Methods for spectral image analysis by exploiting spatial simplicity
Several full-spectrum imaging techniques have been introduced in recent years that promise to provide rapid and comprehensive chemical characterization of complex samples. One of the remaining obstacles to adopting these techniques for routine use is the difficulty of reducing the vast quantities of raw spectral data to meaningful chemical information. Multivariate factor analysis techniques, such as Principal Component Analysis and Alternating Least Squares-based Multivariate Curve Resolution, have proven effective for extracting the essential chemical information from high dimensional spectral image data sets into a limited number of components that describe the spectral characteristics and spatial distributions of the chemical species comprising the sample. There are many cases, however, in which those constraints are not effective and where alternative approaches may provide new analytical insights. For many cases of practical importance, imaged samples are “simple” in the sense that they consist of relatively discrete chemical phases. That is, at any given location, only one or a few of the chemical species comprising the entire sample have non-zero concentrations. The methods of spectral image analysis of the present invention exploit this simplicity in the spatial domain to make the resulting factor models more realistic. Therefore, more physically accurate and interpretable spectral and abundance components can be extracted from spectral images that have spatially simple structure.
1. A method for analyzing a measured spectral image of a sample by factoring a matrix of spectral image data D according to D≅CS T , where C is a concentration matrix describing the distribution of pure components at each spatial location of the sample and S is a spectral shapes matrix representation of the pure-component spectra, comprising:
a) measuring a spectral image of the sample with a spectroscopic imaging system, whereby the spectral image is represented by the matrix of spectral image data D;
b) obtaining an initial estimate for the spectral shapes matrix S;
c) selecting elements of zero abundance in the concentration matrix C;
d) obtaining a least squares estimate for C, subject to the zero-equality constraints at the zero abundance matrix elements, according to
C
←
min
C
S
D
-
CS
T
F
;
and
e) obtaining a least squares estimate for S, subject to constraints, according to
S
←
min
S
C
D
-
CS
T
F
.
2. The method of claim 1 , further comprising repeating steps c) to e) at least once.
3. The method of claim 1 , wherein the data matrix D comprises a factored data matrix, according to D=AB T .
4. The method of claim 1 , further comprising weighting the data matrix D prior to step b).
5. The method of claim 1 , wherein step c) comprises obtaining an unconstrained solution for C, according to
C
←
min
C
S
D
-
CS
T
F
,
to select the elements of zero abundance.
6. The method of claim 1 , wherein step c) comprises selecting the location of elements to be zero that have abundance estimates smaller than a threshold value.
7. The method of claim 6 , wherein the threshold value is fixed at an absolute abundance level.
8. The method of claim 6 , wherein the threshold value is relative to a certain fraction of the total abundance.
9. The method of claim 6 , wherein the threshold value is a global value that applies equally to all components.
10. The method of claim 6 , wherein the threshold value is applied differently to each component.
11. The method of claim 1 , wherein step c) comprises selecting the number of elements of zero abundance to be fixed.
12. The method of claim 11 , wherein the number of elements of zero abundance is fixed globally.
13. The method of claim 11 , wherein the number of elements of zero abundance is fixed individually for each component.
14. The method of claim 1 , wherein step d) further comprises subject the least squares estimate to other constraints applied to the full matrix.