## Blind source unmixing in multi-spectral optoacoustic tomography |

Optics Express, Vol. 19, Issue 4, pp. 3175-3184 (2011)

http://dx.doi.org/10.1364/OE.19.003175

Acrobat PDF (1032 KB)

### Abstract

Multispectral optoacoustic (photoacoustic) tomography (MSOT) is a hybrid modality that can image through several millimeters to centimeters of diffuse tissues, attaining resolutions typical of ultrasound imaging. The method can further identify tissue biomarkers by decomposing the spectral contributions of different photo-absorbing molecules of interest. In this work we investigate the performance of blind source unmixing methods and spectral fitting approaches in decomposing the contributions of fluorescent dyes from the tissue background, based on MSOT measurements in mice. We find blind unmixing as a promising method for accurate MSOT decomposition, suitable also for spectral unmixing in fluorescence imaging. We further demonstrate its capacity with temporal unmixing on real-time MSOT data obtained *in-vivo* for enhancing the visualization of absorber agent flow in the mouse vascular system.

© 2011 Optical Society of America

## 1. Introduction

1. H. Zhang, K. Maslov, G. Stoica, and L. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. **24**, 848–851 (2006). [CrossRef] [PubMed]

2. M. Xu and L. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. **77**, 041101 (2006). [CrossRef]

3. V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods **7**, 603–614 (2010). [CrossRef] [PubMed]

4. C. G. A. Hoelen, F. F. M. de Mul, R. Pongers, and A. Dekker, “Three-dimensional photoacoustic imaging of blood vessels in tissue,” Opt. Lett. **23**, 648–650 (1998). [CrossRef]

5. H. Fang, K. Maslov, and L. V. Wang, “Photoacoustic doppler effect from flowing small light-absorbing particles,” Phys. Rev. Lett. **99**, 184501 (2007). [CrossRef] [PubMed]

6. P.-C. Li, S.-W. Huang, C.-W. Wei, Y.-C. Chiou, C.-D. Chen, and C.-R. C. Wang, “Photoacoustic flow measurements by use of laser-induced shape transitions of gold nanorods,” Opt. Lett. **30**, 3341–3343 (2005). [CrossRef]

7. V. Ntziachristos and D. Razansky, “Molecular imaging by means of multispectral optoacoustic tomography (MSOT),” Chem. Rev. **110**, 2783–2794 (2010). [CrossRef] [PubMed]

8. H.-P. Brecht, R. Su, M. Fronheiser, S. A. Ermilov, A. Conjusteau, and A. A. Oraevsky, “Whole-body three-dimensional optoacoustic tomography system for small animals,” J. Biomed. Opt. **14**, 064007 (2009). [CrossRef]

9. J. Gamelin, A. Maurudis, A. Aguirre, F. Huang, P. Guo, L. V. Wang, and Q. Zhu, “A real-time photoacoustic tomography system for small animals,” Opt. Express **17**, 10489–10498 (2009). [CrossRef] [PubMed]

10. A. Buehler, E. Herzog, D. Razansky, and V. Ntziachristos, “Video rate optoacoustic tomography of mouse kidney perfusion,” Opt. Lett. **35**, 2475–2477 (2010). [CrossRef] [PubMed]

11. G. Busse and A. Rosencwaig, “Subsurface imaging with photoacoustics,” Appl. Phys. Lett. **36**, 815–816 (1980). [CrossRef]

12. A. Rosencwaig, “Potential clinical applications of photoacoustics,” Clin. Chem. **28**, 1878–1881 (1982). [PubMed]

13. X. Wang, X. Xie, G. Ku, L. V. Wang, and G. Stoica, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic tomography,” J. Biomed. Opt. **11**, 024015 (2006). [CrossRef] [PubMed]

14. D. Razansky, C. Vinegoni, and V. Ntziachristos, “Multispectral photoacoustic imaging of fluorochromes in small animals,” Opt. Lett. **32**, 2891–2893 (2007). [CrossRef] [PubMed]

