CN113255799A - Atlas clustering method for mask characteristic graph - Google Patents

Atlas clustering method for mask characteristic graph Download PDF

Info

Publication number
CN113255799A
CN113255799A CN202110614308.5A CN202110614308A CN113255799A CN 113255799 A CN113255799 A CN 113255799A CN 202110614308 A CN202110614308 A CN 202110614308A CN 113255799 A CN113255799 A CN 113255799A
Authority
CN
China
Prior art keywords
matrix
clustering
mask
elements
similarity
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Granted
Application number
CN202110614308.5A
Other languages
Chinese (zh)
Other versions
CN113255799B (en
Inventor
马旭
张胜恩
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Beijing Institute of Technology BIT
Original Assignee
Beijing Institute of Technology BIT
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Beijing Institute of Technology BIT filed Critical Beijing Institute of Technology BIT
Priority to CN202110614308.5A priority Critical patent/CN113255799B/en
Publication of CN113255799A publication Critical patent/CN113255799A/en
Application granted granted Critical
Publication of CN113255799B publication Critical patent/CN113255799B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F18/00—Pattern recognition
    • G06F18/20—Analysing
    • G06F18/23—Clustering techniques
    • G06F18/232—Non-hierarchical techniques
    • G06F18/2323—Non-hierarchical techniques based on graph theory, e.g. minimum spanning trees [MST] or graph cuts
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F18/00—Pattern recognition
    • G06F18/20—Analysing
    • G06F18/23—Clustering techniques
    • G06F18/232—Non-hierarchical techniques
    • G06F18/2321—Non-hierarchical techniques using statistics or function optimisation, e.g. modelling of probability density functions
    • G06F18/23213—Non-hierarchical techniques using statistics or function optimisation, e.g. modelling of probability density functions with fixed number of clusters, e.g. K-means clustering

Landscapes

  • Engineering & Computer Science (AREA)
  • Data Mining & Analysis (AREA)
  • Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • General Engineering & Computer Science (AREA)
  • Mathematical Physics (AREA)
  • Bioinformatics & Cheminformatics (AREA)
  • Pure & Applied Mathematics (AREA)
  • Evolutionary Biology (AREA)
  • Evolutionary Computation (AREA)
  • Bioinformatics & Computational Biology (AREA)
  • Artificial Intelligence (AREA)
  • Life Sciences & Earth Sciences (AREA)
  • Computer Vision & Pattern Recognition (AREA)
  • Computational Mathematics (AREA)
  • Mathematical Optimization (AREA)
  • Mathematical Analysis (AREA)
  • Computing Systems (AREA)
  • Probability & Statistics with Applications (AREA)
  • Discrete Mathematics (AREA)
  • Algebra (AREA)
  • Databases & Information Systems (AREA)
  • Software Systems (AREA)
  • Image Analysis (AREA)

Abstract

The invention provides a atlas clustering method of a mask characteristic graph, which is characterized in that only part of elements of a similar matrix of the atlas clustering method are directly calculated from a mask characteristic graph set, but not all elements of the similar matrix are directly calculated, and then the rest unknown elements in the similar matrix are quickly supplemented by a low-rank matrix supplementing method with the minimum division Brazilianman-SCp norm, so that the supplementing error of the similar matrix is reduced, the calculating efficiency of the similar matrix is improved, and the calculating speed of the whole mask characteristic graph clustering process is further improved; therefore, the method applies the matrix completion method to the processes of mask characteristic pattern similarity matrix fast calculation and mask characteristic pattern spectral clustering for the first time, greatly improves the calculation efficiency and accuracy of mask characteristic pattern spectral clustering, reduces the calculation time, and can keep higher classification accuracy and reconstruction accuracy under a certain sampling rate.

