between motion information δ and unrelated unknown variables η is as follows:

[0035] δ (N×T) =E (N×r) η (r×T) (2)

[0036] In formula (2), δ (N×T) represents the N×T dimensional motion information matrix, η (r×T)represents the uncorrelated matrix after the r×T-dimensional principal component analysis;

[0037] The covariance matrix C of motion information δ δδ(N×N) for:

[0038] C δδ(N×N) =E(δ (N×T) δ T (T×N) ) (3)

[0039] In formula (3), δ T (T×N) represents the transpose of motion information δ, E(C δδ(N×N) δ T (T×N) ) indicates that C δδ(N×N) δ T (T×N) Seek expectations.

[0040] S12. Calculate the covariance matrix C δδ The r non-zero eigenvalues ​​λ of i , and sorted by size, as follows:

[0041] Since the covariance matrix C in formula (3) δδ(N×N) It is a symmetric square matrix, so it can be decomposed into eigenvalues:

[0042] C δδ(Ν×Ν) u i =λ i u i (4)

[0043] In formula (4), u i is the eigenvalue λ i The corresponding eigenvector;

[0044] According to the matrix eigenvalue property, C δδ(N×N) Can be uniquely decomposed into:

[0045]

[0046] In formula (5), Λ (r×r) Represents C δδ(N×N) The r non-zero eigenvalues ​​λ of i ≠0 (i=1,2,...,r) is a diagonal square matrix in descending order, that is:

[0047]

[0048] In formula (6), R r×r represents an r×r dimensional real matrix;

[0049] In formula (3), U rT (r×N) Indicates U r(N×r) The transpose of U r(N×r) represents the r×N dimensional eigenvalue decomposition matrix, U r =[u1,...,u r ], and the following relationship is satisfied:

[0050] U r T (r×N) U r(N×r) =I (r×r) (7)

[0051] In formula (7), I (r×r) is the r×r dimensional identity matrix;

[0052] So δ (N×T) Can be uniquely decomposed into:

[0053] δ (N×T) =U r(N×r) (U r T (r×N) δ (N×T) ) (8)

[0054] Comparing equations (1) and (7) with equations (2) and (8), we can see that the principal component decomposition result of motion information δ is unique, and the U required by equation (7) is r(N×r) That is, the linear transformation matrix E, U required for the principal component decomposition of motion information δ r T (r×N) δ (N×T) That is, the r unrelated unknown variables η required for the principal component decomposition of motion information δ;

[0055] Therefore, based on step S11 and step S12, the picture frame of the video image can be processed to obtain the motion information of N pixels in the structure δ=[δ1,δ2,…δ i ,...,δ N ] T ∈R N×T , see Figure 2 By applying the phase motion estimation method to the captured video stream, the N-point motion information δ of N pixels in the image can be obtained. The motion information δ of each pixel i (i=1,2,...,N) are all vectors of size 1×T.

[0056] S13, calculate the modal response q i (t) and natural frequency u r , as follows:

[0057] The motion information δ of point x on the image structure can be expressed by the modal coordinates q(t):

[0058]

[0059] In formula (9), r is the mode number, is the mode shape matrix, q(t)=[q1,q2,...,q r ] T ∈R r×T is the modal coordinate vector, represents the i-th order mode shape, q i (t) (l×T) represents the coordinates of the modal response of the i-th order;

[0060] Based on vibration theory, the mode shape vector are mutually orthogonal, so:

[0061]

[0062] In formula (10), represents the i-th mode shape The transpose of represents the jth mode shape;

[0063] For civil engineering structures, the modes are sparse, and the first several orders can meet the engineering accuracy requirements. Therefore, the number of pixel motion information δ obtained by the image phase motion estimation method is much higher than the number of modes, that is, N>>r. Since equation (9) is an overcomplete model from high dimension to low dimension, it cannot be directly used for mode extraction. Therefore, principal component analysis is introduced to realize the conversion of data samples from high dimension to low dimension. Comparing equations (1) and (2) with equations (9) and (10), it can be seen that equation (10) is a special case of principal component analysis decomposition. Therefore, the regularized mode vibration shape Φ(x) (N×r) It can be regarded as the linear aliasing matrix E in equation (2), and the modal response matrix q of each order is equal to η in equation (2) (r×T) ,Right now:

[0064]

[0065] In formula (11), U r(N×r) T Indicates U r(N×r) The transpose of U r(N×r) represents the r×N dimensional eigenvalue decomposition matrix;

[0066] After obtaining the modal response q of each order i (t), the frequency corresponding to the peak value of the fast Fourier transform is taken as the natural frequency u r .

