Structural vibration measurement method based on dual-tree complex wavelet transform and empirical central wavelength

Abstract

The present application relates to a structure vibration measurement method based on dual-tree complex wavelet transform and empirical central wavelength, and belongs to the technical field of structure health monitoring. The directional complex wavelet basis of dual-tree complex wavelet transform is used to obtain the amplitude-phase decoupling representation under multi-resolution in the region of interest. The amplitude and phase are calculated to obtain the amplitude weight and phase stability weight in different scales and different directions, and the time sequence energy weight is introduced to form the comprehensive weight. The weighted phase difference is obtained. The weighted phase difference and the comprehensive weight are up-sampled to the pixel level by using the bilinear interpolation, and the effective pixels are obtained after screening. The empirical central wavelength is introduced to extract the vibration displacement from the effective pixels combined with the weighted phase difference, and the vibration frequency of the structure is obtained by the fast Fourier transform. The present application effectively overcomes the limitations of phase wrapping and filter parameter mismatch of the traditional phase method, and has good application prospect in practical engineering.

Description

Technical Field

[0001] This invention relates to the field of structural health monitoring technology, specifically to a method for measuring structural vibration based on dual-tree complex wavelet transform and empirical center wavelength. Background Technology

[0002] Monitoring structural vibration plays a crucial role in assessing the condition and safety of infrastructure such as bridges and buildings. Vibration response provides a wealth of information about dynamic characteristics, such as natural frequencies and mode shapes, which are highly sensitive to changes in structural stiffness, mass distribution, and support conditions. By analyzing changes in structural parameters, engineers can detect anomalies that indicate damage or degradation. This dynamic analysis helps in early detection of damage, optimizing maintenance strategies, thereby improving structural resilience, reducing life-cycle costs, and preventing catastrophic accidents.

[0003] Currently, structural vibration measurement methods are mainly divided into two types: contact and non-contact. Contact methods acquire structural responses by installing sensors such as accelerometers and strain gauges at specific locations on the structure. However, this method suffers from drawbacks such as high cost, difficulty in sensor installation, and limited measurement points, making it difficult to meet the needs of real-time monitoring of structural dynamic responses. To overcome the shortcomings of contact methods, researchers have proposed several non-contact methods, such as Laser Doppler Vibrometer (LDV), microwave radar, and computer vision (CV)-based methods. While LDV and microwave radar offer high accuracy, their equipment is expensive and complex to operate. CV-based methods offer advantages such as low cost, simple operation, and full-field measurement, effectively replacing LDV and microwave radar. CV techniques mainly include intensity methods and phase methods. Intensity methods acquire structural vibration information by analyzing the dynamic changes in image grayscale distribution and pixel intensity. Mainstream methods include optical flow (OF), template matching algorithms, and digital image correlation (DIC). However, under conditions of strong noise or varying lighting, the gray-level gradient of the image becomes unstable, easily leading to error accumulation and affecting the robustness of the algorithm. In contrast, the phase method is insensitive to changes in image gray-level, effectively overcoming the aforementioned shortcomings of the intensity method. This method extracts the local phase of the image through wavelet transform (WT) or filtering decomposition, and then uses the phase information to obtain the vibration displacement of the structure. Furthermore, for the measurement of minute vibrations of structures, the phase method has better measurement accuracy than the intensity method. The phase method mainly includes two types: phase-based motion magnification (PMM) and phase-based motion estimation (PVE). Despite the advantages of the phase method, it still faces two major challenges in practical applications. One is the phase unwrapping problem: phase information is usually wrapped in... Within the specified interval, the signal needs to be unwrapped to recover a continuous signal, a process susceptible to noise, singularities, and occlusion. Secondly, there is the issue of filter parameter mismatch: most existing filters lack translation invariance and require manual parameter setting. When dealing with complex structural textures or scenarios with minute displacements, they are prone to phase information distortion or incomplete capture of directional features, often necessitating the use of targets to improve accuracy. Therefore, improving the identification accuracy of structural vibration measurements is a pressing issue that needs to be addressed. Summary of the Invention

