Abstract

This paper proposes a class-based spectral estimation method for high-resolution red, green, and blue (RGB) images and corresponding low-resolution spectral data. Each spectrum in the low-resolution data is assumed to be a mixture of spectra of different classes. Then, the spectral estimation matrix for every class is derived using a regression approach, in which the clustering results of the high-resolution RGB image are used to incorporate spectral unmixing. Experiments confirm reduced normalized root mean squared error for the spectral images if the number of classes in the clustering is appropriately selected. In addition, the experimental results show that the spectra are accurately reconstructed even when they are observed as mixed spectra in the low-resolution data.

© 2011 Optical Society of America

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2010 (1)

2009 (3)

2008 (3)

R. Zurita-Milla, J. G. P. W. Clevers, and M. E. Schaepman, “Unmixing-based Landsat TM and MERIS FR data fusion,” IEEE Geosci. Remote Sens. Lett. 5, 453–457 (2008).
[CrossRef]

M. Yamaguchi, H. Hideaki, and N. Ohyama, “Beyond red-green-blue (RGB): spectrum-based color imaging technology,” J. Imaging Sci. Technol. 52, 010201 (2008).
[CrossRef]

X. Zhang and H. Xu, “Reconstructing spectral reflectance by dividing spectral space and extending the principal components in principal component analysis,” J. Opt. Soc. Am. A 25, 371–378(2008).
[CrossRef]

2007 (4)

2006 (2)

2005 (2)

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, “Development of 16-band multispectral image archiving system,” Proc. SPIE 5667, 136–145 (2005).
[CrossRef]

M. T. Eismann and R. C. Hardie, “Hyperspectral resolution enhancement using high-resolution multispectral imagery with arbitrary response functions,” IEEE Trans. Geosci. Remote Sens. 43, 455–465 (2005).
[CrossRef]

2004 (1)

M. T. Eismann and R. C. Hardie, “Application of the stochastic mixing model to hyperspectral resolution enhancement,” IEEE Trans. Geosci. Remote Sens. 42, 1924–1933 (2004).
[CrossRef]

2003 (1)

2002 (5)

A. Minghelli-Roman, M. Mangolini, M. Petit, and L. Polodori, “Spatial resolution improvement of MeRIS images by fusion with TM images,” IEEE Trans. Geosci. Remote Sens. 39, 1533–1536 (2002).
[CrossRef]

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, “Nonlinear estimation of spectral reflectance based on Gaussian mixture distribution for color image reproduction,” Appl. Opt. 41, 4840–4847 (2002).
[CrossRef] [PubMed]

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

D. Dupont, “Study of the reconstruction of reflectance curves based on tristimulus values: comparison of methods of optimization,” Color Res. Appl. 27, 88–99 (2002).
[CrossRef]

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2002).
[CrossRef]

2000 (2)

1999 (3)

M. Hauta-Kasari, K. Miyazawa, S. Toyooka, and J. Parkkinen, “Spectral vision system for measuring color images,” J. Opt. Soc. Am. A 16, 2352–2362 (1999).
[CrossRef]

B. Zhukov, D. Oertel, F. Lanzl, and G. Reinhackel, “Unmixing-based multisensory multiresolution image fusion,” IEEE Trans. Geosci. Remote Sens. 37, 1212–1226 (1999).
[CrossRef]

B. Hill, “Color capture, color management and the problem of metamerism,” Proc. SPIE 3963, 2–14 (1999).
[CrossRef]

1997 (1)

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, “Natural color reproduction in the television system for telemedicine,” Proc. SPIE 3031, 482–289 (1997).
[CrossRef]

1994 (1)

D. P. Filiberti, S. E. Marsh, and R. A. Schowengerdt, “Synthesis of imagery with high spatial and spectral resolution from multiple image sources,” Opt. Eng. 33, 2520–2528 (1994).
[CrossRef]

1987 (1)

J. C. Price, “Combining panchromatic and multispectral imagery from dual resolution satellite instruments,” Remote Sens. Environ. 21, 119–128 (1987).
[CrossRef]

Averbuch, A.