15. D. Razansky, M. Distel, C. Vinegoni, R. Ma, M. Perrimon, R. W. Koster, and V. Ntziachristos, “Multispectral opto-acoustic tomography of deep-seated fluorescent proteins in vivo,” Nat. Photonics **3**, 412–417 (2009). [CrossRef]

16. P.-C. Li, C.-R. C. Wang, D.-B. Shieh, C.-W. Wei, C.-K. Liao, C. Poe, S. Jhan, A.-A. Ding, and Y.-N. Wu, “In vivo photoacoustic molecular imaging withsimultaneous multiple selective targeting using antibody-conjugated gold nanorods,” Opt. Express **16**, 18605–18615 (2008). [CrossRef]

17. A. Taruttis, E. Herzog, D. Razansky, and V. Ntziachristos, “Real-time imaging of cardiovascular dynamics and circulating gold nanorods with multispectral optoacoustic tomography,” Opt. Express **18**, 19592–19602 (2010). [CrossRef] [PubMed]

*in-vivo*tissue imaging. In addition, the spectral signature of the agent of interest may also be not accurately known, for instance the absorption spectrum may change in different biochemical environments. Moreover, light attenuation and ultrasonic dispersion in tissues leads to a corresponding non-linear relationship between the measured optoacoustic signals and the corresponding target concentration, as a function of depth, or target size [18

18. D. Razansky, J. Baeten, and V. Ntziachristos, “Sensitivity of molecular target detection by multispectral optoacoustic tomography (MSOT),” Med. Phys. **36**, 939–945 (2009). [CrossRef] [PubMed]

19. A. Rosenthal, D. Razansky, and V. Ntziachristos, “Quantitative optoacoustic signal extraction using sparse signal representation,” IEEE Trans. Med. Imaging **28**, 1997–2006 (2009). [CrossRef] [PubMed]

21. A. Cichocki, R. Zdunek, A. H. Phan, and S. I. Amari, *Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation*, 1st ed. (Wiley, 2009). [PubMed]

22. R. Tauler, B. Kowalski, and S. Fleming, “Multivariate curve resolution applied to spectral data from multiple runs of an industrial process,” Anal. Chem. **65**, 2040–2047 (1993). [CrossRef]

24. M. Funaro, E. Oja, and H. Valpola, “Independent component analysis for artefact separation in astrophysical images,” Neural Networks **16**, 469–478 (2003). [CrossRef] [PubMed]

25. B. A. Draper, K. Baek, M. S. Bartlett, and J. R. Beveridge, “Recognizing faces with pca and ica,” Comput. Vis. Image Underst. **91**, 115 – 137 (2003). [CrossRef]

26. J. Yang, D. Zhang, A. F. Frangi, and J. Y. Yang, “Two-dimensional pca: a new approach to appearance-based face representation and recognition,” IEEE Trans. Pattern Anal. Mach. Intell. **26**, 131–137 (2004). [CrossRef] [PubMed]

27. E. M. C. Hillman and A. Moore, “All optical anatomical co registration for molecular imaging of small animals using dynamic contrast,” Nat. Photonics **1**(9) 526–530 (2007). [CrossRef]

28. H. Xu and B. W. Rice, “In-vivo fluorescence imaging with a multivariate curve resolution spectral unmixing technique,” J. Biomed. Opt. **14**, 064011 (2009). [CrossRef]

29. A.-S. Montcuquet, L. Hervé, F. Navarro, J.-M. Dinten, and J. I. Mars, “Nonnegative matrix factorization: a blind spectra separation method for in vivo fluorescent optical imaging,” J. Biomed. Opt. **15**, 056009 (2010). [CrossRef] [PubMed]

30. S. Clémençon and S. Slim, “On portfolio selection under extreme risk measure: the heavy-tailed ICA Model,” Int. J. Theor. Appl. Finance **10**, 449–474 (2007). [CrossRef]

*in-vivo*and

*ex-vivo*experiments.

