Abstract

We introduce and implement interferometric near-infrared spectroscopy (iNIRS), which simultaneously extracts optical and dynamical properties of turbid media through analysis of a spectral interference fringe pattern. The spectral interference fringe pattern is measured using a Mach-Zehnder interferometer with a frequency-swept narrow linewidth laser. Fourier analysis of the detected signal is used to determine time-of-flight (TOF)-resolved intensity, which is then analyzed over time to yield TOF-resolved intensity autocorrelations. This approach enables quantification of optical properties, which is not possible in conventional, continuous-wave near-infrared spectroscopy (NIRS). Furthermore, iNIRS quantifies scatterer motion based on TOF-resolved autocorrelations, which is a feature inaccessible by well-established diffuse correlation spectroscopy (DCS) techniques. We prove this by determining TOF-resolved intensity and temporal autocorrelations for light transmitted through diffusive fluid phantoms with optical thicknesses of up to 55 reduced mean free paths (approximately 120 scattering events). The TOF-resolved intensity is used to determine optical properties with time-resolved diffusion theory, while the TOF-resolved intensity autocorrelations are used to determine dynamics with diffusing wave spectroscopy. iNIRS advances the capabilities of diffuse optical methods and is suitable for in vivo tissue characterization. Moreover, iNIRS combines NIRS and DCS capabilities into a single modality.

© 2016 Optical Society of America

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References

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2014 (8)

A. Torricelli, D. Contini, A. Pifferi, M. Caffini, R. Re, L. Zucchelli, and L. Spinelli, “Time domain functional NIRS imaging for human brain mapping,” Neuroimage 85, 28–50 (2014).
[Crossref]

F. Scholkmann, S. Kleiser, A. J. Metz, R. Zimmermann, J. M. Pavia, U. Wolf, and M. Wolf, “A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology,” Neuroimage 85, 6–27 (2014).
[Crossref]

T. Durduran and A. G. Yodh, “Diffuse correlation spectroscopy for non-invasive, micro-vascular cerebral blood flow measurement,” Neuroimage 85, 51–63 (2014).
[Crossref]

V. Jain, E. M. Buckley, D. J. Licht, J. M. Lynch, P. J. Schwab, M. Y. Naim, N. A. Lavin, S. C. Nicolson, L. M. Montenegro, A. G. Yodh, and F. W. Wehrli, “Cerebral oxygen metabolism in neonates with congenital heart disease quantified by MRI and optics,” J. Cereb. Blood Flow. Metab. 34, 380–388 (2014).
[Crossref]

N. Roche-Labarbe, A. Fenoglio, H. Radhakrishnan, M. Kocienski-Filip, S. A. Carp, J. Dubb, D. A. Boas, P. E. Grant, and M. A. Franceschini, “Somatosensory evoked changes in cerebral oxygen consumption measured noninvasively in premature neonates,” Neuroimage 85, 279–286 (2014).
[Crossref]

L. Mei, G. Somesfalean, and S. Svanberg, “Frequency-modulated light scattering interferometry employed for optical properties and dynamics studies of turbid media,” Biomed. Opt. Express 5, 2810–2822 (2014).
[Crossref] [PubMed]

T. E. Matthews, M. Medina, J. R. Maher, H. Levinson, W. J. Brown, and A. Wax, “Deep tissue imaging using spectroscopic analysis of multiply scattered light,” Optica 1, 105–111 (2014)
[Crossref]

L. Spinelli, M. Botwicz, N. Zolek, M. Kacprzak, D. Milej, P. Sawosz, A. Liebert, U. Weigel, T. Durduran, F. Foschum, A. Kienle, F. Baribeau, S. Leclair, J.-P. Bouchard, I. Noiseux, P. Gallant, O. Mermut, A. Farina, A. Pifferi, A. Torricelli, R. Cubeddu, H.-C. Ho, M. Mazurenka, H. Wabnitz, K. Klauenberg, O. Bodnar, C. Elster, M. Bénazech-Lavoué, Y. Bérubé-Lauzière, F. Lesage, D. Khoptyar, A. A. Subash, S. Andersson-Engels, P. D. Ninni, F. Martelli, and G. Zaccanti, “Determination of reference values for optical properties of liquid phantoms based on intralipid and india ink,” Biomed. Opt. Express 5, 2037–2053 (2014).
[Crossref] [PubMed]

