White light interference signal denoising method based on VMD combined wavelet threshold improvement

Abstract

The invention discloses a white light interference signal denoising method based on VMD combined wavelet threshold improvement. The method comprises the following steps: decomposing a white light interference signal f by using variational mode decomposition to obtain k intrinsic mode components; solving a correlation coefficient Rj of the obtained intrinsic mode component and the original noise-containing interference signal, and finding a maximum correlation coefficient Rmax; a useful component and a noise component are distinguished by comparing the size relation between the correlation coefficient Rj and Rmax / k of each intrinsic mode component; selecting a wavelet basis, a wavelet decomposition layer number and an improved threshold function according to the characteristics of the current actual white light interference signal; carrying out improved wavelet threshold noise reduction processing on the noise component to obtain an intrinsic mode component after noise reduction; and performing signal reconstruction on the intrinsic mode component and the useful component after noise reduction to obtain a denoised interference signal, and performing surface topography reconstruction by using the denoised interference signal to identify the fine defects of the device. According to the invention, the distortion degree of the interference signal is reduced, the denoising performance is improved, and the interference signal with high signal-to-noise ratio and low root-mean-square error is obtained.

Description

Technical Field

[0001] The invention relates to the technical field of white light interference signal denoising, and in particular to a white light interference signal denoising method based on VMD (variational mode decomposition) and a combined improved wavelet threshold. Background Art

[0002] White Light Interferometry (WLI) technology has unique advantages in measuring discontinuous surfaces, such as micro-electromechanical system components and step height components. However, WLI technology is very sensitive to external interference, which will affect the key information in the interference image, resulting in inaccurate measurement results. Various interference sources such as environmental noise, machine noise, electronic noise and optical noise will affect the accuracy and reliability of WLI measurement results. These noises come from vibrations, temperature changes, air flow, etc. in the laboratory or production environment, or from mechanical components, electronic components and optical components in the WLI system. These noises can cause instability or drift of the measurement signal, seriously affecting the high accuracy and reliability of WLI measurement results. Therefore, effective denoising methods and techniques are needed to reduce or eliminate the impact of these noises.

[0003] In practical applications, WLI technology is combined with denoising methods to ensure the reliability of measurement results. To this end, researchers have proposed a variety of denoising methods, such as windowed Fourier denoising, wavelet threshold denoising and Bayesian denoising, which can be combined with traditional zero optical path difference positioning algorithms to improve measurement accuracy. However, the parameter settings of these methods depend on the specific measurement environment. Some researchers have tried to introduce deep learning technology to compensate for the original measurement results, but this requires a lot of experimental data and training time. Some researchers have proposed using a gap matching algorithm that fully inherits empirical mode decomposition to optimize the measurement results, but this method may introduce additional noise. Therefore, developing an efficient denoising method is of great significance to improving measurement accuracy. Summary of the invention

[0004] The present invention provides a white light interference signal denoising method based on VMD combined with improved wavelet threshold. The present invention reduces the distortion of the interference signal, improves the denoising performance, and obtains an interference signal with a high signal-to-noise ratio and a low root mean square error. The details are described below:

[0005] A VMD combined with improved wavelet threshold white light interference signal denoising method, the method comprising:

[0006] The white light interference signal f is decomposed into k intrinsic modal components using variational mode decomposition; the correlation coefficient R between the obtained intrinsic modal components and the original noisy interference signal is solved. j , and find the maximum correlation coefficient R max;

[0007] By comparing the correlation coefficient R of each intrinsic modal component j With R max / k size relationship, distinguishing useful components from noise components;

[0008] According to the characteristics of the current actual white light interference signal, the wavelet basis, the number of wavelet decomposition layers and the improved threshold function are selected; the noise component is subjected to improved wavelet threshold denoising to obtain the denoised intrinsic mode component;

[0009] The denoised intrinsic modal components and useful components are reconstructed to obtain denoised interference signals, which are then used to reconstruct the surface morphology and identify subtle defects of the device.