[0004] The purpose of this invention is to provide a structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength to solve the problems mentioned in the background art.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution: a structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength, comprising the following steps: For the region of interest, a directional complex wavelet basis of dual-tree complex wavelet transform is used to obtain an amplitude-phase decoupled representation under multiple resolutions, and the phase information is extracted from it. Based on the obtained amplitude and phase, intra-frame amplitude weights and phase stability weights at different scales and in different directions are calculated, and temporal energy weights are introduced to form a comprehensive weight; The phase difference is calculated based on the acquired phase information and combined with the comprehensive weight to obtain the weighted phase difference; Bilinear interpolation is used to upsample the weighted phase difference and the comprehensive weight to the pixel level. Each pixel obtains a corresponding scale weight based on the amplitude energy of each scale. The comprehensive weight and the scale weight are then fused and compared with a preset effective pixel threshold to select effective pixels. By introducing the empirical center wavelength and combining it with the weighted phase difference, the vibration displacement of the structure is extracted from the effective pixels, and the vibration frequency of the structure is obtained by fast Fourier transform.

[0006] Preferably, the dual-tree complex wavelet transform employs two real wavelet transform filter banks for parallel processing, extracting the real and imaginary parts of the complex wavelet coefficients respectively, using the four-quadrant arctangent function to calculate and extract phase information, and determining the corresponding amplitude.

[0007] Preferably, when the phase difference exceeds the range of values... At that time, the phase difference is mapped to a continuous interval: ; in, This represents the phase difference in the t-th frame. To decompose the scale, For the direction of 2D wavelets, This indicates the position of the subband coefficients, and mod represents the modulo operation.

[0008] Preferably, the intra-frame amplitude weight, phase stability weight, and temporal energy weight are weighted and summed to obtain the comprehensive weight, wherein the temporal energy weight... for: ; ; ; in, The sub-band position in frame t grayscale value at that location Sub-band position in image sequence The average gray value at that location, To prevent small constants from being divided by zero.

[0009] Preferably, the first frame is used as the reference frame, the phase difference of each frame is obtained, and the phase difference is multiplied by the normalized comprehensive weight to obtain the initial weighted phase difference. The average value of the initial weighted phase differences in multiple directions is used as the weighted phase difference.

[0010] Preferably, the mapping relationship between the sub-band coefficients and the pixel blocks of the original image in the region of interest is established by rounding operations; For any pixel in the original image of the region of interest, at scale In the sub-band, find the positions of the four nearest sub-band coefficients to construct the interpolation function; Bilinear interpolation is performed based on the interpolation function and the weights of the target pixel coordinates in the horizontal and vertical directions, and the weighted phase difference and the comprehensive weight are both upsampled to the pixel level.

[0011] Preferably, the overall weight and the scale weight are combined as the confidence level for each pixel. : ; in, For scale weights, The overall weights after upsampling are... J represents the pixel coordinates; J represents the total scale of the decomposition. The filtering includes: Set the effective pixel threshold: ; Filter to obtain valid pixels: ; in, This represents the Pth percentile.

[0012] Preferably, the complex domain gradient method is used for calculation. scale, Directional complex coefficients Complex phase gradient ; Based on complex phase gradient Calculate the spatial wavelength; Calculate the median of the spatial wavelength in all directions and use the median as... The empirical center wavelength of the scale.

[0013] Preferably, amplitude is introduced. The empirical center wavelength is obtained by filtering and calculating the median. : ; .

[0014] Preferably, the weighted phase difference at each scale is converted into pixel displacement using the empirical center wavelength at each scale: ; We fuse the displacements at different scales using a weighted average to obtain the pixel displacement of frame t: ; Vibration displacement calculated based on effective pixels: ; After removing the linear baseline drift from the displacement signal, the detrended vibration displacement is obtained: ; in, Weighted phase difference; This is a linear detrending operator.

