Abstract

We present a general stochastic model for hyperspectral imaging data and derive analytical expressions for the Fisher information matrix for the underlying spectral unmixing problem. We investigate the linear mixing model as a special case and define a linear unmixing performance bound by using the Cramer-Rao inequality. As an application, we consider fluorescence imaging and show how the performance bound provides a spectral resolution limit that predicts how accurately a pair of spectrally similar fluorescent labels can be spectrally unmixed. We also report a novel result that shows how the spectral resolution limit can be overcome by exploiting the phenomenon of anti-Stokes shift fluorescence. In addition, we investigate how photon statistics, channel addition and channel splitting affect the performance bound. Finally by using the performance bound as a benchmark, we compare the performance of the least squares and the maximum likelihood estimators for spectral unmixing. For the imaging conditions tested here, our analysis shows that both estimators are unbiased and that the standard deviation of the maximum likelihood estimator is consistently closer to the performance bound than that of the least squares estimator. The results presented here are based on broad assumptions regarding the underlying data model and are applicable to hyperspectral data acquired with point detectors, sCMOS, CCD and EMCCD imaging detectors.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (2)

F. Cutrale, V. Trivedi, L. A. Trinh, C. Chiu, J. M. Choi, M. S. Artiga, and S. E. Fraser, “Hyperspectral phasor analysis enables multiplexed 5D in vivo imaging,” Nat. Methods 14, 149–152 (2017).
[Crossref] [PubMed]

S. Wilhelm, “Perspectives for upconverting nanoparticles,” ACS Nano,  2017,10644–10653 (2017).
[Crossref] [PubMed]

2014 (5)

Y. Li, G. Dong, M. Peng, L. Wondraczek, and J. Qiu,“Anti-Stokes Fluorescent Probe with Incoherent Excitation,” Sci. Rep. 4, 4059 (2014).
[Crossref]

T. Zimmermann, J. Marrison, K. Hogg, and P. O’Toole, “Clearing up the signal: spectral imaging and linear unmixing in fluorescence microscopy,” Methods Mol. Biol. 1075, 129–148 (2014).
[Crossref]

P. Favreau, C. Hernandez, A. S. Lindsey, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Thin-film tunable filters for hyperspectral fluorescence microscopy,” J. Biomed. Opt. 19, 011017, 1–11 (2014).

P. Favreau, C. Hernandez, T. Heaster, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Excitation scanning hyperspectral-imaging microscope,” J. Biomed. Opt. 19, 046010, 1–11 (2014).
[Crossref]

E. Stack, C. Wang, K. A. Roman, and C. C. Hoyt, “Multiplexed immunohistochemistry, imaging, and quantitation: A review, with an assessment of tyramide signal amplification, multispectral imaging and multiplex analysis,” Methods 70, 46–58 (2014).
[Crossref] [PubMed]

2013 (4)

E. Chouzenoux, M. Legendre, S. Moussaoui, and J. Idier, “Fast constrained least squares spectral unmixing using primal-dual interior-point optimization,” IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens. 7, 59–69 (2013).
[Crossref]

A. Esposito, M. Popleteeva, and A. R. venkitaraman, “Maximizing the biochemical resolving power of fluorescence microscopy,” PLoS ONE 8, e77392 (2013).
[Crossref] [PubMed]

P. J. Cutler, M. D. Malik, S. Liu, J. M. Byars, D. S. Lidke, and K. A. Lidke, “Multicolor quantum dot tracking using a high-speed hyperspectral line-scanning microscope,” PLoS ONE 8, e64320 (2013).
[Crossref]

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10, 335–338 (2013).
[Crossref] [PubMed]

2012 (3)

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidimens. Syst. Signal Process. 23, 349–379 (2012).
[Crossref] [PubMed]

A. Gnach and A. Bednarkiewicz, “Lanthanide-doped up-converting nanoparticles: merits and challenges,” Nano Today 47, 532–563 (2012).
[Crossref]

F. Fereidouni, A. N. Bader, and H. C. Gerritsen, “Spectral phasor analysis allows rapid and reliable unmixing of fluorescence microscopy spectral images,” Opt. Express 20, 12729 (2012).
[Crossref] [PubMed]

2009 (1)

2008 (1)

J. R. Mansfield, C. Hoyt, and R. M. Levenson, “Visualization of microscopy-based spectral imaging data from multi-label tissue sections,” Curr. Protoc. Mol. Biol. 84, 14 (2008).

2007 (1)

C. A. Taylan, C. Fevotte, and S. J. Godsill, “Variational and stochastic inference for Bayesian source separation,” Digit. Signal Process. 17, 891–913 (2007).
[Crossref]

2006 (4)

G. McNamara, A. Gupta, J. Reynaert, T. D. Coates, and C. Boswell, “Spectral imaging microscopy web sites and data,” Cytom. Part A 5, 121–132 (2006).

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–58 (2006).
[Crossref]

Y. Garini, I. T. Young, and G. McNamara, “Spectral imaging: principles and applications,” Cytom. Part A 69A, 735–747 (2006).
[Crossref]

T. J. Fountaine, S. M. Wincovitch, D. H. Geho, S. H. Garfield, and S. Pittaluga, “Multispectral imaging of clinically relevant cellular targets in tonsil and lymphoid tissue using semiconductor quantum dots,” Mod. Pathol. 19, 1181–1191 (2006).
[Crossref] [PubMed]

2004 (3)

R. Neher and E. Neher, “Optimizing imaging parameters for the separation of multiple labels in a fluorescence image,” J. Microsc. 213, 46–62 (2004).
[Crossref]

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, and D. L. Snyder, “Spectrum estimation from quantum limited interferograms,” IEEE Transactions on signal processing 52, 950–961 (2004).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

2001 (2)

D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Transactions on Geosci. Remote. Sens. 39, 529–545 (2001).
[Crossref]

M. E. Dickinson, G. Bearman, S. Tille, R. Lansford, and S. E. Fraser, “Multispectral imaging and linear unmixing add a whole new dimension to laser scanning fluorescence microscopy,” Biotechniques 31, 1272–1278 (2001).
[Crossref]

2000 (1)

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

1998 (1)

1990 (1)

J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimation with constraints,” IEEE Transactions on Inf. Theorey 26, 1285–1301 (1990).
[Crossref]

Abraham, A. V.

Achard, V.