[0067] Therefore, principal component analysis can be used to extract structural modal frequencies, and the principal components of principal component analysis have the advantage of retaining high spatial resolution vibration information. Compared with the traditional method of obtaining structural frequencies based on a small number of discrete points, this improves the algorithm's fault tolerance and increases the accuracy of frequency identification.

[0068] S2. High-resolution modal vibration shape derivation is carried out according to the following steps:

[0069] S21, decomposing the video image into an amplitude spectrum and a phase spectrum through a two-dimensional Fourier transform. Correspondingly, the amplitude spectrum and the phase spectrum can also be reconstructed into the original image. Therefore, combining the corresponding relationship between the image phase and the structural motion, the original image structure can be changed by changing the phase spectrum.

[0070] S22, see Figure 3 , by different spatial frequencies ω i The phase spectrum of the video image is processed by an adjustable pyramid filter composed of Gabor waves to obtain a reconstructed phase spectrum. The filtered response obtained by fusing the reconstructed phase spectrum with the amplitude spectrum can be reconstructed into the new video image motion information δ`, as follows:

[0071] Assuming that I(x+δ) is the motion information δ of point x in the image structure, when an adjustable pyramid filter is used to represent the image I(x+δ), it can be written as the superposition of the filter response on the adjustable pyramid filter:

[0072]

[0073] In formula (12), R ω (x, t) represents the filtering response of the image I(x+δ) on the adjustable pyramid filter of scale ω, that is:

[0074] R ω (x,t)=ρ ω (x,t)e j(2πux+δ) (13)

[0075] In formula (13), j is an imaginary unit, ρ ω (x, t) is the corresponding image amplitude, 2πωx+δ is the image phase;

[0076] After applying the phase motion estimation method to the motion information δ of the video image, the principal component analysis is applied, and the modal response coordinates η = [η1, η2, ..., η r ] T ∈R r×T On this basis, the structural modal response coordinates of the motion information δ are respectively amplified and attenuated: the i-th order modal response coordinate η i (t) is multiplied by the magnification factor α, and the remaining modal response coordinates η l(t)(l≠i) is multiplied by the attenuation factor β=-1 and substituted into equation (11) to obtain the new motion information δ`:

[0077]

[0078] In formula (14), u i (x) represents the matrix U r The i-th column of

[0079] The motion information δ of the video image corresponds to the filter response R on the adjustable pyramid filter ω (x,t) can be written as:

[0080]

[0081] In formula (15), u l (x) represents the matrix U r All columns except the i-th column, where l = 1, 2, ... i-1, i+1, ... r;

[0082] Comparing equations (14) and (15), it can be seen that for the motion information δ, the new motion information δ' is equivalent to the i-th order modal motion in the motion information δ amplified by 1+α times, while removing other order modal motions. Therefore, the new motion information δ' represents the motion that only amplifies the i-th order mode of the structure by 1+α times after removing other order modal motions, thereby realizing the separation of single-order modes;

[0083] Referring to equation (12), by performing phase processing of equations (14) to (15) on the adjustable pyramid filter of each scale ω, the reconstruction of the new image I`(x+δ`(x,t),t) corresponding to the new motion information δ`(x,t) can be achieved:

[0084]

[0085] In formula (16), R ω `(x, t) represents the filter response of the new motion information δ` corresponding to the adjustable pyramid filter;

[0086] I`(x+δ`,t) is the new motion information δ` that only amplifies the i-th mode of the structure by 1+α times.

[0087] S23. Since step S22 realizes the visualization of the deflection shape of the i-th order modal motion of the structure, normalization processing is used to separate and obtain high-resolution modal vibration shapes of each order.

[0088] The wind turbine tower model is used for verification:

[0089] This section uses a 1:25 scale model of the H160-3.4MW-100mHH offshore wind turbine tower as the experimental object, and verifies the high-resolution modal identification method proposed in this chapter by recording the vibration information of the tower in the Y direction during the shaking table test.