[0015] Beneficial effects: This invention utilizes the translation invariance and multi-scale, multi-directional decomposition characteristics of dual-tree complex wavelet transform, and calculates a comprehensive weight using time-series energy weight, intra-frame amplitude weight, and phase stability weight to obtain a weighted phase difference. After upsampling to the pixel level and combining with effective pixel filtering, the extraction accuracy of phase information is improved. It effectively overcomes the limitations of phase entanglement and filter parameter mismatch in traditional phase methods, and has good application prospects in practical engineering. Attached Figure Description

[0016] Figure 1 This is a flowchart of the structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength in this invention; Figure 2 This is a schematic diagram of the multi-scale decomposition based on dual-tree complex wavelet transform in this invention; Figure 3 This is a schematic diagram of the wavelet kernel for the dual-tree complex wavelet transform in this invention; Figure 4 This is a schematic diagram of the complex wavelet kernel of the dual-tree complex wavelet transform in this invention; Figure 5 This is a schematic diagram illustrating the mapping from sub-band phase difference to the original pixel in this invention; Figure 6 This is a schematic diagram of the measuring point setup for the main beam in an embodiment of the present invention; Figure 7 This is a phase diagram of the suspension bridge in an embodiment of the present invention; Figure 8 This is a amplitude diagram of the suspension bridge in an embodiment of the present invention; Figure 9 This is a diagram showing the vibration measurement results of the main beam in an embodiment of the present invention. Detailed Implementation

[0017] To make the objectives and advantages of this invention clearer, the invention will be specifically described below with reference to embodiments. It should be understood that the following text is merely used to describe one or more specific embodiments of the invention and does not strictly limit the scope of protection specifically claimed by the invention.

[0018] Example: A structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength, referenced Figure 1 As shown, it includes the following steps: Step 1, Phase extraction based on dual-tree complex wavelet transform: The directional complex wavelet basis of dual-tree complex wavelet transform is used to obtain the amplitude-phase decoupled representation under multiple resolution for the region of interest, and the phase information is extracted from it. Traditional Discrete Wavelet Transform (DWT) lacks translation invariance. When analyzing minute structural vibrations, this deficiency leads to significant fluctuations in the energy distribution of wavelet coefficients across different scales, affecting the stability of phase extraction. Furthermore, DWT can only provide high-frequency subbands in the horizontal, vertical, and diagonal (45° and 135°) directions, offering limited directional selectivity and making it difficult to fully capture the complex texture features of structural surfaces. To address these issues, a dual-tree complex wavelet transform is introduced. This transform not only possesses the time-frequency locality of wavelet transform but also translation invariance and selectivity in six directions (15°, 45°, 75°, 105°, 135°, 165°). It can reflect changes in the image at different scales and in different directions, thus obtaining more accurate phase information from the image.

[0019] In this embodiment, the dual-tree complex wavelet transform employs two real wavelet transform filter banks (tree A and tree B) in parallel to extract the real and imaginary parts of the complex wavelet coefficients, respectively. During the decomposition process, tree A is used to process the real part, and its low-pass and high-pass filters are denoted as follows: and wavelet basis is Tree B is used to process the imaginary part, and its low-pass and high-pass filters are denoted as follows: and wavelet basis is ; In this embodiment, to suppress frequency aliasing and phase distortion, the wavelet functions of tree A and tree B need to form a Hilbert transform (HT) pair; the coefficients of each sub-band domain after the dual-tree complex wavelet transform decomposition are... .

[0020] when At this time, the low-pass filter of tree B is a half-sampled shift of tree A. The wavelet bases of the two trees then form an approximate 2D HT pair, i.e.: ; in, For 2D HT; refer to Figure 2 As shown, 'a' represents the low-frequency coefficients (i.e., approximate subband coefficients) of each layer, and 'd' represents the high-frequency coefficients (i.e., detail subband coefficients) of each layer. Let the 2D image be... , where x = 1,2,…m, y = 1,2,…n, and m and n are the number of rows and columns of pixels in the image, respectively. The values ​​represent the pixel coordinates of the image. After performing a J-scale 2D dual-tree complex wavelet transform decomposition on the image, the coefficients of each sub-band are obtained; the positions of the sub-band coefficients are... The corresponding spatial domain has a size of The pixel blocks (j is the decomposition scale); perform 2D dual-tree complex wavelet transform decomposition on the t-th frame of the video at scale J.

