Method of multivariate spectral analysis
First Claim
1. A method of determining the properties of a sample from measured spectral data collected from the sample;
- comprising;
a) providing a two-dimensional matrix A containing measured spectral data;
b) generating a weighted spectral data matrix D by weighting matrix A;
c) factoring D into the product of two matrices, C and ST, by performing a constrained alternating least-squares analysis of D=CST, where C is a concentration intensity matrix and S is a spectral shapes matrix;
d) unweighting C and S by applying the inverse of the weighting used in step b); and
e) determining the properties of the sample by inspecting C and S.
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Abstract
A method of determining the properties of a sample from measured spectral data collected from the sample by performing a multivariate spectral analysis. The method can include: generating a two-dimensional matrix A containing measured spectral data; providing a weighted spectral data matrix D by performing a weighting operation on matrix A; factoring D into the product of two matrices, C and ST, by performing a constrained alternating least-squares analysis of D=CST, where C is a concentration intensity matrix and S is a spectral shapes matrix; unweighting C and S by applying the inverse of the weighting used previously; and determining the properties of the sample by inspecting C and S. This method can be used to analyze X-ray spectral data generated by operating a Scanning Electron Microscope (SEM) with an attached Energy Dispersive Spectrometer (EDS).
206 Citations
106 Claims
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1. A method of determining the properties of a sample from measured spectral data collected from the sample;
- comprising;
a) providing a two-dimensional matrix A containing measured spectral data;
b) generating a weighted spectral data matrix D by weighting matrix A;
c) factoring D into the product of two matrices, C and ST, by performing a constrained alternating least-squares analysis of D=CST, where C is a concentration intensity matrix and S is a spectral shapes matrix;
d) unweighting C and S by applying the inverse of the weighting used in step b); and
e) determining the properties of the sample by inspecting C and S. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 82)
a) arranging the eigenvalues E in descending order;
b) taking a numerical derivative of E with respect to the eigenvalue number, n, and plotting-dE/dn versus n to create a curve;
c) fitting a line to a portion of this curve over which the eigenvalues are known to describe noise;
d) identifying the first eigenvalue encountered while ascending that exceeds the value predicted by the fitted line by a specified margin; and
e) setting q to be equal to the eigenvalue number n of the first eigenvalue encountered in step d).
- comprising;
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48. The method of claim 47, wherein the set of eigenvalues that are known to describe noise in step c) is equal to the set of eigenvalues whose eigenvalue numbers n are greater than twenty-five, and are less than or equal to 100.
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49. The method of claim 48, wherein the specified margin in step d) is equal to a sensitivity constant times the standard deviation of the prediction residuals of the best-fit line fitted to the set of eigenvalues whose eigenvalue numbers n are greater than twenty-five.
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50. The method of claim 49, wherein the sensitivity constant equals twenty-five.
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51. The method of claim 44, wherein an initial estimate for S is generated by compressing the dimensions of S to have a rank=q, after selecting the proper number of pure components q.
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52. The method of claim 51, further comprising, after compressing the dimensions of S to have a rank=q, the steps of:
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a) performing a general eigenvector transformation on the compressed eigenvector matrix S;
b) selecting the sign of the transformed eigenvectors so that they are predominantly positive; and
c) setting all negative values of the transformed eigenvectors to zero.
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53. The method of claim 51, wherein performing the general eigenvector transformation comprises performing an orthogonal rotation selected from the group consisting of VARIMAX, QUARTIMAX, and ORTHOMAX orthogonal rotations.
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54. The method of claim 51, wherein performing the general eigenvector transformation comprises performing an non-orthogonal rotation selected from the group consisting of OBLIMAX and PROMAX non-orthogonal rotations.
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55. The method of claim 53, further comprising using VARIMAX rotated eigenvectors as an initial estimate for S.
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56. The method of claim 1, further comprising preserving active constraint sets upon successive alternating least-squares iterations to enhance convergence speed by providing better iteration starting points.