2013 (4)

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E 88, 042312 (2013)
[Crossref]

T. Svensson, R. Savo, E. Alerstam, K. Vynck, M. Burresi, and D. S. Wiersma, “Exploiting breakdown of the similarity relation for diffuse light transport: simultaneous retrieval of scattering anisotropy and diffusion constant,” Opt. Lett. 38, 437–439 (2013)
[Crossref] [PubMed]

V. Duc Nguyen, D. J. Faber, E. van der Pol, T. G. van Leeuwen, and J. Kalkman, “Dependent and multiple scattering in transmission and backscattering optical coherence tomography,” Opt. Express 21, 29145–29156 (2013)
[Crossref]

L. Mei, S. Svanberg, and G. Somesfalean, “Frequency-modulated light scattering in colloidal suspensions,” Appl. Phys. Lett. 102, 061104 (2013).
[Crossref]

2012 (1)

M. Ferrari and V. Quaresima, “A brief review on the history of human functional near-infrared spectroscopy (fNIRS) development and fields of application,” Neuroimage 63, 921–935 (2012).
[Crossref] [PubMed]

2011 (1)

P. D. Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. 56, N21 (2011).
[Crossref]

2010 (5)

P. Katherine, H. Taber, P. Elizabeth, M. C. Hillman, M. Robin, and A. Hurley, “Optical imaging: A new window to the adult brain,” The Journal of Neuropsychiatry and Clinical Neurosciences 22, 357–360 (2010).

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15, 011109 (2010).
[Crossref] [PubMed]

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73, 076701 (2010).
[Crossref] [PubMed]

M. Wojtkowski, “High-speed optical coherence tomography: basics and applications,” Appl. Opt. 49, D30–D61 (2010).
[Crossref] [PubMed]

J. Kalkman, R. Sprik, and T. G. van Leeuwen, “Path-Length-Resolved Diffusive Particle Dynamics in Spectral-Domain Optical Coherence Tomography,” Phys. Rev. Lett. 105, 198302 (2010)
[Crossref]

2008 (1)

2006 (1)

2005 (1)

2003 (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography – principles and applications,” Rep. Prog. Phys. 66, 239 (2003).
[Crossref]

2002 (1)

2001 (1)

J.-M. Tualle, E. Tinet, and S. Avrillier, “A new and easy way to perform time-resolved measurements of the light scattered by a turbid medium,” Opt. Commun. 189, 211–220 (2001).
[Crossref]

1999 (2)

1997 (3)

1995 (2)

L. Wang, S. L. Jacques, and L. Zheng, “MCML – Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[Crossref] [PubMed]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[Crossref] [PubMed]

1994 (3)

1993 (4)

S. Venkatesh and W. Sorin, “Phase noise considerations in coherent optical FMCW reflectometry,” J. Lightwave Technol. 11, 1694–1700 (1993).
[Crossref]

J. B. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[Crossref] [PubMed]

J. M. Schmitt, A. Knüttel, and R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 30, 6032–6042 (1993)
[Crossref]

M. R. Hee, J. A. Izatt, J. M. Jacobson, and J. G. Fujimoto, “Femtosecond transillumination optical coherence tomography,” Opt. Lett. 18, 950–952 (1993)
[Crossref] [PubMed]

1992 (1)

B. Ackerson, R. Dougherty, N. Reguigui, and U. Nobbman, “Correlation transfer: application of radiative transfer solution methods to photon correlation problems,” J. Thermophys. Heat Transf. 6, 577 (1992).
[Crossref]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

1990 (2)

A. G. Yodh, P. D. Kaplan, and D. J. Pine, “Pulsed diffusing-wave spectroscopy: High resolution through nonlinear optical gating,” Phys. Rev. B 42, 4744–4747 (1990).
[Crossref]

