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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 16 — Aug. 11, 2014
  • pp: 19469–19483
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Attenuation-corrected fluorescence spectra unmixing for spectroscopy and microscopy

Hayato Ikoma, Barmak Heshmat, Gordon Wetzstein, and Ramesh Raskar  »View Author Affiliations


Optics Express, Vol. 22, Issue 16, pp. 19469-19483 (2014)
http://dx.doi.org/10.1364/OE.22.019469


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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

Fluorescence is a phenomenon where a molecule absorbs light at one wavelength and subsequently emits light at longer wavelength. Since fluorescence spectra reflect the state and type of molecules, the analysis of fluorescence spectra non-destructively provides molecular information of a sample. Therefore, fluorescence spectroscopy is widely used in fundamental sciences, such as analytical chemistry, environmental sciences and medicine. Furthermore, the analysis of autofluorescence, which is naturally emitted by a sample, has been applied for many applications such as the detection of cancer, contamination in water and degradation of food [1

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]

].

The separation of fluorescence signals is of great importance in the analysis of fluorescence spectra. Most analyzed samples for fluorescence spectroscopy, including blood, water and food, have multiple fluorescence species. It is also common for fluorescence microscopy to label biological samples with multiple types of fluorophores. In these samples, if their individual spectra do not overlap, the fluorescence light from mixed fluorescence species can be separated by optical filters. However, overlapping fluorescence spectra cannot be separated optically. Although quantum dots have been emerging as narrow-bandwidth fluorescence dyes for avoiding spectral overlap, they cannot substitute for intrinsic fluorophores in autofluorescence diagnosis and most genetically-encoded fluorescence proteins in biological research [2

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]

]. In those cases, a mathematical technique, called linear unmixing (LU), is often used to separate the overlapping spectra.

LU is an algorithm which decomposes measured fluorescence into each fluorophore’s contribution [3

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

]. This algorithm works under the assumption that the sample’s absorbance is negligible. Under this assumption, the measured signals can be modeled as a linear combination of each fluorescence spectrum. Therefore, the pre-measured emission spectrum for each fluorophore allows for the linear equations to be solved when the number of channels recorded is larger or equal to the number of fluorescence species. This method has been successfully applied to fluorescence spectroscopy and fluorescence spectral imaging systems [4

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]

]. In addition to the decomposition of spectrally-overlapping fluorescence signals, LU allows the removal of undesired signals and the analysis of fluorescence resonance energy transfer (FRET) [5

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]

]. Furthermore, in fluorescence microscopy, the combination of LU and a stochastic expression of multiple fluorescence proteins allows for the labeling of chromosomes, neurons and bacteria with up to approximately 100 distinguishable colors [6

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

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]

].

In addition to LU, under the assumption of negligible absorption, blind unmixing methods have been developed to unmix the measured spectra without knowing single fluorophore’s spectra. Most blind unmixing methods require the measurement of an excitation-emission matrix (EEM). An EEM is composed of a series of emission spectra measured with different excitation wavelengths. For a single fluorophore, its EEM can be described as an outer product of its excitation and emission spectrum, while its intensity is proportional to its amount. Therefore, an EEM of a sample containing several fluorophores has trilinearity. By exploiting this multilinearity of fluorescence signals, the use of advanced mathematical tools, such as nonnegative matrix factorization (NMF) [9

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]

] and parallel factor analysis (PARAFAC) [10

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]

], allows the estimation of molecular fraction and individual spectra of fluorophores without reference fluorescence spectra as prior information.

When absorption of a sample is not negligible, however, both excitation and emission light are absorbed through a measurement. The absorption is caused by dissolved species, including fluorophores themselves. As a result, the measured fluorescence spectra are distorted in a wavelength-dependent manner, which is called inner filter effect. This nonlinear effect violates the assumptions of linear unmixing algorithms and has been an inherent problem for fluorescence spectroscopy.

To overcome this problem, two major methods are often used for liquid samples. The first approach is to dilute a sample to the extent where absorption is negligible enough to measure intrinsic fluorescence spectra [11

11. J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Springer, 2007).

]. The second approach is to measure the absorbance spectrum of a sample and mathematically recover intrinsic fluorescence spectra from the Beer-Lambert law. In spite of their popularity and success, both of the two methods have drawbacks. The dilution approach requires an additional experimental procedure and reduces the signal to noise ratio significantly, while the absorbance correction approach requires additional procedures and equipment.

