A method for generating wafer golden map based on statistical iteration

Abstract

A method for generating a wafer golden map based on statistical iteration, which relates to a method for generating a wafer golden map. The present invention aims to solve the problem that the traditional method based on spatial morphological processing performs size analysis on the wafer to be inspected, but the calculation result of this method is easily affected by factors such as scratches and uneven illumination, and the robustness is not high. The steps of the present invention include step 1, analyzing the grain morphology; step 2, using an adaptive multi-scale filtering method to achieve uniform illumination; step 3, performing feature-based registration on the image sequence to achieve image alignment, and using a statistical iteration method to remove singular points in the image sequence; step 4, generating a defect-free wafer golden map. The present invention belongs to the field of image processing technology.

Description

Technical Field

[0001] The invention relates to a method for generating a wafer golden image, and belongs to the technical field of image processing. Background Art

[0002] The current implementation methods of graphical wafer defect detection methods are mainly divided into two categories. One is to use deep learning methods in large sample conditions to achieve one-stop detection and classification of wafer defects. For example, Mask RCNN target detection network, YOLO network and other methods are used to directly realize the location and type recognition of wafer defects. Such methods usually introduce residual networks, attention mechanisms, autoencoders and other methods to improve the performance and versatility of the network. Such wafer defect detection methods based on deep learning generally require massive data sets to ensure the performance of the network model. Some deep learning networks also combine the ideas of meta-learning and transfer learning to fine-tune the models trained on other wafer data sets and apply them to wafer defect detection tasks. However, such methods also require a large number of labeled wafer data sets to support them. Another method is to use image processing methods in small sample conditions to decompose the wafer defect detection task into several subtasks, and solve the wafer defect detection task in a sub-station manner. Such methods can generally be implemented in three steps. The first is a series of preprocessing steps for the image, including filtering and image enhancement. The purpose of this step is to perform appropriate image processing on the wafer to be inspected so that it meets the requirements of defect detection. For example, when using bright field wafer images for defect detection, homomorphic filtering, Retinex filtering and other methods are generally used to filter out the influence of illumination. There are also deep learning methods to build an adaptive filter. For the usual wafer defect detection task, a defect-free wafer golden image template is usually required. Generally, multiple wafer images are used to synthesize a defect-free image. There are also style transfer methods to convert wafer design images into realistic style wafer golden images, but this type of method is only applicable to the case of wafer design images. The second is the identification and positioning of wafer defects. There are deep learning methods to learn the tolerable degree of change between the wafer golden image and the image to be detected, so as to identify the position that exceeds the tolerable degree of change. This type of method also requires a large number of labeled wafer data sets. The use of traditional image difference and other methods to identify defects also has the problem of low robustness, especially the lateral error at the edge of the image structure will greatly reduce the difficulty of defect identification. There are also some defect detection methods based on sparse optical flow, which can better tolerate lateral errors, but this method requires the defect to have a clear corner point structure. For the classification of wafer defects, different defects can be classified by setting various feature thresholds, but this method is more suitable for wafers with a single background such as bare wafers, and the classification accuracy is low for complex backgrounds. The method of using convolutional networks to extract features of wafer defects and then classifying them according to the features is also widely used. It has high accuracy and versatility when there is a large amount of data as training samples. There are also machine learning methods to classify wafer images in different dimensions. These methods can have good classification capabilities in the case of small samples, but their feature extraction capabilities are usually not strong.The traditional method based on spatial morphological processing performs dimensional analysis on the wafer to be inspected. However, the calculation results of this method are easily affected by factors such as scratches and uneven lighting, and are not very robust. Summary of the invention

[0003] In order to solve the problem that the traditional method based on spatial morphological processing performs size analysis on the wafer to be inspected, but the calculation result of this method is easily affected by factors such as scratches and uneven lighting, and has low robustness, a method for generating a wafer golden map based on statistical iteration is proposed.

[0004] The technical solution adopted by the present invention to solve the above-mentioned problem is: the steps of the present invention include:

[0005] Step 1, analyzing the grain morphology;

[0006] Step 2: Adopting the adaptive multi-scale filtering method to achieve the unification of illumination;

[0007] Step 3: Perform feature-based registration on the image sequence to achieve image alignment, and use a statistical iteration method to remove singular points in the image sequence;

[0008] Step 4: Generate a defect-free wafer golden map.