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[Crossref] [PubMed]

1989 (2)

F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[Crossref]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[Crossref] [PubMed]

1988 (3)

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[Crossref] [PubMed]

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1–5 (1988).
[Crossref]

K. K. Bizheva, A. M. Siegel, and D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: The transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664 (1988)
[Crossref]

1987 (3)

1981 (1)

1977 (1)

F. Jobsis, “Noninvasive, infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[Crossref] [PubMed]

1976 (1)

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

1970 (1)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[Crossref]

1965 (1)

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231 (1965).
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Figures (11)

Fig. 1
Fig. 1 Geometry for the calculation of the autocorrelation function Γ rs ( iNIRS ) of the reference and multiply scattered sample light fields. The real part of the cross-spectral density function is measured by a detector at the exit of the Mach-Zehnder interferometer and used to determine the autocorrelation function [see Eq. (4)].
Fig. 2
Fig. 2 Dependence of measured photon TOF distribution on the instrument response function (IRF), proportional to the modulus-squared of the mutual coherence function (field autocorrelation) of the incident optical field, Γinc(τs ). (A) For a very narrow mutual coherence function (infinite-width spectrum of incident light), the measured TOF distribution I s ( iNIRS ) ( r , τ s ) precisely reflects the actual photon DTOF Is (r, τs ). (B) The measured photon DTOF is blurred when the mutual coherence function of the incident optical field is not a delta function (finite-width spectrum of incident light).
Fig. 3
Fig. 3 Geometry for the calculation of the time-resolved transmittance through a homogeneous turbid slab (after [7, 57]). The locations of the first positive (blue dots) and negative (red dots) isotropic point sources are shown.
Fig. 4
Fig. 4 Measurement of the spectral interference pattern and a simple illustration of a linear versus nonlinear frequency sweep in iNIRS. A) The sample placed in the interferometer sample arm delays the optical field from the frequency-swept light source by a time τ s,0. Practically, this delay must be shorter than the coherence time, which in turn is less than the sweep time of our laser. B) Due to the fact that the laser frequency is changed over time, or ”swept”, photons which take a longer time to travel through one interferometer arm will generate a larger beat frequency upon interference. Hence for a linear sweep, iNIRS encodes time delay, τ s,0 as beat frequency. C) For a nonlinear sweep, delay is still encoded as beat frequency; however the beat frequency is no longer constant versus time. This nonlinearity in the beat frequency can be accounted for through a numerical recalibration procedure described in section 4. Because time delay is encoded in the beat frequency, the value of τ s,0 can be determined by inverse Fourier-transforming the electronic interference signal (D).
Fig. 5
Fig. 5 Dependence of measurement range and resolution on laser tuning parameters. The instantaneous linewidth, δν (A) is related to the coherence time, τc or time-of-flight (TOF) range (B) of the measurement (dashed lines), while the overall bandwidth over which the laser frequency is tuned, Δν (A), is related to the FWHM of the TOF resolution, δτs,Is (B), of the measurement (solid lines) [33]. The TOF resolution, determined by the tuning spectral range, in principle, can be made much higher than is possible with time-domain NIRS, without losing efficiency. In time-domain NIRS methods that use gating, improvement in resolution comes at the cost of reduced efficiency.
Fig. 6