Ayala, F.

Berns, P. D.

P. D. Berns and R. S. Berns, “Analysis of multispectral image capture,” in Proceedings of the 4th Color Imaging Conference (IS&T, 1996), pp. 19–22.

Berns, R. S.

P. D. Berns and R. S. Berns, “Analysis of multispectral image capture,” in Proceedings of the 4th Color Imaging Conference (IS&T, 1996), pp. 19–22.

Brady, D.

Brettel, H.

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

Cai, P.

Chinea, J. D.

A. Santos-Garcia, M. Velez-Reyez, S. Rosario-Torres, and J. D. Chinea, “A comparison of unmixing algorithms for hyperspectral imagery,” Proc. SPIE 7334, 73341N (2009).
[CrossRef]

Clevers, J. G. P. W.

R. Zurita-Milla, J. G. P. W. Clevers, and M. E. Schaepman, “Unmixing-based Landsat TM and MERIS FR data fusion,” IEEE Geosci. Remote Sens. Lett. 5, 453–457 (2008).
[CrossRef]

Dávila, C. A.

DiCarlo, J. M.

Dupont, D.

D. Dupont, “Study of the reconstruction of reflectance curves based on tristimulus values: comparison of methods of optimization,” Color Res. Appl. 27, 88–99 (2002).
[CrossRef]

Echavarri, J. F.

Eismann, M. T.

M. T. Eismann and R. C. Hardie, “Hyperspectral resolution enhancement using high-resolution multispectral imagery with arbitrary response functions,” IEEE Trans. Geosci. Remote Sens. 43, 455–465 (2005).
[CrossRef]

M. T. Eismann and R. C. Hardie, “Application of the stochastic mixing model to hyperspectral resolution enhancement,” IEEE Trans. Geosci. Remote Sens. 42, 1924–1933 (2004).
[CrossRef]

Filiberti, D. P.

D. P. Filiberti, S. E. Marsh, and R. A. Schowengerdt, “Synthesis of imagery with high spatial and spectral resolution from multiple image sources,” Opt. Eng. 33, 2520–2528 (1994).
[CrossRef]

Finlayson, G. D.

Fukuda, H.

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, “Development of 16-band multispectral image archiving system,” Proc. SPIE 5667, 136–145 (2005).
[CrossRef]

Gao, L.

Gehm, M. E.

Golub, M. A.

Hagen, N.

Haneishi, H.

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, “Development of 16-band multispectral image archiving system,” Proc. SPIE 5667, 136–145 (2005).
[CrossRef]

H. Haneishi, T. Hasegawa, A. Hosoi, Y. Yokoyama, N. Tsumura, and Y. Miyake, “System design for accurately estimating spectral reflectance of art paintings,” Appl. Opt. 39, 6621–6632 (2000).
[CrossRef]

Hardeberg, J. Y.

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

Hardie, R. C.

M. T. Eismann and R. C. Hardie, “Hyperspectral resolution enhancement using high-resolution multispectral imagery with arbitrary response functions,” IEEE Trans. Geosci. Remote Sens. 43, 455–465 (2005).
[CrossRef]

M. T. Eismann and R. C. Hardie, “Application of the stochastic mixing model to hyperspectral resolution enhancement,” IEEE Trans. Geosci. Remote Sens. 42, 1924–1933 (2004).
[CrossRef]

Hasegawa, T.

Hauta-Kasari, M.

Heikkinen, V.

Hideaki, H.

M. Yamaguchi, H. Hideaki, and N. Ohyama, “Beyond red-green-blue (RGB): spectrum-based color imaging technology,” J. Imaging Sci. Technol. 52, 010201 (2008).
[CrossRef]

Hill, B.

B. Hill, “Color capture, color management and the problem of metamerism,” Proc. SPIE 3963, 2–14 (1999).
[CrossRef]

Hosoi, A.

Hunt, B. R.

Ietomi, K.