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57. The method of claim 1, further comprising using the BLAS routine “
- DGEMM”
or “
DSYRK”
, to compute large matrix crossproducts.
- DGEMM”
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58. The method of claim 57, further comprising using BLAS routines from the Intel Math Kernel Library.
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59. The method of claim 1, further comprising applying a convergence criterion based on the sum of the squared differences of S between successive ALS iterations, including computing the criterion separately for each of the columns of S.
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60. The method of claim 59, further comprising removing individual converged components from S during subsequent ALS iterations.
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61. The method of claim 1, further comprising identifying the prominent spectral peaks of a selected chemical species from the converged matrices C and S by comparing the prominent spectral peaks with known spectrum signatures of known elements, phases, or alloys.
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62. The method of claim 1, implemented in a parallel-processing computational architecture.
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63. The method of claim 1, further comprising generating an initial estimate for the matrix C, prior to factoring D.
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64. The method of claim 63, further comprising weighting the rows of D, followed by performing the corresponding unweighting of the converged results for C by multiplying each column of C element-wise by the square root of a mean image.
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82. A computer readable medium having thereon instructions for causing a computer to perform the method of claim 1.
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65. A method of determining the properties of a sample from measured spectral data collected from the sample;
- comprising;
a) providing a two-dimensional matrix of measured spectral data A;
b) generating a weighted spectral data matrix D by weighting matrix A;
c) generating an initial estimate for a spectral shapes matrix S;
d) using the initial estimate for S, factoring D into the product of two matrices C and ST, by performing a non-negativity constrained alternating least-squares analysis of D=CST, where C is a concentration intensity matrix;
e) unweighting the matrix S generated in step d) by applying the inverse of the weighting used in step b);
f) estimating the spectral background B;
g) calculating P=S−
B and SB=B, and redefine S=[P SB], where P is a matrix of pure spectral components without background;
h) reweighting S and B, using the same weighting that was used in step b);
i) solving D=[CP CB]·
[P SB]T by performing alternating least squares analysis, subject to CP, CB, P≧
0 and SB=B;
j) repeating steps e) through i) until acceptable convergence is achieved;
k) unweighting CP and S by applying the inverse of the weighting used in step b);
l) reformatting the columns of CP as a two-dimensional image; and
m) determining the properties of the sample by inspecting CP and P;
wherein SB is the background spectral shapes matrix, CP is the concentration intensity matrix without background, and CB is the background concentration intensity matrix. - View Dependent Claims (66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79)
a) selecting a pure component;
b) identifying at least one major element that comprise the selected pure component by inspecting S;
c) applying an automated analysis program to generate a physically accurate background shape for the particular combination of elements or element identified in step b); and
d) fitting the background shape generated in from step b) to the pure component spectrum of the selected pure component.
- comprising;
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69. The method of claim 68, wherein the automated analysis program used in step c) comprises the Desktop Spectrum Analyzer program (DTSA).
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70. The method of claim 65, wherein the spectral background is estimated using linear combinations of shapes based on a physical model.
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71. The method of claim 70, wherein the physical model is given by:
-
wherein I=background intensity, Z=atomic number, E0=primary beam energy, and E=variable photon energy.
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72. The method of claim 65, wherein estimating the spectral background B comprises executing a fitting algorithm, wherein the algorithm comprises:
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a) selecting a number of points, r, where r≧
p/2+1, and where p is the total number of energy channels;
b) performing non-negativity constrained least squares analysis to fit to the background shapes;
c) selecting the number of points, r, having the smallest residuals; and
d) if the set from step c) is the same set of points as we fit in step b), then stop, otherwise go back to step b) and repeat steps b), c) and d).
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73. The method of claim 72, wherein r=p−
- s, where s equals the number of channels corresponding to the spectral peaks in a spectrum.