Fig. 6 Layout of the iNIRS optical system. The waveform generator sinusoidally modulates the current (with repetition frequency fr = 50 kHz) supplied by the current controller which drives the DFB laser. The frequency-swept near infrared light is collimated by an aspheric lens (L1) and the shape of the beam is changed from elliptical to circular as it passes through the anamorphic prism pair (APP). Mirrors M1 and M2 are used to level the beam with the optical table and guide it through the isolator. Mirrors M3 and M4 serve to couple the beam through lens L2 into the single mode fiber. The beam is then divided into reference and sample arms (1% and 99% respectively) via a 99/1 fiber coupler. The beam in the sample arm is collimated with lens L3 and irradiates the sample. The scattered light from the sample is collected by L4 and fiber coupled. The light in the reference arm undergoes a path delay and is then split into two arms with a 50/50 fiber coupler, where one arm is detected by a photodetector (PD) to monitor power and the other arm passes through a polarization controller. The reference and sample arms are finally combined by the 50/50 fiber coupler and the resulting interference pattern is detected by a dual balanced detector (DBD), digitized with a GaGe digitizer, then stored and processed by the PC.
Fig. 7
Fig. 7 (A,B) Photon time-of-flight distribution under varying concentrations of Intralipid 20%, denoted by c. As the Intralipid 20% concentration is increased, the time-integrated intensity signal is attenuated (C), and the mean arrival time is increased (D). Theoretical points in subfigures C and D were obtained using Eq. (26) and the values of μa = 4.53 × 10−2 cm−1 (water absorption due to negligible lipid absorption at near-infrared wavelengths) and μ′s (c) = cμ′s,t , where μ′s,t = 170.27 cm−1 is theoretical estimation of the reduced scattering coefficient of Intralipid 20% [65]. The attenuation plots are normalized with respect to c = 4.8%, since the diffusion approximation is invalid for smaller concentrations.
Fig. 8
Fig. 8 Optical properties determined by iNIRS in fluid tissue phantoms with varying Intralipid 20% concentrations, c: μ′s (A), μa (B). Reduced scattering coefficients are compared to data reported in [66] by assuming a linear dependence of μ′s on c (orange line in subfigure A). Furthermore in subfigure A, the green solid line depicts a linear fit to the experimental data, i.e., μ′s (c) = ηc with η = 171.19 ± 1.92 cm−1. The error bars denote the standard errors of the extracted parameter values.
Fig. 9
Fig. 9 Diffusion theory validation. Plots depict the experimental transmittance and fits to diffusion theory [cf. Eq. (26)] under varying Intralipid 20% concentrations: (A) c = 2.4%, (B) c = 4.8%, (C) c = 13.0%, and (D) c = 21.6%. For low concentrations (c < 4.8%) significant deviations from the diffusion approximation are noticed. For c ≥ 4.8% diffusion theory agrees well with experimental data. The mean squared error, MSE for each fit, after thresholding normalized intensity values below 0.08, is: (A) MSE = 0.448, (B) MSE = 0.050, (C) MSE = 0.054, and (D) MSE = 0.038.
Fig. 10
Fig. 10 Time-of-flight-resolved scatterer dynamics. (A) The absolute value of the field autocorrelation function, | g ˜ 2 ( iNIRS ) ( τ s , τ d ) | , obtained from the temporal intensity autocorrelation g ˜ 2 ( iNIRS ) ( τ s , τ d ) using Eq. (23), is plotted for three different path lengths (l = υτs ), corresponding to time-of-flight values, τs , and TOF windows marked on the iNIRS DTOF (B). The solid lines in subfigure A denote the fits of the experimental normalized field autocorrelation function to the exponential decay predicted by DWS theory [cf. Eq. (27)].
Fig. 11
Fig. 11 Diffusing wave spectroscopy validation. The decay rate, ξ of the first-order autocorrelation function, extracted from intensity autocorrelation function using Siegert relation [Eq. (23)], is plotted for increasing values of τs and fit assuming a linear dependence of ξ on τs for τs > 200 ps, as predicted by DWS theory (ξ (τs ) = 1.02 × 10−4 τs ). The error bars denote the standard error of ξ extracted by fitting experimental data using Eq. (27) and Eq. (37). The time-of-flight through a water-filled cuvette is τw = 44.33 ps.

Tables (1)

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Table 1 Comparison of optical properties determined by iNIRS with literature values.