[0021] In this embodiment, tree A uses a real wavelet basis. Multi-scale decomposition of the image yields real wavelet coefficients at different scales and in different directions. for: ; in, This represents the inner product operation; t is the t-th frame of the video; j is the decomposition scale, where a larger j corresponds to low-frequency information in the image, and a smaller j corresponds to high-frequency information in the image. The directions of the 2D wavelet are defined as follows: In this embodiment, the 2D dual-tree complex wavelet transform includes 6 directions (i.e., 15°, 45°, 75°, 105°, 135°, 165°), which can cover the global features of the image. Representing the j-th scale of tree A, 2D scaling and translation wavelet basis in the direction, and its corresponding wavelet kernel, such as Figure 3 As shown in (a) of the diagram.

[0022] exist Figure 3 In (a), LL is the low-frequency approximate subband, LH and HL are the horizontal and vertical detail subbands, respectively, and HH is the diagonal detail subband; The expression is, ; in, Here is the rotation transformation matrix; and This refers to the position translation parameter.

[0023] Tree B uses a real wavelet basis that forms an HT pair with tree A. Images at the same scale and orientation Decomposition yields the imaginary wavelet coefficients. for: ; in, The scale, orientation, and translation parameters of tree A Completely identical, except that the wavelet basis functions satisfy the HT relation, and their corresponding wavelet kernels are as follows: Figure 3 As shown in (b) of the diagram.

[0024] Using the same scale j and the same direction same location By combining the real wavelet coefficients of tree A with the imaginary wavelet coefficients of tree B, the complex wavelet coefficients of the dual-tree complex wavelet transform can be obtained. The corresponding complex wavelet kernel is as follows: Figure 4 As shown, (+) and (-) are the positive and negative responses of the complex wavelet kernel, respectively.

[0025] Complex wavelet coefficients It is the information carrier for phase extraction, that is: ; conjugate for: ; Where i is the imaginary unit.

[0026] The phase of the complex wavelet coefficients directly reflects the phase shift of the image at the corresponding scale, direction, and position; in one embodiment, to avoid quadrant ambiguity of the arctangent function, a four-quadrant arctangent function is used. Calculate the phase; in the t-th frame image, scale j, Complex coefficients of direction phase for: ; in, For the j-th frame image, The edge offset or texture direction information, its value range is: It can fully characterize the phase information of an image.

[0027] Amplitude The calculation formula is: ; Using frame 1 as the reference frame, the phase difference of frame t for: ; Since the range of phase values ​​is When the phase difference exceeds this range, phase blurring will occur. To maintain phase continuity, in one embodiment, the following processing method is introduced to map the phase difference to a continuous interval, and its calculation formula is as follows: ; Here, mod represents the modulo operation.