Description

Atlas clustering method for mask characteristic graph
Technical Field
The invention belongs to the field of computational lithography, mask classification and graph signal processing, and particularly relates to a graph clustering method for mask characteristic graphs.
Background
In the current era of electrification, digitalization and networking, electronic equipment plays a very important role in life and work. Chips are the core of many electronic devices, and breakthroughs in chip manufacturing technology have led to the development of the semiconductor industry. In the chip manufacturing process, the photolithography process is an important step. Optical lithography is based on the principle of projection imaging, in which a mask layout printed with an electronic circuit is exposed to a light source, so that a circuit pattern on the mask is transferred onto a wafer coated with photoresist, thereby forming an etched electronic circuit pattern on the wafer.
In a lithographic process, a mask is an important optical element as a template. Since the line width of the mask pattern is very fine, a large number of regularly shaped geometric figures, called features, are distributed in the mask layout, and these features constitute the basic cells of the electronic circuit. The geometric figures on the mask can be divided into a plurality of types according to different characteristics of space structures, frequency domain distribution and the like, and mask characteristic figures of the same type have relatively similar space structure characteristics or frequency spectrum distribution characteristics. In some important lithography optimization processes, such as the light source mask joint optimization process, it is necessary to cluster a plurality of mask feature patterns into different groups, and to pick out a representative feature pattern in each group, which is called a representative feature pattern. These representative feature patterns contain features common to all feature patterns in the group for use in subsequent optimization processes.
Spectral clustering is a graph signal-based clustering method, can be used for clustering on sample spaces with any shapes, and can be converged to a global optimal solution, so that the method is suitable for the classification problem of mask feature graphs. And spectral clustering calculates the Laplace matrix of the graph through the similar matrix, further calculates the characteristic column vectors corresponding to the first K minimum characteristic values, forms the K characteristic column vectors into a matrix, and finally clusters all K dimensional row vectors of the matrix.
The first step in spectral clustering of mask feature patterns is to construct a graph structure containing all the feature patterns and to compute a similarity matrix for the graph structure. The conventional method is to calculate the elements in the similarity matrix one by one, i.e. each mask feature needs to calculate the similarity with all other mask features, and then put them into the similarity matrix. However, when the full mask includes a large number of feature pattern structures, the corresponding pattern signals also include a large number of vertices, so that the process of calculating the similarity matrix of the feature pattern structures introduces a large amount of calculation, and the operation time of the spectral clustering process is increased.
In summary, it is necessary to provide a fast mask pattern clustering method, which improves the completion accuracy of the mask pattern similarity matrix and the efficiency and accuracy of mask pattern spectrum clustering, aiming at the characteristic of large mask pattern data volume.
Disclosure of Invention
In order to solve the problems, the invention provides a map clustering method of mask characteristic graphs, which applies a matrix completion method to the mask characteristic graph map clustering for the first time, converts the matrix completion problem into the SCp norm minimization problem, solves the problem by utilizing a split Brazilian algorithm, and greatly improves the efficiency and the accuracy of the mask characteristic graph map clustering.
A method for clustering a mask feature pattern atlas includes the following steps:
s1: obtaining similar matrixes corresponding to N two-dimensional mask characteristic graphs to be classified
Figure BDA0003097402180000021
Wherein the similarity matrix
Figure BDA0003097402180000022
The system is composed of known elements and unknown elements;
s2: similarity matrix pair by split-based Brageman-SCp norm minimization method
Figure BDA0003097402180000023
Make up and calculate
Figure BDA0003097402180000024
To obtain a similarity matrix with all known elements
Figure BDA0003097402180000025
S3: obtaining a similarity matrix
Figure BDA0003097402180000031
Corresponding normalized Laplace matrix L', and forming eigenvectors corresponding to the first K minimum eigenvalues of the normalized Laplace matrix LAnd clustering the N row vectors of the feature matrix, wherein the clustering result corresponds to the classification result of the N two-dimensional mask feature patterns.
Further, the similarity matrix
Figure BDA0003097402180000032
The completion method comprises the following steps:
s201: the following objective function is constructed based on the split Brazilian method:
Figure BDA0003097402180000033
wherein ρ is a weight coefficient, C is an auxiliary variable matrix, | | · | | computationally |, C is a vectorSCpFor the SCp norm, the exponent p e (0, 1)]X is an optimized variable matrix, PΩ(. DEG) represents an orthogonal projection operator, omega is a set of positions of known elements, | | ·| purple windFIs Frobenius norm, and lambda is residual error term coefficient;
s202: an iterative equation set for obtaining an optimized variable matrix X, an auxiliary variable matrix C and a residual B from an objective function by adopting a split Brahman iterative algorithm is as follows:
Figure BDA0003097402180000034
wherein k represents the number of iterations, and the residual B represents the residual between the optimized variable matrix X and the auxiliary variable matrix C;