## 2. Theory

### 2.1. Spectral Fitting

*n*×

*m*) multispectral measurement matrix

**M**, where

*n*is the number of image pixels and

*m*is the number of measurements, as well as the (

*k*×

*m*) spectral matrix

**S**with the absorption coefficients of the

*k*components at the

*m*measurement wavelengths, the data can be unmixed with the Moore-Penrose pseudoinverse [32, 33

33. R. Penrose, “A generalized inverse for matrices,” in Proceedings of the Cambridge Philosophical Society , (1955), Vol. **51**, pp. 406–412. [CrossRef]

**S**

^{+}. With this generalized inverse, the source component reconstruction

**R**

_{pinv}, which best fits the spectra in

**S**can be determined as The performance of the spectral fitting method depends on the accuracy and completeness of the absorption spectra and the absence of systematic errors in the data. It therefore becomes challenging to apply the method when reliable spectral information for all potential contributions in the signal is not available, for instance for

*in-vivo*tissue measurements. In such cases, approaches that do not require

*a priori*spectral information may be beneficial.

### 2.2. Principal Component Analysis

### 2.3. Independent Component Analysis

**R**′

_{PCA}is the reduced PCA subset, the unmixing is denoted by Equation (5) also shows that the estimated spectral characteristics of the unmixed components can be retrieved as the product of the two mixing matrices

## 3. Experimental Methods

### 3.1. Experimental system and data processing

10. A. Buehler, E. Herzog, D. Razansky, and V. Ntziachristos, “Video rate optoacoustic tomography of mouse kidney perfusion,” Opt. Lett. **35**, 2475–2477 (2010). [CrossRef] [PubMed]

*μ*V/Pa and are shaped to create a cylindrical focus at 40 mm. The detected signals are digitized at a sampling frequency of 60 MHz by 8 multi-channel analog to digital converters (Model PXI5105, National Instruments, Austin, TX, USA) with noise floor of

*z*direction. The data acquisition is synchronized so that the signals are acquired only when the stage comes to a complete rest. For all experiments the optoacoustic pressure distribution was reconstructed with a filtered backprojection algorithm [39

39. M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E **71**, 016706 (2005). [CrossRef]

### 3.2. Animal preparation and imaging

*μ*M of ICG (peak absorption ≈ 800 nm, Pulsion Medical, Munich, Germany) and 3.7

*μ*M of Cy7 Cyanine Dye (peak absorption ≈ 750 nm, GE Healthcare, Little Chalfont, UK) were prepared to both have 2 cm

^{−1}optical density at their absorption peaks. At 35 °C, 50

*μ*L of each solution was injected subcutaneously in the upper back / neck area of a euthanized mouse and were left to equilibrate with room temperature and solidify to a shape that is similar to a subcutaneous tumor.

39. M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E **71**, 016706 (2005). [CrossRef]

*μ*L of black india ink saline solution as contrast medium. An axial slice in the pelvic region, approximately 2 cm away from the catheter towards the heart, was imaged with MSOT for 80 s, using herein data at 1 Hz frame rate, after performing 10 signal averages at a wavelength of 800 nm. After imaging, the mouse was euthanized

*in situ*.

## 4. Spectral domain unmixing

*ρ*, the correlation between the known spectra and the ones estimated by ICA was calculated to be

*ρ*= 0.91 for the background,

*ρ*= 0.83 for ICG and

*ρ*= 0.96 for Cy7. In order to evaluate the performance of the unmixing methods the signal-to-background ratios (SBR) were calculated using the standard deviation of the pixel values for the two fluorochrome inclusions. The results, as shown in Table 1 coincide with the visual examination that the ICA unmixing method outperforms fitting and PCA.

## 5. Time domain unmixing

*t*= 10 s leading to a notably increasing and then gradually decreasing signal in the right vein. A few seconds later the signal in the left vein showed an oscillatory pattern that eventually faded out, indicating that the absorber was fully diluted in the blood stream after 50 s. To investigate the unmixing methods in the time domain, we examined the ability to separate the tissue components based on the dynamic changes of the images. The first three principle components from PCA on the full 80-frame time-series and the corresponding temporal profiles (eigenvectors) are shown in Figs. 5a–d.