[0028] Step 2, Effective Pixel Filtering Based on Comprehensive Weight and Scale Weight: In this embodiment, after obtaining phase information, an effective pixel filtering strategy based on comprehensive weight and scale weight is introduced, including: First, calculating intra-frame amplitude weight and phase stability weight at different scales and directions based on the obtained amplitude and phase, and introducing temporal energy weight to form a comprehensive weight. Then, calculating the phase difference based on the obtained phase information and combining it with the comprehensive weight to obtain a weighted phase difference, thereby achieving zero-phase difference processing for noise signals; Next, upsampling the weighted phase difference and comprehensive weight at the pixel level through bilinear interpolation, uniformly mapping them to the same size as the spatial domain, thus achieving a one-to-one correspondence between detail sub-band features in each direction and image pixels; Finally, each pixel obtains a corresponding assigned scale weight based on the amplitude energy at each scale, and after fusing the comprehensive weight and scale weight, comparing it with a preset effective pixel threshold, effective pixels are filtered out; specifically as follows: The grayscale value of a moving target fluctuates significantly over time, and its temporal energy... The image background grayscale value fluctuates less over time, resulting in lower temporal energy; temporal energy weights vary at different scales and orientations. Represented as: ; in, , , Let be the grayscale value at position (u, v) in the subband of frame t. This represents the average gray value at position (u, v) in the sub-band of the image sequence. To prevent division by zero of small constants, this embodiment... ; Amplitude of complex coefficients in dual-tree complex wavelet transform This reflects the strength of local texture energy at a pixel; the larger the coefficient amplitude, the more reliable the phase information at the corresponding location. Intra-frame amplitude weights vary at different scales and in different directions. for: ; The spatial gradient of the phase difference of moving targets is small, while the spatial gradient of the phase difference of noisy pixels is large; phase stability weights at different scales and in different directions. for: ; in, and For gradient operators; The sub-band position in frame t The phase difference gradient in the horizontal direction at the j-th scale; The sub-band position in frame t The phase difference gradient in the vertical direction at the j-th scale; The combined weight is obtained by weighting and summing the three weights mentioned above. for: ; The constraints are Normalized composite weight for: ; Weighted phase differences at different scales and in different directions are obtained by multiplying the phase difference by the normalized comprehensive weight. : ; In this embodiment, the average of the weighted phase differences in the six directions is calculated to obtain the weighted phase difference of the t-th frame image at scale j: ; Vibration signals are spatially continuous, and pixel-based filtering can maintain the spatial correlation of multi-scale information, which is beneficial for subsequent displacement transformation and wavelength estimation. In contrast, phase difference filtering destroys the spatial structure and forms a circular dependency with wavelength estimation, leading to unstable results. Therefore, this invention maps the weighted phase difference and comprehensive weight from the sub-band domain to the pixel domain through upsampling to achieve pixel-level vibration signal extraction. The specific steps are as follows: After extracting the phase using dual-tree complex wavelet transform, the phase difference Included Within the sub-band of the direction, the size is ,Right now: ; in, The rounding up symbol; In this embodiment, the size of the region of interest is set to W×H, and the sub-band phase difference is mapped to the pixel region of the original image. Since the dual-tree complex wavelet transform uses binary downsampling, different scales correspond to different resolutions (i.e., the larger the scale, the lower the resolution); the position at scale j... The subband coefficients at that location correspond to a set of pixel domains in the original image. rather than a single pixel: ; in, The subband coefficients correspond to a region size of approximately [missing information - likely a rounding down sign] in the region of interest. Pixel; In this embodiment, bilinear interpolation is used to upsample the sub-band phase difference to the pixel space, and the sub-band position is... The weighted phase difference at point is: ; For any position At point j, the interpolation function is constructed by finding the four nearest coefficient positions in the subband of scale j: ; in, , , and This is the index variable for the sub-band coefficients.

[0029] The bilinear interpolation formula is: ; Where i and k are index variables; The weight of the target pixel's x-coordinate in the horizontal direction reflects the relationship between pixel x and the sub-band index. Spatial distance; The weight of the target pixel's y-coordinate in the vertical direction reflects the relationship between the pixel's y-coordinate and the sub-band index. Spatial distance; Sub-band position The weighted phase difference at each point can be mapped to each pixel through upsampling; each subband Corresponding to the original region of interest Pixel region, when j=1, each Corresponding to the 2×2 pixel region in the original region of interest, such as Figure 5 As shown; The weighting function is: ; in, and These are the weights of two adjacent coefficients in the subband; The coordinates of the target pixel can be represented as x or y; Map the target pixel to the position and index of the sub-band space The offset.

[0030] The upsampling process can be represented using linear operators as follows: ; in, It is a bilinear upsampling operator.

[0031] To ensure pixel-level alignment of phase difference and weights, the overall weights also need to be upsampled, i.e.: ; After upsampling, for each pixel : ; The scale weights are obtained based on the amplitude energy at each scale. for: ; in, |ROI| represents the total number of pixels in the region of interest; The combined comprehensive weight and scale weight are used as the confidence score for each pixel. : ; Set the effective pixel threshold: ; in, This represents the Pth percentile.