M. Cubero-Castan, J. Chanussot, X. Briottet, M. Shimoni, and V. Achard, “An unmixing-based method for the analysis of thermal hyperspectral images,” in IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2014), pp. 7859–7863.

Alvarez, D. F.

P. Favreau, C. Hernandez, T. Heaster, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Excitation scanning hyperspectral-imaging microscope,” J. Biomed. Opt. 19, 046010, 1–11 (2014).
[Crossref]

P. Favreau, C. Hernandez, A. S. Lindsey, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Thin-film tunable filters for hyperspectral fluorescence microscopy,” J. Biomed. Opt. 19, 011017, 1–11 (2014).

Artiga, M. S.

F. Cutrale, V. Trivedi, L. A. Trinh, C. Chiu, J. M. Choi, M. S. Artiga, and S. E. Fraser, “Hyperspectral phasor analysis enables multiplexed 5D in vivo imaging,” Nat. Methods 14, 149–152 (2017).
[Crossref] [PubMed]

Atkinson, A.

A. Atkinson, A. Donev, and R. Tobias, Optimum Experimental Design (Oxford University Press, 2007).

Axelrod, D. E.

M. Kimmel and D. E. Axelrod, Branching processes in biology.(Springer, 2002).
[Crossref]

Bader, A. N.

Bearman, G.

M. E. Dickinson, G. Bearman, S. Tille, R. Lansford, and S. E. Fraser, “Multispectral imaging and linear unmixing add a whole new dimension to laser scanning fluorescence microscopy,” Biotechniques 31, 1272–1278 (2001).
[Crossref]

Bednarkiewicz, A.

A. Gnach and A. Bednarkiewicz, “Lanthanide-doped up-converting nanoparticles: merits and challenges,” Nano Today 47, 532–563 (2012).
[Crossref]

Bialkowski, S. E.

Boswell, C.

G. McNamara, A. Gupta, J. Reynaert, T. D. Coates, and C. Boswell, “Spectral imaging microscopy web sites and data,” Cytom. Part A 5, 121–132 (2006).

Briottet, X.

M. Cubero-Castan, J. Chanussot, X. Briottet, M. Shimoni, and V. Achard, “An unmixing-based method for the analysis of thermal hyperspectral images,” in IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2014), pp. 7859–7863.

Byars, J. M.

P. J. Cutler, M. D. Malik, S. Liu, J. M. Byars, D. S. Lidke, and K. A. Lidke, “Multicolor quantum dot tracking using a high-speed hyperspectral line-scanning microscope,” PLoS ONE 8, e64320 (2013).
[Crossref]

Chang, C.-I.

D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Transactions on Geosci. Remote. Sens. 39, 529–545 (2001).
[Crossref]

Chanussot, J.

M. Cubero-Castan, J. Chanussot, X. Briottet, M. Shimoni, and V. Achard, “An unmixing-based method for the analysis of thermal hyperspectral images,” in IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2014), pp. 7859–7863.

Chao, J.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10, 335–338 (2013).
[Crossref] [PubMed]

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidimens. Syst. Signal Process. 23, 349–379 (2012).
[Crossref] [PubMed]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17, 23352–23373 (2009).
[Crossref]

Chiu, C.

F. Cutrale, V. Trivedi, L. A. Trinh, C. Chiu, J. M. Choi, M. S. Artiga, and S. E. Fraser, “Hyperspectral phasor analysis enables multiplexed 5D in vivo imaging,” Nat. Methods 14, 149–152 (2017).
[Crossref] [PubMed]

Choi, J. M.

F. Cutrale, V. Trivedi, L. A. Trinh, C. Chiu, J. M. Choi, M. S. Artiga, and S. E. Fraser, “Hyperspectral phasor analysis enables multiplexed 5D in vivo imaging,” Nat. Methods 14, 149–152 (2017).
[Crossref] [PubMed]

Chouzenoux, E.

E. Chouzenoux, M. Legendre, S. Moussaoui, and J. Idier, “Fast constrained least squares spectral unmixing using primal-dual interior-point optimization,” IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens. 7, 59–69 (2013).
[Crossref]

Coates, T. D.

G. McNamara, A. Gupta, J. Reynaert, T. D. Coates, and C. Boswell, “Spectral imaging microscopy web sites and data,” Cytom. Part A 5, 121–132 (2006).

Cubero-Castan, M.

M. Cubero-Castan, J. Chanussot, X. Briottet, M. Shimoni, and V. Achard, “An unmixing-based method for the analysis of thermal hyperspectral images,” in IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2014), pp. 7859–7863.

Cutler, P. J.

P. J. Cutler, M. D. Malik, S. Liu, J. M. Byars, D. S. Lidke, and K. A. Lidke, “Multicolor quantum dot tracking using a high-speed hyperspectral line-scanning microscope,” PLoS ONE 8, e64320 (2013).
[Crossref]

Cutrale, F.

F. Cutrale, V. Trivedi, L. A. Trinh, C. Chiu, J. M. Choi, M. S. Artiga, and S. E. Fraser, “Hyperspectral phasor analysis enables multiplexed 5D in vivo imaging,” Nat. Methods 14, 149–152 (2017).
[Crossref] [PubMed]

Dickinson, M. E.

M. E. Dickinson, G. Bearman, S. Tille, R. Lansford, and S. E. Fraser, “Multispectral imaging and linear unmixing add a whole new dimension to laser scanning fluorescence microscopy,” Biotechniques 31, 1272–1278 (2001).
[Crossref]

Donev, A.

A. Atkinson, A. Donev, and R. Tobias, Optimum Experimental Design (Oxford University Press, 2007).

Dong, G.

Y. Li, G. Dong, M. Peng, L. Wondraczek, and J. Qiu,“Anti-Stokes Fluorescent Probe with Incoherent Excitation,” Sci. Rep. 4, 4059 (2014).
[Crossref]

Esposito, A.

A. Esposito, M. Popleteeva, and A. R. venkitaraman, “Maximizing the biochemical resolving power of fluorescence microscopy,” PLoS ONE 8, e77392 (2013).
[Crossref] [PubMed]

Favreau, P.

P. Favreau, C. Hernandez, T. Heaster, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Excitation scanning hyperspectral-imaging microscope,” J. Biomed. Opt. 19, 046010, 1–11 (2014).
[Crossref]

P. Favreau, C. Hernandez, A. S. Lindsey, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Thin-film tunable filters for hyperspectral fluorescence microscopy,” J. Biomed. Opt. 19, 011017, 1–11 (2014).