[0010] Among them, the correlation coefficient R j for:

[0011]

[0012] In the formula, cov(f,IMF j ) indicates that the IMF component and the original interference signal f(z) are covariances, D(f) and D(IMF j ) represent the variance of the original interference signal f(z) and the IMF component, respectively.

[0013] The method for distinguishing useful components from noise components is as follows:

[0014] k=max(R j ) / k

[0015] R j The natural modal components with a value less than k are set as useful components, and R j The natural modal components of >k are set as noise components.

[0016] The noise component is subjected to improved wavelet threshold denoising processing as follows:

[0017] Biorthogonal wavelet basis function and 5 wavelet decomposition layers are used to perform wavelet decomposition on the noise components respectively, and 5 layers of high-frequency wavelet coefficients and 1 layer of low-frequency wavelet coefficients are obtained; Stein's unbiased risk estimation principle is used to select the adaptive threshold to improve the wavelet threshold;

[0018] And all high-frequency wavelet coefficients are processed in combination with the improved wavelet threshold function; the processed high-frequency wavelet coefficients and low-frequency wavelet coefficients are used to reconstruct the signal, that is, through the inverse wavelet transform, the noise component after the improved wavelet threshold denoising is obtained.

[0019] Wherein, the improved threshold function is:

[0020] SURE(λ)=σ 2 -2σ 2 λ+λ 2

[0021] Among them, λ is the threshold, σ 2 is the variance of the signal, SURE(λ) is Stein’s unbiased risk estimator;

[0022] The optimal threshold

[0023]

[0024] The optimal threshold for the wavelet threshold is selected using Stein's unbiased risk estimation principle.

[0025] The beneficial effects of the technical solution provided by the present invention are:

[0026] 1. The present invention combines VMD decomposition with an improved wavelet threshold denoising method to obtain an effective white light interference signal denoising method, which can effectively retain the envelope profile characteristics of the interference signal and improve the quality of the white light interference signal;

[0027] 2. The present invention improves the signal-to-noise ratio of the white light interference signal after noise reduction processing and reduces the root mean square error, so that the surface morphology obtained during white light interference measurement is closer to the true value, and can be effectively applied to many fields such as semiconductor detection and optical component detection;

[0028] 3. This method is more adaptive and robust than the traditional wavelet threshold denoising method and can be extended to different signal processing fields, such as audio signals, image signals, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 This is a flowchart of a white light interference signal denoising method based on VMD combined with improved wavelet threshold;

[0030] Figure 2 is a schematic diagram of a simulated white light interference signal;

[0031] Among them, (a) is a schematic diagram of the interference signal without noise; (b) is a schematic diagram of the interference signal containing Gaussian white noise.

[0032] Figure 3 Schematic diagram of the IMF (intrinsic modal component) component of the interference signal containing Gaussian white noise after VMD decomposition;

[0033] Figure 4 This is the interference signal image after VMD joint improved wavelet threshold denoising;

[0034] Figure 5is the interference signal diagram after other denoising methods;

[0035] Among them, (a) is the improved wavelet threshold denoising interference signal image; (b) is the VMD partial reconstruction denoising interference signal image. DETAILED DESCRIPTION

[0036] In order to make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention are described in further detail below.

[0037] like Figure 1 As shown, the embodiment of the present invention provides a white light interference signal denoising method using VMD combined with improved wavelet threshold, which starts with a noisy interference signal f as input and finally outputs a denoised interference signal. Specifically, the following steps are included:

[0038] Step 1: Use a CCD camera to collect interference image data, perform variational modal decomposition on the interference signal f, and obtain k inherent modal components;

[0039] In WLI, the captured white light interference signal can be expressed by the following formula:

[0040]

[0041] Where I0 is the background intensity, h0 is the surface height, λ0 is the equivalent central wavelength of the white light source, is the initial phase of reflection, z is the scanning position, and g is usually the envelope of the Gaussian distribution.

[0042] In the time domain, the coherent signal is sinusoidally modulated. In practical applications, the coherent signal is usually interfered by various noises, and its expression can be written as:

[0043] f(z)= I(z)+e(z) (2)

[0044] Among them, e(z) is regarded as random noise. At this time, the envelope of the interference signal is distorted, which seriously affects the extraction of the peak position. Therefore, denoising is very necessary to improve the accuracy of surface topography measurement.

