Spatial compression algorithm for the analysis of very large multivariate images
View Patent ↗A method for spatially compressing data sets enables the efficient analysis of very large multivariate images. The spatial compression algorithms use a wavelet transformation to map an image into a compressed image containing a smaller number of pixels that retain the original image's information content. Image analysis can then be performed on a compressed data matrix consisting of a reduced number of significant wavelet coefficients. Furthermore, a block algorithm can be used for performing common operations more efficiently. The spatial compression algorithms can be combined with spectral compression algorithms to provide further computational efficiencies.
1. A method for analyzing multivariate images, comprising:
a) providing a data matrix D containing measured spectral data,
b) applying at least one wavelet transform to the data matrix D to obtain a transformed data matrix {tilde over (D)} comprising an approximation matrix {tilde over (D)} a and a detail matrix {tilde over (D)} d ,
c) performing an image analysis on the approximation matrix {tilde over (D)} a to obtain an approximation concentration matrix {tilde over (C)} a and a spectral shapes matrix S, and
d) computing a concentration matrix C from the approximation concentration matrix {tilde over (C)} a , the detail matrix {tilde over (D)} d , and the spectral shapes matrix S.
2. The method of claim 1 , wherein the at least one wavelet transform comprises a Haar transform.
3. The method of claim 1 , wherein the image analysis of step c) comprises an alternating least squares analysis and the approximation concentration matrix {tilde over (C)} a and the spectral shapes matrix S are obtained from a constrained least squares solution of min∥{tilde over (D)} a −{tilde over (C)} a S T ∥ F .
4. The method of claim 3 , wherein the alternating least squares analysis comprises a transformed non-negativity constraint {tilde over (C)} a ≧0.
5. The method of claim 1 , wherein the computing step d) comprises:
computing a detail concentration matrix {tilde over (C)} d from the detail matrix {tilde over (D)} d and the spectral shapes matrix S;
combining the approximation concentration matrix {tilde over (C)} a and the detail concentration matrix {tilde over (C)} d to provide a transformed concentration matrix {tilde over (C)}; and
applying an inverse wavelet transform to the transformed concentration matrix {tilde over (C)} to provide the concentration matrix C.
6. A method for analyzing multivariate images, comprising:
a) providing a data factor matrix A and a data factor matrix B obtained from a factorization of measured spectral data D,
b) transforming the data factor matrix A, using a wavelet transform, to obtain a transformed data factor matrix Ã,
c) thresholding the wavelet coefficients of the transformed data factor matrix Ã,
d) performing an image analysis on the transformed data factor matrix à and data factor matrix B to obtain a spatially compressed concentration matrix {tilde over (C)} and a spectral shapes matrix S, and
e) computing a concentration matrix C from the spatially compressed concentration matrix {tilde over (C)}.
7. The method of claim 6 , wherein the data factor matrix A comprises a total of j blocks of data factors A i and the data factor matrix B comprises k blocks of data factors B i , thereby providing a concentration block C i in step e), and wherein steps a) through e) are repeated sequentially until the concentration matrix C is accumulated blockwise, according to C=[C 1 C 2 . . . C j−1 C j ].
8. The method of claim 6 , wherein the wavelet transform comprises a Haar transform.
9. The method of claim 6 , wherein the thresholding comprises decimating the detail coefficients.
10. The method of claim 6 , wherein the image analysis of step d) comprises an alternating least squares analysis and the spatially compressed concentration matrix {tilde over (C)} and the spectral shapes matrix S are obtained from a constrained least squares solution of
min
C
~
,
S
A
~
B
T
-
C
~
S
T
F
.
11. The method of claim 10 , wherein the alternating least squares analysis comprises a transformed non-negativity constraint.
12. The method of claim 6 , wherein the computing step e) comprises applying an inverse wavelet transform to the spatially compressed concentration matrix {tilde over (C)} to provide the concentration matrix C.
13. The method of claim 6 , wherein the computing step e) comprises projecting the product of the data factor matrix A and the data factor matrix B from step a) onto the spectral shapes matrix S from step d), according to
min
C
AB
T
-
CS
T
F
and subject to appropriate constraints.