## 6. Discussion and Conclusion

10. A. Buehler, E. Herzog, D. Razansky, and V. Ntziachristos, “Video rate optoacoustic tomography of mouse kidney perfusion,” Opt. Lett. **35**, 2475–2477 (2010). [CrossRef] [PubMed]

17. A. Taruttis, E. Herzog, D. Razansky, and V. Ntziachristos, “Real-time imaging of cardiovascular dynamics and circulating gold nanorods with multispectral optoacoustic tomography,” Opt. Express **18**, 19592–19602 (2010). [CrossRef] [PubMed]

*a priori*information is needed, making the technique suitable for a wide range of applications. It can separate contrast agents of a common spectral or temporal bio-distribution from background absorbers such as hemoglobin. The unmixing quality of PCA and ICA demonstrated herein on mouse measurements was found to be superior to spectral fitting, further generating components that can lead to spectral identification of specific tissue biomarkers such as hemoglobin or the contributions of intrinsically expressed or administered molecular probes. Additionally, the use of blind unmixing was successfully demonstrated in the time domain by separation of blood veins based on dynamic changes of an optoacoustic image sequence. It is evident, that the sensitivity of the imaging systems directly benefits from an improved unmixing performance.

*a priori*knowledge of temporal profiles of biological processes or the contrast agent’s temporal biodistribution in living organisms. Therefore fitting procedures are not common in dynamic measurements. In response, we found that temporal unmixing based on blind decomposition methods required a combination of PCA and ICA to lead to clearly perceived time components.

*a priori*information that may be available. Applications of these unmixing methods are envisioned also in the field of optical fluorescence imaging, where they can help to increase detection sensitivity, and in autofluorescence removal.

## Acknowledgments

## References and links

1. | H. Zhang, K. Maslov, G. Stoica, and L. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. |

2. | M. Xu and L. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. |

3. | V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods |

4. | C. G. A. Hoelen, F. F. M. de Mul, R. Pongers, and A. Dekker, “Three-dimensional photoacoustic imaging of blood vessels in tissue,” Opt. Lett. |

5. | H. Fang, K. Maslov, and L. V. Wang, “Photoacoustic doppler effect from flowing small light-absorbing particles,” Phys. Rev. Lett. |

6. | P.-C. Li, S.-W. Huang, C.-W. Wei, Y.-C. Chiou, C.-D. Chen, and C.-R. C. Wang, “Photoacoustic flow measurements by use of laser-induced shape transitions of gold nanorods,” Opt. Lett. |

7. | V. Ntziachristos and D. Razansky, “Molecular imaging by means of multispectral optoacoustic tomography (MSOT),” Chem. Rev. |

8. | H.-P. Brecht, R. Su, M. Fronheiser, S. A. Ermilov, A. Conjusteau, and A. A. Oraevsky, “Whole-body three-dimensional optoacoustic tomography system for small animals,” J. Biomed. Opt. |

9. | J. Gamelin, A. Maurudis, A. Aguirre, F. Huang, P. Guo, L. V. Wang, and Q. Zhu, “A real-time photoacoustic tomography system for small animals,” Opt. Express |

10. | A. Buehler, E. Herzog, D. Razansky, and V. Ntziachristos, “Video rate optoacoustic tomography of mouse kidney perfusion,” Opt. Lett. |

11. | G. Busse and A. Rosencwaig, “Subsurface imaging with photoacoustics,” Appl. Phys. Lett. |

12. | A. Rosencwaig, “Potential clinical applications of photoacoustics,” Clin. Chem. |

13. | X. Wang, X. Xie, G. Ku, L. V. Wang, and G. Stoica, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic tomography,” J. Biomed. Opt. |

14. | D. Razansky, C. Vinegoni, and V. Ntziachristos, “Multispectral photoacoustic imaging of fluorochromes in small animals,” Opt. Lett. |

