## Attenuation-corrected fluorescence spectra unmixing for spectroscopy and microscopy |

Optics Express, Vol. 22, Issue 16, pp. 19469-19483 (2014)

http://dx.doi.org/10.1364/OE.22.019469

Acrobat PDF (5100 KB)

### Abstract

In fluorescence measurements, light is often absorbed and scattered by a sample both for excitation and emission, resulting in the measured spectra to be distorted. Conventional linear unmixing methods computationally separate overlapping spectra but do not account for these effects. We propose a new algorithm for fluorescence unmixing that accounts for the attenuation-related distortion effect on fluorescence spectra. Using a matrix representation, we derive forward measurement formation and a corresponding inverse method; the unmixing algorithm is based on nonnegative matrix factorization. We also demonstrate how this method can be extended to a higher-dimensional tensor form, which is useful for unmixing overlapping spectra observed under the attenuation effect in spectral imaging microscopy. We evaluate the proposed methods in simulation and experiments and show that it outperforms a conventional, linear unmixing method when absorption and scattering contributes to the measured signals, as in deep tissue imaging.

© 2014 Optical Society of America

## 1. Introduction

1. J. Zhang, S. Liu, J. Yang, M. Song, J. Song, H. Du, and Z. Chen, “Quantitative spectroscopic analysis of heterogeneous systems: chemometric methods for the correction of multiplicative light scattering effects,” Rev. Anal. Chem **32**, 113–125 (2013). [CrossRef]

2. U. Resch-Genger, M. Grabolle, S. Cavaliere-Jaricot, R. Nitschke, and T. Nann, “Quantum dots versus organic dyes as fluorescent labels,” Nat. Methods **5**, 763–775 (2008). [CrossRef] [PubMed]

3. Y. Garini, I. T. Young, and G. McNamara, “Spectral imaging: Principles and applications,” Cytometry A **69A**, 735–747 (2006). [CrossRef]

4. 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]

5. T. Zimmermann, J. Rietdorf, A. Girod, V. Georget, and R. Pepperkok, “Spectral imaging and linear un-mixing enables improved FRET efficiency with a novel GFP2-YFP FRET pair,” FEBS lett. **531**, 245–249 (2002). [CrossRef] [PubMed]

6. E. Schröck, S. du Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science **273**, 494–497 (1996). [CrossRef] [PubMed]

8. A. M. Valm, J. L. M. Welch, C. W. Rieken, Y. Hasegawa, M. L. Sogin, R. Oldenbourg, F. E. Dewhirst, and G. G. Borisy, “Systems-level analysis of microbial community organization through combinatorial labeling and spectral imaging,” Proc. Natl. Acad. Sci. U.S.A. **108**, 4152–4157 (2011). [CrossRef] [PubMed]

9. R. A. Neher, M. Mitkovski, F. Kirchhoff, E. Neher, F. J. Theis, and A. Zeug, “Blind source separation techniques for the decomposition of multiply labeled fluorescence images,” Biophys. J. **96**, 3791–3800 (2009). [CrossRef] [PubMed]

10. K. R. Murphy, C. A. Stedmon, D. Graeber, and R. Bro, “Fluorescence spectroscopy and multi-way techniques. PARAFAC,” Anal. Methods **5**, 6557–6566 (2013). [CrossRef]

12. R. S. Bradley and M. S. Thorniley, “A review of attenuation correction techniques for tissue fluorescence,” J. R. Soc. Interface **3**, 1–13 (2006). [CrossRef] [PubMed]

13. M. Ducros, L. Moreaux, J. Bradley, P. Tiret, O. Griesbeck, and S. Charpak, “Spectral unmixing: analysis of performance in the olfactory bulb in vivo,” PloS ONE **4**, e4418 (2009). [CrossRef] [PubMed]

## 2. Attenuation-corrected fluorescence unmixing

### 2.1. Attenuation-corrected fluorescence matrix unmixing

12. R. S. Bradley and M. S. Thorniley, “A review of attenuation correction techniques for tissue fluorescence,” J. R. Soc. Interface **3**, 1–13 (2006). [CrossRef] [PubMed]

14. X. Luciani, S. Mounier, R. Redon, and A. Bois, “A simple correction method of inner filter effects affecting FEEM and its application to the PARAFAC decomposition,” Chemometr. Intell. Lab. **96**, 227–238 (2009). [CrossRef]