Equations (50)

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U ( r , t s ) = 𝒰 ( r , ν ) exp [ 2 π i ν t s ] d ν .
𝒮 ( r , ν ) = 𝒮 r ( r , ν ) + 𝒮 s ( r , ν ) + 2 Re [ 𝒲 rs ( r , ν ) ] ,
𝒲 i j ( r , ν ) δ ( ν ν ) = 𝒰 i * ( r , ν ) 𝒰 j ( r , ν )
𝒲 i j ( r , ν ) = Γ i j ( r , τ s ) exp [ 2 π i ν τ s ] d τ s
Γ i j ( r , τ s ) = U i * ( r , t s ) U j ( r , t s + τ s ) t s = lim T 1 2 T T T U i * ( r , t s ) U j ( r , t s + τ s ) d t s ,
I ( r , τ s ) = Γ r ( r ) + Γ s ( r ) + Γ rs ( r , τ s ) + Γ sr ( r , τ s ) ,
U r ( r , t s , t d ) = α r exp [ i φ r ( r , t d ) ] U inc ( t s τ r ) ,
U s ( r , t s , t d ) = n = 1 N α s , n ( r ) exp [ i φ s , n ( r , t d ) ] U inc ( t s τ s , n ) .
Γ rs ( iNIRS ) ( r , t s , t d ) = α r n = 1 N α s , n ( r ) exp [ i Δ φ r s , n ( r , t d ) ] Γ inc ( τ s τ s , n ) ,
Δ φ r s , n ( r , t d ) = φ r ( r , t d ) φ s , n ( r , t d )
Γ inc ( τ s ) = U inc * ( t s ) U inc ( t s + τ s ) t s ,
I s ( iNIRS ) ( r , τ s ) = | Γ rs ( iNIRS ) ( r , τ s , t d ) | 2 t d ,
I s ( iNIRS ) ( r , τ s ) = α r 2 n = 1 N m = 1 N α s , n ( r ) α s , m ( r ) Γ inc * ( τ s τ s , n ) Γ inc ( τ s τ s , m ) × exp { i [ Δ φ r s , n ( r , t d ) Δ φ r s , m ( r , t d ) } t d .
I s ( iNIRS ) ( r , τ s ) = n = 1 N I s , n ( r ) | Γ inc ( τ s τ s , n ) | 2 ,
I s ( iNIRS ) ( r , τ s ) = d τ s I s ( r , τ s ) I 0 ( τ s τ s ) ,
I 0 ( τ s ) = | Γ inc ( τ s ) | 2
Γ inc ( τ s ) = 1 { 𝒮 inc ( ν ) } ,
g 1 ( iNIRS ) ( r , τ s , τ d ) = G 1 ( iNIRS ) ( r , τ s , τ d ) G 1 ( iNIRS ) ( r , τ s , 0 ) ,
G 1 ( iNIRS ) ( r , τ s , τ d ) = Γ rs * ( iNIRS ) ( r , τ s , τ d ) Γ rs ( iNIRS ) ( r , τ s , t d + τ d ) t d .
G 1 ( iNIRS ) ( r , τ s , τ d ) = α r 2 n = 1 N α s , n 2 ( r ) exp [ i Δ Φ r s , n ( r , t d , τ d ) ] t d | Γ inc ( τ s τ s , n ) | 2 ,
Δ Φ r s , n ( r , t d , τ d ) = Δ φ r s , n ( r , t d ) Δ φ r s , n ( r , t d + τ d ) .
G 1 , n ( r , τ d ) = I s , n ( r ) exp [ i Δ Φ r s , n ( r , t d , τ d ) ] t d
G 1 ( iNIRS ) ( r , τ s , τ d ) = n = 1 N G 1 , n ( r , τ d ) | Γ inc ( τ s τ s , n ) | 2 .
G 1 ( iNIRS ) ( r , τ s , τ d ) = d τ s G 1 ( r , τ s , τ d ) I 0 ( τ s τ s ) .