15. | D. Razansky, M. Distel, C. Vinegoni, R. Ma, M. Perrimon, R. W. Koster, and V. Ntziachristos, “Multispectral opto-acoustic tomography of deep-seated fluorescent proteins in vivo,” Nat. Photonics |

16. | P.-C. Li, C.-R. C. Wang, D.-B. Shieh, C.-W. Wei, C.-K. Liao, C. Poe, S. Jhan, A.-A. Ding, and Y.-N. Wu, “In vivo photoacoustic molecular imaging withsimultaneous multiple selective targeting using antibody-conjugated gold nanorods,” Opt. Express |

17. | A. Taruttis, E. Herzog, D. Razansky, and V. Ntziachristos, “Real-time imaging of cardiovascular dynamics and circulating gold nanorods with multispectral optoacoustic tomography,” Opt. Express |

18. | D. Razansky, J. Baeten, and V. Ntziachristos, “Sensitivity of molecular target detection by multispectral optoacoustic tomography (MSOT),” Med. Phys. |

19. | A. Rosenthal, D. Razansky, and V. Ntziachristos, “Quantitative optoacoustic signal extraction using sparse signal representation,” IEEE Trans. Med. Imaging |

20. | I. T. Jolliffe, |

21. | A. Cichocki, R. Zdunek, A. H. Phan, and S. I. Amari, |

22. | R. Tauler, B. Kowalski, and S. Fleming, “Multivariate curve resolution applied to spectral data from multiple runs of an industrial process,” Anal. Chem. |

23. | A. Hyvärinen, J. Karhunen, and E. Oja, |

24. | M. Funaro, E. Oja, and H. Valpola, “Independent component analysis for artefact separation in astrophysical images,” Neural Networks |

25. | B. A. Draper, K. Baek, M. S. Bartlett, and J. R. Beveridge, “Recognizing faces with pca and ica,” Comput. Vis. Image Underst. |

26. | J. Yang, D. Zhang, A. F. Frangi, and J. Y. Yang, “Two-dimensional pca: a new approach to appearance-based face representation and recognition,” IEEE Trans. Pattern Anal. Mach. Intell. |

27. | E. M. C. Hillman and A. Moore, “All optical anatomical co registration for molecular imaging of small animals using dynamic contrast,” Nat. Photonics |

28. | H. Xu and B. W. Rice, “In-vivo fluorescence imaging with a multivariate curve resolution spectral unmixing technique,” J. Biomed. Opt. |

29. | A.-S. Montcuquet, L. Hervé, F. Navarro, J.-M. Dinten, and J. I. Mars, “Nonnegative matrix factorization: a blind spectra separation method for in vivo fluorescent optical imaging,” J. Biomed. Opt. |

30. | S. Clémençon and S. Slim, “On portfolio selection under extreme risk measure: the heavy-tailed ICA Model,” Int. J. Theor. Appl. Finance |

31. | N. Keshava, “A survey of spectral unmixing algorithms,” Lincoln Lab. J. |

32. | E. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Am. Math. Soc. |

33. | R. Penrose, “A generalized inverse for matrices,” in Proceedings of the Cambridge Philosophical Society , (1955), Vol. |

34. | K. Pearson, “On lines and planes of closest fit to a system of points in space,” |

35. | S. M. Kay, |

36. | J. Nash, “The singular-value decomposition and its use to solve least-squares problems,” in |

37. | L. Le Cam, “The central limit theorem around 1935,” Stat. Sci. |

38. | A. Hyvrinen, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Trans. Neural Netw. |

39. | M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E |

**OCIS Codes**

(170.5120) Medical optics and biotechnology : Photoacoustic imaging

(170.6280) Medical optics and biotechnology : Spectroscopy, fluorescence and luminescence