Fereidouni, F.

Fevotte, C.

C. A. Taylan, C. Fevotte, and S. J. Godsill, “Variational and stochastic inference for Bayesian source separation,” Digit. Signal Process. 17, 891–913 (2007).
[Crossref]

Fountaine, T. J.

T. J. Fountaine, S. M. Wincovitch, D. H. Geho, S. H. Garfield, and S. Pittaluga, “Multispectral imaging of clinically relevant cellular targets in tonsil and lymphoid tissue using semiconductor quantum dots,” Mod. Pathol. 19, 1181–1191 (2006).
[Crossref] [PubMed]

Fraser, S. E.

F. Cutrale, V. Trivedi, L. A. Trinh, C. Chiu, J. M. Choi, M. S. Artiga, and S. E. Fraser, “Hyperspectral phasor analysis enables multiplexed 5D in vivo imaging,” Nat. Methods 14, 149–152 (2017).
[Crossref] [PubMed]

M. E. Dickinson, G. Bearman, S. Tille, R. Lansford, and S. E. Fraser, “Multispectral imaging and linear unmixing add a whole new dimension to laser scanning fluorescence microscopy,” Biotechniques 31, 1272–1278 (2001).
[Crossref]

Fuhrmann, D. R.

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, and D. L. Snyder, “Spectrum estimation from quantum limited interferograms,” IEEE Transactions on signal processing 52, 950–961 (2004).
[Crossref]

Garfield, S. H.

T. J. Fountaine, S. M. Wincovitch, D. H. Geho, S. H. Garfield, and S. Pittaluga, “Multispectral imaging of clinically relevant cellular targets in tonsil and lymphoid tissue using semiconductor quantum dots,” Mod. Pathol. 19, 1181–1191 (2006).
[Crossref] [PubMed]

Garini, Y.

Y. Garini, I. T. Young, and G. McNamara, “Spectral imaging: principles and applications,” Cytom. Part A 69A, 735–747 (2006).
[Crossref]

Gat, N.

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

Geho, D. H.

T. J. Fountaine, S. M. Wincovitch, D. H. Geho, S. H. Garfield, and S. Pittaluga, “Multispectral imaging of clinically relevant cellular targets in tonsil and lymphoid tissue using semiconductor quantum dots,” Mod. Pathol. 19, 1181–1191 (2006).
[Crossref] [PubMed]

Gerritsen, H. C.

Gnach, A.

A. Gnach and A. Bednarkiewicz, “Lanthanide-doped up-converting nanoparticles: merits and challenges,” Nano Today 47, 532–563 (2012).
[Crossref]

Godsill, S. J.

C. A. Taylan, C. Fevotte, and S. J. Godsill, “Variational and stochastic inference for Bayesian source separation,” Digit. Signal Process. 17, 891–913 (2007).
[Crossref]

Gorman, J. D.

J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimation with constraints,” IEEE Transactions on Inf. Theorey 26, 1285–1301 (1990).
[Crossref]

Gupta, A.

G. McNamara, A. Gupta, J. Reynaert, T. D. Coates, and C. Boswell, “Spectral imaging microscopy web sites and data,” Cytom. Part A 5, 121–132 (2006).

Heaster, T.

P. Favreau, C. Hernandez, T. Heaster, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Excitation scanning hyperspectral-imaging microscope,” J. Biomed. Opt. 19, 046010, 1–11 (2014).
[Crossref]

Heinz, D. C.

D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Transactions on Geosci. Remote. Sens. 39, 529–545 (2001).
[Crossref]

Hernandez, C.

P. Favreau, C. Hernandez, T. Heaster, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Excitation scanning hyperspectral-imaging microscope,” J. Biomed. Opt. 19, 046010, 1–11 (2014).
[Crossref]

P. Favreau, C. Hernandez, A. S. Lindsey, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Thin-film tunable filters for hyperspectral fluorescence microscopy,” J. Biomed. Opt. 19, 011017, 1–11 (2014).

Hero, A. O.

J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimation with constraints,” IEEE Transactions on Inf. Theorey 26, 1285–1301 (1990).
[Crossref]

Hogg, K.

T. Zimmermann, J. Marrison, K. Hogg, and P. O’Toole, “Clearing up the signal: spectral imaging and linear unmixing in fluorescence microscopy,” Methods Mol. Biol. 1075, 129–148 (2014).
[Crossref]

Hoyt, C.

J. R. Mansfield, C. Hoyt, and R. M. Levenson, “Visualization of microscopy-based spectral imaging data from multi-label tissue sections,” Curr. Protoc. Mol. Biol. 84, 14 (2008).

Hoyt, C. C.

E. Stack, C. Wang, K. A. Roman, and C. C. Hoyt, “Multiplexed immunohistochemistry, imaging, and quantitation: A review, with an assessment of tyramide signal amplification, multispectral imaging and multiplex analysis,” Methods 70, 46–58 (2014).
[Crossref] [PubMed]

Idier, J.

E. Chouzenoux, M. Legendre, S. Moussaoui, and J. Idier, “Fast constrained least squares spectral unmixing using primal-dual interior-point optimization,” IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens. 7, 59–69 (2013).
[Crossref]

Janesick, J. R.

J. R. Janesick, Scientific charge-coupled devices.(SPIE Press, 2000).

Kimmel, M.

M. Kimmel and D. E. Axelrod, Branching processes in biology.(Springer, 2002).
[Crossref]

Lakowicz, J.

J. Lakowicz, Principles of fluorescence spectroscopy(Springer, 2006).
[Crossref]

Lansford, R.

M. E. Dickinson, G. Bearman, S. Tille, R. Lansford, and S. E. Fraser, “Multispectral imaging and linear unmixing add a whole new dimension to laser scanning fluorescence microscopy,” Biotechniques 31, 1272–1278 (2001).
[Crossref]

Leavesley, S. J.

P. Favreau, C. Hernandez, A. S. Lindsey, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Thin-film tunable filters for hyperspectral fluorescence microscopy,” J. Biomed. Opt. 19, 011017, 1–11 (2014).

P. Favreau, C. Hernandez, T. Heaster, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Excitation scanning hyperspectral-imaging microscope,” J. Biomed. Opt. 19, 046010, 1–11 (2014).
[Crossref]

Legendre, M.