g 2 ( iNIRS ) ( r , τ s , τ d ) = G 2 ( iNIRS ) ( r , τ s , τ d ) [ G 1 ( iNIRS ) ( r , τ s , 0 ) ] 2 ,
G 2 ( iNIRS ) ( r , τ s , τ d ) = | Γ rs ( iNIRS ) ( r , τ s , t d ) | 2 | Γ rs ( iNIRS ) ( r , τ s , t d + τ d ) | 2 t d .
g 2 ( iNIRS ) ( r , τ s , τ d ) = 1 + β | g 1 ( iNIRS ) ( r , τ s , τ d ) | 2 ,
β = g 2 ( iNIRS ) ( r , τ s , 0 ) 1 .
β = G 2 ( iNIRS ) ( r , τ s , 0 ) G 2 ( iNIRS ) ( r , τ s , ) G 2 ( iNIRS ) ( r , τ s , ) ,
I s ( DE ) ( ρ , L , τ s ) = exp [ μ a υ τ s ] 2 π σ t 2 τ s 4 D π υ τ s exp [ ρ 2 2 σ t 2 ] m = { ( L z + , m ) exp [ ( L z + , m ) 2 4 D υ τ s ] ( L z , m ) exp [ ( L z , m ) 2 4 D υ τ s ] } ,
z + , m = 2 m ( L + 2 z e ) + z 0 , z , m = 2 m ( L + 2 z e ) 2 z e z 0
A = 1 + R ( n ) 1 R ( n ) ,
R ( n ) = 1.4399 n 2 + 0.7099 n 1 + 0.6681 + 0.0636 n ,
g 1 ( DWS ) ( τ s , τ d ) = exp [ 1 3 k 2 μ s Δ r 2 ( τ d ) υ τ s ] ,
Γ rs ( iNIRS ) ( r , τ s , t d ) = Γ rs ( r , τ s , t d ) * Γ inc ( τ s ) ,
Γ rs ( r , τ s , t d ) = α r n = 1 N α s , n ( r ) exp [ i Δ φ r s , n ( r , t d ) ] δ ( τ s τ s , n )
𝒮 inc ( ν ) = exp [ 4 ln ( 2 ) ( ν ν c Δ ν ) 2 ] ,
Γ inc ( τ s ) = π Δ ν 2 ln ( 2 ) exp [ ( π Δ ν τ s 2 ln ( 2 ) ) 2 ] exp [ 2 π i ν c τ s ] .
FWHM = 2 2 ln ( 2 ) σ ,
δ τ s = 4 ln ( 2 ) π Δ ν = 4 ln ( 2 ) π λ c 2 υ Δ λ ,
I s ( iNIRS ) ( r , τ s ) = I s ( r , τ s ) * I 0 ( τ s ) ,
δ τ s , I s = δ τ s 2 .
τ c = 4 ln ( 2 ) π δ ν = 4 ln ( 2 ) π λ c 2 υ δ λ ,
I ¯ ^ s ( iNIRS ) ( τ s ) = 1 N n = 0 N 1 I ^ s ( iNIRS ) ( τ s , n × t m ) ,
I ^ s ( iNIRS ) ( τ s , n × t m ) = | Γ ^ rs ( iNIRS ) ( τ s , n × t m ) | 2 .
I ¯ ˜ s ( iNIRS ) ( τ s ) = I ¯ ^ s ( iNIRS ) ( τ s ) I ¯ ^ bg ( iNIRS ) ( τ s ) ,
g ^ 2 ( iNIRS ) ( τ s , τ d ) = 1 ( N m ) [ I ˜ s ( iNIRS ) ( τ s ) ] 2 k = 0 N m 1 I ˜ s ( iNIRS ) ( τ s , k × t m ) I ˜ s ( iNIRS ) ( τ s , k × t m + τ d ) ,
τ ¯ s ( iNIRS ) = τ s I ¯ ˜ s ( iNIRS ) ( τ s ) d τ s I ¯ ˜ s ( iNIRS ) ( τ s ) d τ s
min ( μ a , μ s ) log [ I s ( DE ) ( τ s ) * I 0 ( τ s ) ] log [ α I ¯ ˜ s ( iNIRS ) ( τ s ) ] 2 ,
min ξ | g 1 ( DWS ) ( τ s , τ d ) | | g ˜ 1 ( iNIRS ) ( τ s , τ d ) | 2 ,

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