[0009] Furthermore, in step 1, a grain morphology analysis method based on frequency domain analysis is used to analyze the grain morphology, and the specific steps are as follows:

[0010] Step 101, performing frequency domain exchange on the wafer and performing a two-dimensional Fourier transform;

[0011]

[0012] Among them, f(x,y) represents the grayscale value of the input image at (x,y), F(u,v) represents the value of its frequency domain at (u,v), M represents the width of the input image, and N represents the height of the input image;

[0013] Step 102, i.e., transform the image in step 101 as follows:

[0014]

[0015] Among them, F′(u,v) represents the value of the frequency domain at (u,v) after the shift.

[0016] Furthermore, the specific steps of using the adaptive multi-scale filtering method in step 2 to achieve illumination unification are:

[0017] Step 201, performing multi-scale decomposition on the original two-dimensional grayscale wafer image I (x, y), decomposing the original M×N size image into three scales of M×N, 2M×2N, and 1.5M×1.5N;

[0018] Step 202: Perform frequency domain conversion on the wafer image of each scale, and first take the natural logarithm of the image of each scale:

[0019] I log (x,y)=log(I(x,y)+1),

[0020] Perform discrete Fourier transform on the logarithmic transformed wafer image to obtain:

[0021]

[0022] After the image is subjected to discrete Fourier transform, the low-frequency component is moved to the center of the image. For an image of size M×N, the low-frequency component is moved to the center as follows:

[0023] I dft_shift (u,v)=I dft (uM / 2,vn / 2);

[0024] Step 203: Design an adaptive filter for each scale image. In the frequency domain image F(u,v) of each scale, use a local adaptive window to process the low-frequency components. For each scale image, calculate other energy distributions based on its frequency domain graph. The total energy E is:

[0025]

[0026] Arrange the energy in the frequency domain image in ascending order and calculate its cumulative energy:

[0027]

[0028] Where E sorted Represents the descending order of energy, C(i) represents the cumulative energy from 0 to i in descending order. When the cumulative energy reaches 90% of the total energy, the frequency value D0 at this position is used as the adaptive cutoff frequency of the scale image, and the Gaussian filter is constructed with this frequency:

[0029]

[0030] The frequency domain images of each scale are filtered according to the above filters:

[0031] I filtered_dft_shift (u,v)=I dft_shift (u,v)·H(u,v),

[0032] Perform an inverse frequency shift operation on the filtered image:

[0033] I filtered_dft (u,v)=I filtered_dft_shift (M-u+1,N-v+1),

[0034] Perform a two-dimensional inverse discrete Fourier transform on the frequency domain image after inverse frequency shift:

[0035]

[0036] Perform an inverse logarithmic transform on the image after inverse Fourier transformation to obtain images after adaptive filtering of each size:

[0037]

[0038] Step 204: Combine the adaptively filtered images of all scales into a final adjusted image.

[0039] Furthermore, in step 204, the weighted summation method is used to combine the images of all scales after adaptive filtering into the final adjusted image: for the images of three different scales after adaptive filtering, for each scale i, a weight w is defined i , ensuring that the sum of the weights of all scales is 1, that is: Each scaled image is resized to the same size as the original image by bicubic interpolation. i Represents the i-th scale image after resizing, and the final fused image I f It can be calculated by the following weighted summation formula:

[0040]

[0041] Furthermore, in step 3, the specific steps of performing feature-based registration on the image sequence to achieve image alignment are:

[0042] Step 301: For two images I1 and I2, I2 needs to be aligned to the coordinate system of I1. First, the two images are grayed, and then the feature points in I1 and I2 are extracted by the Hessian algorithm.