For turbid media, such as biological tissues, the distortion can also be caused by scattering along with absorption. This effect has been recognized in the analysis of autofluorescence for clinical applications, and numerous techniques have been proposed to compensate the effect in the analysis. Based on [12

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]

], those methods can be mainly categorized into three groups. The first approach utilizes empirical models. Although this approach usually requires only a simple implementation, its application is strictly limited. The second approach is based on a measurement where photons propagating through a medium for short distances are collected. This method requires a specially-designed device and also lacks the applicability to other measurement systems. The third approach is based on theoretical models. In this approach, the propagation of light in a turbid medium is physically modeled. Since all of the models are based on various approximations, their application is also often limited to specific types of samples. Moreover, most of them require additional measurements other than fluorescence spectra. On the other hand, the distortion of emission spectra in a thick biological sample is also reported in fluorescence spectral imaging [13

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]

]. While there is a growing need to quantitatively analyze images of neuronal activities through significant depths of the brain, the unmixing algorithm considering attenuation in such a depth has not been developed for fluorescence microscopy so far.

In this paper, we propose a novel attenuation-corrected fluorescence unmixing framework for fluorescence spectroscopy and microscopy to estimate abundance of fluorophores in a sample. In particular, we make the following contributions. We introduce a model of fluorescence spectra from a light attenuating sample as a matrix representation. We also model spectral fluorescence imaging of a light attenuating sample as a tensor representation. Based on the matrix and tensor representations, we introduce new attenuation-corrected fluorescence unmixing methods by using nonnegative matrix and tensor factorization. This method does not require any additional measurements other than conventional fluorescence spectroscopy and imaging. We evaluate the performance of the unmixing methods on simulated fluorescence spectra and spectral images. We also show that the attenuation-corrected unmixing methods experimentally outperform the conventional linear unmixing method.

2. Attenuation-corrected fluorescence unmixing

This section presents the framework for unmixing distorted fluorescence EEMs. First, the model of fluorescence spectra from a light-attenuating material in matrix representation is introduced. Second, by using this matrix representation and NMF, an algorithm is derived to estimate molecular fraction from each fluorophore’s pure EEM by utilizing NMF. Third, the matrix model is extended to the model of a fluorescence spectral image in tensor representation. In this representation, an image is represented as a stack of distorted EEMs of all pixels. Forth, an algorithm is proposed from the tensor representation to estimate molecular fraction only from each fluorophore’s emission spectrum, instead of its EEM, with nonnegative tensor factorization (NTF). Based on known line shapes (each intrinsic fluorescence spectrum) and measured attenuation-deformed spectra, our method simultaneously estimates contributions from each fluorescence species and how the measured fluorescence spectra are deformed.

2.1. Attenuation-corrected fluorescence matrix unmixing

There have been many attempts to model fluorescence spectra affected by absorption effects in fluorometer measurements [12

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]

]. Standard spectrofluorometers hold a cuvette for a sample solution with a specific dimension, which defines the path length of excitation and emission light. Based on this optical setup and the Beer-Lambert law, Luciani et al. [14

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]

] mathematically described the absorption effect in a fluorescence spectroscopy measurement with right-angle geometry in detail. Following their formulation, the fluorescence intensity F(λex, λem) of a solution containing N types of fluorophores can be described as:
F(λex,λem)=M110(A(λex)+A(λem))/2n=1NI0(λex)ϕncnεn(λex)γn(λem)+e,
(1)
where λex and λem are the excitation and emission wavelength, I0(λ) is the intensity of the illumination, A(λ) is the absorption spectrum of the solution, M1 is a wavelength-independent constant factor, e is the measurement noise, and ϕn, cn, εn and γn are the quantum yield, the concentration, the molar extinction coefficient and the emission spectrum of the fluorophore n (n = 1, 2, 3,..., N). The dependency of geometrical factors, such as the size of slits and a sample cell, is incorporated in M1.