[0045] The noisy interference signal f(z) is the input signal of the variational mode decomposition VMD. The constrained variational model is constructed and solved as follows:

[0046]

[0047] In the formula, when D reaches the minimum value, the optimal IMF can be determined, u k ={u1,u2,…,u k},ω k ={ω1,ω2,…,ω k} are all IMF modes and their corresponding center frequencies, δ(z) represents the unit pulse function, is the gradient operator with respect to the spatial sequence z, f(z) is the interference signal to be decomposed, and k is the sequence number of the decomposed IMF.

[0048] From equation (3), it can be concluded that VMD is used to decompose the input signal f(z) into multiple sub-signals u k , the superposition of these IMFs constitutes the original signal. The above variational problem is to find the optimal solution of equation (3) through the quadratic penalty function and the Lagrange operator. The above constrained variational problem is transformed into an unconstrained variational problem to find the optimal solution. Introduce the Lagrange multiplier operator:

[0049]

[0050] In the formula, α is the bandwidth parameter and λ is the Lagrange multiplier operator. By introducing the alternating direction method of multipliers (ADMM), the decomposition process is realized in the frequency domain, as shown in formula (5):

[0051]

[0052] in, represents the spectrum of the IMF mode, represents the spectrum of the original signal, and n is the number of iterations. In order to update the Lagrange multiplier, the paired ascent method is used in the iterative process of solving the variational model. The center frequency and bandwidth of each IMF component are continuously updated until the convergence condition is reached, as shown in formula (6):

[0053]

[0054] Where τ is the parameter of noise tolerance, ε is the discrimination accuracy, and ε>0.

[0055] Step 2: Calculate the correlation coefficient R between the acquired intrinsic modal component and the original noisy interference signal j , and find the maximum correlation coefficient R max ;

[0056] The jth intrinsic mode component IMF j Correlation coefficient R with the original interference signal f(z) j The calculation process is as follows:

[0057]

[0058] In the formula, cov(f,IMF j ) indicates that the IMF component and the original interference signal f(z) are covariances, D(f) and D(IMF j ) represent the variance of the original interference signal f(z) and the IMF component, respectively.

[0059] Step 3: By comparing the correlation coefficient R of each intrinsic modal component j With R max / k size relationship, distinguishing useful components from noise components;

[0060] The noise modal component and the effective modal component are determined by distinguishing the correlation coefficient through the demarcation formula:

[0061] κ=max(R j ) / k (8)

[0062] R j The natural modal components with a value less than k are set as useful components, and R j The natural mode components of >k are set as noise components.

[0063] Step 4: According to the characteristics of the current actual white light interference signal, the wavelet basis, the wavelet decomposition layer number and the improved threshold function are selected; the above noise component is subjected to improved wavelet threshold denoising to obtain the denoised intrinsic mode component;

[0064] Biorthogonal wavelet basis function and 5-layer wavelet decomposition levels are used to perform wavelet decomposition on the noise components, and 5 layers of high-frequency wavelet coefficients and 1 layer of low-frequency wavelet coefficients are obtained. Then, Stein's unbiased risk estimation principle (SURE) is used to select the adaptive threshold to improve the wavelet threshold. All high-frequency wavelet coefficients are processed in combination with the improved wavelet threshold function. The processed high-frequency wavelet coefficients and low-frequency wavelet coefficients are used to reconstruct the signal, that is, through inverse wavelet transform, the intrinsic mode components after improved wavelet threshold denoising are obtained.

[0065] Furthermore, the improved wavelet threshold calculation formula is as follows:

[0066] SURE(λ)=σ 2 -2σ 2 λ+λ 2 (9)

[0067] Among them, λ is the threshold, σ 2 is the variance of the signal, SURE(λ) is Stein’s unbiased risk estimate. The goal is to find the optimal threshold that minimizes the SURE estimate

[0068]

[0069] By using Stein's unbiased risk estimation principle, the optimal threshold of the wavelet threshold can be adaptively selected, thereby improving the denoising performance and noise resistance.