E. Chouzenoux, M. Legendre, S. Moussaoui, and J. Idier, “Fast constrained least squares spectral unmixing using primal-dual interior-point optimization,” IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens. 7, 59–69 (2013).
[Crossref]

Levenson, R. M.

J. R. Mansfield, C. Hoyt, and R. M. Levenson, “Visualization of microscopy-based spectral imaging data from multi-label tissue sections,” Curr. Protoc. Mol. Biol. 84, 14 (2008).

Li, Y.

Y. Li, G. Dong, M. Peng, L. Wondraczek, and J. Qiu,“Anti-Stokes Fluorescent Probe with Incoherent Excitation,” Sci. Rep. 4, 4059 (2014).
[Crossref]

Lidke, D. S.

P. J. Cutler, M. D. Malik, S. Liu, J. M. Byars, D. S. Lidke, and K. A. Lidke, “Multicolor quantum dot tracking using a high-speed hyperspectral line-scanning microscope,” PLoS ONE 8, e64320 (2013).
[Crossref]

Lidke, K. A.

P. J. Cutler, M. D. Malik, S. Liu, J. M. Byars, D. S. Lidke, and K. A. Lidke, “Multicolor quantum dot tracking using a high-speed hyperspectral line-scanning microscope,” PLoS ONE 8, e64320 (2013).
[Crossref]

Lindsey, A. S.

P. Favreau, C. Hernandez, A. S. Lindsey, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Thin-film tunable filters for hyperspectral fluorescence microscopy,” J. Biomed. Opt. 19, 011017, 1–11 (2014).

Liu, S.

P. J. Cutler, M. D. Malik, S. Liu, J. M. Byars, D. S. Lidke, and K. A. Lidke, “Multicolor quantum dot tracking using a high-speed hyperspectral line-scanning microscope,” PLoS ONE 8, e64320 (2013).
[Crossref]

Malik, M. D.

P. J. Cutler, M. D. Malik, S. Liu, J. M. Byars, D. S. Lidke, and K. A. Lidke, “Multicolor quantum dot tracking using a high-speed hyperspectral line-scanning microscope,” PLoS ONE 8, e64320 (2013).
[Crossref]

Mansfield, J. R.

J. R. Mansfield, C. Hoyt, and R. M. Levenson, “Visualization of microscopy-based spectral imaging data from multi-label tissue sections,” Curr. Protoc. Mol. Biol. 84, 14 (2008).

Marrison, J.

T. Zimmermann, J. Marrison, K. Hogg, and P. O’Toole, “Clearing up the signal: spectral imaging and linear unmixing in fluorescence microscopy,” Methods Mol. Biol. 1075, 129–148 (2014).
[Crossref]

McNamara, G.

Y. Garini, I. T. Young, and G. McNamara, “Spectral imaging: principles and applications,” Cytom. Part A 69A, 735–747 (2006).
[Crossref]

G. McNamara, A. Gupta, J. Reynaert, T. D. Coates, and C. Boswell, “Spectral imaging microscopy web sites and data,” Cytom. Part A 5, 121–132 (2006).

Miller, M. I.

D. L. Snyder and M. I. Miller, Random point processes in time and space.(Springer, 1991), 2nd ed.
[Crossref]

Moussaoui, S.

E. Chouzenoux, M. Legendre, S. Moussaoui, and J. Idier, “Fast constrained least squares spectral unmixing using primal-dual interior-point optimization,” IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens. 7, 59–69 (2013).
[Crossref]

Neher, E.

R. Neher and E. Neher, “Optimizing imaging parameters for the separation of multiple labels in a fluorescence image,” J. Microsc. 213, 46–62 (2004).
[Crossref]

Neher, R.

R. Neher and E. Neher, “Optimizing imaging parameters for the separation of multiple labels in a fluorescence image,” J. Microsc. 213, 46–62 (2004).
[Crossref]

O’Sullivan, J. A.

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, and D. L. Snyder, “Spectrum estimation from quantum limited interferograms,” IEEE Transactions on signal processing 52, 950–961 (2004).
[Crossref]

O’Toole, P.

T. Zimmermann, J. Marrison, K. Hogg, and P. O’Toole, “Clearing up the signal: spectral imaging and linear unmixing in fluorescence microscopy,” Methods Mol. Biol. 1075, 129–148 (2014).
[Crossref]

Ober, R. J.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10, 335–338 (2013).
[Crossref] [PubMed]

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidimens. Syst. Signal Process. 23, 349–379 (2012).
[Crossref] [PubMed]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17, 23352–23373 (2009).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–58 (2006).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

Pawley, J. B.

J. B. Pawley, Handbook of biological confocal microscopy(Springer, 2006), 3rd ed.
[Crossref]

Peng, M.

Y. Li, G. Dong, M. Peng, L. Wondraczek, and J. Qiu,“Anti-Stokes Fluorescent Probe with Incoherent Excitation,” Sci. Rep. 4, 4059 (2014).
[Crossref]

Pittaluga, S.

T. J. Fountaine, S. M. Wincovitch, D. H. Geho, S. H. Garfield, and S. Pittaluga, “Multispectral imaging of clinically relevant cellular targets in tonsil and lymphoid tissue using semiconductor quantum dots,” Mod. Pathol. 19, 1181–1191 (2006).
[Crossref] [PubMed]

Popleteeva, M.

A. Esposito, M. Popleteeva, and A. R. venkitaraman, “Maximizing the biochemical resolving power of fluorescence microscopy,” PLoS ONE 8, e77392 (2013).
[Crossref] [PubMed]

Prabhat, P.

P. Favreau, C. Hernandez, T. Heaster, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Excitation scanning hyperspectral-imaging microscope,” J. Biomed. Opt. 19, 046010, 1–11 (2014).
[Crossref]

P. Favreau, C. Hernandez, A. S. Lindsey, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Thin-film tunable filters for hyperspectral fluorescence microscopy,” J. Biomed. Opt. 19, 011017, 1–11 (2014).

Preza, C.

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, and D. L. Snyder, “Spectrum estimation from quantum limited interferograms,” IEEE Transactions on signal processing 52, 950–961 (2004).
[Crossref]

Qiu, J.

Y. Li, G. Dong, M. Peng, L. Wondraczek, and J. Qiu,“Anti-Stokes Fluorescent Probe with Incoherent Excitation,” Sci. Rep. 4, 4059 (2014).
[Crossref]

Ram, S.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10, 335–338 (2013).
[Crossref] [PubMed]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17, 23352–23373 (2009).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–58 (2006).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

S. Ram, “Resolution and localization in single molecule microscopy,” Ph.D. thesis, University of Texas at Arlington/University of Texas Southwestern Medical Center at Dallas (2007).