By scanning I excitation channels and J emission channels, the normalized fluorescence intensity F(λex, λem)/I0(λex) constructs a two-dimensional array F ∈ ℝI×J. Let N columns of the matrix A = [a1, a2,...,aN] ∈ ℝI×N and B = [b1, b2,...,bN] ∈ ℝJ×N be defined as the pure emission and excitation spectra of fluorophores, where an(i) = ϕnεn(λi) for i = 1, 2, 3,...,I and bn(j) = γn(λj) for j = 1, 2, 3,...,J. With these notations, Eq. (1) can be rewritten in discretized matrix representation as:
F=M1(w1w2)ADB+E,
(2)
where Fi,j = F(λi, λj)/I0(λi), w1(i) = 10A(λi)/2, w2(j) = 10A(λj)/2, D = diag([c1, c2,...,cN]) and Ei,j ∈ ℝI×J represents measurement noise. PQ denotes the Hadamard product (elementwise product) of P ∈ ℝL1×L2 and Q ∈ ℝL1×L2, and diag(·) indicates an operator for the conversion of a column vector to a diagonal matrix. Equation (2) is illustrated in Fig. 1.

Fig. 1 Noiseless EEM model affected by attenuation. The top and bottom rows are the visualizations of Eqs. (2) and (5), respectively.

To estimate D from the measurement F in Eq. (2), the following minimization problem is formulated:
minw^1,w^2,D^F(w^1w^2)AD^BFsubjectto0w^1,w^21,D^0.
(3)
‖ · ‖F represents the Frobenius norm. ŵr (r = 1, 2) and = diag(ĉ1, ĉ2,..., ĉN) are the estimations of wr, D and cn, respectively. The scaling factor M1 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 ŵr and 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

The fluorescence matrix model Eq. (2) can be extended to a tensor model by stacking EEMs in another dimension. A fluorescence spectral imaging system captures images at multiple channels with different excitation wavelengths, which provides an EEM at each pixel. The series of the captured images is the convolution of the wavelength-dependent point spread function (PSF) of the imaging system and the distribution of fluorophores in a sample [15

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

]. Therefore, if the bandwidth of exploited excitation and emission wavelength are wide, deconvolution should be performed on the captured images beforehand to analyze spectra [16

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]

]. Hereafter, we base our discussion on deconvolved images and images in which the wavelength dependency of PSF is negligible.

Fluorescence from turbid media, such as biological tissues, was modeled with photon migration theory by Muller et al. [17

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]

]. Since their model has been successful in a wide range of optical properties of various specimens and can be adopted to any optical setup, we extend their model to spectral imaging and p-photon process. In a K1 by K2 pixel image, the intensity of fluorescence from N types of fluorophores contributing to the signal at pixel k (k = 1, 2, 3,..., K1K2) is formulated as follows:
fk(λex,λem)=M2E0,k(λex)pg1,k(λex)g2,k(λem)n=1Nsn(λex)tn(λem)un,k,
(4)
where E0,k is the illumination intensity at pixel k, M2 is a wavelength-independent factor, and sn(λex), tn(λem) and un,k are the excitation spectrum, the emission spectrum, the abundance at pixel k of the fluorophore n. g1,k(λex) and g2,k(λem) are attenuation factors for excitation and emission light at pixel k, respectively. The dependency of geometrical factors of the measurement system is incorporated in M2. 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: g1,k(λex) = G1,k v1(λex) and g2,k(λem) = G2,k v2(λem).

When the spectral image is captured with I excitation and J emission channels, the normalized fluorescence intensity fk(λex, λem)/(E0,k(λex)pG1,kG2,k) constructs a three-dimensional array ∈ ℝI×J×K1K2. Following the strategy for the formulation of Eq. (2), let N columns of the matrix S = [s1, s2,...,sN] ∈ ℝI×K1K2 and T = [t1, t2,...,tN] ∈ ℝJ×K1K2 be defined as the pure excitation and emission spectra of fluorophores, where sn(i) = sn(λi) for i = 1, 2, 3,...,I and tn(j) = tn(λj) for j = 1, 2, 3,..., J. With these notations, Eq. (4) can be rewritten in tensor representation as:
=M2𝒱n=1Nsn°tn°un+
(5)
=M2𝒱[[S,T,U]]+,
(6)
where ijk = fk(λi, λj)/(E0,k(λi)pG1,kG2,k), v1(i) = v1(λi), v2(j) = v2(λj), 𝒱 is a I × J × K1K2 third-order tensor all of whose frontal slices 𝒱ij: = v1(i)v2(j), un(k) = un,k, U = [u1, u2,..., uN], and ∈ ℝI×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]

]. Equation 5 is visualized in Fig. 1.