Y. Murakami, K. Ietomi, M. Yamaguchi, and N. Ohyama, “MAP estimation of spectral reflectance from color image and multipoint spectral measurements,” Appl. Opt. 46, 7068–7082(2007).
[CrossRef] [PubMed]

Y. Murakami, K. Ietomi, A. Tadano, M. Yamaguchi, and N. Ohyama, “Comparison of spectral image reconstruction methods using multipoint spectral measurements,” in Proceedings of the 4th European Conference on Colour in Graphics, Imaging, and Vision (IS&T, 2008), pp. 591–596.

K. Ietomi, Y. Murakami, M. Yamaguchi, and N. Ohyama, “MAP estimation for spectral image reconstruction using 3-band image and multipoint spectral measurements,” in Proceedings of the 9th International Symposium on Multispectral Colour Science and Application (IS&T, 2007), pp. 53–59.

Iwama, R.

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, “Natural color reproduction in the television system for telemedicine,” Proc. SPIE 3031, 482–289 (1997).
[CrossRef]

Jaaskelainen, T.

Jetsu, T.

John, R.

Kester, R. T.

Kohonen, O.

O. Kohonen, “Multiresolution-based pansharpening in spectral color images,” in Proceedings of the 5th European Conference on Colour in Graphics, Imaging, and Vision (IS&T, 2010), pp. 535–540.

Komiya, Y.

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, “Natural color reproduction in the television system for telemedicine,” Proc. SPIE 3031, 482–289 (1997).
[CrossRef]

Lanzl, F.

B. Zhukov, D. Oertel, F. Lanzl, and G. Reinhackel, “Unmixing-based multisensory multiresolution image fusion,” IEEE Trans. Geosci. Remote Sens. 37, 1212–1226 (1999).
[CrossRef]

Lavi, E.

Lee, S. D.

Mangolini, M.

A. Minghelli-Roman, M. Mangolini, M. Petit, and L. Polodori, “Spatial resolution improvement of MeRIS images by fusion with TM images,” IEEE Trans. Geosci. Remote Sens. 39, 1533–1536 (2002).
[CrossRef]

Marsh, S. E.

D. P. Filiberti, S. E. Marsh, and R. A. Schowengerdt, “Synthesis of imagery with high spatial and spectral resolution from multiple image sources,” Opt. Eng. 33, 2520–2528 (1994).
[CrossRef]

Minghelli-Roman, A.

A. Minghelli-Roman, M. Mangolini, M. Petit, and L. Polodori, “Spatial resolution improvement of MeRIS images by fusion with TM images,” IEEE Trans. Geosci. Remote Sens. 39, 1533–1536 (2002).
[CrossRef]

Miyake, Y.

Miyazawa, K.

Morovic, P.

Motomura, H.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2002).
[CrossRef]

Murakami, Y.

Y. Murakami, M. Yamaguchi, and N. Ohyama, “Piecewise Wiener estimation for reconstruction of spectral reflectance image by multipoint spectral measurements,” Appl. Opt. 48, 2188–2202(2009).
[CrossRef] [PubMed]

Y. Murakami, K. Ietomi, M. Yamaguchi, and N. Ohyama, “MAP estimation of spectral reflectance from color image and multipoint spectral measurements,” Appl. Opt. 46, 7068–7082(2007).
[CrossRef] [PubMed]

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, “Nonlinear estimation of spectral reflectance based on Gaussian mixture distribution for color image reproduction,” Appl. Opt. 41, 4840–4847 (2002).
[CrossRef] [PubMed]

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2002).
[CrossRef]

K. Ietomi, Y. Murakami, M. Yamaguchi, and N. Ohyama, “MAP estimation for spectral image reconstruction using 3-band image and multipoint spectral measurements,” in Proceedings of the 9th International Symposium on Multispectral Colour Science and Application (IS&T, 2007), pp. 53–59.

Y. Murakami, K. Ietomi, A. Tadano, M. Yamaguchi, and N. Ohyama, “Comparison of spectral image reconstruction methods using multipoint spectral measurements,” in Proceedings of the 4th European Conference on Colour in Graphics, Imaging, and Vision (IS&T, 2008), pp. 591–596.

Nathan, M.