Rao, C. R.

C. R. Rao, Linear statistical inference and its applications.(Wiley, 1965).

Reynaert, J.

G. McNamara, A. Gupta, J. Reynaert, T. D. Coates, and C. Boswell, “Spectral imaging microscopy web sites and data,” Cytom. Part A 5, 121–132 (2006).

Rich, T.

P. Favreau, C. Hernandez, T. Heaster, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Excitation scanning hyperspectral-imaging microscope,” J. Biomed. Opt. 19, 046010, 1–11 (2014).
[Crossref]

P. Favreau, C. Hernandez, A. S. Lindsey, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Thin-film tunable filters for hyperspectral fluorescence microscopy,” J. Biomed. Opt. 19, 011017, 1–11 (2014).

Roman, K. A.

E. Stack, C. Wang, K. A. Roman, and C. C. Hoyt, “Multiplexed immunohistochemistry, imaging, and quantitation: A review, with an assessment of tyramide signal amplification, multispectral imaging and multiplex analysis,” Methods 70, 46–58 (2014).
[Crossref] [PubMed]

Saleh, B.

B. Saleh, Photoelectron statistics(Springer, 1978).
[Crossref]

Shimoni, M.

M. Cubero-Castan, J. Chanussot, X. Briottet, M. Shimoni, and V. Achard, “An unmixing-based method for the analysis of thermal hyperspectral images,” in IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2014), pp. 7859–7863.

Snyder, D. L.

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, and D. L. Snyder, “Spectrum estimation from quantum limited interferograms,” IEEE Transactions on signal processing 52, 950–961 (2004).
[Crossref]

D. L. Snyder and M. I. Miller, Random point processes in time and space.(Springer, 1991), 2nd ed.
[Crossref]

Stack, E.

E. Stack, C. Wang, K. A. Roman, and C. C. Hoyt, “Multiplexed immunohistochemistry, imaging, and quantitation: A review, with an assessment of tyramide signal amplification, multispectral imaging and multiplex analysis,” Methods 70, 46–58 (2014).
[Crossref] [PubMed]

Taylan, C. A.

C. A. Taylan, C. Fevotte, and S. J. Godsill, “Variational and stochastic inference for Bayesian source separation,” Digit. Signal Process. 17, 891–913 (2007).
[Crossref]

Tille, S.

M. E. Dickinson, G. Bearman, S. Tille, R. Lansford, and S. E. Fraser, “Multispectral imaging and linear unmixing add a whole new dimension to laser scanning fluorescence microscopy,” Biotechniques 31, 1272–1278 (2001).
[Crossref]

Tobias, R.

A. Atkinson, A. Donev, and R. Tobias, Optimum Experimental Design (Oxford University Press, 2007).

Trinh, L. A.

F. Cutrale, V. Trivedi, L. A. Trinh, C. Chiu, J. M. Choi, M. S. Artiga, and S. E. Fraser, “Hyperspectral phasor analysis enables multiplexed 5D in vivo imaging,” Nat. Methods 14, 149–152 (2017).
[Crossref] [PubMed]

Trivedi, V.

F. Cutrale, V. Trivedi, L. A. Trinh, C. Chiu, J. M. Choi, M. S. Artiga, and S. E. Fraser, “Hyperspectral phasor analysis enables multiplexed 5D in vivo imaging,” Nat. Methods 14, 149–152 (2017).
[Crossref] [PubMed]

venkitaraman, A. R.

A. Esposito, M. Popleteeva, and A. R. venkitaraman, “Maximizing the biochemical resolving power of fluorescence microscopy,” PLoS ONE 8, e77392 (2013).
[Crossref] [PubMed]

Wang, C.

E. Stack, C. Wang, K. A. Roman, and C. C. Hoyt, “Multiplexed immunohistochemistry, imaging, and quantitation: A review, with an assessment of tyramide signal amplification, multispectral imaging and multiplex analysis,” Methods 70, 46–58 (2014).
[Crossref] [PubMed]

Ward, E. S.

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10, 335–338 (2013).
[Crossref] [PubMed]

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidimens. Syst. Signal Process. 23, 349–379 (2012).
[Crossref] [PubMed]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17, 23352–23373 (2009).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–58 (2006).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

Wilhelm, S.

S. Wilhelm, “Perspectives for upconverting nanoparticles,” ACS Nano,  2017,10644–10653 (2017).
[Crossref] [PubMed]

Wincovitch, S. M.

T. J. Fountaine, S. M. Wincovitch, D. H. Geho, S. H. Garfield, and S. Pittaluga, “Multispectral imaging of clinically relevant cellular targets in tonsil and lymphoid tissue using semiconductor quantum dots,” Mod. Pathol. 19, 1181–1191 (2006).
[Crossref] [PubMed]

Wondraczek, L.

Y. Li, G. Dong, M. Peng, L. Wondraczek, and J. Qiu,“Anti-Stokes Fluorescent Probe with Incoherent Excitation,” Sci. Rep. 4, 4059 (2014).
[Crossref]

Young, I. T.

Y. Garini, I. T. Young, and G. McNamara, “Spectral imaging: principles and applications,” Cytom. Part A 69A, 735–747 (2006).
[Crossref]

Zimmermann, T.

T. Zimmermann, J. Marrison, K. Hogg, and P. O’Toole, “Clearing up the signal: spectral imaging and linear unmixing in fluorescence microscopy,” Methods Mol. Biol. 1075, 129–148 (2014).
[Crossref]

ACS Nano (1)

S. Wilhelm, “Perspectives for upconverting nanoparticles,” ACS Nano,  2017,10644–10653 (2017).
[Crossref] [PubMed]

Appl. Spectrosc. (1)

Biophys. J. (1)

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

Biotechniques (1)

M. E. Dickinson, G. Bearman, S. Tille, R. Lansford, and S. E. Fraser, “Multispectral imaging and linear unmixing add a whole new dimension to laser scanning fluorescence microscopy,” Biotechniques 31, 1272–1278 (2001).
[Crossref]

Curr. Protoc. Mol. Biol. (1)

J. R. Mansfield, C. Hoyt, and R. M. Levenson, “Visualization of microscopy-based spectral imaging data from multi-label tissue sections,” Curr. Protoc. Mol. Biol. 84, 14 (2008).