As for the case of fluorescence spectrophotometer, we formulate the following nonlinear least squares problem to estimate the abundance matrix U from the fluorescence measurement in Eq. (6):
min𝒱^,S^,U^𝒱^[[S^,T,U^]]Fsubjectto0𝒱^1,S^0,U^0.
(7)
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 M2 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]

]. To solve this least-squares problem, we developed another nonlinear unmixing method: attenuation-corrected fluorescence tensor unmixing (AFTU), whose detail is described in Appendix B.

3. Simulation

This section analyzes the performance of our nonlinear unmixing algorithms, AFMU and AFTU, through simulated datasets. First, we assess AFMU on simulated spectroscopic datasets. Second, we quantify AFTU on simulated spectral imaging datasets. The robustness of our un-mixing methods are discussed with respect to signal-to-noise ratio (SNR), the number of channels, the number of fluorophores and the degree of spectral overlap. The generation of simulated datasets and their analysis were performed with MATLAB Version 8.2 software. The models and algorithms were implemented with MATLAB Tensor Toolbox Version 2.5 [20

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

, 21

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

Following Eq. (2), the datasets of EEMs of samples containing several fluorophores in a light-absorbing medium are generated for the analysis of AFMU. Fluorescence excitation and emission spectra are simulated as Gaussian distributions 𝒩 (μn, σn2) with their peak position μn and standard deviation σn (n = 1...N). Since the shape of fluorescence spectra is not important in our algorithm, all of the standard deviation σn 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.

As a standard dataset for the performance evaluations, the following parameters are used for the generation of the spectroscopic data: intrinsic fluorescence spectra are simulated with 1 nm resolution, the sampling bandwidth and interval for excitation and emission channels are 5 nm, the number of fluorophores N is 2, spectral overlap Δμn = μn+1μn = 11.7nm for both excitation and emission spectra and σn = 20. The standard deviation of the absorption spectrum is 120. This standard dataset is shown in Fig. 2.

Fig. 2 Simulated fluorescence emission and excitation spectra of two fluorophores and an absorption spectrum of the standard dataset are shown on the left. Continuous and dotted lines represent emission and excitation spectra, respectively. The contribution of each fluorophore to signals are simulated to be the same. The estimated absorption spectrum is also shown in the bottom-left figure. The performance of AFMU is evaluated on various (a) SNR levels in measurements, (b) sampling intervals, (c) numbers of fluorophores and (d) spectral overlaps of the two fluorophores.

To quantify the estimation error, we define the normalized mean squared error (NRMSE) for the estimation of the contribution ratio:
NRMSE=c^c^2cc222/cc222.
(8)
For each condition, NRMSE was evaluated thirty times. The parameters for generating datasets are changed for each performance evaluation in the following ways; a) White Gaussian noise is added to the standard dataset to generate a range of SNR in measurements. b) Sampling interval for both excitation and emission channels is changed from 1 nm to 40 nm. c) The number of fluorophores is increased from 2 to 10, while all Δμn are set to be equal. d) Δμn is changed from 1 nm to 20 nm.

Figure 2 summarizes the performance of the proposed method. As shown in the estimated absorption spectrum in the left-bottom figure of Fig. 2, it is worth noting that our method does not estimate absorption spectrum completely. AFMU accurately estimates absorption spectrum only at wavelengths where fluorescence signal is detected. Figures 2(a) and 2(b) show that as SNR and sampling resolution increase, the proposed method estimates the contribution ratio with higher accuracy. Figure 2(d) illustrates that the performance is degraded when spectral overlap Δμn 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.

3.2. Performance analysis of AFTU

Fluorescence spectral image datasets are simulated for the evaluation of AFTU. As in the simulation of spectroscopy measurements, excitation and emission spectra and absorption spectra are simulated in the same way. In a 32 by 32 pixel image, each fluorophore is simulated to have a different spatial distribution. Since AFTU is not dependent on spatial parameters, any arbitrary two dimensional patterns with varying concentration and overlap can be used for the performance evaluation of AFTU. Here, we use the real part of the two dimensional discrete Fourier transform basis functions to produce more than three different patterns for increased number of fluorophores. This simple function can conveniently generate various values in different pixels. The representative examples of the simulated distribution patterns are shown in Fig. 3. For excitation and emission channels, the sampling bandwidth are set to be 1 nm and 10 nm, while the sampling intervals are set to be 40 nm and 10 nm, respectively. These parameters are close to the commonly-used measurement parameters for a conventional fluorescence spectral imaging system.