s203: continuously carrying out iterative computation on the optimized variable matrix X, the auxiliary variable matrix C and the residual error B according to the iterative equation set until a set iteration termination condition is met, wherein the finally obtained optimized variable matrix is a complete similar matrix
Figure BDA0003097402180000035
Further, for the optimized variable matrix X: if position (i, j) ∈ Ω, PΩ(X) is defined as PΩ(Xi,j)＝Xi,jWhether or notThen P isΩ(Xi,j) 0; for similarity matrix
Figure BDA0003097402180000041
If the position (i, j) ∈ Ω, then
Figure BDA0003097402180000042
Is defined as
Figure BDA0003097402180000043
Otherwise
Figure BDA0003097402180000044
Further, the SCp norm is defined as
Figure BDA0003097402180000045
Wherein σiIs the ith large singular value of X, tau > 0 is the threshold, and the exponent p epsilon (0, 1)]。
Further, the iteration termination condition is that the iteration number reaches a set upper limit value or
Figure BDA0003097402180000046
Less than a given margin of error.
Further, the similarity matrix
Figure BDA0003097402180000047
The acquisition method comprises the following steps:
s101: determining similarity matrices
Figure BDA0003097402180000048
Number N of known elementss＝ceil(q·N2) Wherein q is a similarity matrix
Figure BDA00030974021800000412
Ceil (-) is an upward rounding function;
s102: determining a set omega of positions of the known elements, wherein the acquisition mode of the set omega is as follows:
assume a blue noise template of
Figure BDA0003097402180000049
All 1 matrices are
Figure BDA00030974021800000410
And E (i, j) and T (i, j) are the elements at position (i, j) in the full 1 matrix E and the blue noise template T, respectively, i, j equals 1, 2.
Sequentially judging whether the positions (i, j) meet alpha E (i, j) + beta is more than or equal to T (i, j), wherein alpha and beta are parameters for controlling the number of the positions, if so, adding the positions (i, j) into the set omega until the number of the positions in the set omega is Ns；
S103: and respectively calculating the absolute value of the Pearson correlation coefficient between the two-dimensional mask characteristic patterns corresponding to each position (i, j) in the set omega, and taking the absolute value of the Pearson correlation coefficient as the known element value at the position (i, j).
Further, the absolute value of the Pearson correlation coefficient between two-dimensional mask feature patterns
Figure BDA00030974021800000411
The calculation formula of (a) is as follows:
Figure BDA0003097402180000051
wherein s isiAnd sjThe scan vectors, s, of the two-dimensional mask feature patterns corresponding to the positions (i, j), respectivelyipAnd sjpAre respectively siAnd sjThe p-th component of (a) is,
Figure BDA0003097402180000052
and
Figure BDA0003097402180000053
are respectively siAnd sjAverage value of all elements in (M) is siAnd sjThe number of elements contained in (1).
Further, according to the setOmega-closed construction of sampling matrix PΩWherein if the position (i, j) ∈ Ω, the sampling matrix PΩElement P in position (i, j)Ω(i, j) is 1, otherwise PΩ(i,j)＝0。
Further, the calculation formula of the normalized laplacian matrix L' is as follows:
Figure BDA0003097402180000054
where D is a degree matrix, D ═ diag { D1,d2,…,dN}，
Figure BDA0003097402180000055
As a similarity matrix
Figure BDA0003097402180000056
The element at position (i, j), i, j, is 1, 2.
Further, a K-means clustering algorithm is adopted to carry out feature matrix pair
Figure BDA0003097402180000057
The N row vectors are clustered to obtain a clustering result { C1,…,CKIn which v is1～vKIs the eigenvector corresponding to the first K minimum eigenvalues of the normalized Laplace matrix L', and the clustering result C1～CKThe two-dimensional mask feature patterns corresponding to the row vectors contained in each mask feature pattern are classified into one type.
Has the advantages that:
1. the invention provides a atlas clustering method of a mask characteristic graph, which is characterized in that only part of elements of a similar matrix of the atlas clustering method are directly calculated from a mask characteristic graph set, but not all elements of the similar matrix are directly calculated, and then the rest unknown elements in the similar matrix are quickly supplemented by a low-rank matrix supplementing method with a minimum split Brazilianman-SCp norm, so that the supplementing error of the similar matrix is reduced, the calculation efficiency of the similar matrix is improved, the quick calculation method of the similar matrix is applied to a spectral clustering algorithm, and the calculation speed of the whole mask characteristic graph clustering process is further improved; therefore, the method applies the matrix completion method to the processes of mask characteristic pattern similarity matrix fast calculation and mask characteristic pattern spectral clustering for the first time, greatly improves the calculation efficiency and accuracy of mask characteristic pattern spectral clustering, reduces the calculation time, and can keep higher classification accuracy and reconstruction accuracy under a certain sampling rate.
2. The invention provides a mask characteristic graph atlas clustering method, which applies a rapid calculation method of a similar matrix to a spectral clustering algorithm, effectively improves the calculation efficiency of mask characteristic graph spectral clustering, reduces the calculation time, and can improve the calculation efficiency and accuracy of mask characteristic graph clustering under a certain sampling rate.
3. The invention provides a map clustering method of mask characteristic patterns, which adopts a blue noise template to select corresponding positions of known elements from a similar matrix, and utilizes a blue noise sampling method to calculate partial real element values of the mask layout similar matrix.
Drawings
FIG. 1 is a flow chart of a fast spectral clustering method for mask feature patterns according to the present invention;
FIG. 2 is a sample example of a two-dimensional mask feature pattern for spectral clustering experiments according to the present invention, wherein the size of the mask feature pattern is 512 × 512 pixels, and 50 mask features are divided into 5 groups, and two mask feature patterns with strong correlation in each group are given as an example;