[0043] Step 302: For each feature point, calculate its SURF descriptor to represent the local features of the point; the SURF descriptor is formed by calculating the Haar wavelet response, the direction histogram and the feature vector of the local image block around the feature point. For a 4×4 sub-region, it is divided into 16 small sub-blocks, each of which has a gradient amplitude and direction; the SURF descriptor is a 128-dimensional vector formed by connecting these 16 amplitudes and directions in a certain order;

[0044] D=[d1,d2,...,d 128 ],

[0045] where d i Represents the vector consisting of the gradient magnitude and direction within the i-th sub-block;

[0046] Step 303: Match the feature points of the two images to be matched one by one, find all the corresponding feature point pairs in the two images, and use the nearest neighbor search method based on the kd tree;

[0047] Step 304: Use the Random Sample Consensus algorithm to remove feature points that were incorrectly matched in the previous feature matching process; the algorithm randomly selects a set of feature point pairs, calculates the transformation matrix between them, and uses the transformation matrix to transform the feature points in I2 to I1; then counts the distance between the transformed feature points and the feature points in I1, and determines which feature points are correctly matched based on the distance; repeat this process multiple times, and finally retain the correctly matched feature points;

[0048] Step 305: Estimation of transformation matrix: Based on the correctly matched feature point pairs, the least squares algorithm is used to estimate the affine transformation matrix.

[0049] Step 306: Use the transformation matrix estimated in the previous step to align I2 to the coordinate system of I1, thereby achieving image registration.

[0050] Furthermore, in step 3, the steps of removing singular points in the image sequence by using the statistical iteration method are as follows:

[0051] Step 1: Input image sequence;

[0052] Step 2: Solve the vertical statistics of each coordinate;

[0053] Step 3: Create a coordinate corresponding vertical storage table;

[0054] Step 4: traverse the linked list;

[0055] Step 5: Determine whether the i-th node meets the screening condition. If so, update the node and statistics and return to step 4. If not, execute step 6.

[0056] Step 6: Determine whether the longitudinal statistics have converged. If so, the process ends. If not, the process returns to step 4.

[0057] Among them, for the pixel at the coordinate point (r, c) in the image, if the gray value f of the pixel at the corresponding position in the i-th image i (r,c) satisfies the following conditions: The i-th position of the linked list in the image sequence can be considered as the location of the singular point. In this case, it is only necessary to delete the singular point and update the horizontal statistics of the point in the manner mentioned above.

[0058] The beneficial effects of the present invention are as follows: the present invention takes into account both the local details and the overall structure of the image, designs an adaptive multi-scale filtering method, and compares it with the homomorphic filtering algorithm and the Retinex algorithm; finally, the image sequence after the adaptive multi-scale filtering is aligned based on feature registration, the image sequences are pixel-aligned, and then a method based on statistical iteration is used to remove singular points in the grayscale sequence of each point in the image sequence, and a wafer golden map is generated using wafer data that does not contain singular points. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 It is a schematic diagram of the wafer to be inspected and its central area. Figure 1 a is the image of the wafer to be inspected, Figure 1 b is a schematic diagram of the central area;

[0060] Figure 2 is the wafer spectrum diagram, Figure 2 a is the wafer spectrum diagram, Figure 2 b is the three-dimensional diagram of the wafer spectrum;

[0061] Figure 3 It is the adaptive multi-scale filtering flow chart;

[0062] Figure 4 are the images in the image sequence before filtering;

[0063] Figure 5 are the images in the filtered image sequence;

[0064] Figure 6 This is the effect diagram of Retinex filtering, homomorphic filtering and adaptive multi-scale filtering. Figure 6 a is the original image, Figure 6 b is a schematic diagram of Retinex filtering. Figure 6 c is a schematic diagram of homomorphic filtering, Figure 6 d is a schematic diagram of adaptive multi-scale filtering;

[0065] Figure 7 are the images in the image sequence after registration;

[0066] Figure 8 It is a flow chart of the threshold screening algorithm based on statistics;

[0067] Fig. 9 This is a flow chart of the threshold screening algorithm based on statistics. Fig. 9 a is a schematic diagram of the statistics of a certain position in the image sequence, Fig. 9 b is the wafer gold map. DETAILED DESCRIPTION

[0068] Specific implementation method 1: Combination Figures 1 to 9 To illustrate this embodiment, the steps of a method for generating a wafer golden map based on statistical iteration described in this embodiment include:

[0069] Step 1, analyzing the grain morphology;

[0070] Step 2: Adopting the adaptive multi-scale filtering method to achieve the unification of illumination;

[0071] Step 3: Perform feature-based registration on the image sequence to achieve image alignment, and use a statistical iteration method to remove singular points in the image sequence;

[0072] Step 4: Generate a defect-free wafer golden map.