Obi, T.

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, “Nonlinear estimation of spectral reflectance based on Gaussian mixture distribution for color image reproduction,” Appl. Opt. 41, 4840–4847 (2002).
[CrossRef] [PubMed]

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, “Natural color reproduction in the television system for telemedicine,” Proc. SPIE 3031, 482–289 (1997).
[CrossRef]

Oertel, D.

B. Zhukov, D. Oertel, F. Lanzl, and G. Reinhackel, “Unmixing-based multisensory multiresolution image fusion,” IEEE Trans. Geosci. Remote Sens. 37, 1212–1226 (1999).
[CrossRef]

Ohsawa, K.

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2002).
[CrossRef]

Ohya, Y.

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, “Natural color reproduction in the television system for telemedicine,” Proc. SPIE 3031, 482–289 (1997).
[CrossRef]

Ohyama, N.

Y. Murakami, M. Yamaguchi, and N. Ohyama, “Piecewise Wiener estimation for reconstruction of spectral reflectance image by multipoint spectral measurements,” Appl. Opt. 48, 2188–2202(2009).
[CrossRef] [PubMed]

M. Yamaguchi, H. Hideaki, and N. Ohyama, “Beyond red-green-blue (RGB): spectrum-based color imaging technology,” J. Imaging Sci. Technol. 52, 010201 (2008).
[CrossRef]

Y. Murakami, K. Ietomi, M. Yamaguchi, and N. Ohyama, “MAP estimation of spectral reflectance from color image and multipoint spectral measurements,” Appl. Opt. 46, 7068–7082(2007).
[CrossRef] [PubMed]

H. Fukuda, T. Uchiyama, H. Haneishi, M. Yamaguchi, and N. Ohyama, “Development of 16-band multispectral image archiving system,” Proc. SPIE 5667, 136–145 (2005).
[CrossRef]

M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on the multispectral and multiprimary imaging: experimental evaluation,” Proc. SPIE 4663, 15–26 (2002).
[CrossRef]

Y. Murakami, T. Obi, M. Yamaguchi, and N. Ohyama, “Nonlinear estimation of spectral reflectance based on Gaussian mixture distribution for color image reproduction,” Appl. Opt. 41, 4840–4847 (2002).
[CrossRef] [PubMed]

M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, “Natural color reproduction in the television system for telemedicine,” Proc. SPIE 3031, 482–289 (1997).
[CrossRef]

K. Ietomi, Y. Murakami, M. Yamaguchi, and N. Ohyama, “MAP estimation for spectral image reconstruction using 3-band image and multipoint spectral measurements,” in Proceedings of the 9th International Symposium on Multispectral Colour Science and Application (IS&T, 2007), pp. 53–59.

Y. Murakami, K. Ietomi, A. Tadano, M. Yamaguchi, and N. Ohyama, “Comparison of spectral image reconstruction methods using multipoint spectral measurements,” in Proceedings of the 4th European Conference on Colour in Graphics, Imaging, and Vision (IS&T, 2008), pp. 591–596.

Parkkinen, J.

Petit, M.

A. Minghelli-Roman, M. Mangolini, M. Petit, and L. Polodori, “Spatial resolution improvement of MeRIS images by fusion with TM images,” IEEE Trans. Geosci. Remote Sens. 39, 1533–1536 (2002).
[CrossRef]

Polodori, L.

A. Minghelli-Roman, M. Mangolini, M. Petit, and L. Polodori, “Spatial resolution improvement of MeRIS images by fusion with TM images,” IEEE Trans. Geosci. Remote Sens. 39, 1533–1536 (2002).
[CrossRef]

Price, J. C.

J. C. Price, “Combining panchromatic and multispectral imagery from dual resolution satellite instruments,” Remote Sens. Environ. 21, 119–128 (1987).
[CrossRef]

Reinhackel, G.

B. Zhukov, D. Oertel, F. Lanzl, and G. Reinhackel, “Unmixing-based multisensory multiresolution image fusion,” IEEE Trans. Geosci. Remote Sens. 37, 1212–1226 (1999).
[CrossRef]

Renet, P.