Fig. 3 The left column shows the simulated arbitrary distribution patterns of two fluorophores used for the standard dataset. The pattern for each fluorophore is generated based on the two dimensional discrete Fourier transform basis functions. The performance of AFTU is evaluated in respect of normalized root mean squared error of the estimated contributions. The robustness of AFTU is evaluated with (a) SNR levels in measurements, (b) sampling interval of emission channel, (c) number of fluorophores and (d) spectral overlap of the two fluorophores.

Similar to the evaluation of AFMU, NRMSE is defined as follows:
NRMSE=U˜U¨22U¨22.
(9)
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. Δμn is set to be all equal. d) Spectral overlap between fluorophores Δμn is changed from 1 nm to 20 nm.

Figures 3 and 4 summarize the performance of AFTU in terms of the estimation error of molecular fractions. As can be seen, the minimum average NRMSE of AFTU for each parameter is slightly larger than of AFMU, which is due to the larger interval of excitation channel. On the other hand, as opposed to AFMU, Figs. 3(a) and 3(b) show that the estimation error of AFTU does not increase significantly along with the increase of measurement noise and measurement interval of emission channels. This robustness of AFTU benefits from the third linearity of fluorescence signal to the amount of the analytes, which is also reported in the application of PARAFAC to fluorescence analysis [10

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]

]. In addition, as shown in Fig. 4, even though the estimation error has dependency on the relative concentration, the figure of NRMSE shows that AFTU is accurate for a wide range of molecular fraction (NRMSE < 0.01 in all pixels). As for the perfomance evaluation of AFMU, it should be also noted that the variance of NRMSE decreases correspondingly with the average of NRMSE.

Fig. 4 Molecular fraction of Fluorophore 1 of the standard dataset for the simulation of fluorescence spectral imaging. The original fraction, its estimation by AFTU and its pixel-by-pixel estimation error (NRMSE) are shown from left to right, respectively. Similar to Eq. 8, NRMSE is computed at each pixel.

4. Experimental results

This section assesses the performance of AFMU and AFTU on experimental datasets by comparing the results with LU. We first describe the experimental procedures of fluorescence spectroscopy. Then, we apply AFMU and LU to the measured EEM of a solution containing two fluorophores. Second, we describe the acquisition of fluorescence spectral images of micro-spheres and biological samples stained with two spectrally-overlapping fluorophores. Afterward, we unmix the fluorescence signals of the captured spectral images with AFTU and LU. We show that both of the two methods outperform LU.

4.1. Fluorometer

Fluorescence solutions were prepared by dissolving two fluorophores, Rhodamine 123 and 4-(Dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran (DCM) in ethanol. While the concentration of Rhodamine 123 was prepared to be the same in every solution, eight different concentrations of DCM were prepared to achieve different extent of the inner filter effect. The EEMs of the solutions were measured with a conventional fluorescence spectrophotometer, Varian Cary Eclipse Fluorescence Spectrophotometer, at room temperature. Excitation spectra were recorded from 400 nm to 600 nm with 10 nm sampling interval and 5 nm bandwidth. Emission spectra were measured from 500 nm to 750 nm with 10 nm sampling interval and 2.5 nm bandwidth. The reference excitation and emission spectra of DCM and Rhodamine 123 were obtained separately as shown in Fig. 5(a). For each measurement, SNR was calculated as the ratio of the mean intensity of signal to the standard deviation of background signal. The computed SNRs were between 25 dB to 55 dB. By using these reference spectra, AFMU and LU were applied to the EEMs of the solutions to estimate their molecular fraction. In the unmixing process, scatter peaks of EEMs were handled as missing data [10

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]

]. With a 2.0 GHz Intel Core i7 processor and 8GB of RAM, the processing time was about 3 seconds for AFMU and less than one second for LU.