*F*(

*λ*

_{ex},

*λ*

_{em}) of a solution containing

*N*types of fluorophores can be described as: where

*λ*

_{ex}and

*λ*

_{em}are the excitation and emission wavelength,

*I*

_{0}(

*λ*) is the intensity of the illumination,

*A*(

*λ*) is the absorption spectrum of the solution,

*M*

_{1}is a wavelength-independent constant factor,

*e*is the measurement noise, and

*ϕ*,

_{n}*c*,

_{n}*ε*and

_{n}*γ*are the quantum yield, the concentration, the molar extinction coefficient and the emission spectrum of the fluorophore

_{n}*n*(

*n*= 1, 2, 3,...,

*N*). The dependency of geometrical factors, such as the size of slits and a sample cell, is incorporated in

*M*

_{1}.

*I*excitation channels and

*J*emission channels, the normalized fluorescence intensity

*F*(

*λ*

_{ex},

*λ*

_{em})/

*I*

_{0}(

*λ*

_{ex}) constructs a two-dimensional array

**F**∈ ℝ

^{I×J}. Let

*N*columns of the matrix

**A**= [

**a**

_{1},

**a**

_{2},...,

**a**

*] ∈ ℝ*

_{N}^{I×N}and

**B**= [

**b**

_{1},

**b**

_{2},...,

**b**

*] ∈ ℝ*

_{N}^{J×N}be defined as the pure emission and excitation spectra of fluorophores, where

**a**

_{n(i)}=

*ϕ*(

_{n}ε_{n}*λ*) for

_{i}*i*= 1, 2, 3,...,

*I*and

**b**

_{n(j)}=

*γ*(

_{n}*λ*) for

_{j}*j*= 1, 2, 3,...,

*J*. With these notations, Eq. (1) can be rewritten in discretized matrix representation as: where

**F**

*=*

_{i,j}*F*(

*λ*,

_{i}*λ*)/

_{j}*I*

_{0}(

*λ*),

_{i}**w**

_{1(i)}= 10

^{−A(λi)/2},

**w**

_{2(j)}= 10

^{−A(λj)/2},

**D**= diag([

*c*

_{1},

*c*

_{2},...,

*c*]

_{N}^{⊤}) and

**E**

*∈ ℝ*

_{i,j}^{I×J}represents measurement noise.

**P**⊛

**Q**denotes the Hadamard product (elementwise product) of

**P**∈ ℝ

^{L1×L2}and

**Q**∈ ℝ

^{L1×L2}, and diag(·) indicates an operator for the conversion of a column vector to a diagonal matrix. Equation (2) is illustrated in Fig. 1.

**D**from the measurement

**F**in Eq. (2), the following minimization problem is formulated: ‖ · ‖

*represents the Frobenius norm.*

_{F}**ŵ**

*(*

_{r}*r*= 1, 2) and

**D̂**= diag(

*ĉ*

_{1},

*ĉ*

_{2},...,

*ĉ*) are the estimations of

_{N}**w**

*,*

_{r}**D**and

*c*, respectively. The scaling factor

_{n}*M*

_{1}is dropped in the formulation of this minimization problem. Here, we assume the emission and excitation spectra of fluorophores, the matrix

**A**and

**B**, can be measured separately. On this assumption, we developed a nonlinear unmixing method: attenuation corrected fluorescence matrix unmixing (AFMU). The detail of the method is described in Appendix A. Even though the estimated values

**ŵ**

*and*

_{r}**D̂**are not the absolute absorbance spectrum and concentrations, this does not usually become a problem because most chemical analysis demand only the ratio of the components. Instead of comparing absolute concentration, our method compares the dimensionless ratio of each fluorophore’s concentration between different samples.

### 2.2. Attenuation-corrected fluorescence tensor unmixing

15. J. B. Pawley, *Handbook of Biological Confocal Microscopy* (Springer, 2006). [CrossRef]

16. S. Henrot, C. Soussen, M. Dossot, and D. Brie, “Does deblurring improve geometrical hyperspectral unmixing?” IEEE Trans. Image Process. **23**, 1169–1180 (2014). [CrossRef] [PubMed]

17. M. G. Müller, I. Georgakoudi, Q. Zhang, J. Wu, and M. S. Feld, “Intrinsic fluorescence spectroscopy in turbid media: disentangling effects of scattering and absorption,” Appl. Opt. **40**, 4633–4646 (2001). [CrossRef]