FIG. 3 is a graph of the variation of the calculation error with the sampling rate for various similar matrix completion methods;
FIG. 4 is a curve showing the change of the accuracy of clustering classification of mask feature patterns with the sampling rate by using various similar matrix completion methods.
Detailed Description
In order to make the technical solutions better understood by those skilled in the art, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application.
The elements in the similarity matrix represent the similarity between the data samples of the map nodes, each map node corresponds to one mask feature pattern, and the elements in each row (or column) of the similarity matrix represent the similarity between a certain mask feature pattern and all other mask feature patterns, and the similarity can be calculated by using the normalized inner product of the two mask feature patterns, such as a Pearson correlation coefficient and the like; therefore, the rows or columns in the similarity matrix have high linear correlation, so the similarity matrix of the mask feature patterns can be regarded as a low-rank matrix; based on the method, partial elements of the similar matrix can be directly calculated from the mask characteristic pattern set (instead of directly calculating all elements of the similar matrix), and then the rest unknown elements are quickly supplemented by using a low-rank matrix supplementing method with the minimum splitting Brazimann-SCp norm, so that the calculation efficiency of the similar matrix is improved, and the calculation speed of the whole mask characteristic pattern clustering process is further improved.
Specifically, as shown in fig. 1, a method for clustering a mask feature pattern atlas includes the following steps:
s1: obtaining similar matrixes corresponding to N two-dimensional mask characteristic graphs to be classified
Figure BDA0003097402180000071
Wherein the similarity matrix
Figure BDA0003097402180000072
Is composed of known elements and unknown elements.
Further, the similarity matrix
Figure BDA0003097402180000073
The acquisition method comprises the following steps:
s101: determining similarity matrices
Figure BDA0003097402180000074
Number N of known elementss＝ceil(q·N2) Wherein q is a similarity matrix
Figure BDA0003097402180000075
Ceil (-) is an upward rounding function;
s102: determining a set omega of positions of the known elements, wherein the acquisition mode of the set omega is as follows:
assume a blue noise template of
Figure BDA0003097402180000076
Having the same dimension as the similarity matrix, all 1 matrices being
Figure BDA0003097402180000077
And E (i, j) and T (i, j) are the elements at position (i, j) in the full 1 matrix E and the blue noise template T, respectively, i, j equals 1, 2.
Sequentially judging whether the positions (i, j) meet alpha E (i, j) + beta is more than or equal to T (i, j), wherein alpha and beta are parameters for controlling the number of the positions, if so, adding the positions (i, j) into the set omega until the number of the positions in the set omega is Ns；
It should be noted that, in the location screening process, the number of selected locations is increasing, the locations are approximately evenly distributed, the first selected location is determined by the blue noise template, with macroscopic randomness, and when the number of selected locations reaches NsAfter that, the positions that subsequently do not participate in the selection are not considered.
S103: and respectively calculating the absolute value of the Pearson correlation coefficient between the two-dimensional mask characteristic patterns corresponding to each position (i, j) in the set omega, and taking the absolute value of the Pearson correlation coefficient as the known element value at the position (i, j). Wherein, the absolute value of the Pearson correlation coefficient
Figure BDA0003097402180000081
The calculation formula of (a) is as follows:
Figure BDA0003097402180000082
wherein, the vectors obtained by scanning N mask characteristic patterns are marked as s1,s2,…,sNThen siAnd sjThe scan vectors, s, of the two-dimensional mask feature patterns corresponding to the positions (i, j), respectivelyipAnd sjpAre respectively siAnd sjThe p-th component of (a) is,
Figure BDA0003097402180000083
and
Figure BDA0003097402180000084
are respectively siAnd sjAverage value of all elements in (M) is siAnd sjThe number of elements contained in (1).
That is, the invention provides a sampling calculation method of a similarity matrix truth value based on blue noise down-sampling. Blue noise down-sampling has a more uniform distribution of sampling points than random sampling. For an initial matrix, all elements of the initial matrix are zero, a proper sampling rate is selected, some positions of the matrix elements are subjected to screening sampling according to blue noise distribution, all sampled positions, namely the positions selected to be used as known elements, are marked as a set omega, and similar matrix elements contained in the set omega are calculated by using the absolute value of the Pearson correlation coefficient and are used as real element values of the to-be-complemented similar matrix. The blue noise down-sampling method enables the real element values in the similarity matrix obtained through direct calculation to be distributed more uniformly, and the similarity matrix estimation result with lower error can be obtained more easily in the subsequent matrix completion process.
S2: similarity matrix minimization method (SCp + split Bregman for short) based on split Brageman-SCp norm
Figure BDA0003097402180000091
Make up and calculate
Figure BDA0003097402180000092
Obtaining all elements of the other unknown elementsPixelized similarity matrix
Figure BDA0003097402180000093
It should be noted that the low-rank matrix completion method can be mainly divided into: a completion method based on nuclear norm relaxation, a completion method based on matrix decomposition, and a matrix completion method based on non-convex function relaxation. The existing SCp norm minimization method belongs to a non-convex function, the non-convex function can be degenerated into the existing nuclear norm, Schatten p norm and capped norm under different conditions, and the advantages of a rank function and the nuclear norm can also be combined. In the prior art, an alternating direction multiplier (ADMM) method is used for solving the SCp norm minimization problem, but the ADMM algorithm has high complexity, so that the calculation efficiency of a similar matrix is low, and in addition, a matrix completion method is not applied to the rapid calculation of the similar matrix of the mask characteristic pattern and the process of spectral clustering of the mask characteristic pattern in the technology.