[0032] Once the threshold is determined, the effective pixels can be represented as: ; By following the steps above, high-confidence effective pixels can be obtained; Step 3: Structural vibration measurement based on empirical center wavelength: Introducing the empirical center wavelength, combining it with weighted phase difference to extract vibration displacement from effective pixels, and obtaining the vibration frequency of the structure through fast Fourier transform; specifically including: Calculate using the complex field gradient method scale, Directional complex coefficients Complex phase gradient ; ; in, This refers to partial differential operations; To avoid the problems caused by directly differentiating the phase For jump problems, the complex domain gradient method is used, i.e.: ; in, ; For imaginary part operators; Local spatial frequency The gradient magnitude can be represented as: ; in, It is the Euclidean norm.

[0033] Spatial wavelength for: ; In one embodiment, an amplitude is introduced. filter: ; In this embodiment, the median wavelength is calculated in six directions within the measurement area and used as the empirical center wavelength at scale j, i.e.: ; Using the empirical center wavelength at each scale, the weighted phase difference at the corresponding scale is converted into pixel displacement, i.e.: ; in, For the t-th frame, at scale j, the pixel point Displacement at; Corresponding to a complete wavelength .

[0034] We fuse the displacements at different scales using a weighted average, resulting in the pixel displacement of frame t: ; Vibration displacement calculated based on effective pixels: ; After removing the linear baseline drift from the displacement signal, the detrended vibration displacement is obtained: ; in, For linear detrending operators After obtaining the vibration displacement of the structure, the vibration frequency of the structure is obtained by using Fast Fourier Transform (FFT).

[0035] This application also provides a specific embodiment that comprehensively analyzes the effectiveness and superiority of the method provided above; the following settings are made: Taking a long-span suspension bridge as an example, the vibration of the main girder was measured using the method proposed in this invention. A fixed camera with a resolution of 1920×1080 pixels and a sampling frequency of 30 Hz was used to record the vibration video of the main girder. Based on the location of the lighting equipment on the main girder, 51 measuring points were selected at equal intervals along the lower edge of the main girder, arranged as follows: Figure 6 As shown, the complex wavelet coefficients of the suspension bridge at four scales and six directions (i.e., 15°, 45°, 75°, 105°, 135°, 165°) are obtained using dual-tree complex wavelet transform, and the corresponding phase and amplitude are extracted, as shown. Figure 7 and Figure 8 As shown; from Figure 7 and Figure 8It can be seen that the horizontal texture of the suspension bridge is mainly concentrated in the sub-bands in the 15° and 165° directions, the vertical texture is mainly concentrated in the sub-bands in the 75° and 105° directions, and the diagonal texture is mainly concentrated in the sub-bands in the 45° and 135° directions.

[0036] To verify the accuracy of the main beam frequency obtained using this invention, measurement points 16, 29, and 41 were analyzed. The texture of the regions where these three measurement points are located is concentrated in the horizontal direction, and its phase is mainly contained in the sub-bands at 15° and 165°. However, the sub-band at 165° contains yellow spots, which may introduce noise interference; therefore, the phase of the sub-band at 15° was selected for displacement identification. The displacement and frequency of the three measurement points are as follows: Figure 9 As shown; from Figure 9 It can be seen that the frequency of the main beam identified by displacement at three different measuring points is 0.3684 Hz, which verifies the applicability of the proposed method.

[0037] To further quantify the frequency identification accuracy of the proposed method, the main beam frequency obtained using the present invention was compared with the reference value, as shown in Table 1. Table 1 Comparison of main beam frequencies obtained by different methods

[0038] As shown in Table 1, the main beam frequency obtained using this invention matches the reference value well, with an error within 0.60%. The results indicate that this invention has high frequency identification accuracy in practical engineering applications and provides a simple, widely applicable target-free method for structural vibration measurement.

[0039] The embodiments of the present invention have been described in detail above with reference to the examples. However, the present invention is not limited to the above embodiments. For those skilled in the art, after learning the contents described in the present invention, several equivalent changes and substitutions can be made without departing from the principle of the present invention. These equivalent changes and substitutions should also be considered to fall within the protection scope of the present invention.