*p*-photon process. In a

*K*

_{1}by

*K*

_{2}pixel image, the intensity of fluorescence from

*N*types of fluorophores contributing to the signal at pixel

*k*(

*k*= 1, 2, 3,...,

*K*

_{1}

*K*

_{2}) is formulated as follows: where

*E*

_{0,k}is the illumination intensity at pixel

*k*,

*M*

_{2}is a wavelength-independent factor, and

*s*(

_{n}*λ*

_{ex}),

*t*(

_{n}*λ*

_{em}) and

*u*are the excitation spectrum, the emission spectrum, the abundance at pixel

_{n,k}*k*of the fluorophore

*n*.

*g*

_{1,k}(

*λ*

_{ex}) and

*g*

_{2,k}(

*λ*

_{em}) are attenuation factors for excitation and emission light at pixel

*k*, respectively. The dependency of geometrical factors of the measurement system is incorporated in

*M*

_{2}. On the assumption that the spectral shape of attenuation is the same at every pixel, the attenuation factors can be decomposed into wavelength-dependent and wavelength-independent factors:

*g*

_{1,k}(

*λ*

_{ex}) =

*G*

_{1,k}

*v*

_{1}(

*λ*

_{ex}) and

*g*

_{2,k}(

*λ*

_{em}) =

*G*

_{2,k}

*v*

_{2}(

*λ*

_{em}).

*I*excitation and

*J*emission channels, the normalized fluorescence intensity

*f*(

_{k}*λ*

_{ex},

*λ*

_{em})/(

*E*

_{0,k}(

*λ*

_{ex})

^{p}G_{1,k}

*G*

_{2,k}) constructs a three-dimensional array

**ℱ**∈ ℝ

^{I×J×K1K2}. Following the strategy for the formulation of Eq. (2), let

*N*columns of the matrix

**S**= [

**s**

_{1},

**s**

_{2},...,

**s**

*] ∈ ℝ*

_{N}^{I×K1K2}and

**T**= [

**t**

_{1},

**t**

_{2},...,

**t**

*] ∈ ℝ*

_{N}^{J×K1K2}be defined as the pure excitation and emission spectra of fluorophores, where

**s**

_{n(i)}=

*s*(

_{n}*λ*) for

_{i}*i*= 1, 2, 3,...,

*I*and

**t**

_{n(j)}=

*t*(

_{n}*λ*) for

_{j}*j*= 1, 2, 3,...,

*J*. With these notations, Eq. (4) can be rewritten in tensor representation as: where

**ℱ**

*=*

_{ijk}*f*(

_{k}*λ*,

_{i}*λ*)/(

_{j}*E*

_{0,k}(

*λ*)

_{i}

^{p}G_{1,k}

*G*

_{2,k}),

**v**

_{1(i)}=

*v*

_{1}(

*λ*),

_{i}**v**

_{2(j)}=

*v*

_{2}(

*λ*),

_{j}**𝒱**is a

*I*×

*J*×

*K*

_{1}

*K*

_{2}third-order tensor all of whose frontal slices

**𝒱**

*: =*

_{ij}**v**

_{1(i)}

**v**

_{2(j)},

**u**

_{n(k)}=

*u*,

_{n,k}**U**= [

**u**

_{1},

**u**

_{2},...,

**u**

*], and*

_{N}**ℰ**∈ ℝ

^{I×J×K1K2}represents measurement noise. The outer product operator is defined as ○, and [[·]] represents the Kruskal operator, which performs the summation of the outer products of the columns of matrices [18

18. T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,” SIAM Rev. **51**, 455–500 (2009). [CrossRef]

**U**from the fluorescence measurement

**ℱ**in Eq. (6): Here, as is the case with LU, we assume that the series of emission spectra of fluorophores, namely the matrix

**T**, is available.

**𝒱̂**,

**Ŝ**, and

**Û**are the estimated values of

**𝒱**,

**S**and

**U**. As is the case in Eq. (3), the scaling factor

*M*

_{2}of Eq. (7) is removed in the formulation of this minimization problem Eq. (6). Therefore, it is noteworthy that, instead of comparing absolute numbers, our method compares the ratio of contribution to signal from each fluorescence species between different pixels. This is also not a concern since absolute values are typically not important in fluorescence microscopy applications [19

19. H. Shirakawa and S. Miyazaki, “Blind spectral decomposition of single-cell fluorescence by parallel factor analysis,” Biophys. J. **86**, 1739–1752 (2004). [CrossRef] [PubMed]

## 3. Simulation

20. B. W. Bader and T. G. Kolda, “Matlab tensor toolbox version 2.5,” available at http://www.sandia.gov/tgkolda/TensorToolbox/ (2012).