Cytom. Part A (2)

Y. Garini, I. T. Young, and G. McNamara, “Spectral imaging: principles and applications,” Cytom. Part A 69A, 735–747 (2006).
[Crossref]

G. McNamara, A. Gupta, J. Reynaert, T. D. Coates, and C. Boswell, “Spectral imaging microscopy web sites and data,” Cytom. Part A 5, 121–132 (2006).

Digit. Signal Process. (1)

C. A. Taylan, C. Fevotte, and S. J. Godsill, “Variational and stochastic inference for Bayesian source separation,” Digit. Signal Process. 17, 891–913 (2007).
[Crossref]

IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens. (1)

E. Chouzenoux, M. Legendre, S. Moussaoui, and J. Idier, “Fast constrained least squares spectral unmixing using primal-dual interior-point optimization,” IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens. 7, 59–69 (2013).
[Crossref]

IEEE Transactions on Geosci. Remote. Sens. (1)

D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Transactions on Geosci. Remote. Sens. 39, 529–545 (2001).
[Crossref]

IEEE Transactions on Inf. Theorey (1)

J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimation with constraints,” IEEE Transactions on Inf. Theorey 26, 1285–1301 (1990).
[Crossref]

IEEE Transactions on signal processing (1)

D. R. Fuhrmann, C. Preza, J. A. O’Sullivan, and D. L. Snyder, “Spectrum estimation from quantum limited interferograms,” IEEE Transactions on signal processing 52, 950–961 (2004).
[Crossref]

J. Biomed. Opt. (2)

P. Favreau, C. Hernandez, A. S. Lindsey, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Thin-film tunable filters for hyperspectral fluorescence microscopy,” J. Biomed. Opt. 19, 011017, 1–11 (2014).

P. Favreau, C. Hernandez, T. Heaster, D. F. Alvarez, T. Rich, P. Prabhat, and S. J. Leavesley, “Excitation scanning hyperspectral-imaging microscope,” J. Biomed. Opt. 19, 046010, 1–11 (2014).
[Crossref]

J. Microsc. (1)

R. Neher and E. Neher, “Optimizing imaging parameters for the separation of multiple labels in a fluorescence image,” J. Microsc. 213, 46–62 (2004).
[Crossref]

Methods (1)

E. Stack, C. Wang, K. A. Roman, and C. C. Hoyt, “Multiplexed immunohistochemistry, imaging, and quantitation: A review, with an assessment of tyramide signal amplification, multispectral imaging and multiplex analysis,” Methods 70, 46–58 (2014).
[Crossref] [PubMed]

Methods Mol. Biol. (1)

T. Zimmermann, J. Marrison, K. Hogg, and P. O’Toole, “Clearing up the signal: spectral imaging and linear unmixing in fluorescence microscopy,” Methods Mol. Biol. 1075, 129–148 (2014).
[Crossref]

Mod. Pathol. (1)

T. J. Fountaine, S. M. Wincovitch, D. H. Geho, S. H. Garfield, and S. Pittaluga, “Multispectral imaging of clinically relevant cellular targets in tonsil and lymphoid tissue using semiconductor quantum dots,” Mod. Pathol. 19, 1181–1191 (2006).
[Crossref] [PubMed]

Multidimens. Syst. Signal Process. (2)

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–58 (2006).
[Crossref]

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidimens. Syst. Signal Process. 23, 349–379 (2012).
[Crossref] [PubMed]

Nano Today (1)

A. Gnach and A. Bednarkiewicz, “Lanthanide-doped up-converting nanoparticles: merits and challenges,” Nano Today 47, 532–563 (2012).
[Crossref]

Nat. Methods (2)

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “Ultrahigh accuracy imaging modality for super-localization microscopy,” Nat. Methods 10, 335–338 (2013).
[Crossref] [PubMed]

F. Cutrale, V. Trivedi, L. A. Trinh, C. Chiu, J. M. Choi, M. S. Artiga, and S. E. Fraser, “Hyperspectral phasor analysis enables multiplexed 5D in vivo imaging,” Nat. Methods 14, 149–152 (2017).
[Crossref] [PubMed]

Opt. Express (2)

PLoS ONE (2)

A. Esposito, M. Popleteeva, and A. R. venkitaraman, “Maximizing the biochemical resolving power of fluorescence microscopy,” PLoS ONE 8, e77392 (2013).
[Crossref] [PubMed]

P. J. Cutler, M. D. Malik, S. Liu, J. M. Byars, D. S. Lidke, and K. A. Lidke, “Multicolor quantum dot tracking using a high-speed hyperspectral line-scanning microscope,” PLoS ONE 8, e64320 (2013).
[Crossref]

Proc. SPIE (1)

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

Sci. Rep. (1)

Y. Li, G. Dong, M. Peng, L. Wondraczek, and J. Qiu,“Anti-Stokes Fluorescent Probe with Incoherent Excitation,” Sci. Rep. 4, 4059 (2014).
[Crossref]

Other (10)

J. Lakowicz, Principles of fluorescence spectroscopy(Springer, 2006).
[Crossref]

M. Kimmel and D. E. Axelrod, Branching processes in biology.(Springer, 2002).
[Crossref]

J. R. Janesick, Scientific charge-coupled devices.(SPIE Press, 2000).

B. Saleh, Photoelectron statistics(Springer, 1978).
[Crossref]

D. L. Snyder and M. I. Miller, Random point processes in time and space.(Springer, 1991), 2nd ed.
[Crossref]

J. B. Pawley, Handbook of biological confocal microscopy(Springer, 2006), 3rd ed.
[Crossref]

S. Ram, “Resolution and localization in single molecule microscopy,” Ph.D. thesis, University of Texas at Arlington/University of Texas Southwestern Medical Center at Dallas (2007).

M. Cubero-Castan, J. Chanussot, X. Briottet, M. Shimoni, and V. Achard, “An unmixing-based method for the analysis of thermal hyperspectral images,” in IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2014), pp. 7859–7863.