Claims

1. A structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength, characterized in that: Includes the following steps: For the region of interest, a directional complex wavelet basis of dual-tree complex wavelet transform is used to obtain an amplitude-phase decoupled representation under multiple resolutions, and the phase information is extracted from it. Based on the acquired amplitude and phase, intra-frame amplitude weights and phase stability weights at different scales and in different directions are calculated, and temporal energy weights are introduced to form a comprehensive weight. The phase difference is calculated based on the acquired phase information and combined with the comprehensive weight to obtain the weighted phase difference; Bilinear interpolation is used to upsample the weighted phase difference and the comprehensive weight to the pixel level. Each pixel obtains a corresponding scale weight based on the amplitude energy of each scale. The comprehensive weight and the scale weight are then fused and compared with a preset effective pixel threshold to select effective pixels. By introducing the empirical center wavelength and combining it with the weighted phase difference, the vibration displacement of the structure is extracted from the effective pixels, and the vibration frequency of the structure is obtained by fast Fourier transform.

2. The structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength according to claim 1, characterized in that: The dual-tree complex wavelet transform employs two real wavelet transform filter banks for parallel processing, extracting the real and imaginary parts of the complex wavelet coefficients respectively. The phase information is then calculated and extracted using the four-quadrant arctangent function, and the corresponding amplitude is determined.

3. The structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength according to claim 2, characterized in that: When the phase difference exceeds the range of values At that time, the phase difference is mapped to a continuous interval: ； in, This represents the phase difference in the t-th frame. To decompose the scale, For the direction of 2D wavelets, This indicates the position of the subband coefficients, and mod represents the modulo operation.

4. The structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength according to claim 3, characterized in that: The comprehensive weight is obtained by weighting and summing the intra-frame amplitude weight, phase stability weight, and temporal energy weight, wherein the temporal energy weight... for: ； ； ； in, The sub-band position in frame t grayscale value at that location Sub-band position in image sequence The average gray value at that location, To prevent small constants from being divided by zero.

5. The structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength according to claim 4, characterized in that: Using the first frame as the reference frame, the phase difference of each frame is obtained, and the initial weighted phase difference is obtained by multiplying the phase difference with the normalized comprehensive weight. The average value of the initial weighted phase differences in multiple directions is used as the weighted phase difference.

6. The structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength according to claim 1, characterized in that: The mapping relationship between sub-band coefficients and pixel blocks of the original image in the region of interest is established by rounding operations; For any pixel in the original image of the region of interest, at scale In the sub-band, find the positions of the four nearest sub-band coefficients to construct the interpolation function; Bilinear interpolation is performed based on the interpolation function and the weights of the target pixel coordinates in the horizontal and vertical directions, and the weighted phase difference and the comprehensive weight are both upsampled to the pixel level.

7. The structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength according to claim 6, characterized in that: The combined comprehensive weight and scale weight are used as the confidence level for each pixel. : ； in, For scale weights, The overall weights after upsampling are... J represents the pixel coordinates; J represents the total scale of the decomposition. The filtering includes: Set the effective pixel threshold: ； Filter to obtain valid pixels: ； in, This represents the Pth percentile.

8. The structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength according to claim 7, characterized in that: Calculate using the complex field gradient method scale, Directional complex coefficients Complex phase gradient ; Based on complex phase gradient Calculate the spatial wavelength; Calculate the median of the spatial wavelength in all directions and use the median as... The empirical center wavelength of the scale.

9. The structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength according to claim 8, characterized in that: Introducing amplitude The empirical center wavelength is obtained by filtering and calculating the median. : ； 。 10. The structural vibration measurement method based on dual-tree complex wavelet transform and empirical center wavelength according to claim 9, characterized in that: Using the empirical center wavelength at each scale, the weighted phase difference at the corresponding scale is converted into pixel displacement: ； We fuse the displacements at different scales using a weighted average to obtain the pixel displacement of frame t: ； Vibration displacement calculated based on effective pixels: ； After removing the linear baseline drift from the displacement signal, the detrended vibration displacement is obtained: ； in, Weighted phase difference; This is a linear detrending operator.

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