[0073] Specific implementation method 2: Combination Figures 1 to 9 To illustrate this embodiment, in step 1 of a method for generating a wafer golden map based on statistical iteration described in this embodiment, a grain morphology analysis method based on frequency domain analysis is used to analyze the grain morphology, and the specific steps are as follows:

[0074] Step 101, performing frequency domain exchange on the wafer and performing a two-dimensional Fourier transform;

[0075]

[0076] Among them, f(x,y) represents the grayscale value of the input image at (x,y), F(u,v) represents the value of its frequency domain at (u,v), M represents the width of the input image, and N represents the height of the input image;

[0077] Step 102, i.e., transform the image in step 101 as follows:

[0078]

[0079] Among them, F′(u,v) represents the value of the frequency domain at (u,v) after the shift.

[0080] Specific implementation method three: Combination Figures 1 to 9 To illustrate this embodiment, the specific steps of using the adaptive multi-scale filtering method to achieve illumination unification in step 2 of a method for generating a wafer golden map based on statistical iteration described in this embodiment are as follows:

[0081] Step 201, performing multi-scale decomposition on the original two-dimensional grayscale wafer image I (x, y), decomposing the original M×N size image into three scales of M×N, 2M×2N, and 1.5M×1.5N;

[0082] Step 202: Perform frequency domain conversion on the wafer image of each scale, and first take the natural logarithm of the image of each scale:

[0083] I log (x,y)=log(I(x,y)+1),

[0084] Perform discrete Fourier transform on the logarithmic transformed wafer image to obtain:

[0085]

[0086] After the image is subjected to discrete Fourier transform, the low-frequency component is moved to the center of the image. For an image of size M×N, the low-frequency component is moved to the center as follows:

[0087] I dft_shift (u,v)=I dft (uM / 2,vn / 2);

[0088] Step 203: Design an adaptive filter for each scale image. In the frequency domain image F(u,v) of each scale, use a local adaptive window to process the low-frequency components. For each scale image, calculate other energy distributions based on its frequency domain graph. The total energy E is:

[0089]

[0090] Arrange the energy in the frequency domain image in ascending order and calculate its cumulative energy:

[0091]

[0092] Where E sorted Represents the descending order of energy, C(i) represents the cumulative energy from 0 to i in descending order. When the cumulative energy reaches 90% of the total energy, the frequency value D0 at this position is used as the adaptive cutoff frequency of the scale image, and the Gaussian filter is constructed with this frequency:

[0093]

[0094] The frequency domain images of each scale are filtered according to the above filters:

[0095] I filtered_dft_shift (u,v)=I dft_shift (u,v)·H(u,v),

[0096] Perform an inverse frequency shift operation on the filtered image:

[0097] I filtered_dft (u,v)=I filtered_dft_shift (M-u+1,N-v+1),

[0098] Perform a two-dimensional inverse discrete Fourier transform on the frequency domain image after inverse frequency shift:

[0099]

[0100] Perform an inverse logarithmic transform on the image after inverse Fourier transformation to obtain images after adaptive filtering of each size:

[0101]

[0102] Step 204: Combine the adaptively filtered images of all scales into a final adjusted image.

[0103] Specific implementation method four: Combination Figures 1 to 9 In step 204 of the method for generating a wafer golden map based on statistical iteration described in this embodiment, a weighted summation method is used to combine the images of all scales after adaptive filtering into the final adjusted image: for the images of three different scales after adaptive filtering, for each scale i, a weight w is defined i , ensuring that the sum of the weights of all scales is 1, that is: Each scaled image is resized to the same size as the original image by bicubic interpolation. i Represents the i-th scale image after resizing, and the final fused image I f It can be calculated by the following weighted summation formula:

[0104]

[0105] Specific implementation method five: Combination Figures 1 to 9 To illustrate this embodiment, the specific steps of performing feature-based registration on the image sequence to achieve image alignment in step 3 of a method for generating a wafer golden map based on statistical iteration described in this embodiment are:

[0106] Step 301: For two images I1 and I2, I2 needs to be aligned to the coordinate system of I1. First, the two images are grayed, and then the feature points in I1 and I2 are extracted by the Hessian algorithm.