21. B. W. Bader and T. G. Kolda, “Algorithm 862: MATLAB tensor classes for fast algorithm prototyping,” ACM Trans. Math. Software **32**, 635–653 (2006). [CrossRef]

### 3.1. Performance analysis of AFMU

*𝒩*(

*μ*,

_{n}*μ*and standard deviation

_{n}*σ*(

_{n}*n*= 1...

*N*). Since the shape of fluorescence spectra is not important in our algorithm, all of the standard deviation

*σ*are set to be equal. An absorption spectrum is also simulated as a Gaussian distribution which has larger bandwidth than fluorescence spectra. The contribution from each fluorophore is set to be equal.

_{n}*N*is 2, spectral overlap Δ

*μ*=

_{n}*μ*

_{n+1}−

*μ*= 11.7nm for both excitation and emission spectra and

_{n}*σ*= 20. The standard deviation of the absorption spectrum is 120. This standard dataset is shown in Fig. 2.

_{n}*μ*are set to be equal. d) Δ

_{n}*μ*is changed from 1 nm to 20 nm.

_{n}*μ*is less than 5 nm. This corresponds to the sampling resolution of excitation and emission channels. In addition, Fig. 2(c) remarkably shows that the number of fluorophores does not deteriorate the performance of AFMU. It should be noted that the variance of NRMSE decrease correspondingly with the average of NRMSE.

_{n}### 3.2. Performance analysis of AFTU

**Ũ**and

**Ü**are a row-wise normalized matrix of

**Û**and

**U**, respectively. NRMSE is evaluated ten times. Similar to the simulation of fluorescence spectroscopy, the parameters for generating datasets are modified from the standard spectral image dataset for the performance evaluations in the following ways: a) White Gaussian noise is added to the standard dataset to generate a range of SNR for measurements. b) The sampling bandwidth and interval of emission channels are increased from 1 nm to 50 nm, while the sampling bandwidth and interval of excitation channels remain the same. c) The number of fluorescence species is increased from 2 to 10. Δ

*μ*is set to be all equal. d) Spectral overlap between fluorophores Δ

_{n}*μ*is changed from 1 nm to 20 nm.

_{n}10. K. R. Murphy, C. A. Stedmon, D. Graeber, and R. Bro, “Fluorescence spectroscopy and multi-way techniques. PARAFAC,” Anal. Methods **5**, 6557–6566 (2013). [CrossRef]

## 4. Experimental results

### 4.1. Fluorometer

10. K. R. Murphy, C. A. Stedmon, D. Graeber, and R. Bro, “Fluorescence spectroscopy and multi-way techniques. PARAFAC,” Anal. Methods **5**, 6557–6566 (2013). [CrossRef]

### 4.2. Fluorescence microscopy

13. M. Ducros, L. Moreaux, J. Bradley, P. Tiret, O. Griesbeck, and S. Charpak, “Spectral unmixing: analysis of performance in the olfactory bulb in vivo,” PloS ONE **4**, e4418 (2009). [CrossRef] [PubMed]

22. R. M. Zucker, P. Rigby, I. Clements, W. Salmon, and M. Chua, “Reliability of confocal microscopy spectral imaging systems: Use of multispectral beads,” Cytometry Part A **71A**, 174–189 (2007). [CrossRef]

23. S. Schlachter, S. Schwedler, A. Esposito, G. S. Kaminski Schierle, G. D. Moggridge, and C. F. Kaminski, “A method to unmix multiple fluorophores in microscopy images with minimal a priori information,” Opt. Express **17**, 22747–22760 (2009). [CrossRef]

*β*-tubulins and actin filaments in the cellular samples were stained with Alexa Fluor 568 and TRITC, respectively. Their emission spectra are shown in Fig. 6(c).