C. R. Rao, Linear statistical inference and its applications.(Wiley, 1965).

A. Atkinson, A. Donev, and R. Tobias, Optimum Experimental Design (Oxford University Press, 2007).

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Figures (7)

Fig. 1
Fig. 1 Hyperspectral imaging in fluorescence microscopy. Panel A shows a typical configuration of a fluorescence microscope for multicolor imaging application. The sample is selectively illuminated at distinct excitation passbands (excitation filter wheel) and the fluorescence signal is then collected at matched emission passbands (emission filter wheel) on an imaging detector. Panel B shows the emission spectra of fluorescent labels exhibiting significant spectral overlap. Panel C illustrates the spectral unmixing problem where given a hyperspectral data set, i.e. the input image cube, which consists of Nch spectral images each with Np pixels, the goal is to obtain an estimate of the output image cube that represents the relative abundance θk of Ns different fluorescent labels at each pixel in the sample. Here I θ , k is a random vector with probability density pθ,k that is a function of νθ,k, which describes the signal at the kth pixel block in the input image cube (see Section 2.2 for details).
Fig. 2
Fig. 2 The effect of changing the spectral distance on spectral resolution. Panel A shows the normalized excitation (red lines) and emission spectra (blue lines) of two fluorescent labels that are spectrally separated by a distance ds. The panel also shows the spectral emissivity of a metal hallide broadband light source that is used to excite the two labels (black dotted line). Here, for both labels we consider the traditional Stokes shift for their fluorescence emission spectra. Panels B and C show the nLUP bound for labels 1 and 2 at a pixel, respectively, for the Poisson and the Poisson + Gaussian data models and also for the best case scenario pertaining to the Poisson data model. Panel C also shows the mixing matrix pertaining to different values of ds. In panels B and C, the expected photon count for both labels is set to be 500 photons per pixel and for the Poisson + Gaussian data model we consider the mean and standard deviation of the Gaussian noise to be 0 e/pixel and 8 e/pixel, respectively.
FIg. 3
FIg. 3 Improving the spectral resolution by using anti-Stokes shift fluorescences. Panel A shows the normalized excitation (red lines) and emission spectra (blue lines) of two fluorescent labels that are spectrally separated by a distance ds. The panel also shows the spectral emissivity of a metal hallide broadband light source that is used to excite the two labels (black dotted line). Here, label 1 is the same as that shown in Fig. 2, while label 2 is a fluorophore with anti-Stokes shift fluorescence emission, where the peak of its emission spectra is at a shorter wavelength (higher energy) than the peak of its excitation spectra. Panels B and C show the nLUP bounds for labels 1 and 2, respectively, at a pixel for the Poisson and the Poisson + Gaussian data models and also for the best case scenario pertaining to the Poisson data model. Panel C also shows the mixing matrix pertaining to different values of ds. The numerical values used to generate the above plots are identical to those used in Fig. 2.
Fig. 4
Fig. 4 Effect of photon count on the nLUP bound. Panels A and B show the normalized excitation (red lines) and emission (blue lines) spectra for Cy3.5 and Cy3 fluorescent labels, respectively, along with the spectral emissivity of a metal hallide light source (black dotted line). The panels also show the corresponding excitation and emission passbands for each fluorescent label. Panels C and D show the behavior of the nLUP bound for Cy3.5 and Cy3, respectively, at a single pixel as a function of the expected photon count for the Poisson and the Poisson + Gaussian data models. As reference, the panel also shows the nLUP bound for the best case scenario pertaining to the Poisson data model. Here, we assume the expected photon count to be the same for both fluorophores and for the Poisson + Gaussian data model, the mean and standard deviation of the readout noise is set to be 0 e/pixel and 8e/pixel, respectively.
Fig. 5
Fig. 5 Effect of channel addition on the LUP bound. Panels A and B show the normalized excitation (red lines) and emission (blue lines) spectra for Cy3.5 and Cy3 fluorescent labels, respectively, along with the spectral emissivity of a metal hallide light source (black dotted line). The panels also show the excitation passband and the emission passbands pertaining to the different spectral channels. Panel C shows the behavior of the LUP bound for Cy3.5 at a single pixel as a function of the number of channels after channel addition for different data models. Panel D shows the same for Cy3. In Panels C and D, the expected photon count is set to be 3000 for both labels, and the numerical values of the mean and standard deviation of the Gaussian noise component are identical to those used in Fig. 4.
Fig. 6
Fig. 6 The effect of channel splitting on the LUP bound. Panel A shows the behavior of the LUP bound for Cy3.5 at a single pixel as a function of the number of channels after channel splitting for the Poisson and the Poisson + Gaussian data models. Panel B shows the same for Cy3. In Panels A and B, the expected photon count is set to be 3000 for both labels, and for the Poisson + Gaussian data model we assume the mean of the Gaussian noise to be 0 e/pixel and different values of standard deviation as indicated in the legend in panel A.
Fig. 7
Fig. 7 Performance of spectral unmixing algorithms on hyperspectral data. The figure shows the behavior of the LS and ML estimators on simulated hyperspectral data. Panels A and B show the % of relative error of the estimators for Cy3.5 and Cy3 labels,respectively, for different expected photon counts. For both algorithms, we do not impose any non-negativity constraint for the photon counts during estimation. Here we assume that the expected photon count is the same for both fluorescence labels. Panels C and D show the bias of the estimators for Cy3.5 amd Cy3, respectively, for different expected photon counts.

Equations (42)