Fig. 5 Experimental results of fluorescence spectroscopy. The fluorescence spectra are measured with the right angle geometry. Eight samples were prepared with different concentrations of DCM in ethanol, while all of them had the same concentration of Rhodamine 123. The relative concentration of DCM in Sample 1–8 is set to be 1, 2, 4, 8, 20, 40, 80 and 120. (a) Measured emission and excitation spectra of 4-(Dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran (DCM) and Rhodamine 123. (b) Relative concentrations of DCM in the eight solutions are estimated by AFMU and LU from the measured EEM, and the estimated values are compared with experimentally-measured relative concentrations. (c) Reconstruction errors of the eight samples are calculated from the Euclidean distance between the measured EEM and the reconstructed EEM from the estimated concentration and the absorption spectrum. The reconstruction errors are compared between AFMU and LU. (d, e) The measured EEM and the residuals of AFMU and LU for Sample 1 and 8, respectively. Residuals are computed from the difference between a measured EEM and a corresponding reconstructed EEM.

Figure 5 summarizes the comparison of the estimation by LU and AFMU and the experimental values of DCM’s relative concentrations. Figure 5(b) shows that when the concentration of DCM is so low that the inner filter effect is negligible, both AFMU and LU can estimate the concentration accurately. However, as the relative concentration of DCM increases, the estimated relative concentration by LU does not match the experimentally-measured relative concentration due to the inner filter effect. On the other hand, the estimations by AFMU closely match the experimental values throughout a wide range of DCM’s relative concentrations. Although the estimated values by AFMU are further off from the experimental values for Sample 7 and 8, Fig. 5(c) shows that the reconstruction error of AFMU remains low at all samples compared to LU. Moreover, as is the case for both AFMU and LU on Sample 1, the randomness and smallness of the residual values by AFMU on Sample 8 in Fig. 5(e) proves that AFMU accurately separates the distorted EEMs. It should be noted that the experimentally-measured relative concentration was calculated from the amount of DCM stock solution added to ethanol; therefore, it is subject to minor experimental errors.

4.2. Fluorescence microscopy

A microsphere and a biological sample stained with two spectrally-overlapping fluorophores were employed in this experiment. The FocalCheck fluorescence Microscope Test Slide #2 (Life Technologies), which is commonly used for testing fluorescence unmixing algorithms, was used [13

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

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

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]

]. This test slide has microspheres stained differently at its shell and core as illustrated in Fig. 6(a). Their individual emission spectra are shown in Fig. 6(b). The diameter of the microspheres is 6 μm. For the biological specimen, Xenopus S3 cells cultured on cover slips were used. After formaldehyde fixation and permeabilization of cell membranes with Triton-X, 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).

Fig. 6 Spectrally unmixed images of the microsphere and the cellular specimen. (a) Schematic illustration of the microsphere. The core and shell are stained with different fluorophores. (b) Emission spectra of the two fluorophores staining the microsphere. (c) Emission spectra of the two fluorophores, TRITC and Alexa Fluor 568, staining the cellular sample. (d) Unmixed images of the stained microsphere. The second and third rows show the unmixed images of the microsphere. The first row shows the merged images of the second and third rows. The left and middle columns show the unmixed results on the same spectral image dataset captured through a hemoglobin layer. The right column shows the unmixed results by LU of the datasets captured without a hemoglobin layer. The fourth row shows the contribution from each fluorophore on the yellow line of the top-row images. The white bar in the top-left image is a scale bar whose length is 5 nm. (e) Unmixed images of the cellular specimen. Actin filaments and β tubulin are stained with Alexa Fluor 568 (Fluorophore 1, represented in red) and TRITC (Fluorophore 2, represented in green). The first, second and third rows are shown in the same way as for (d). The images in the fourth row are the magnified images of the yellow box region of the third-row images. The white scale bar in the top-left image is 20 nm. All images are represented in pseudocolor.

To imitate the environment of deep tissue imaging, hemoglobin from bovine blood was used as fluorescence attenuator as follows. Lyophilized powder of hemoglobin (Sigma-Aldrich) was dissolved in distilled water and placed on a coverslip. After evaporation of the hemoglobin solution, the coverslip was placed on top of each sample. Then, its fluorescence spectral images were captured through the two stacked coverslips with a Zeiss LSM 710 laser scanning confocal microscope, where the bandwidth and interval of emission channels were set to be 10 nm and the size of a pinhole was set to be 1 airy unit. A Plan-Apo 63×/1.4 NA oil immersion objective was used throughout all acquisitions. Although this objective is not designed to capture images through two stacked coverslips, an oil immersion objective can capture a sharp image in this setting without suffering from spherical aberration. The wavelengths of the utilized excitation lasers were 488 nm and 514 nm for the microspheres and 514 nm and 561 nm for the cellular samples. For each excitation wavelength, the spectral images were recorded with emission wavelengths from 488 nm to 628 nm for the microsphere and from 498 nm to 718 nm for the cellular sample. SNR (the ratio of the mean intensity of signal to the standard deviation of signal) was about 25 dB for the microsphere and about 6 dB for the cellular sample. Then, AFTU and LU were directly applied to the captured spectral images without deconvolution. Since the sampling ranges of excitation and emission wavelength are as narrow as about 200 nm, the wavelength-dependency of PSF was assumed to be negligible. With a 2.0 GHz Intel Core i7 processor and 8GB of RAM, the processing time was about 5 minutes for AFTU and 2 minutes for LU.