[0107] Step 302: For each feature point, calculate its SURF descriptor to represent the local features of the point; the SURF descriptor is formed by calculating the Haar wavelet response, the direction histogram and the feature vector of the local image block around the feature point. For a 4×4 sub-region, it is divided into 16 small sub-blocks, each of which has a gradient amplitude and direction; the SURF descriptor is a 128-dimensional vector formed by connecting these 16 amplitudes and directions in a certain order;

[0108] D=[d1,d2,...,d 128 ],

[0109] where d i Represents the vector consisting of the gradient magnitude and direction within the i-th sub-block;

[0110] Step 303: Match the feature points of the two images to be matched one by one, find all the corresponding feature point pairs in the two images, and use the nearest neighbor search method based on the kd tree;

[0111] Step 304: Use the Random Sample Consensus algorithm to remove feature points that were incorrectly matched in the previous feature matching process; the algorithm randomly selects a set of feature point pairs, calculates the transformation matrix between them, and uses the transformation matrix to transform the feature points in I2 to I1; then counts the distance between the transformed feature points and the feature points in I1, and determines which feature points are correctly matched based on the distance; repeat this process multiple times, and finally retain the correctly matched feature points;

[0112] Step 305: Estimation of transformation matrix: Based on the correctly matched feature point pairs, the least squares algorithm is used to estimate the affine transformation matrix.

[0113] Step 306: Use the transformation matrix estimated in the previous step to align I2 to the coordinate system of I1, thereby achieving image registration.

[0114] Specific implementation method six: Combination Figures 1 to 9 To illustrate this embodiment, in step 3 of the method for generating a wafer golden map based on statistical iteration described in this embodiment, the step of removing singular points in the image sequence by using the statistical iteration method is as follows:

[0115] Step 1: Input image sequence;

[0116] Step 2: Solve the vertical statistics of each coordinate;

[0117] Step 3: Create a coordinate corresponding vertical storage table;

[0118] Step 4: traverse the linked list;

[0119] Step 5: Determine whether the i-th node meets the screening condition. If so, update the node and statistics and return to step 4. If not, execute step 6.

[0120] Step 6: Determine whether the longitudinal statistics have converged. If so, the process ends. If not, the process returns to step 4.

[0121] Among them, for the pixel at the coordinate point (r, c) in the image, if the gray value f of the pixel at the corresponding position in the i-th image i (r,c) satisfies the following conditions: The i-th position of the linked list in the image sequence can be considered as the location of the singular point. In this case, it is only necessary to delete the singular point and update the horizontal statistics of the point in the manner mentioned above.

[0122] The above is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement made to the above embodiments without departing from the content of the technical solution of the present invention, based on the technical essence of the present invention, within the spirit and principles of the present invention, still fall within the protection scope of the technical solution of the present invention.