14. The method of claim 6 , wherein the data factor matrix A comprises a scores matrix T and the data factor matrix B comprises a loadings matrix P, and wherein T and P are obtained from a principal components analysis of the measured spectral data D, according to D=TP T .
15. The method of claim 14 , wherein T and P represent the significant components of the principal components.
16. The method of claim 1 , wherein the data matrix D is weighted.
17. The method of claim 6 , wherein the data factor matrix A and the data factor matrix B are weighted.
18. The method of claim 1 , wherein the wavelet transform applying step b) comprises:
folding the data matrix D into a (x+1)-dimensional multiway array D consisting of x spatial dimensions and 1 spectral dimension comprising p spectral channels, wherein x=1, 2, or 3;
applying an independent wavelet transform to each of the x spatial dimensions for each of the p spectral channels to provide a transformed multiway array {tilde over (D)} ;
partitioning the transformed multiway array {tilde over (D)} into a multiway approximation array {tilde over (D)} a and a multiway detail array {tilde over (D)} d ; and
unfolding the multiway approximation array {tilde over (D)} a to obtain the approximation matrix {tilde over (D)} a and the detail matrix {tilde over (D)} d .
19. The method of claim 18 , wherein the computing step d) comprises:
computing a detail concentration matrix {tilde over (C)} d from the detail matrix {tilde over (D)} d and the spectral shapes matrix S;
combining the approximation concentration matrix {tilde over (C)} a and the detail concentration matrix {tilde over (C)} d to provide a transformed concentration matrix {tilde over (C)};
folding the transformed concentration matrix {tilde over (C)} into an (x+1)-dimensional transformed concentration matrix {tilde over (C)} ;
applying an independent inverse wavelet transform to each of the x spatial dimensions of the transformed concentration array {tilde over (C)} to obtain the multiway concentration array C ; and
unfolding the multiway concentration array C to obtain the concentration matrix C.
20. The method of claim 1 , wherein D is a 2D data matrix comprising m rows and n columns and the wavelet transforms W are applied according to (W n {circle around (x)}W m )×D={tilde over (D)}.
21. A method for analyzing multivariate images, comprising:
a) providing a data matrix D containing measured spectral data,
b) applying at least one wavelet transform to the data matrix D to obtain a transformed data matrix {tilde over (D)} comprising an approximation matrix {tilde over (D)} a ,
c) performing an image analysis on the approximation matrix {tilde over (D)} a to obtain a spectral shapes matrix S, and
d) computing a concentration matrix C from the data matrix D, and the spectral shapes matrix S.
22. The method of claim 21 , wherein the at least one wavelet transform comprises a Haar transform.
23. The method of claim 21 , wherein the image analysis of step c) comprises an alternating least squares analysis and an approximation concentration matrix {tilde over (C)} a and the spectral shapes matrix S are obtained from a constrained least squares solution of min∥{tilde over (D)} a −{tilde over (C)} a S T ∥ F .
24. The method of claim 23 , wherein the alternating least squares analysis comprises a transformed non-negativity constraint {tilde over (C)} a ≧0.
25. The method of claim 21 , wherein the computing step d) comprises projecting the data matrix D from step a) onto the spectral shapes matrix S from step c), according to
min
C
D
-
CS
T
F
,
subject to constraints.
26. The method of claim 21 , wherein the wavelet transforms applying step b) comprises:
folding the data matrix D into a (x+1)-dimensional multiway array D consisting of x spatial dimensions and 1 spectral dimension comprising p spectral channels, wherein x=1, 2, or 3;
applying an independent wavelet transform to each of the x spatial dimensions for each of the p spectral channels to provide a transformed multiway array {tilde over (D)};
decimating the detail coefficients of the transformed multiway array {tilde over (D)} to obtain a multiway approximation array {tilde over (D)} a ; and
unfolding the multiway approximation array {tilde over (D)} a to obtain the approximation matrix {tilde over (D)} a .