Rosario-Torres, S.

A. Santos-Garcia, M. Velez-Reyez, S. Rosario-Torres, and J. D. Chinea, “A comparison of unmixing algorithms for hyperspectral imagery,” Proc. SPIE 7334, 73341N (2009).
[CrossRef]

Santos-Garcia, A.

A. Santos-Garcia, M. Velez-Reyez, S. Rosario-Torres, and J. D. Chinea, “A comparison of unmixing algorithms for hyperspectral imagery,” Proc. SPIE 7334, 73341N (2009).
[CrossRef]

Schaepman, M. E.

R. Zurita-Milla, J. G. P. W. Clevers, and M. E. Schaepman, “Unmixing-based Landsat TM and MERIS FR data fusion,” IEEE Geosci. Remote Sens. Lett. 5, 453–457 (2008).
[CrossRef]

Schclar, A.

Schmitt, F.

J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. 41, 2532–2548 (2002).
[CrossRef]

Schowengerdt, R. A.

D. P. Filiberti, S. E. Marsh, and R. A. Schowengerdt, “Synthesis of imagery with high spatial and spectral resolution from multiple image sources,” Opt. Eng. 33, 2520–2528 (1994).
[CrossRef]

Schulz, T. J.

Shao, S.

Shen, H.

Tadano, A.

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Figures (13)

Fig. 1
Fig. 1

Two types of observations are assumed in this paper: B-band image without spatial deterioration and low-resolution spectral image without spectral deterioration. The virtual low-resolution B-band image is obtained from the B-band image, which appears in Section 3B.

Fig. 2
Fig. 2

Conceptual diagram of spectra belonging to classes and measured low-resolution spectral data in two-dimensional spectral space. (a) Two classes span one-dimensional subspaces, respectively. (b) Accurate estimation is possible if class identification is available and the subspace of each class is known. (c) Pure-spectrum case: low-resolution spectral data distributed in either subspace of the two classes. (d) Mixed-spectrum case: low-resolution spectral data distributed between subspaces of the two classes.

Fig. 3
Fig. 3

Conceptual diagram of the proposed method in a model case of M = 3 , B = 1 , and K = 2 . Star symbols are low-resolution spectral data, closed circles (class #1) and diamonds (class #2) are component spectra, and the open circles/diamonds are one-dimensional observations of component spectra. The unknown subspace of each class is determined by fitting low-resolution spectral data subject to the observations with different mixture ratios.

Fig. 4
Fig. 4

Spectral reflectance images of “Toys” (left) and “Scarf” (right) used in the simulations are presented in color images: original whole image (top), three 112 × 112 test images (middle), and three 7 × 7 low-resolution spectral images (bottom). The rectangles in the original whole images indicate the locations of the three test images.

Fig. 5
Fig. 5

NRMSE of the spectral reflectance (left column) and the average CIELAB error under D65 illuminant (right column) for the three test images from “Toys.” Closed and open circles are the averages and standard deviations over 50 trials, respectively. Large open circles indicate the selected number of classes for the results in Fig. 9.

Fig. 6
Fig. 6

NRMSE of the spectral reflectance (left column) and the average CIELAB error under D65 illuminant (right column) for the three test images from “Scarf.” Closed and open circles are the averages and standard deviations over 50 trials, respectively. Large open circles indicate the selected number of classes for the results in Fig. 10.

Fig. 7
Fig. 7

Visualized NRMSE images (bottom) and the corresponding clustering results (top) of “Toys-1” with the proposed method while changing the number of classes from one to six. The result of “ Number of Classes = 1 ” corresponds to the normal regression-based estimation. The NRMSE value of each image is indicated, and the minimum NRMSE is obtained at “ Number of Classes = 3 ”. The original image in the sRGB representation is shown in upper left for reference.

Fig. 8
Fig. 8

Visualized NRMSE images (bottom) and the corresponding clustering results (top) of “Scarf-2” with the proposed method while changing the number of classes from one to six. The result of “ Number of Classes = 1 ” corresponds to the normal regression-based estimation. The NRMSE value of each image is indicated, and the minimum NRMSE is obtained at “ Number of Classes = 4 ”. The original image in the sRGB representation is shown in the upper left as reference.