Based on the method, the invention provides a minimization method based on the split Brazimann-SCp norm, which is applied to matrix completion for the first time, and specifically comprises the following steps:
s201: the following objective function is constructed based on the split Brazilian method:
Figure BDA0003097402180000094
wherein ρ is a weight coefficient, C is an auxiliary variable matrix, | | · | | computationally |, C is a vectorSCpIs the SCp norm, which is defined as
Figure BDA0003097402180000095
Wherein σiIs the ith large singular value of X, tau > 0 is the threshold, and the exponent p epsilon (0, 1)]X is an optimized variable matrix, and the initialized optimized variable matrix
Figure BDA0003097402180000096
Is an all-0 matrix, PΩ(. DEG) represents an orthogonal projection operator, omega is a set of positions of known elements, | | ·| purple windFIs Frobenius norm, and lambda is residual error term coefficient; for the optimized variable matrix X: if position (i, j) ∈ Ω, PΩ(X) is defined as PΩ(Xi,j)＝Xi,jElse PΩ(Xi,j) 0; for similarity matrix
Figure BDA0003097402180000101
If the position (i, j) ∈ Ω, then
Figure BDA0003097402180000102
Is defined as
Figure BDA0003097402180000103
Otherwise
Figure BDA0003097402180000104
S202: an iterative equation set for obtaining an optimized variable matrix X, an auxiliary variable matrix C and a residual B from an objective function by adopting a split Brahman iterative algorithm is as follows:
Figure BDA0003097402180000105
wherein k represents the number of iterations, and the residual B represents the residual between the optimized variable matrix X and the auxiliary variable matrix C;
s203: continuously carrying out iterative computation on the optimized variable matrix X, the auxiliary variable matrix C and the residual error B according to the iterative equation set until a set iteration termination condition is met, such as the iteration times reach a set upper limit value or
Figure BDA0003097402180000106
Less than the given error limit, the optimized variable matrix obtained by the final iteration is the similar matrix completed by completion
Figure BDA0003097402180000107
It should be noted that, in order to facilitate iteration, the iteration equation set may be optimizedIterative formula of variable matrix X
Figure BDA0003097402180000108
The form from solving the minimum function is converted to the following form:
Figure BDA0003097402180000109
wherein, PΩIs a sampling matrix constructed from a set Ω, where if a position (i, j) ∈ Ω, then the sampling matrix PΩElement P in position (i, j)Ω(i, j) is 1, otherwise PΩ(i, j) ═ 0. It is worth mentioning that PΩ(. o) represents an operator, and PΩRepresenting a matrix, the relationship of which is: pΩ(X)＝PΩ⊙X。
S3: obtaining a similarity matrix
Figure BDA0003097402180000118
And forming a feature matrix by using feature vectors corresponding to the first K minimum feature values of the normalized Laplacian matrix L', clustering N row vectors of the feature matrix, wherein the clustering result corresponds to the classification result of the N two-dimensional mask feature patterns.
The calculation formula of the normalized laplacian matrix L' is as follows:
Figure BDA0003097402180000111
where D is a degree matrix, D ═ diag { D1,d2,…,dN}，
Figure BDA0003097402180000112
As a similarity matrix
Figure BDA0003097402180000113
The element at position (i, j), i, j, is 1, 2.
Using K-means clustering algorithm pairFeature matrix
Figure BDA0003097402180000114
The N row vectors are clustered to obtain a clustering result { C1,…,CKIn which the m-th row vector of the matrix V is denoted as
Figure BDA0003097402180000115
Where m is 1, …, N, v1～vKIs the eigenvector corresponding to the first K minimum eigenvalues of the normalized Laplace matrix L', and the clustering result C1～CKThe two-dimensional mask feature patterns corresponding to the row vectors contained in the two-dimensional mask feature patterns are classified into one type, K is the number of clusters, and the value range of K is related to the number of the two-dimensional mask feature patterns, for example, if the number of the two-dimensional mask feature patterns is 50, K can be 2-50, that is, the two-dimensional mask feature patterns can be classified into 2 types-50 types.
A detailed derivation of the iterative equation set is given below.
The objective function of the matrix completion constructed in the conventional method is as follows:
Figure BDA0003097402180000116
the present invention solves the above optimization problem using a split brazieman method. Introducing an auxiliary variable C ≈ X, and converting the target function into the following formula:
Figure BDA0003097402180000117
wherein λ is a residual term coefficient, it is obvious that the present invention decomposes the variable of X in the conventional objective function into two variables of X and C to optimize.
Order to
Figure BDA0003097402180000121
The brageman iterative algorithm of the improved objective function is as follows:
Figure BDA0003097402180000122
wherein
Figure BDA0003097402180000123
And
Figure BDA0003097402180000124
the sub-gradients of X and C at k +1,<·,·>representing an inner product operation, instruction
Figure BDA0003097402180000125
And will be
Figure BDA0003097402180000126
And
Figure BDA0003097402180000127
substituting in the first equation, the above equation can be transformed into:
Figure BDA0003097402180000128
this can thus be converted into the following:
Figure BDA0003097402180000129
it should be noted that the residual B represents the residual between the optimized variable matrix X and the auxiliary variable matrix C, Ck-XkRepresenting the residual error between the actual values obtained by the iteration of the optimized variable matrix and the auxiliary variable matrix in the kth iteration, BkRepresenting the cumulative value of the residuals from the first k iterations.