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I θ = { I θ , 1 , I θ , 2 , , I θ , N p } ,     θ Θ ,
p θ , k j ( z ) = 1 2 π σ k j exp   ( ( z ( ν θ , k j + η k j ) ) 2 2 ( σ k j ) 2 ) ,   z ,
p θ , k j ( z ) = e ν θ , k j ( ν θ , k j ) z z ! ,   z = 0 , 1 , 2 , ,
p θ , k j ( z ) = 1 2 π σ k j l = 0 e ν θ , k j ( ν θ , k j ) l l ! e 1 2 ( z l η k j σ k j ) 2 ,
p θ , k j ( z ) = e ν θ , k j A B 2 π σ k j [ e ( z η k j 2 σ k j ) 2 + l = 1 e ( z l η k j 2 σ k j ) 2 × h = 0 l 1 ( ( l 1 ) h ) C l 1 h ( D ν θ , k j ) h + 1 ( h + 1 ) ! B h + l + 1 ] ,   z ,
θ = ( θ 1 , 1 , θ 1 , 2 , , θ 1 , N s θ 1 T , θ 2 , 1 , θ 2 , 2 , , θ 2 , N s θ 2 T , , θ N p , 1 , θ N p , 2 , , θ N p , N s θ N p T ) ,
I ( θ ) = k = 1 N p j = 1 N c h α θ , k j ( ν θ , k j θ ) T ν θ , k j θ ,     θ Θ ,
α θ , k j : = E [ ( l n   ( p θ , k j ( z k j ) ) ν θ , k j ) 2 ]
I ( θ ) = D i a g [ I 1 ( θ ) , I 2 ( θ ) , , I N p ( θ ) ] ,     θ Θ ,
I k ( θ ) : = j = 1 N c h α θ , k j ( ν θ , k j θ k ) T ν θ , k j θ k .
ν θ , k : = A θ k ,     k = 1 , , N p ,     θ Θ ,
I ( θ ) = D i a g [ A T G 1 ( θ ) A , A T G 2 ( θ ) A , , A T G N p ( θ ) A ] ,     θ Θ ,
G k ( θ ) : = D i a g ( α θ , k 1 , α θ , k 2 , , α θ , k N c h ) ,
I b c ( θ ) = D i a g [ G 1 ( θ ) , G 2 ( θ ) , , G N p ( θ ) ] ,     θ Θ ,
ν θ , k j = l = 1 N s a j l θ k , l = a j 1 θ k , 1 + a j 2 θ k , 2 + + a j N s θ k , N s .
ν θ , k j θ m = [ ν θ , k j θ m , 1   ν θ , k j θ m , 2     ν θ , k j θ m , N s ] = 0 ,   θ Θ ,
I k ( θ ) = j = 1 N c h α θ , k j ( ν θ , k j θ k ) T ν θ , k j θ k = j = 1 N c h α θ , k j ( a j 1 a j 2 a j N s ) ( a j 1   a j 2     a j N s )
= j = 1 N c h α θ , k j ( a j 1 2 a j 1 a j 2 a j 1 a j N s a j 2 a j 1 a j 2 2 a j 2 a j N s a j N s a j 1 a j N s a j 2 a j N s 2 ) = A T G k ( θ ) A ,    θ Θ ,   k = 1 , , N p .
ν θ , k 1 = a 11 θ k , 1 + a 12 θ k , 2 ,     ν θ , k 2 = a 21 θ k , 1 + a 22 θ k , 2 ,     k = 1 , , N p
I ( θ ) = D i a g [ A T G 1 g A , A T G 2 g A , , A T G N p g A ] ,
G k g : = D i a g ( 1 ( σ k 1 ) 2 , 1 ( σ k 2 ) 2 , , 1 ( σ k N c h ) 2 )
I ( θ ) = D i a g [ A T G 1 p ( θ ) A , A T G 2 p ( θ ) A , , A T G N p p ( θ ) A ] ,
G k p ( θ ) : = D i a g ( 1 ν θ , k 1 , 1 ν θ , k 2 , , 1 ν θ , k N c h ) .
I ( θ ) = D i a g [ A T G 1 p g ( θ ) A , A T G 2 p g ( θ ) A , , A T G N p p g ( θ ) A ]     θ Θ ,
Γ θ , k j : = ( ζ θ , k j ( z ) ) 2 p θ , k j ( z ) d z 1 ,
ζ θ , k j ( z ) : = l = 1 [ ν θ , k j ] l 1 e ν θ , k j ( l 1 ) ! 1 2 π σ k j e 1 2 ( z l η k j σ k j ) 2 ,     z .
α θ , k j = E [ ( ln   ( p θ , k j ( z k j ) ) ν θ , k j ) 2 ] = 1 ( σ k j ) 4 E [ ( z k j ) 2 2 z k j ( η k j + ν θ , k j ) + ( η k j + ν θ , k j ) 2 ] = 1 ( σ k j ) 2 ,
α θ , k j = E [ ( l n   ( p θ , k j ( z k j ) ) ν θ , k j ) 2 ] = E [ ( z k j ν θ , k j 1 ) 2 ] = E [ ( z k j ) 2 ] ( ν θ , k j ) 2 1 = 1 ν θ , k j ,
α θ , k j = Γ θ , k j = ( ζ θ , k j ( z ) ) 2 p θ , k j ( z ) d z 1 ,
I k ( θ | N c h + 1 ) = A ˜ T G ˜ k ( θ ) A ˜ ,
G ˜ k ( θ ) = ( G k ( θ ) 0 0 T α θ , k N c h + 1 ) ,   θ Θ ,   k = 1 , , N p ,
I k ( θ | N c h + 1 ) = ( A T   R T ) ( G k ( θ ) 0 0 T α θ , k N c h + 1 ) ( A R ) = ( A T   R T ) ( G k ( θ ) A α θ , k N c h + 1 R ) = I k ( θ | N c h ) + α θ , k N c h + 1 R T R ,
I ( θ | N c h + 1 ) > I ( θ | N c h ) ,     θ Θ .
a i j = ξ e x ( i , j ) ξ e m ( i , j ) ,       i = 1 , , N c h ,     j = 1 , , N s ,
ξ e x ( i , j ) : = b ( i , j ) m a x ( b ( 1 , j ) , b ( 2 , j ) , , b ( N c h , j ) ) ,
b ( i , j ) : = λ i , m i n e x λ i , m a x e x t j , e x ( λ ) E L S ( λ ) d λ ,
ξ e m ( i , j ) : = λ i , m i n e m λ i , m a x e m t j , e m ( λ ) t o p t ( λ ) t c a m ( λ ) d λ .
θ ^ k ( L S ) : = ( A T A ) 1 A T z k ,     k = 1 , , N p ,
θ ^ k ( M L ) = max θ Θ ( j = 1 N c h l n   ( p θ , k j ( z k j ) ) ) ,     k = 1 , , N p ,
E [ θ ^ k ( L S ) ] = ( A T A ) 1 A T E [ z k ] = ( A T A ) 1 A T ν θ , k = ( A T A ) 1 ( A T A ) θ k = θ k ,   θ Θ ,   k = 1 , , N p ,
E [ θ ^ k ( L S ) ] = ( A T A ) 1 A T E [ z k ] = ( A T A ) 1 A T ( ν θ , k + η k ) = ( A T A ) 1 ( A T ( A θ k + η k ) = θ k + ( A T A ) 1 A T η k ,
Bias for j t h label = mean ( θ ^ j ) θ j , % of relative error for j t h label = 100 × std ( θ ^ j ) mean ( θ ^ j ) δ j θ j δ j θ j ,

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