13. M. Ducros, L. Moreaux, J. Bradley, P. Tiret, O. Griesbeck, and S. Charpak, “Spectral unmixing: analysis of performance in the olfactory bulb in vivo,” PloS ONE **4**, e4418 (2009). [CrossRef] [PubMed]

## 5. Discussion

**5**, 6557–6566 (2013). [CrossRef]

24. I. Urbančič, Z. Arsov, A. Ljubetič, D. Biglino, and J. Štrancar, “Bleaching-corrected fluorescence microspectroscopy with nanometer peak position resolution,” Opt. Express **21**, 25291–25306 (2013). [CrossRef]

23. S. Schlachter, S. Schwedler, A. Esposito, G. S. Kaminski Schierle, G. D. Moggridge, and C. F. Kaminski, “A method to unmix multiple fluorophores in microscopy images with minimal a priori information,” Opt. Express **17**, 22747–22760 (2009). [CrossRef]

## 6. Conclusion

## Appendix A: update rules for AFMU

25. D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature **401**, 788–791 (1999). [CrossRef] [PubMed]

**Ŵ**

*(*

_{r}*r*= 1, 2) is a

*N*-column matrix with columns that are identical horizontal copies of the column vector

**ŵ**

*and*

_{r}**ĉ**= [

*ĉ*

_{1},

*ĉ*

_{2},...,

*ĉ*]

_{N}^{⊤}. The vectorization of an

*L*

_{1}×

*L*

_{2}matrix

**X**to an

*L*

_{1}

*L*

_{2}× 1 column vector is denoted by vec(

**X**). During the updates of

**ĉ**and

**ŵ**

*,*

_{r}**D̂**and

**Ŵ**

*are updated accordingly. The Khatri-Rao product operator ⊙ is defined as for matrices*

_{r}**P**∈ ℝ

^{L1×L3}and

**Q**∈ ℝ

^{L2×L3}, where ⊗ represents the Kronecker product operator, and

**p**

_{l1}and

**q**

_{l2}denote the

**P**and

**Q**. Here, we name this method attenuation corrected fluorescence matrix unmixing, which decomposes a nonlinearly-distorted EEM.

## Appendix B: update rules for AFTU

26. G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: Compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graphics **31**, 80 (2012). [CrossRef]

**v̂**

*(*

_{r}*r*= 1, 2) is the estimate of

**v**

*,*

_{r}**V̂**

*is a*

_{r}*N*-column matrix whose columns are identical horizontal copies of

**v̂**

*,*

_{r}**ṽ**

*is a column vector which has*

_{r}*K*

_{1}

*K*

_{2}vertical copies of

**v̂**

*. The mode-*

_{r}*l*matricization of a 3-way tensor

**𝒳**is denoted by

**𝒳**

_{(l)}; this matrix have columns that are the mode-

*l*fibers of the original tensor

**𝒳**(for example, the first column of

**𝒳**

_{(1)}is

**𝒳**

_{:11}). A more general definition of notations and operations for a tensor, including matricization, is available in [18

18. T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,” SIAM Rev. **51**, 455–500 (2009). [CrossRef]

**v̂**

*,*

_{r}**V̂**

*and*

_{r}**ṽ**

*are updated accordingly. Similar to AFMU, we name this method attenuation-corrected fluorescence tensor unmixing. This method decomposes a fluorescence spectral image, namely a three-way fluorescence dataset, affected by wavelength-dependent attenuation.*

_{r}## Acknowledgments

## References and links

1. | J. Zhang, S. Liu, J. Yang, M. Song, J. Song, H. Du, and Z. Chen, “Quantitative spectroscopic analysis of heterogeneous systems: chemometric methods for the correction of multiplicative light scattering effects,” Rev. Anal. Chem |

2. | U. Resch-Genger, M. Grabolle, S. Cavaliere-Jaricot, R. Nitschke, and T. Nann, “Quantum dots versus organic dyes as fluorescent labels,” Nat. Methods |

3. | Y. Garini, I. T. Young, and G. McNamara, “Spectral imaging: Principles and applications,” Cytometry A |

4. | 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. |

5. | T. Zimmermann, J. Rietdorf, A. Girod, V. Georget, and R. Pepperkok, “Spectral imaging and linear un-mixing enables improved FRET efficiency with a novel GFP2-YFP FRET pair,” FEBS lett. |

6. | E. Schröck, S. du Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science |

7. | J. Livet, T. A. Weissman, H. Kang, R. W. Draft, J. Lu, R. A. Bennis, J. R. Sanes, and J. W. Lichtman, “Transgenic strategies for combinatorial expression of fluorescent proteins in the nervous system,” Nature |