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74. The method of claim 72 wherein r=p/2+1.
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75. The method of claim 72, wherein the rth point of a spectrum is assumed to lie on the background.
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76. The method of claim 72, further comprising selecting the r points about the median point of a spectrum, to provide a better starting point than using a random selection.
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77. The method of claim 72, wherein the channels selected for fitting are partially constrained to ensure relatively uniform coverage over the entire energy range.
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78. The method of claim 72, wherein half of the points are selected from each of the low-energy and the high-energy halves of the spectrum, to assure a relatively uniform distribution of points.
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79. The method of claim 65, further comprising refolding CP and P into a three-dimensional data array, after step j).
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80. A method of determining the properties of a sample from measured spectral data collected from the sample;
- comprising
a) providing, and then unfolding, a three-dimensional spectral data array X to form a two-dimensional matrix of measured spectral data A;
b) generating a weighted spectral data matrix D by weighting matrix A, using Poisson statistics;
c) generating an initial estimate for a spectral shapes matrix S by obtaining eigenvectors V of the cross product matrix DTD;
automatically selecting the proper number of pure components q;
compressing V to have rank=q; and
performing a VARIMAX rotation on the reduced matrix V to produce the initial estimate for S;
d) using the initial estimate for S, factoring D into the product of two matrices C and ST, by performing a non-negativity constrained alternating least-squares analysis of D=CST, where C is a concentration intensity matrix;
e) unweighting S by applying the inverse of the weighting used in step b);
f) estimating the spectral background B;
g) reweighting S and B, using the same weighting that was used in step b);
h) solving D=CST for C and S by performing an alternating least-squares analysis subject to the constraints C≧
0 and S≧
B;
i) repeating steps e) through i) until acceptable convergence is achieved;
j) unweighting C and S by applying the inverse of the weighting used in step b);
k) calculating P=S−
B, where P is a matrix of pure spectral components without background;
l) reformatting the columns of C as a two-dimensional image; and
m) determining the properties of the sample by inspecting C and P. - View Dependent Claims (81)
- comprising
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83. A method of determining the properties of a sample from measured spectral data collected from the sample;
- comprising
a) providing, and then unfolding, a three-dimensional spectral data array X to form a two-dimensional matrix of measured spectral data A;
b) generating an initial estimate for a spectral shapes matrix S by obtaining eigenvectors V of the cross product matrix ATA;
automatically selecting the proper number of pure components q;
compressing V to have rank=q; and
performing a VARIMAX rotation on the reduced V to produce the initial estimate for S;
c) using the initial estimate for S, factoring A into the product of two matrices C and ST, by performing a non-negativity constrained alternating least-squares analysis of A=CST, where C is a concentration intensity matrix;
d) estimating the spectral background B using a robust least squares statistical method based on a linear combination of shapes based on a physical model;
e) solving A=CST for C and S by performing an alternating least-squares analysis subject to the constraints C≧
0 and S≧
B;
f) repeating steps d) through f) until acceptable convergence is achieved;
g) subtracting B from S to generate a matrix P of pure spectral components without background;
h) reformatting the columns of C as a two-dimensional image; and
determining the properties of the sample by inspecting C and P.
- comprising
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84. A method of determining the properties of a sample, comprising:
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a) detecting radiation emitted by the sample;
b) generating measured spectral data by processing the detected radiation;
c) performing a multivariate spectral analysis of the measured spectral data, comprising;
i) providing a two-dimensional matrix A containing the measured spectral data;
ii) generating a weighted spectral data matrix D by weighting matrix A;
iii) factoring D into the product of two matrices C and ST, by performing a constrained alternating least-squares analysis of D=CST, where C is a concentration intensity matrix and S is a spectral shapes matrix;
iv) unweighting C and S by applying the inverse of the weighting used in step ii); and
d) determining the properties of the sample by inspecting C and S. - View Dependent Claims (85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106)
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Specification