Subsequently, the variable matrix X, the auxiliary variable matrix C and the residual error B can be updated in an iterative mode, and then the X is solved, wherein the Singular Value Decomposition process is calculated by a random Singular Value Decomposition algorithm (RSVD), the calculation time required by the method is short, and the calculation efficiency of the matrix completion and spectral clustering algorithm is improved.
Further, the solving process of the optimization problem of the auxiliary variable matrix C is as follows:
will iterate the formula
Figure BDA0003097402180000131
Performing variable substitution to obtain the following minimization problem for solving the following SCp norm and Frobenius norm:
Figure BDA0003097402180000132
where ζ ═ ρ/λ; y ═ Ck；G＝Xk+1+Bk。
Step 1, performing singular value decomposition on G to obtain G ═ U Δ VTWhere Δ ═ diag ({ δ)i}1≤i≤N) The singular value decomposition process can be calculated using RSVD.
Step 2, calculate v ═ 2 ζ (1-p)]1/2-pAnd v' ═ v + ζ pvp-1。
Step 3, comparing deltaiAnd v ', then x ' is determined by 'iWhere i ═ 1, …, N:
Figure BDA0003097402180000133
wherein is v'i＜δiThe time can be calculated by a gradient descent method
Figure BDA0003097402180000134
See steps 301-303.
Step 4, calculating
Figure BDA0003097402180000135
Step 5, comparison
Figure BDA0003097402180000136
And τ, and then determined by
Figure BDA0003097402180000137
Where τ is the threshold in the SCp norm:
Figure BDA0003097402180000138
step 6, obtaining the result by calculation
Figure BDA0003097402180000139
Form a diagonal matrix, denoted
Figure BDA00030974021800001310
Step 7, calculating Y ═ U ∑*VTWhere U and V are the left and right singular matrices of G, the resulting Y is the solution to the minimization problem of the SCp norm and the Frobenius norm.
Further, the solving process of the optimization problem in step 3 of the present invention is as follows:
step 301-303 is used for calculating a formula
Figure BDA0003097402180000141
Step 301, initialization
Figure BDA0003097402180000142
Step 302, update
Figure BDA0003097402180000143
Step 303, if the condition is satisfied, stopping iteration to obtain
Figure BDA0003097402180000144
Otherwise, the iteration is continued by returning to step 302.
Referring to fig. 2 to 4 and table 1, a fast spectral clustering method for mask feature patterns according to an embodiment of the present invention is described.
Fig. 3 shows error curves of five matrix completion methods, which are respectively an SCp + splitbegman + RSVD method, an SCp + splitbegman method, an SVT method, a nuclear norm + splitbegman method, and an SCp + ADMM method. The SCp + split Bregman + RSVD method uses a random singular value decomposition algorithm to calculate singular values, and SVD is adopted in the other four methods to calculate the singular values. The error calculation method comprises the following steps: and (4) subtracting the completed similar matrix from the similar matrix with all elements being real values, and then taking the square of the Frobenius norm. In view of the whole, the completion error is reduced along with the increase of the sampling rate, because the higher the sampling rate is, the fewer the unknown elements in the matrix to be completed are, which is more beneficial to passing through the algorithm or the completion result with higher precision, thereby reducing the error. As can be seen from FIG. 3, the completion errors of the SCp + split Bregman + RSVD method and the SCp + split Bregman method are very low and are very close to each other, and the errors of the two methods are also lower than those of the other three methods, wherein the SCp + split Bregman + RSVD method and the SCp + split Bregman method are both shown by black solid lines. The error of the SCp + splitbegman + RSVD method is only a little higher than that of the SCp + splitbegman method, because the RSVD method maps a high-dimensional matrix to a low-dimensional matrix using a gaussian distribution matrix, thereby introducing a slight error compared to the SCp + splitbegman method, but as can be seen from table 1, the former has a faster calculation speed than the latter.
Fig. 4 shows a spectral clustering accuracy curve based on five matrix completion methods. It can be seen from the figure that as the sampling rate is gradually reduced, the accuracy of spectral clustering is gradually reduced. The SCp + split Bregman + RSVD method has the highest accuracy. It should be noted that the accuracy of the SCp + splitbegman + RSVD method is completely the same as that of the SCp + splitbegman method, and is higher than that of the other three methods.
Table 1: time consumed by mask layout clustering operation of different spectral clustering methods under different sampling rates
Figure BDA0003097402180000151
Table 1 shows the operating time of the spectral clustering method based on five matrix completion algorithms at different sampling rates, and the operating time of the spectral clustering method without the matrix completion algorithm (instead, all the truth values of the similar matrix elements are directly calculated). Therefore, the time consumed by the spectral clustering method adopting matrix completion is obviously shorter than that consumed by the spectral clustering method not adopting matrix completion. In addition, under different sampling rates, the spectral clustering of the SCp + split Bregman + RSVD method takes the shortest time. That is to say, the invention provides a map clustering method of mask characteristic graphs, which takes the calculation of a similar matrix in a map clustering algorithm as the primary step, applies a split Brazilian-SCp norm minimization low-rank matrix completion method to the calculation of the mask characteristic graph similar matrix, then adopts the completed similar matrix to execute the subsequent operation of the map clustering, finally obtains the clustering result of the mask characteristic graphs, and can quickly realize the clustering of the mask characteristic graphs.
The present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof, and it will be understood by those skilled in the art that various changes and modifications may be made herein without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (10)