8. | A. M. Valm, J. L. M. Welch, C. W. Rieken, Y. Hasegawa, M. L. Sogin, R. Oldenbourg, F. E. Dewhirst, and G. G. Borisy, “Systems-level analysis of microbial community organization through combinatorial labeling and spectral imaging,” Proc. Natl. Acad. Sci. U.S.A. |

9. | R. A. Neher, M. Mitkovski, F. Kirchhoff, E. Neher, F. J. Theis, and A. Zeug, “Blind source separation techniques for the decomposition of multiply labeled fluorescence images,” Biophys. J. |

10. | K. R. Murphy, C. A. Stedmon, D. Graeber, and R. Bro, “Fluorescence spectroscopy and multi-way techniques. PARAFAC,” Anal. Methods |

11. | J. R. Lakowicz, |

12. | R. S. Bradley and M. S. Thorniley, “A review of attenuation correction techniques for tissue fluorescence,” J. R. Soc. Interface |

13. | M. Ducros, L. Moreaux, J. Bradley, P. Tiret, O. Griesbeck, and S. Charpak, “Spectral unmixing: analysis of performance in the olfactory bulb in vivo,” PloS ONE |

14. | X. Luciani, S. Mounier, R. Redon, and A. Bois, “A simple correction method of inner filter effects affecting FEEM and its application to the PARAFAC decomposition,” Chemometr. Intell. Lab. |

15. | J. B. Pawley, |

16. | S. Henrot, C. Soussen, M. Dossot, and D. Brie, “Does deblurring improve geometrical hyperspectral unmixing?” IEEE Trans. Image Process. |

17. | M. G. Müller, I. Georgakoudi, Q. Zhang, J. Wu, and M. S. Feld, “Intrinsic fluorescence spectroscopy in turbid media: disentangling effects of scattering and absorption,” Appl. Opt. |

18. | T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,” SIAM Rev. |

19. | H. Shirakawa and S. Miyazaki, “Blind spectral decomposition of single-cell fluorescence by parallel factor analysis,” Biophys. J. |

20. | B. W. Bader and T. G. Kolda, “Matlab tensor toolbox version 2.5,” available at http://www.sandia.gov/tgkolda/TensorToolbox/ (2012). |

21. | B. W. Bader and T. G. Kolda, “Algorithm 862: MATLAB tensor classes for fast algorithm prototyping,” ACM Trans. Math. Software |

22. | R. M. Zucker, P. Rigby, I. Clements, W. Salmon, and M. Chua, “Reliability of confocal microscopy spectral imaging systems: Use of multispectral beads,” Cytometry Part A |

23. | S. Schlachter, S. Schwedler, A. Esposito, G. S. Kaminski Schierle, G. D. Moggridge, and C. F. Kaminski, “A method to unmix multiple fluorophores in microscopy images with minimal a priori information,” Opt. Express |

24. | I. Urbančič, Z. Arsov, A. Ljubetič, D. Biglino, and J. Štrancar, “Bleaching-corrected fluorescence microspectroscopy with nanometer peak position resolution,” Opt. Express |

25. | D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature |

26. | G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: Compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graphics |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(110.0180) Imaging systems : Microscopy

(170.2520) Medical optics and biotechnology : Fluorescence microscopy

(300.1030) Spectroscopy : Absorption

(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence

(110.4234) Imaging systems : Multispectral and hyperspectral imaging

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: June 9, 2014

Revised Manuscript: July 24, 2014

Manuscript Accepted: July 25, 2014

Published: August 5, 2014

**Virtual Issues**

Vol. 9, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Hayato Ikoma, Barmak Heshmat, Gordon Wetzstein, and Ramesh Raskar, "Attenuation-corrected fluorescence spectra unmixing for spectroscopy and microscopy," Opt. Express **22**, 19469-19483 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-19469

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### References

- J. Zhang, S. Liu, J. Yang, M. Song, J. Song, H. Du, and Z. Chen, “Quantitative spectroscopic analysis of heterogeneous systems: chemometric methods for the correction of multiplicative light scattering effects,” Rev. Anal. Chem32, 113–125 (2013). [CrossRef]
- U. Resch-Genger, M. Grabolle, S. Cavaliere-Jaricot, R. Nitschke, and T. Nann, “Quantum dots versus organic dyes as fluorescent labels,” Nat. Methods5, 763–775 (2008). [CrossRef] [PubMed]
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