Figure 6(d) summarizes the analyzed images of the microsphere. Based on the distinct core and shell structure of the microsphere, the fluorescence light from Fluorophore 1 should dominate at the shell. This phenomenon is observed in the unmixing by LU of the spectral image captured without the hemoglobin layer, while the signal from Fluorophore 2 dominates at the core. The fluorescence signals both at the core and at the shell have contribution from both fluorophores because of out-of-focus light. However, LU does not show the dominance of Fluorophore 1 at the shell on the spectral image captured through the attenuation layer. On the other hand, in spite of the existence of the attenuation layer, AFTU clearly shows the dominance of Fluorophore 2 at the shell. The AFTU unmixed image of the cellular sample shows clearer actin bundle structures than the LU unmixed image as shown in Fig. 6(e).

Figures 6(d) and 6(e) may give an impression that AFTU is not outperforming LU [13

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]

]; however this is only because AFTU shows better performance in mixed pixels where signals from both fluorescence species exist. Therefore, at a region where multiple fluorescence species spatially overlap, the contrast of these two techniques can be apparent. As shown in the fourth row of Figs. 6(d) and 6(e), AFTU provides convincing results in a sense of ratio of intensity (microspheres) and shape (actin bundles). In addition, the reconstruction error is compared between AFTU and LU in terms of PSNR. For both samples, AFTU provides about 4 dB higher PSNR than LU.

5. Discussion

AFMU and AFTU have several limitations inherent to EEM measurements and fluorescence spectral imaging. Since both methods require scanning of wavelength with narrow bandwidth, there is an inevitable trade-off between scanning time, noise level and sampling resolution. Since this trade-off depends on the properties of fluorophores, a general guideline for deciding the measurement parameters is required. Therefore, we plan to develop an optimization framework for finding the best measurement parameters for given fluorophores. In addition, as is the case with LU and PARAFAC, AFMU and AFTU cannot be applied when fluorescence spectral properties change due to other nonlinearities, such as bleaching and pH dependence [10

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]

]. To overcome this disadvantage, a new method combining our technique with spectral fitting models considering the other nonlinearities should be developed [24

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]

].

While our algorithms, AFMU and AFTU, require spectral information of fluorophores as a priori knowledge, the exploration of a blind unmixing algorithm of the fluorescence models Eq. (2) and Eq. (6) can be considered as future work. As is incorporated in several blind unmixing methods [23

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]

], fluorescence lifetime would help to solve the problem. Furthermore, the evaluation of AFMU will be compared with existing inner-filtering correction methods. While we examined our algorithms on experimentally-controlled samples, AFMU and AFTU will expand the possibility of fluorescence spectroscopy and imaging on real world problems, such as the analysis of dissolved organic matter and deep tissue imaging.

Here, we emphasize that AFTU requires only the same prerequisite information as LU. Although we show the performance of both AFMU and AFTU on samples containing two fluorophores in experiments, they can be applied to unmixing spectra of samples containing more than two fluorophores, as we show in the simulation. AFTU is a natural extension of AFMU. However, in contrast to AFMU, AFTU utilizes the benefit of large numbers of EEMs that are acquired in one image. With this advantage, AFTU exploits an additional linearity of fluorescence signals, which makes AFTU much more robust than AFMU.