Fig. 9
Fig. 9

Comparison between the proposed method (multiclass regression) and the conventional method (regression) for the RMSE of the spectral reflectance (top) and the average CIELAB error under D65 illuminant (bottom) for three test images from “Toys.” Both are calculated for the pixels assigned to every class, and the optimum number of classes is selected approximately. The number of pixels in each class is shown by bar graph (right axis).

Fig. 10
Fig. 10

Comparison between the proposed method (multiclass regression) and the conventional method (regression) for the RMSE of the spectral reflectance (top) and the average CIELAB error under D65 illuminant (bottom) for three test images from “Scarf.” Both are calculated for the pixels assigned to every class, and the optimum number of classes is selected approximately. The number of pixels in each class is shown by the bar graph (right axis).

Fig. 11
Fig. 11

Average spectral reflectance functions estimated by the proposed method (multiclass regression) and the conventional method (regression) for class # 3 / 3 ( B ) (middle) and # 2 / 3 ( G ) (bottom) of test image “Toys-1.” Original color image (top left) and clustering results (top right) are shown for reference.

Fig. 12
Fig. 12

Average spectral reflectance functions estimated by the proposed method (multiclass regression) and the conventional method (regression) for class # 1 / 4 ( R ) (middle) and # 3 / 4 ( B ) (bottom) of test image “Scarf-2.” Original color image (top left) and clustering results (top right) are shown for reference.

Fig. 13
Fig. 13

Average spectral reflectance functions estimated by the proposed method (multiclass regression) and the conventional method (regression) for class # 1 / 5 ( R ) (middle) and # 5 / 5 ( C ) (bottom) of test image “Scarf-3”. Original color image (top left) and clustering results (top right) are shown as reference.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

F = [ f ( 1 ) , f ( 2 ) , , f ( i ) , , f ( N ) ] .
g ( i ) = H G f ( i ) + ε G ( i ) ,
G = H G F + E G ,
s ( j ) = i Ω j f ( i ) + ε S ( j ) ,
[ H S ] i j = { 1 i Ω j 0 else ,
S = F H S + E S ,
f ^ ( i ) = A g ( i ) .
T = G H S ,
T = ( H G F + E G ) H S = H G ( S E S ) + E G H S = H G S + ( E G H S H G E S ) .
Φ regression = j = 1 M s ( j ) A regression t ( j ) 2 .
A regression = S T T ( T T T ) 1 .
f ^ regression ( i ) = A regression g ( i ) .
f ^ # k ( i ) = A # k g ( i ) | k ,
g ˜ ( i ) = ( δ i 1 g ( i ) δ i K g ( i ) ) ,
δ i k = { 1 if     g ( i ) class # k 0 else ,
G ˜ = [ g ˜ ( 1 ) , , g ˜ ( N ) ] .
f ^ # k ( i ) = A ˜ g ˜ ( i ) ,
A ˜ = [ A # 1 A # K ] .
T ˜ = G ˜ H S ,
Φ unmix = j = 1 M s ( j ) A ˜ t ˜ ( j ) 2 .
A ˜ = S T ˜ T ( T ˜ T ˜ T ) 1 .
G ˜ example = [ g ( 1 ) 0 g ( 3 ) g ( 4 ) 0 0 g ( 7 ) 0 0 0 g ( 2 ) 0 0 g ( 5 ) g ( 6 ) 0 g ( 8 ) g ( 9 ) ] .
T ˜ example = [ g ( 1 ) + g ( 3 ) g ( 4 ) g ( 7 ) g ( 2 ) g ( 5 ) + g ( 6 ) g ( 8 ) + g ( 9 ) ] ,
H example = [ 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 ] T .
T ˜ T ˜ T | pure = [ T # 1 T # 1 T T # 2 T # 2 T T # K T # K T ] ,
A # k | pure = S # k T # k ( T # k T # k T ) 1 .

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