(170.6960) Medical optics and biotechnology : Tomography

(100.1455) Image processing : Blind deconvolution

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: November 2, 2010

Revised Manuscript: January 25, 2011

Manuscript Accepted: January 30, 2011

Published: February 3, 2011

**Virtual Issues**

Vol. 6, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Jürgen Glatz, Nikolaos C. Deliolanis, Andreas Buehler, Daniel Razansky, and Vasilis Ntziachristos, "Blind source unmixing in multi-spectral optoacoustic tomography," Opt. Express **19**, 3175-3184 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3175

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### References

- H. Zhang, K. Maslov, G. Stoica, and L. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24, 848–851 (2006). [CrossRef] [PubMed]
- M. Xu, and L. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101 (2006). [CrossRef]
- V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7, 603–614 (2010). [CrossRef] [PubMed]
- C. G. A. Hoelen, F. F. M. de Mul, R. Pongers, and A. Dekker, “Three-dimensional photoacoustic imaging of blood vessels in tissue,” Opt. Lett. 23, 648–650 (1998). [CrossRef]
- H. Fang, K. Maslov, and L. V. Wang, “Photoacoustic doppler effect from flowing small light-absorbing particles,” Phys. Rev. Lett. 99, 184501 (2007). [CrossRef] [PubMed]
- P.-C. Li, S.-W. Huang, C.-W. Wei, Y.-C. Chiou, C.-D. Chen, and C.-R. C. Wang, “Photoacoustic flow measurements by use of laser-induced shape transitions of gold nanorods,” Opt. Lett. 30, 3341–3343 (2005). [CrossRef]
- V. Ntziachristos, and D. Razansky, “Molecular imaging by means of multispectral optoacoustic tomography (MSOT),” Chem. Rev. 110, 2783–2794 (2010). [CrossRef] [PubMed]
- H.-P. Brecht, R. Su, M. Fronheiser, S. A. Ermilov, A. Conjusteau, and A. A. Oraevsky, “Whole-body threedimensional optoacoustic tomography system for small animals,” J. Biomed. Opt. 14, 064007 (2009). [CrossRef]
- J. Gamelin, A. Maurudis, A. Aguirre, F. Huang, P. Guo, L. V. Wang, and Q. Zhu, “A real-time photoacoustic tomography system for small animals,” Opt. Express 17, 10489–10498 (2009). [CrossRef] [PubMed]
- A. Buehler, E. Herzog, D. Razansky, and V. Ntziachristos, “Video rate optoacoustic tomography of mouse kidney perfusion,” Opt. Lett. 35, 2475–2477 (2010). [CrossRef] [PubMed]
- G. Busse, and A. Rosencwaig, “Subsurface imaging with photoacoustics,” Appl. Phys. Lett. 36, 815–816 (1980). [CrossRef]
- A. Rosencwaig, “Potential clinical applications of photoacoustics,” Clin. Chem. 28, 1878–1881 (1982). [PubMed]
- X. Wang, X. Xie, G. Ku, L. V. Wang, and G. Stoica, “Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic tomography,” J. Biomed. Opt. 11, 024015 (2006). [CrossRef] [PubMed]
- D. Razansky, C. Vinegoni, and V. Ntziachristos, “Multispectral photoacoustic imaging of fluorochromes in small animals,” Opt. Lett. 32, 2891–2893 (2007). [CrossRef] [PubMed]
- D. Razansky, M. Distel, C. Vinegoni, R. Ma, M. Perrimon, R. W. Koster, and V. Ntziachristos, “Multispectral opto-acoustic tomography of deep-seated fluorescent proteins in vivo,” Nat. Photonics 3, 412–417 (2009). [CrossRef]
- P.-C. Li, C.-R. C. Wang, D.-B. Shieh, C.-W. Wei, C.-K. Liao, C. Poe, S. Jhan, A.-A. Ding, and Y.-N. Wu, “In vivo photoacoustic molecular imaging withsimultaneous multiple selective targeting using antibody-conjugated gold nanorods,” Opt. Express 16, 18605–18615 (2008). [CrossRef]
- A. Taruttis, E. Herzog, D. Razansky, and V. Ntziachristos, “Real-time imaging of cardiovascular dynamics and circulating gold nanorods with multispectral optoacoustic tomography,” Opt. Express 18, 19592–19602 (2010). [CrossRef] [PubMed]