6. Conclusion

In summary, we have presented new nonlinear fluorescence unmixing algorithms with attenuation correction, AFMU for fluorescence spectroscopy and AFTU for fluorescence microscopy, which use modified multiplicative update rules for nonnegative matrix and tensor factorization. From an EEM measurement of a sample, AFMU estimates the abundance ratio of fluorophores by using each fluorophore’s pure EEM. On the other hand, AFTU unmixes spectral images of a multiply-stained sample acquired with several excitation wavelengths by using only emission spectra of fluorophores. To the best of our knowledge, AFTU is the only unmixing algorithm which separates fluorescence spectral images affected by attenuation. Even though our methods are computationally more expensive than LU, our estimation technique showed superior performance based on estimated relative concentration and reconstruction error for fluorescence spectroscopy and based on PSNR for fluorescence microscopy.

Appendix A: update rules for AFMU

To solve Eq. (3), we derive new update rules by modifying the multiplicative update rules typically employed for nonnegative matrix factorization [25

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]

]:
c^c^((W^2B)(W^1A))vec(F)((W^2B)(W^1A))(W^2B)(W^1A))c^,
(10)
w^1((AD^B)F)w^2((AD^B)(AD^B))(w^2w^2),
(11)
w^2((AD^B)F)w^1((AD^B)(AD^B))(w^1w^1),
(12)
where Ŵr (r = 1, 2) is a N-column matrix with columns that are identical horizontal copies of the column vector ŵr and ĉ = [ĉ1, ĉ2,..., ĉN]. The vectorization of an L1 × L2 matrix X to an L1L2 × 1 column vector is denoted by vec(X). During the updates of ĉ and ŵr, and Ŵr are updated accordingly. The Khatri-Rao product operator ⊙ is defined as
PQ=[p1q1p2q2pL3qL3]
(13)
for matrices P ∈ ℝL1×L3 and Q ∈ ℝL2×L3, where ⊗ represents the Kronecker product operator, and pl1 and ql2 denote the l1th and l2th columns of 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

The solver for Eq. (7) is also derived by modifying the multiplicative update rules for nonnegative tensor decomposition [26

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]

]:
S^S^(1)(U^(V^2T))((V^1S^)(U^(V^2T)))(U^(V^2T)),
(14)
U^U^(3)((V^2T^)(V^1S))(U^(V^2T^)(V^1S)))((V^2T^)(V^1S)),
(15)
v^1((1)(𝒱^[[S^,T,U^]])(1))v^2((𝒱^[[S^,T,U^]])(1)(𝒱^[[S^,T,U^]])(1))(V^2V^2),
(16)
v^2((2)(𝒱^[[S^,T,U^]])(2))v^1((𝒱^[[S^,T,U^]])(2)(𝒱^[[S^,T,U^]])(2))(V^1V^1),
(17)
where r (r = 1, 2) is the estimate of vr, r is a N-column matrix whose columns are identical horizontal copies of r, r is a column vector which has K1K2 vertical copies of r. The mode-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]

]. During the updates of r, r and r 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.

Acknowledgments

We thank Keisuke Ishihara for the preparation of the biological sample. This work was conducted utilizing the W. M. Keck Foundation Biological Imaging Facility at the Whitehead Institute.

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 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]

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 450, 56–62 (2007). [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]

11.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Springer, 2007).

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]

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]

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]

18.

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

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]

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]

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]

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]

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]

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]

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

  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. Chem32, 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. Methods5, 763–775 (2008). [CrossRef] [PubMed]
  3. Y. Garini, I. T. Young, and G. McNamara, “Spectral imaging: Principles and applications,” Cytometry A69A, 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,” Science273, 494–497 (1996). [CrossRef] [PubMed]
  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,” Nature450, 56–62 (2007). [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. Methods5, 6557–6566 (2013). [CrossRef]
  11. J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Springer, 2007).
  12. R. S. Bradley and M. S. Thorniley, “A review of attenuation correction techniques for tissue fluorescence,” J. R. Soc. Interface3, 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 ONE4, e4418 (2009). [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]
  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]
  18. T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,” SIAM Rev.51, 455–500 (2009). [CrossRef]
  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]
  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. Software32, 635–653 (2006). [CrossRef]
  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 A71A, 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. Express17, 22747–22760 (2009). [CrossRef]
  24. I. Urbančič, Z. Arsov, A. Ljubetič, D. Biglino, and J. Štrancar, “Bleaching-corrected fluorescence microspectroscopy with nanometer peak position resolution,” Opt. Express21, 25291–25306 (2013). [CrossRef]
  25. D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature401, 788–791 (1999). [CrossRef] [PubMed]
  26. G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: Compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graphics31, 80 (2012). [CrossRef]

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