Claims

1. A method for generating a wafer golden map based on statistical iteration, characterized in that: The steps of the method for generating a wafer golden map based on statistical iteration include: Step 1: Analyze the grain morphology; specifically include: Step 101, performing frequency domain exchange on the wafer and performing a two-dimensional Fourier transform; Among them, f(x,y) represents the grayscale value of the input image at (x,y), F(u,v) represents the value of its frequency domain at (u,v), M represents the width of the input image, and N represents the height of the input image; Step 102, i.e., transform the image in step 101 as follows: Among them, F′(u,v) represents the value of the frequency domain at (u,v) after the shift; Step 2: Adopting the adaptive multi-scale filtering method to achieve the uniformity of illumination; specifically including: Step 201, performing multi-scale decomposition on the original two-dimensional grayscale wafer image I (x, y), decomposing the original M×N size image into three scales of M×N, 2M×2N, and 1.5M×1.5N; Step 202: Perform frequency domain conversion on the wafer image of each scale, and first take the natural logarithm of the image of each scale: I log (x,y)=log(I(x,y)+1), Perform discrete Fourier transform on the logarithmic transformed wafer image to obtain: After the image is subjected to discrete Fourier transform, the low-frequency component is moved to the center of the image. For an image of size M×N, the low-frequency component is moved to the center as follows: I dft_shift (u,v)=I dft (u-M / 2,v-N / 2); Step 203: Design an adaptive filter for each scale image. In the frequency domain image F(u,v) of each scale, use a local adaptive window to process the low-frequency components. For each scale image, calculate other energy distributions based on its frequency domain graph. The total energy E is: Arrange the energy in the frequency domain image in descending order and calculate its cumulative energy: Where E sorted Represents the descending order of energy, C(i) represents the cumulative energy from 0 to i in descending order. When the cumulative energy reaches 90% of the total energy, the frequency value D0 at this position is used as the adaptive cutoff frequency of the scale image, and a Gaussian filter is constructed at this frequency: The frequency domain images of each scale are filtered according to the above filters: I filtered_dft_shift (u,v)=I dft_shift (u,v) H(u,v), Perform an inverse frequency shift operation on the filtered image: I filtered_dft (u,v)=I filtered_dft_shift (M-u+1,N-v+1), Perform a two-dimensional inverse discrete Fourier transform on the frequency domain image after inverse frequency shift: Perform an inverse logarithmic transform on the image after inverse Fourier transformation to obtain images after adaptive filtering of each size: Step 204: Combine the images of all scales after adaptive filtering into a final adjusted image; Use weighted summation to combine the images of all scales after adaptive filtering into a final adjusted image: For the images of three different scales after adaptive filtering, for each scale i, define a weight w i , ensuring that the sum of the weights of all scales is 1, that is: Each scaled image is resized to the same size as the original image by bicubic interpolation. i Represents the i-th scale image after resizing, and the final fused image I f It can be calculated by the following weighted summation formula: Step 3: Perform feature-based registration on the image sequence to achieve image alignment, and use a statistical iteration method to remove singular points in the image sequence; Step 4: Generate a defect-free wafer golden map.

2. The method for generating a wafer golden map based on statistical iteration according to claim 1, characterized in that: The specific steps of performing feature-based registration on the image sequence in step 3 to achieve image alignment are: Step 301: For two images I1 and I2, I2 needs to be aligned to the coordinate system of I1. First, the two images are grayed, and then the feature points in I1 and I2 are extracted by the Hessian algorithm. Step 302: For each feature point, calculate its SURF descriptor to represent the local features of the point; The SURF descriptor is constructed by calculating the Haar wavelet response, directional histogram and feature vector of the local image block around the feature point. For a 4×4 sub-region, it is divided into 16 small sub-blocks, each of which has a gradient amplitude and direction. The SURF descriptor is a 128-dimensional vector formed by connecting these 16 amplitudes and directions in a certain order. D=[d1,d2,...,d 128 ], where d i Represents the vector consisting of the gradient magnitude and direction within the i-th sub-block; Step 303: Match the feature points of the two images to be matched one by one, find all the corresponding feature point pairs in the two images, and use the nearest neighbor search method based on the kd tree; Step 304: Use the Random Sample Consensus algorithm to remove feature points that were incorrectly matched in the previous feature matching process; the algorithm randomly selects a set of feature point pairs, calculates the transformation matrix between them, and uses the transformation matrix to transform the feature points in I2 to I1; then counts the distance between the transformed feature points and the feature points in I1, and determines which feature points are correctly matched based on the distance; repeat this process multiple times, and finally retain the correctly matched feature points; Step 305: estimating the transformation matrix, estimating the affine transformation matrix using the least squares algorithm based on the correctly matched feature point pairs; Step 306: Use the transformation matrix estimated in the previous step to align I2 to the coordinate system of I1, thereby achieving image registration.

3. The method for generating a wafer golden map based on statistical iteration according to claim 1, characterized in that: In step 3, the steps of removing singular points in the image sequence by using the statistical iteration method are as follows: Step 1: Input image sequence; Step 2: Solve the vertical statistics of each coordinate; Step 3: Create a coordinate corresponding vertical storage table; Step 4: traverse the linked list; Step 5: Determine whether the i-th node meets the screening condition. If so, update the node and statistics and return to step 4. If not, execute step 6. Step 6: Determine whether the longitudinal statistics have converged. If so, the process ends. If not, the process returns to step 4. Among them, for the pixel at the coordinate point (r, c) in the image, if the gray value f of the pixel at the corresponding position in the i-th image i (r,c) satisfies the following conditions: The i-th position of the linked list in the image sequence can be considered as the location of the singular point. In this case, it is only necessary to delete the singular point and update the longitudinal statistics of the point.

Images