1. A method for clustering a mask feature pattern atlas is characterized by comprising the following steps:
s1: obtaining similar matrixes corresponding to N two-dimensional mask characteristic graphs to be classified
Figure FDA0003097402170000011
Wherein the similarity matrix
Figure FDA0003097402170000012
The system is composed of known elements and unknown elements;
s2: similarity matrix pair by split-based Brageman-SCp norm minimization method
Figure FDA0003097402170000013
Make up and calculate
Figure FDA0003097402170000014
To obtain a similarity matrix with all known elements
Figure FDA0003097402170000015
S3: obtaining a similarity matrix
Figure FDA0003097402170000016
And forming a feature matrix by using feature vectors corresponding to the first K minimum feature values of the normalized Laplacian matrix L', clustering N row vectors of the feature matrix, wherein the clustering result corresponds to the classification result of the N two-dimensional mask feature patterns.
2. The method of claim 1, wherein the similarity matrix comprises a plurality of mask feature patterns
Figure FDA0003097402170000017
The completion method comprises the following steps:
s201: the following objective function is constructed based on the split Brazilian method:
Figure FDA0003097402170000018
wherein ρ is a weight coefficient, C is an auxiliary variable matrix, | | · | | computationally |, C is a vectorSCpFor the SCp norm, the exponent p e (0, 1)]X is an optimized variable matrix, PΩ(. DEG) represents an orthogonal projection operator, omega is a set of positions of known elements, | | ·| purple windFIs Frobenius norm, and lambda is residual error term coefficient;
s202: an iterative equation set for obtaining an optimized variable matrix X, an auxiliary variable matrix C and a residual B from an objective function by adopting a split Brahman iterative algorithm is as follows:
Figure FDA0003097402170000019
wherein k represents the number of iterations, and the residual B represents the residual between the optimized variable matrix X and the auxiliary variable matrix C;
s203: continuously carrying out iterative computation on the optimized variable matrix X, the auxiliary variable matrix C and the residual error B according to the iterative equation set until a set iteration termination condition is met, wherein the finally obtained optimized variable matrix is a complete similar matrix
Figure FDA0003097402170000021
3. The method of claim 2, wherein for the optimization variable matrix X: if position (i, j) ∈ Ω, PΩ(X) is defined as PΩ(Xi,j)＝Xi,jElse PΩ(Xi,j) 0; for similarity matrix
Figure FDA0003097402170000022
If the position (i, j) ∈ Ω, then
Figure FDA0003097402170000023
Is defined as
Figure FDA0003097402170000024
Otherwise
Figure FDA0003097402170000025
4. The method of claim 2, wherein the SCp norm is defined as
Figure FDA0003097402170000026
Wherein σiIs the ith large singular value of X, and is the threshold value when tau is more than 0.
5. The method of claim 2, wherein the iteration termination condition is that the number of iterations reaches a predetermined upper limit value or
Figure FDA0003097402170000027
Less than a given margin of error.
6. The method of claim 1, wherein the similarity matrix comprises a plurality of mask feature patterns
Figure FDA0003097402170000028
The acquisition method comprises the following steps:
s101: determining similarity matrices
Figure FDA0003097402170000029
Number N of known elementss＝ceil(q·N2) Wherein q is a similarity matrix
Figure FDA00030974021700000210
Ceil (-) is an upward rounding function;
s102: determining a set omega of positions of the known elements, wherein the acquisition mode of the set omega is as follows:
assume a blue noise template of
Figure FDA00030974021700000211
All 1 matrices are
Figure FDA00030974021700000212
And E (i, j) and T (i, j) are the elements at position (i, j) in the full 1 matrix E and the blue noise template T, respectively, i, j equals 1, 2.
Sequentially judging whether the position (i, j) meets the condition that alpha E (i, j) + beta is more than or equal to T (i)J), where α and β are parameters for controlling the number of positions, and if satisfied, adding the position (i, j) to the set Ω until the number of positions in the set Ω is Ns；
S103: and respectively calculating the absolute value of the Pearson correlation coefficient between the two-dimensional mask characteristic patterns corresponding to each position (i, j) in the set omega, and taking the absolute value of the Pearson correlation coefficient as the known element value at the position (i, j).
7. The method of claim 6, wherein the absolute value of the Pearson's correlation coefficient between two-dimensional mask patterns is determined by the method of clustering patterns of mask patterns
Figure FDA0003097402170000031
The calculation formula of (a) is as follows:
Figure FDA0003097402170000032
wherein s isiAnd sjThe scan vectors, s, of the two-dimensional mask feature patterns corresponding to the positions (i, j), respectivelyipAnd sjpAre respectively siAnd sjThe p-th component of (a) is,
Figure FDA0003097402170000033
and
Figure FDA0003097402170000034
are respectively siAnd sjAverage value of all elements in (M) is siAnd sjThe number of elements contained in (1).
8. The method of claim 6, wherein the sampling matrix P is constructed according to the set ΩΩWherein if the position (i, j) ∈ Ω, the sampling matrix PΩElement P in position (i, j)Ω(i, j) is 1, otherwise PΩ(i,j)＝0。
9. The method for clustering an atlas of mask feature images according to any one of claims 1 to 8, wherein the calculation formula of the normalized Laplace matrix L' is:
Figure FDA0003097402170000035
where D is a degree matrix, D ═ diag { D1,d2,…,dN}，
Figure FDA0003097402170000041
Figure FDA0003097402170000042
As a similarity matrix
Figure FDA0003097402170000043
The element at position (i, j), i, j, is 1, 2.
10. The method of any one of claims 1 to 8, wherein a K-means clustering algorithm is used to cluster the feature matrix
Figure FDA0003097402170000044
The N row vectors are clustered to obtain a clustering result { C1,…,CKIn which v is1～vKIs the eigenvector corresponding to the first K minimum eigenvalues of the normalized Laplace matrix L', and the clustering result C1～CKThe two-dimensional mask feature patterns corresponding to the row vectors contained in each mask feature pattern are classified into one type.
CN202110614308.5A 2021-06-02 2021-06-02 Atlas clustering method for mask characteristic graph Active CN113255799B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202110614308.5A CN113255799B (en) 2021-06-02 2021-06-02 Atlas clustering method for mask characteristic graph