[0090] like Figure 4 As shown, the height of the scaled wind turbine tower model is 3870mm. The model tower is divided into 5 sections. Each tower section is connected by bolts through a flange. The tower itself and the flange are welded by butt welds, and the bottom surface of the flange is flush with the top surface of each tower section. Before the test, a Huawei mobile phone video device with a sampling frame rate of 60fps and a resolution of 3840px×2160px was set 1.5 meters in front of the wind turbine tower model to capture the movement of the tower model in the Y direction, while ensuring sufficient ambient light. During the video shooting process, the lens was kept as perpendicular to the Y-direction movement of the tower model as possible to reduce the interference of out-of-plane displacement. The laboratory resolution of the tower model is 1.574 pixels / mm. Figure 4 (b) shows a frame in the tower model motion video. For comparison, the experiment uses 5 acceleration sensors (US MEAS accelerometers) and 1 wire displacement sensor (US Unimeasure wire displacement sensor) as reference standards. Figure 4 (a) is a diagram of the layout of different sensors. At the same time, the NI PXIe-1065 slot chassis optoelectronic test and measurement equipment is used to collect acceleration and displacement sensor data.

[0091] The test was completed by frequency sweep loading, and a 25-second motion video of the tower model was recorded using a mobile phone. During this period, the accelerometer and displacement meter collected information synchronously. Figure 5 (a)-(e) show the responses of each acceleration sensor. In this experiment, the frequency domain decomposition method is used to process the acceleration data, and the obtained frequency and vibration mode are used as reference results to verify the proposed modal identification algorithm. Figure 5 (f) is the power spectrum curve obtained using the frequency domain decomposition method.

[0092] Please see Table 1, which shows the various frequencies and mode coefficients of the tower model obtained by the frequency domain decomposition method.

[0093] Table 4.2 Frequency and mode coefficients of the structure

[0094]

[0095] Before processing the 25s motion video of the tower model for modal recognition, the Hilbert transform-based civil engineering structure dynamic monitoring phase evaluation method (RPME algorithm for short) disclosed in Chinese patent CN113076517A is first applied to extract the motion of the displacement sensor measurement point at the top of the tower. By comparing it with the results collected by the displacement meter, the effectiveness of the structural motion estimation algorithm is verified, and the motion recognition basis is provided for the subsequent modal recognition algorithm. When the RPME algorithm is used to extract the Y-direction motion of the tower from the captured video, the single-scale image sequence acquisition method is first applied to obtain the center frequency f of the ideal narrowband filter. s =4, the bandwidth is 1 Hz, and then the motion information of the structure can be obtained according to the RPME algorithm program. Figure 6 A comparison chart of the motion results recognized by RPME and the motion data collected by the wire displacement sensor (represented by D-sensor) is given.

[0096] from Figure 6 It can be seen that the fitting effect of D-sensor and RPME differs around 0s, which is mainly affected by experimental errors. At this time, due to the influence of the ambient noise around the test, the mobile phone may shake up and down. Therefore, the movement or shaking of the camera should be avoided during the test. However, the overall fitting effect of the Y-direction motion information of the tower model identified by RPME and the motion data collected by the cable displacement sensor is excellent, which also shows the effectiveness of the RPME algorithm in identifying motion information.

[0097] Based on the motion information of each pixel point on the tower model structure, the frequency feature extraction method based on principal component analysis is used to extract the structural frequency information, such as Figure 7 As shown, Figure 7 (a) Time response of principal component 1 after principal component analysis, Figure 7 (b) Frequency spectrum of principal component 1 after principal component analysis, Figure 7 (c) The time response of principal component 2 after principal component analysis, Figure 7 (d) Frequency spectrum of principal component 2 after principal component analysis, Figure 7 (e) Time response of principal component 3 after principal component analysis, Figure 7 (f) Frequency spectrum of principal component 3 after principal component analysis.

[0098] Please refer to Table 2, which gives the error comparison between the proposed frequency extraction method based on principal component analysis and the frequency in Table 1.

[0099] Table 4.3 Structural frequency error comparison

[0100]

[0101] It can be seen from Table 2 that the error between the structural frequency obtained by the proposed frequency extraction method based on principal component analysis and the frequency obtained by the frequency domain decomposition method is within 2%, which proves the effectiveness of the proposed modal frequency extraction method.

[0102] In order to obtain the single-order high-resolution modal vibration shape at the natural frequency of the structure, the method of this embodiment is applied to extract the structural vibration shape. After obtaining the single-order high-resolution motion deflection shape of the structure from the image phase angle, the visualization diagram of the single-order high-resolution modal deflection shape of the tower model can be obtained by normalization, as shown in FIG. Figure 8 As shown, Figure 8 (a) is the first-order vibration mode, Figure 8 (b) is the second-order vibration mode, Figure 8 (c) is the third-order vibration mode.

[0103] Finally, it should be noted that the above description is only a preferred embodiment of the present invention. Under the guidance of the present invention, ordinary technicians in this field can make various similar expressions without violating the purpose and claims of the present invention, and such changes all fall within the scope of protection of the present invention.

Claims

1. A structural modal recognition method based on image phase in video stream, characterized in that: Follow these steps: S1. High-precision modal frequency derivation is carried out according to the following steps: S11, calculate the covariance matrix C of the video image motion information δ δδ ; S12. Calculate the covariance matrix C δδ The r non-zero eigenvalues ​​λ of i , and sort by size; S13, calculate the modal response q i (t) and natural frequency u r ; S2. High-resolution mode shape derivation is carried out according to the following steps: S21, decomposing the video image into an amplitude spectrum and a phase spectrum by two-dimensional Fourier transform; S22, by different spatial frequencies ω i The phase spectrum of the video image is processed by an adjustable pyramid filter composed of Gabor waves to obtain a reconstructed phase spectrum. The filtered response obtained by fusing the reconstructed phase spectrum with the amplitude spectrum can be reconstructed into the motion information δ` of the new video image; S23, using normalization processing to separate and obtain high-resolution modal vibration shapes of each order; In step S11, for the space R N The motion information of N pixels in the video image δ=[δ1,δ2,...,δ N ] T ∈R N ×T , where R N×T Represents an N×T dimensional real matrix. According to the principle of principal component analysis, it is assumed that the motion information δ can be obtained from the space R r The last r unrelated unknown variables η=[η1,η2,...,η r ] T ∈R r×T After the linear transformation matrix Ε∈R N×r Mixed, where R r×T represents an r×T-dimensional real matrix, R N×r represents an N×r dimensional real matrix, and the linear transformation matrix E satisfies the conditions: E T (r×N) E (N×r) =I (r×r) (1) In formula (1), E (N×r) represents the N×r dimensional linear transformation real matrix, E T (r×N) represents the transpose of the E matrix, I (r×r) represents the r×r dimensional identity matrix; r unrelated unknown variables η are used as the principal components of motion information δ, and the relationship between motion information δ and unrelated unknown variables η is as follows: d (N×T) =E (N×r) or (r×T) (2) In formula (2), δ (N×T) represents the N×T dimensional motion information matrix, η (r×T) represents the uncorrelated matrix after the r×T-dimensional principal component analysis; The covariance matrix C of motion information δδ δδ(N×N) for: C δδ(N×N) =E(δ (N×T) d T (T×N) ) (3) In formula (3), δ T (T×N) represents the transpose of the motion information δδ, E(C δδ(N×N) δ T (T×N) ) indicates that C δδ(N×N) δ T (T×N) Seek expectations; In step S12, since the covariance matrix C in formula (3) δδ(N×N) It is a symmetric square matrix, so it can be decomposed into eigenvalues: Csss(N×N)u i =λ i you i (4) In formula (4), u i is the eigenvalue λ i The corresponding eigenvector; According to the matrix eigenvalue property, C δδ(N×N) Can be uniquely decomposed into: In formula (5), Λ (r×r) Represents C δδ(N×N) The r non-zero eigenvalues ​​λ of i ≠0 (i=1,2,...,r) is a diagonal square matrix in descending order, that is: In formula (6), R r×r represents an r×r dimensional real matrix; In formula (3), U r T (r×N) Indicates U r(N×r) The transpose of U r(N×r) represents the r×N dimensional eigenvalue decomposition matrix, U r =[u1,...,u r ], and the following relationship is satisfied: In formula (7), I (r×r) is the r×r dimensional identity matrix; So δ (N×T) Can be uniquely decomposed into: Comparing equations (1) and (7) with equations (2) and (8), we can see that the principal component decomposition result of motion information δ is unique, and the U required by equation (7) is r(N×r) That is, the linear transformation matrix E, U required for the principal component decomposition of motion information δ r T (r×N) δ (N×T) That is, the r unrelated unknown variables η required for the principal component decomposition of motion information δ; Therefore, based on step S11 and step S12, the picture frame of the video image can be processed to obtain the motion information of N pixels in the structure δδ=[δ1,δ2,…δ i ,...,δ N ] T ∈R N×T , where the motion information of each pixel is i (i=1,2,...,N) are all vectors of size 1×T.

2. The structural modal recognition method based on image phase in video stream according to claim 1, characterized in that: In step S13, the motion information δ of point x on the image structure can be expressed by the modal coordinates q(t) as follows: In formula (9), r is the mode number, is the mode shape matrix, q(t)=[q1,q2,...,q r ] T ∈R r×T is the modal coordinate vector, represents the i-th order mode shape, q i (t) (l×T) represents the coordinates of the modal response of the i-th order; Based on vibration theory, the mode shape vector are mutually orthogonal, so: In formula (10), represents the i-th mode shape The transpose of represents the jth mode shape; For civil engineering structures, the modes are sparse, and the first several orders can meet the engineering accuracy requirements. Therefore, the number of pixel motion information δ obtained by the image phase motion estimation method is much higher than the number of modes, that is, N>>r. Since equation (9) is an overcomplete model from high dimension to low dimension, it cannot be directly used for mode extraction. Therefore, principal component analysis is introduced to realize the conversion of data samples from high dimension to low dimension. Comparing equations (1) and (2) with equations (9) and (10), it can be seen that equation (10) is a special case of principal component analysis decomposition. Therefore, the regularized mode vibration shape Φ(x) (N×r) It can be regarded as the linear aliasing matrix E in equation (2), and the modal response matrix q of each order is equal to η in equation (2) (r×T) ,Right now: In formula (11), U r(N×r) T Indicates U r(N×r) The transpose of U r(N×r) represents the r×N dimensional eigenvalue decomposition matrix; After obtaining the modal response q of each order i (t), the frequency corresponding to the peak value of the fast Fourier transform is taken as the natural frequency u r .

3. The structural modal recognition method based on image phase in video stream according to claim 2, characterized in that: In step S22, assuming that I(x+δ) is the motion information δ of point x on the image structure, when the image I(x+δ) is represented by an adjustable pyramid filter, it can be written as the superposition of the filter responses on the adjustable pyramid filter: In formula (12), R ω (x, t) represents the filtering response of the image I(x+δ) on the adjustable pyramid filter of scale ω, that is: R ω (x,t)=ρ ω (x,t)e j(2πux+δ) (13) In formula (13), j is an imaginary unit, ρ ω (x, t) is the corresponding image amplitude, 2πωx+δ is the image phase; After applying the phase motion estimation method to the motion information δ of the video image, the principal component analysis is applied, and the modal response coordinates η = [η1, η2, ..., η r ] T ∈R r×T On this basis, the structural modal response coordinates of the motion information δ are respectively amplified and attenuated: the i-th order modal response coordinate η i (t) is multiplied by the magnification factor α, and the remaining modal response coordinates η l (t)(l≠i) is multiplied by the attenuation factor β=-1 and substituted into equation (11) to obtain the new motion information δ`: In formula (14), u i (x) represents the matrix U r The i-th column of The motion information δ of the video image corresponds to the filter response R on the adjustable pyramid filter ω (x,t) can be written as: In formula (15), u l (x) represents the matrix U r All columns except the i-th column, where l = 1, 2, ... i-1, i+1, ... r; Comparing equations (14) and (15), it can be seen that for the motion information δ, the new motion information δ' is equivalent to the i-th order modal motion in the motion information δ amplified by 1+α times, while removing other order modal motions. Therefore, the new motion information δ' represents the motion that only amplifies the i-th order mode of the structure by 1+α times after removing other order modal motions, thereby realizing the separation of single-order modes; Referring to equation (12), by performing phase processing of equations (14) to (15) on the adjustable pyramid filter of each scale ω, the reconstruction of the new image I`(x+δ`(x,t),t) corresponding to the new motion information δ`(x,t) can be achieved: In formula (16), R ω `(x, t) represents the filter response of the new motion information δ` corresponding to the adjustable pyramid filter; I`(x+δ`,t) is the new motion information δ` that only amplifies the i-th mode of the structure by 1+α times.

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