- D. Razansky, J. Baeten, and V. Ntziachristos, “Sensitivity of molecular target detection by multispectral optoacoustic tomography (MSOT),” Med. Phys. 36, 939–945 (2009). [CrossRef] [PubMed]
- A. Rosenthal, D. Razansky, and V. Ntziachristos, “Quantitative optoacoustic signal extraction using sparse signal representation,” IEEE Trans. Med. Imaging 28, 1997–2006 (2009). [CrossRef] [PubMed]
- I. T. Jolliffe, Principal Component Analysis, 2nd ed. (Springer, 2002).
- A. Cichocki, R. Zdunek, A. H. Phan, and S. I. Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation, 1st ed. (Wiley, 2009). [PubMed]
- R. Tauler, B. Kowalski, and S. Fleming, “Multivariate curve resolution applied to spectral data from multiple runs of an industrial process,” Anal. Chem. 65, 2040–2047 (1993). [CrossRef]
- A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis, Adaptive and Learning Systems for Signal Processing, Communications, and Control, 1st ed. (Wiley InterScience, 2002).
- M. Funaro, E. Oja, and H. Valpola, “Independent component analysis for artefact separation in astrophysical images,” Neural Netw. 16, 469–478 (2003). [CrossRef] [PubMed]
- B. A. Draper, K. Baek, M. S. Bartlett, and J. R. Beveridge, “Recognizing faces with pca and ica,” Comput. Vis. Image Underst. 91, 115–137 (2003). [CrossRef]
- J. Yang, D. Zhang, A. F. Frangi, and J. Y. Yang, “Two-dimensional pca: a new approach to appearance-based face representation and recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 131–137 (2004). [CrossRef] [PubMed]
- E. M. C. Hillman, and A. Moore, “All optical anatomical co registration for molecular imaging of small animals using dynamic contrast,” Nat. Photonics 1(9), 526–530 (2007). [CrossRef]
- H. Xu, and B. W. Rice, “In-vivo fluorescence imaging with a multivariate curve resolution spectral unmixing technique,” J. Biomed. Opt. 14, 064011 (2009). [CrossRef]
- A.-S. Montcuquet, L. Herv’e, F. Navarro, J.-M. Dinten, and J. I. Mars, “Nonnegative matrix factorization: a blind spectra separation method for in vivo fluorescent optical imaging,” J. Biomed. Opt. 15, 056009 (2010). [CrossRef] [PubMed]
- S. Clémençon, and S. Slim, “On portfolio selection under extreme risk measure: the heavy-tailed ICA Model,” Int. J. Theor. Appl. Finance 10, 449–474 (2007). [CrossRef]
- N. Keshava, “A survey of spectral unmixing algorithms,” Lincoln Lab. J. 14, 55–78 (2003).
- E. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Am. Math. Soc. 26, 394–395 (1920).
- R. Penrose, “A generalized inverse for matrices,” in Proceedings of the Cambridge Philosophical Society (1955) Vol. 51, pp. 406–412. [CrossRef]
- K. Pearson, “On lines and planes of closest fit to a system of points in space,” London, Edinburgh Dublin Philos, Mag. J. Sci. 6, 559–572 (1901).
- S. M. Kay, Fundamentals of Statistical Signal Processing, 1st ed. (Prentice Hall PTR, 1993),Vol. 1.
- J. Nash, “The singular-value decomposition and its use to solve least-squares problems,” in Compact Numerical Methods for Computers: Linear Algebra and Function Minimization, 2nd ed. (Inst. of Physics Pub., 1990), pp. 30–48.
- L. Le Cam, “The central limit theorem around 1935,” Stat. Sci. 1, 78–91 (1986). [CrossRef]
- A. Hyvrinen, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Trans. Neural Netw. 10, 626–634 (1999). [CrossRef]
- M. Xu, and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71, 016706 (2005). [CrossRef]

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