## Application of maximum likelihood estimator in nano-scale optical path length measurement using spectral-domain optical coherence phase microscopy

Optics Express, Vol. 16, Issue 22, pp. 17186-17195 (2008)

http://dx.doi.org/10.1364/OE.16.017186

Acrobat PDF (332 KB)

### Abstract

Spectral-domain optical coherence phase microscopy (SD-OCPM) measures minute phase changes in transparent biological specimens using a common path interferometer and a spectrometer based optical coherence tomography system. The Fourier transform of the acquired interference spectrum in spectral-domain optical coherence tomography (SD-OCT) is complex and the phase is affected by contributions from inherent random noise. To reduce this phase noise, knowledge of the probability density function (PDF) of data becomes essential. In the present work, the intensity and phase PDFs of the complex interference signal are theoretically derived and the optical path length (OPL) PDF is experimentally validated. The full knowledge of the PDFs is exploited for optimal estimation (Maximum Likelihood estimation) of the intensity, phase, and signal-to-noise ratio (SNR) in SD-OCPM. Maximum likelihood (ML) estimates of the intensity, SNR, and OPL images are presented for two different scan modes using Bovine Pulmonary Artery Endothelial (BPAE) cells. To investigate the phase accuracy of SD-OCPM, we experimentally calculate and compare the cumulative distribution functions (CDFs) of the OPL standard deviation and the square root of the Cramér- Rao lower bound

© 2008 Optical Society of America

## 1. Introduction

1. K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,”Nature **365**, 721–727 (1993). [CrossRef] [PubMed]

3. C. Yang, A. Wax, M. S. Hahn, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Phase-referenced interferometer with subwavelength and subhertz sensitivity applied to the study of cell membrane dynamics,” Opt. Lett. **26**, 1271–1273 (2001). [CrossRef]

4. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. **23**, 817–819 (1998). [CrossRef]

5. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. **30**, 468–470 (2005). [CrossRef] [PubMed]

6. G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. **29**, 2503–2505 (2004). [CrossRef] [PubMed]

7. S. Kostianovski, S. G. Lipson, and E. N. Ribak, “Interference microscopy and Fourier fringe analysis applied to measuring the spatial refractive-index distribution,” Appl. Opt. **32**, 4744- (1993). [CrossRef] [PubMed]

8. T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. **30**, 1165–1167 (2005). [CrossRef] [PubMed]

3. C. Yang, A. Wax, M. S. Hahn, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Phase-referenced interferometer with subwavelength and subhertz sensitivity applied to the study of cell membrane dynamics,” Opt. Lett. **26**, 1271–1273 (2001). [CrossRef]

9. M. A. Choma, A. K. Ellerbee, C. Yang, T. L. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,”Opt. Lett. **30**, 1162–1164 (2005). [CrossRef] [PubMed]

11. D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. **32**, 626–628 (2007). [CrossRef] [PubMed]

12. B. White, M. Pierce, N. Nassif, B. Cense, B. Park, G. Tearney, B. Bouma, T. Chen, and J. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography,” Opt. Express **11**, 3490–3497 (2003). [CrossRef] [PubMed]

10. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. **30**, 2131–2133 (2005). [CrossRef] [PubMed]

13. S. Yazdanfar, C. Yang, M. Sarunic, and J. Izatt, “Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound,” Opt. Express **13**, 410–416 (2005). [CrossRef] [PubMed]

14. B. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm,” Opt. Express **13**, 3931–3944 (2005). [CrossRef] [PubMed]

## 2. Probability density functions of intensity and phase

*ρ*) and phase (

*θ*) at a certain location and time. The actual complex value (

*S*) can be corrupted by the addition of a white (circular symmetric) complex Gaussian variable (

*N*), giving the measured complex value X expressed as,

*ρ*and

*θ*the magnitude and phase of the actual complex value

*S*. The joint probability distribution function of the real parts,

*X*(

_{R}*S*), and the imaginary part,

_{R},σ*X*(

_{I}*S*), of the corrupted data is given by,

_{I},σ*σ*characterizing the magnitude of the noise. To find the PDFs of the amplitude and phase of the corrupted data, we define the real and imaginary parts in the polar coordinate system

*I=R*) is given by [15],

^{2}*0≤ϕ<2π*) is extracted by,

*u=(R-ρ cos(ϕ-θ))/σ*, Eq. (9) can be written as

*SNR*=

*ρ*/(

^{2}*2σ*), and

^{2}*I*(.), and

_{0}*Q*(.) are the zeroth order modified Bessel function and Q function, respectively. Using upper and lower bounds on the Q function [15], we can show the marginal PDF of the phase is bounded by two analytical expressions as follows:

*ϕ-θ*) is equal to

*π/2 or3π/2*, the marginal PDF of the phase is equal to (

*exp*(-

*SNR*))/

*2π*. Using Eq. (12), the marginal PDF of the phase approaches a uniform distribution over [

*0, 2π*] for low SNR values. For high SNR values, the derived inequalities (13-14) show that the marginal PDF of the phase can be approximated as follows:

*ϕ*) in Eq. (12) for different true phase values (

*θ*) at two different SNR values. As shown in Figs. 1(a)-1(b), when either SNR decreases or the true phase value approaches zero or

*2π*, the PDF broadens due to the branch cut at 0 modulo

*2π*and causes an increase in the measured phase variance (or standard deviation). At a SNR equal to 20 dB (CRLB=0.005), the calculated phase variances using Eq. (12) were 0.005 and 8.35 rad

^{2}for the true phase values

*π/4*and

*π/100*, respectively. When the SNR was decreased to 6 dB (CRLB=0.126), the phase variances were calculated to be 0.74 and 4.97 rad

^{2}for the true phase values

*π/4*and

*π/10*, respectively. In this scenario, there is a significant difference between the exact phase sensitivity (standard deviation) and the reported fundamental uncertainty limits on the estimated phase

13. S. Yazdanfar, C. Yang, M. Sarunic, and J. Izatt, “Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound,” Opt. Express **13**, 410–416 (2005). [CrossRef] [PubMed]

## 3. Maximum likelihood estimator

16. A. J. Miller and P. M. Joseph, “The use of power images to perform quantitative analysis on low SNR MR images,” Magn. Reson. Imaging **11**, 1051–1056 (1993). [CrossRef] [PubMed]

17. G. McGibney and M. R. Smith, “An unbiased signal-to-noise ratio measure for magnetic resonance images,” Med Phys. **20**, 1077–1078 (1993). [CrossRef] [PubMed]

*N*independent, Gaussian distributed, complex data points, the joint PDF of the complex data,

*p*is simply the product of the PDFs of these data points:

_{c}*L*[18]. The ML estimate of each parameter is found by maximizing

*L*with respect to that parameter [18]. To simplify maximizing

*L*, we may work with the natural logarithm of

*p*, which is a monotonic function. Thus,

_{c}*Log L*, with respect to these variables. At the maximum, the first order derivatives of

*Log L*with respect to these variables are zero. From the resulting equations, ML estimators of the amplitude, phase, and variance are found to be:

## 4. Experimental and simulation results

*λ*) Kerr-lens mode-locked laser (~130 nm in FWHM, FemtoLasers, Austria) was employed for high resolution OCPM. The OCPM system was described in detail in Ref [10

_{0}10. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. **30**, 2131–2133 (2005). [CrossRef] [PubMed]

*τ*=25 µs). While the mean value of the measured OPLs was 222.396 nm, the histogram of the measured OPLs and the derived PDF were offset to zero nm. Figure 3(b) shows the experimental and theoretical results for the OPL standard deviation as a function of the measured SNR for OPL~222 nm. Figure 3(c) shows the theoretically predicted standard deviation of the OPL as a function of SNR for different OPLs. It is clear the sensitivity degrades when SNR decreases. In addition, when the true value of the OPL approaches an integer number of half the center wavelength (branch cuts={0 nm, 400 nm,…}) the OPL standard deviation increases. At very small OPL values (for example OPL=3 nm), the standard deviation increased with increasing SNR, which can be explained by a PDF of the phase that became narrower around 0 and

*2π*with a mean phase value of approximately

*π*, as depicted in Figs. 1(a)-1(b). The standard deviation gradually decreased again when the PDF of the phase became asymmetric with a mean shifting towards zero when the SNR increased further. Figure 3(d) shows that the OPL standard deviation deviates from the square root of the CRLB at low SNR and OPL values close to an integer number of half the center wavelength.

*M*sets of

*N*data points, and we applied the ML and mean estimators to each set of data. Finally, we calculated the RMSE of each estimator by using the estimated OPLs for each set and the true OPL value for at least 65533 datasets.

*N*). At SNR=10 dB, true OPL value of 25 nm (

*θ=π/8*), and

*N*=100, the RMSE of the ML and mean estimators were 1.4 nm and 18.7 nm, respectively. The simulation results show that the ML method outperforms the mean method. For example, the precision of the ML and mean estimators were 2.2 nm and 58.2 nm at SNR=6 dB, true OPL value of 25 nm, and N=100, respectively.

## 5. Images

## Acknowledgment

## References and links

1. | K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,”Nature |

2. | J. Farinas and A. S. Verkman, “Cell volume and plasma membrane osmotic water permeability in epithelial cell layers measured by interferometry,” Biophys. J. |

3. | C. Yang, A. Wax, M. S. Hahn, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Phase-referenced interferometer with subwavelength and subhertz sensitivity applied to the study of cell membrane dynamics,” Opt. Lett. |

4. | A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. |

5. | P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. |

6. | G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. |

7. | S. Kostianovski, S. G. Lipson, and E. N. Ribak, “Interference microscopy and Fourier fringe analysis applied to measuring the spatial refractive-index distribution,” Appl. Opt. |

8. | T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. |

9. | M. A. Choma, A. K. Ellerbee, C. Yang, T. L. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,”Opt. Lett. |

10. | C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. |

11. | D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. |

12. | B. White, M. Pierce, N. Nassif, B. Cense, B. Park, G. Tearney, B. Bouma, T. Chen, and J. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography,” Opt. Express |

13. | S. Yazdanfar, C. Yang, M. Sarunic, and J. Izatt, “Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound,” Opt. Express |

14. | B. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm,” Opt. Express |

15. | A. Papoulis and S. U. Pillai, |

16. | A. J. Miller and P. M. Joseph, “The use of power images to perform quantitative analysis on low SNR MR images,” Magn. Reson. Imaging |

17. | G. McGibney and M. R. Smith, “An unbiased signal-to-noise ratio measure for magnetic resonance images,” Med Phys. |

18. | B. A. van den, |

**OCIS Codes**

(110.0180) Imaging systems : Microscopy

(110.4500) Imaging systems : Optical coherence tomography

(170.0110) Medical optics and biotechnology : Imaging systems

(180.3170) Microscopy : Interference microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: April 22, 2008

Revised Manuscript: August 20, 2008

Manuscript Accepted: October 6, 2008

Published: October 13, 2008

**Virtual Issues**

Vol. 3, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

S. M. R. Motaghian Nezam, C. Joo, G. J. Tearney, and J. F. de Boer, "Application of maximum likelihood estimator in nano-scale optical path length measurement using spectral-domain optical coherence phase microscopy," Opt. Express **16**, 17186-17195 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-22-17186

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### References

- K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,”Nature 365, 721–727 (1993). [CrossRef] [PubMed]
- J. Farinas, A. S. Verkman, “Cell volume and plasma membrane osmotic water permeability in epithelial cell layers measured by interferometry,” Biophys. J. 71, 3511–3522 (1996). [CrossRef] [PubMed]
- C. Yang, A. Wax, M. S. Hahn, K. Badizadegan, R. R. Dasari, M. S. Feld, “Phase-referenced interferometer with subwavelength and subhertz sensitivity applied to the study of cell membrane dynamics,” Opt. Lett. 26, 1271–1273 (2001). [CrossRef]
- A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998). [CrossRef]
- P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30, 468–470 (2005). [CrossRef] [PubMed]
- G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29, 2503–2505 (2004). [CrossRef] [PubMed]
- S. Kostianovski, S. G. Lipson, E. N. Ribak, “Interference microscopy and Fourier fringe analysis applied to measuring the spatial refractive-index distribution,” Appl. Opt. 32, 4744- (1993). [CrossRef] [PubMed]
- T. Ikeda, G. Popescu, R. R. Dasari, M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1167 (2005). [CrossRef] [PubMed]
- M. A. Choma, A. K. Ellerbee, C. Yang, T. L. Creazzo, J. A. Izatt, “Spectral-domain phase microscopy,”Opt. Lett. 30, 1162–1164 (2005). [CrossRef] [PubMed]
- C. Joo, T. Akkin, B. Cense, B. H. Park, J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. 30, 2131–2133 (2005). [CrossRef] [PubMed]
- D. C. Adler, R. Huber, J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32, 626–628 (2007). [CrossRef] [PubMed]
- B. White, M. Pierce, N. Nassif, B. Cense, B. Park, G. Tearney, B. Bouma, T. Chen, J. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography,” Opt. Express 11, 3490–3497 (2003). [CrossRef] [PubMed]
- S. Yazdanfar, C. Yang, M. Sarunic, J. Izatt, “Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound,” Opt. Express 13, 410–416 (2005). [CrossRef] [PubMed]
- B. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. Tearney, B. Bouma, J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm,” Opt. Express 13, 3931–3944 (2005). [CrossRef] [PubMed]
- A. Papoulis, S. U. Pillai, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, 2002) 4th edition.
- A. J. Miller, P. M. Joseph, “The use of power images to perform quantitative analysis on low SNR MR images,” Magn. Reson. Imaging 11, 1051–1056 (1993). [CrossRef] [PubMed]
- G. McGibney, M. R. Smith, “An unbiased signal-to-noise ratio measure for magnetic resonance images,” Med Phys. 20, 1077–1078 (1993). [CrossRef] [PubMed]
- B. A. van den, Handbook of Measurement Science, (Wiley, Chichester, England, 1982) Vol. 1, pp. 331–377.

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