[0070] Step 5: reconstruct the denoised intrinsic modal components and useful components to obtain a denoised interference signal, wherein the reconstructed signal is a denoised interference signal with an improved signal-to-noise ratio and a reduced noise level;

[0071] Step 6. In the field of semiconductor inspection, using the denoised interference signal to reconstruct the surface morphology can more accurately identify subtle defects on the wafer surface and effectively improve inspection accuracy and reliability.

[0072] In specific applications, the above-mentioned denoised interference signal can also be applied to optical component detection, which can more accurately identify subtle defects in optical components and effectively improve detection accuracy and reliability.

[0073] In order to prove the effectiveness and superiority of the method of the present invention, relevant simulation test experiments were carried out. The signal-to-noise ratio (SNR) and root mean square error (RMSE) evaluation indicators of the reconstructed signal under each denoising method were calculated and compared.

[0074] Embodiment 1:

[0075] Simulation conditions: The simulation provided in this embodiment is performed in a hardware environment of AMD Ryzen 7 5800H with a main frequency of 3.2 GHz, a memory of 16.0 GB, and a software environment of MATLAB R2018b.

[0076] Simulation content: The experiment provided in this embodiment is to compare the denoising effects of the VMD combined with improved wavelet threshold denoising method, improved wavelet threshold function denoising method, and VMD partial reconstruction denoising method in this paper.

[0077] First, construct a white light interference simulation signal: Figure 2 are simulated white light interference signals, where (a) is the interference signal without noise; (b) is the interference signal with Gaussian white noise added, and the signal-to-noise ratio (SNR) is 25 dB.

[0078] Simulation experiment results and analysis:

[0079] First, perform VMD decomposition on the simulated interference signal containing Gaussian white noise to obtain 9 IMF components, such as Figure 3 As shown;

[0080] The correlation coefficients of each IMF component and the original noisy interference signal are calculated, which are 0.062, 0.092, 0.113, 0.629, 0.710, 0.534, 0.115, 0.090, and 0.101 respectively; the maximum correlation coefficient is 0.710, and the discrimination value is 0.079; through the discrimination conditions, the useful component is: IMF1, and the noise components are IMF2, IMF3, IMF4, IMF5, IMF6, IMF7, IMF8, and IMF9 for subsequent processing.

[0081] The screened noise components are subjected to improved wavelet threshold denoising, and the improved wavelet threshold selection adopted uses Stein's unbiased risk estimation (SURE). The SURE principle can provide an unbiased risk estimation and automatically select the threshold without manual adjustment, making the threshold selection more accurate and reliable. Traditional wavelet threshold methods usually use fixed thresholds or experience-based threshold selection methods, which may not be adaptive and robust. Using the SURE principle to select adaptive thresholds can overcome these shortcomings.

[0082] Set the wavelet basis function to Biorthogonal, the wavelet decomposition layer number to 5, and perform improved wavelet threshold denoising on the noise component to obtain the noise component after denoising. The processed noise component and the useful component are superimposed to complete the reconstruction of the denoised signal. Figure 4 This is the simulated interference signal after denoising by VMD and improved wavelet threshold function.

[0083] In order to further evaluate the denoising effect of the method of the present invention, the signal-to-noise ratio and root mean square error are selected as interference signal quality evaluation indicators for analysis. The interference signal containing noise is subjected to VMD combined with improved wavelet threshold function denoising, improved wavelet threshold denoising, and VMD partial reconstruction denoising method. Among them, the improved wavelet threshold denoising method is: taking the noisy interference signal as input, directly subjecting it to the above-mentioned improved wavelet threshold function processing to obtain the denoised signal; the VMD partial reconstruction denoising method is: after VMD decomposition of the noisy interference signal, remove the first three high-frequency IMF components, and use the remaining IMF components to reconstruct the signal to obtain the denoised signal.