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202110614308.5A CN113255799B (en) 2021-06-02 2021-06-02 Atlas clustering method for mask characteristic graph

Publications (2)

Publication Number Publication Date
CN113255799A true CN113255799A (en) 2021-08-13
CN113255799B CN113255799B (en) 2022-10-14

Family

ID=77185993

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202110614308.5A Active CN113255799B (en) 2021-06-02 2021-06-02 Atlas clustering method for mask characteristic graph

Country Status (1)

Country Link
CN (1) CN113255799B (en)

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2024021590A1 (en) * 2022-07-28 2024-02-01 东方晶源微电子科技（北京）有限公司 Position deviation-allowed graphic grouping method, apparatus and device, and storage medium

Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN104134196A (en) * 2014-08-08 2014-11-05 重庆大学 Split Bregman weight iteration image blind restoration method based on non-convex higher-order total variation model

Patent Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN104134196A (en) * 2014-08-08 2014-11-05 重庆大学 Split Bregman weight iteration image blind restoration method based on non-convex higher-order total variation model

Non-Patent Citations (5)

* Cited by examiner, † Cited by third party
Title
AKIFUMI OKUNO等: "Hyperlink Regression via Bregman Divergence", 《ARXIV》 *
GUORUI LI等: "Matrix Completion via Schatten Capped p Norm", 《IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING》 *
TOM GOLDSTEIN等: "The Split Bregman Method for L1-Regularized Problems", 《RESEARCHGATE》 *
李艳凤等: "基于解剖学特征的乳腺X线图像胸肌分割", 《自动化学报》 *
王瑞国: "基于网络嵌入和领袖识别的属性图聚类算法研究", 《中国优秀硕士学位论文全文数据库（电子期刊）》 *

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2024021590A1 (en) * 2022-07-28 2024-02-01 东方晶源微电子科技（北京）有限公司 Position deviation-allowed graphic grouping method, apparatus and device, and storage medium

Also Published As

Publication number Publication date
CN113255799B (en) 2022-10-14

Similar Documents

Publication Publication Date Title
Liu et al. Group fisher pruning for practical network compression
US11106973B2 (en) Method and system for bit-depth reduction in artificial neural networks
CN117313640B (en) Training method, device, equipment and storage medium for lithography mask generation model
CN104133348B (en) A kind of adaptive optical etching system light source optimization method
Jiang et al. Training binary neural network without batch normalization for image super-resolution
CN113759657B (en) Optical proximity correction method
CN109766863B (en) Face image super-resolution method based on local and sparse non-local regularization
KR20200034918A (en) System and method to provide balanced pruning of weights of a deep neuaral network
CN110119805B (en) Convolutional neural network algorithm based on echo state network classification
CN112655071A (en) Learning device, estimation device, and learned model
CN113255799B (en) Atlas clustering method for mask characteristic graph
CN113238460B (en) Deep learning-based optical proximity correction method for extreme ultraviolet
CN106971189B (en) A kind of noisy method for recognising star map of low resolution
Lee et al. Flood fill mean shift: A robust segmentation algorithm
CN111985611A (en) Computing method based on physical characteristic diagram and DCNN machine learning reverse photoetching solution
Hou et al. Nonlinear dimensionality reduction by locally linear inlaying
CN115496983A (en) Class increment learning method and system guided by semantic knowledge
Geng et al. Sparse VLSI layout feature extraction: A dictionary learning approach
TWI801837B (en) Aligning a distorted image
CN106503648A (en) Face identification method and device based on sparse projection binary-coding
Bernard et al. A fast algorithm to find best matching units in self-organizing maps
CN110364829B (en) Sparse linear array antenna
CN111898653A (en) Based on robustness l1,2Norm constrained supervised dimension reduction method
CN116974139A (en) Method, device and equipment for rapidly calculating photoetching mask image
US12086526B2 (en) Methods and devices of correcting layout for semiconductor processes using machine learning

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant