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 12SNR over 100 BPAE images for two different scan modes. The correction to the OPL measurement by applying ML estimation to SD-OCPM for BPAE cells is demonstrated.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2007 (1)

2005 (6)

2004 (1)

2003 (1)

2001 (1)

1998 (1)

1996 (1)

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]

1993 (4)

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]

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]

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]

Adler, D. C.

Akkin, T.

Badizadegan, K.

Barty, A.

Block, S. M.

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]

Bouma, B.

Cense, B.

Chen, T.

Choma, M. A.

Colomb, T.

Creazzo, T. L.

Cuche, E.

Dasari, R. R.

de Boer, J.

de Boer, J. F.

Deflores, L. P.

Depeursinge, C.

Ellerbee, A. K.

Emery, Y.

Farinas, J.

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]

Feld, M. S.

Fujimoto, J. G.

Hahn, M. S.

Huber, R.

Ikeda, T.

Iwai, H.

Izatt, J.

Izatt, J. A.

Joo, C.

Joseph, P. M.

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]

Kostianovski, S.

Lipson, S. G.

Magistretti, P. J.

Marquet, P.

McGibney, G.

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

Miller, A. J.

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]

Mujat, M.

Nassif, N.

Nugent, K. A.

Paganin, D.

Papoulis, A.

A. Papoulis, S. U. Pillai, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, 2002) 4th edition.

Park, B.

Park, B. H.

Pierce, M.

Pierce, M. C.

Pillai, S. U.

A. Papoulis, S. U. Pillai, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, 2002) 4th edition.

Popescu, G.

Rappaz, B.

Ribak, E. N.

Roberts, A.

Sarunic, M.

Schmidt, C. F.

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]

Schnapp, B. J.

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]

Smith, M. R.

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

Svoboda, K.

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]

Tearney, G.

van den, B. A.

B. A. van den, Handbook of Measurement Science, (Wiley, Chichester, England, 1982) Vol. 1, pp. 331–377.

Vaughan, J. C.

Verkman, A. S.

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]

Wax, A.

White, B.

Yang, C.

Yazdanfar, S.

Yun, S. H.

Appl. Opt. (1)

Biophys. J. (1)

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]

Magn. Reson. Imaging (1)

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]

Med Phys. (1)

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

Nature (1)

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]

Opt. Express (3)

Opt. Lett. (8)

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]

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]

Other (2)

A. Papoulis, S. U. Pillai, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, 2002) 4th edition.

B. A. van den, Handbook of Measurement Science, (Wiley, Chichester, England, 1982) Vol. 1, pp. 331–377.

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Figures (8)

Fig. 1.
Fig. 1.

The theoretical PDF of the corrupted phase (ϕ) for different true phase values at (a) SNR=6 dB and (b) SNR=20 dB.

Fig. 2.
Fig. 2.

A schematic diagram of the SD-OCPM system. C: Collimator, PBS: polarizing beam splitter, SL: scan lens, TL: tube lens, DM: dichroic mirror, and PZT: piezo-electric transducer.

Fig. 3.
Fig. 3.

(a). Histogram of the measured OPLs and the theoretical PDF of the OPL. (b) The OPL sensitivity as a function of SNR. (c) The OPL standard deviation as a function of SNR for different OPL values. (d) The theoretical difference between the OPL standard deviation and the square root of the CRLB as a function of SNR.

Fig. 4.
Fig. 4.

The simulated precision of the ML and mean estimators as a function of data points at SNR=10 dB and true OPL=25 nm.

Fig. 5.
Fig. 5.

ML estimated intensities of the BPAE cell in the (a) BM and (b) MB scan modes. ML estimated SNRs of the BPAE cell in the (c) BM and (d) MB scan modes.

Fig. 6.
Fig. 6.

(a). The quantitative ML estimated OPL image of the BPAE cell using the BM scan mode. Cumulative distribution functions of OPL sensitivity based on the square root of the CRLB 1 2 SNR in blue and standard deviation over 100 images in red for (b) BM and (c) MB scans.

Fig. 7.
Fig. 7.

The quantitative OPL standard deviation image of the BPAE cell using the ML estimator in (a) BM and (b) MB scan modes.

Fig. 8.
Fig. 8.

The magnitude of the corrected OPL (|OPL-OPLML| using the ML estimator in (a) BM and (b) MB scan modes.

Equations (23)

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X = S + N = ρ e j θ + N
X R = S R + N R = ρ cos ( θ ) + N R
X I = S I + N I = ρ sin ( θ ) + N I
f ( X R , X I ) = 1 2 π σ 2 e { ( X R S R ) 2 + ( X I S I ) 2 2 σ 2 }
X R = R cos ( ϕ )
X I = R sin ( ϕ )
f ( R , ϕ ) = R 2 π σ 2 e { R 2 + ρ 2 2 R ρ cos ( ϕ θ ) 2 σ 2 } .
f ( I ) = e { I + ρ 2 2 σ 2 } 2 σ 2 I 0 ( ρ I σ 2 ) , I 0 .
f ( ϕ ) = 0 + f ( R , ϕ ) dR = 1 2 π σ 2 0 + R · exp ( ( R ρ cos ( ϕ θ ) ) 2 + ρ 2 sin 2 ( ϕ θ ) 2 σ 2 ) dR .
f ( ϕ ) = exp ( SNR sin 2 ( ϕ θ ) ) 2 π σ 2 ρ cos ( ϕ θ ) σ + [ σ u + ρ cos ( ϕ θ ) ] · σ exp ( u 2 2 ) du
f ( ϕ ) = exp ( SNR sin 2 ( ϕ θ ) ) 2 π σ 2 [ σ 2 exp ( SNR cos 2 ( ϕ θ ) ) + ρ σ cos ( ϕ θ ) ρ cos ( ϕ θ ) σ + exp ( u 2 2 ) du ] .
f ( ϕ ) = { exp ( SNR ) 2 π + SNR π exp ( SNR sin 2 ( ϕ θ ) ) cos ( ϕ θ ) Q ( 2 SNR cos ( ϕ θ ) ) 0 ϕ θ < 2 π 0 otherwise
0 f ( ϕ ) exp ( SNR ) 4 π SNR cos 2 ( ϕ θ ) , π 2 < ϕ θ < 3 π 2
SNR π exp ( SNR sin 2 ( ϕ θ ) ) cos ( ϕ θ ) f ( ϕ ) SNR π exp ( SNR sin 2 ( ϕ θ ) ) cos ( ϕ θ ) + exp ( SNR ) 4 π SNR cos 2 ( ϕ θ )
, 0 ϕ θ < π 2     and     3 π 2 < ϕ θ < 2 π
f ( ϕ ) { SNR π exp ( SNR sin 2 ( ϕ θ ) ) cos ( ϕ θ ) 0 ϕ θ < π 2 , 3 π 2 < ϕ θ < 2 π 0 otherwise .
p c = n = 1 N f ( X R ( n ) , X I ( n ) ) = ( 1 2 π σ 2 ) N e { [ n = 1 N ( X R ( n ) ) 2 + ( X I ( n ) ) 2 2 ρ X R ( n ) cos ( θ ) 2 ρ X I ( n ) sin ( θ ) ] + N ρ 2 2 σ 2 }
L ( ρ , θ , σ ) = 2 N ln ( σ ) n = 1 N ( X R ( n ) ) 2 + ( X I ( n ) ) 2 2 ρ X R ( n ) cos ( θ ) 2 ρ X I ( n ) sin ( θ ) + N ρ 2 2 σ 2
ρ ̂ ML = 1 N ( n = 1 N X R ( n ) ) 2 + ( n = 1 N X I ( n ) ) 2
θ ̂ ML = tan 1 ( n = 1 N X I ( n ) n = 1 N X R ( n ) )
σ ̂ ML 2 = n = 1 N [ ( ρ ̂ ML cos ( θ ̂ ML ) X R ( n ) ) 2 + ( ρ ̂ ML sin ( θ ̂ ML ) X I ( n ) ) 2 ] 2 N
S N ̂ R ML = ρ ̂ ML 2 σ 2 = 2 N ρ ̂ ML 2 n = 1 N [ ( ρ ̂ ML cos ( θ ̂ ML ) X R ( n ) ) 2 + ( ρ ̂ ML sin ( θ ̂ ML ) X I ( n ) ) 2 ]
OPL ( x , y ) = λ 0 4 π ϕ ( x , y ) .

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