[0084] Table 1 is a comparison of the denoising effects of the VMD combined with improved wavelet threshold denoising method, the improved wavelet threshold denoising method and the VMD partial reconstruction denoising method. The larger the SNR value, the smaller the RMSE value, indicating that the denoising effect of the algorithm is better. From the data in the table, it can be concluded that the denoising method proposed by the present invention is better than the other two methods. Figure 5 (a) is the signal image after denoising by improved wavelet threshold denoising method. Figure 5 (b) is the signal diagram after denoising by VMD partial reconstruction denoising method.

[0085] Table 1 Comparison of denoising indicators of three denoising methods

[0086]

[0087] In summary, the embodiment of the present invention realizes denoising of white light interference signals through the method of VMD combined with improved wavelet threshold function. First, the VMD algorithm decomposes the signal into multiple intrinsic modal components IMF, and then applies the improved wavelet threshold algorithm to the noisy IMF for denoising. The improved wavelet threshold algorithm uses Stein's unbiased risk estimation to adaptively select the threshold, thereby ensuring the best denoising performance. This method can effectively remove noise and interference in white light interference signals, improve the signal-to-noise ratio and accuracy of the signal, has good robustness and practicality, and can be applied to the field of optical precision measurement.

[0088] Those skilled in the art will appreciate that the accompanying drawing is only a schematic diagram of a preferred embodiment, and the serial numbers of the embodiments of the present invention are only for description and do not represent the advantages or disadvantages of the embodiments.

[0089] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A white light interference signal denoising method based on VMD combined with improved wavelet threshold, characterized in that: The method comprises: The white light interference signal f is decomposed using variational mode decomposition to obtain k intrinsic mode components; Solve the correlation coefficient R between the acquired intrinsic modal component and the original noisy interference signal j , and find the maximum correlation coefficient R max ; By comparing the correlation coefficient R of each intrinsic modal component j With R max / k size relationship, distinguishing useful components from noise components; According to the characteristics of the current actual white light interference signal, the wavelet basis, the number of wavelet decomposition layers and the improved threshold function are selected; the noise component is subjected to improved wavelet threshold denoising to obtain the denoised intrinsic mode component; The denoised intrinsic modal components and useful components are reconstructed to obtain denoised interference signals, which are then used to reconstruct the surface morphology and identify subtle defects of the device.

2. The method for denoising white light interference signals by VMD combined with improved wavelet threshold according to claim 1, characterized in that: The correlation coefficient R j for: In the formula, cov(f,IMF j ) indicates that the IMF component and the original interference signal f(z) are covariances, D(f) and D(IMF j ) represent the variance of the original interference signal f(z) and the IMF component, respectively.

3. The method for denoising white light interference signals by VMD combined with improved wavelet threshold according to claim 1, characterized in that: The distinction between useful components and noise components is: κ=max(R j ) / k R j The natural modal components with a value less than k are set as useful components, and R j The natural mode components of >k are set as noise components.

4. The method for denoising white light interference signals by VMD combined with improved wavelet threshold according to claim 1, characterized in that: The noise component is subjected to improved wavelet threshold denoising processing as follows: Biorthogonal wavelet basis function and 5 wavelet decomposition layers are used to perform wavelet decomposition on the noise components respectively, and 5 layers of high-frequency wavelet coefficients and 1 layer of low-frequency wavelet coefficients are obtained; Stein's unbiased risk estimation principle is used to select the adaptive threshold to improve the wavelet threshold; And combined with the improved wavelet threshold function, all high-frequency wavelet coefficients are processed; The processed high-frequency wavelet coefficients and low-frequency wavelet coefficients are used to reconstruct the signal, that is, through inverse wavelet transform, the noise component after improved wavelet threshold denoising is obtained.

5. The method for denoising white light interference signals by VMD combined with improved wavelet threshold according to claim 1, characterized in that: The improved threshold function is: SURE(λ)=σ 2 -2s 2 λ+λ 2 Among them, λ is the threshold, σ 2 is the variance of the signal, SURE(λ) is Stein’s unbiased risk estimator; The optimal threshold The optimal threshold for the wavelet threshold is selected using Stein's unbiased risk estimation principle.

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