Abstract

We present a spatial frequency domain (SFD) diffuse optical tomography for simultaneous acquisition of multi-wavelength tomographic images of turbid media. We propose a highly sensitive single-pixel SFD imaging system for simultaneously collecting multi-wavelength spatially modulated reflectance images, instead of using the expensive electron-multiplying charge-coupled device camera that requires switching between the multi-wavelength collections. The single-pixel SFD imaging system using three low-power light sources (455, 532 and 660 nm) that were intensity-modulated by square waves with three different frequencies for frequency encoding, and all the light sources were focused onto one digital micromirror device (DMD) for generating wide-field sinusoidal illumination patterns. Reflected light from the surface of the turbid media was modulated by the other DMD with many sampling patterns before being spatially integrated. Spatially integrated light signals were frequency decoded with a novel highly sensitive lock-in photon counting detection, then multi-wavelength spatially modulated reflectance images were recovered with the single-pixel imaging (SPI) method. We incorporated the two-dimensional discrete cosine transform (DCT) into the SPI method to reduce the number of sampling patterns, and, thereby, the proposed DCT-SPI scheme achieved a fast acquisition of SFD reflectance images that is desired for a dynamic SFD imaging application. Direct current (DC) and alternating current (AC) amplitudes at all the locations on the media surface were extracted from the recovered images. Multi-wavelength tomographic images were reconstructed with an inversion algorithm based on the first-order Rytov approximation of the diffusion equation, using both the extracted DC and AC amplitudes. We performed experiments using a series of tissue simulating phantoms to verify the performances of the proposed approach and compared the experimental results with those using a conventional camera-based SFD imaging system. The results demonstrate that our DCT-SPI based SFD-DOT approach is well suited for simultaneous reconstruction of multi-wavelength tomographic images to pave the way for many SFD imaging applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2018 (1)

X.-M. Ding, B.-Y. Wang, D.-Y. Liu, Y. Zhang, J. He, H. J. Zhao, and F. Gao, “A three-wavelength multi-channel brain functional imager based on digital lock-in photon counting technique,” Proc. SPIE 10480, 104800S (2018).

2017 (6)

B.-L. Liu, Z.-H. Yang, and L.-A. Wu, “Coloured computational imaging with single-pixel detectors based on a 2D discrete cosine transform,” J. Mod. Opt. 64(3), 259–264 (2017).
[Crossref]

S. Jin, W. Hui, Y. Wang, K. Huang, Q. Shi, C. Ying, D. Liu, Q. Ye, W. Zhou, and J. Tian, “Hyperspectral imaging using the single-pixel Fourier transform technique,” Sci. Rep. 7(1), 45209 (2017).
[Crossref] [PubMed]

M. Torabzadeh, I. Y. Park, R. A. Bartels, A. J. Durkin, and B. J. Tromberg, “Compressed single pixel imaging in the spatial frequency domain,” J. Biomed. Opt. 22(3), 030501 (2017).
[Crossref] [PubMed]

Y.-Z. Lu and R.-F. Lu, “Using composite sinusoidal patterns in structured-illumination reflectance imaging (SIRI) for enhanced detection of apple bruise,” J. Food Eng. 199, 54–64 (2017).
[Crossref]

T. Li, Z. Qin, W. Chen, H. Zhao, P. Yan, K. Zhao, and F. Gao, “Wide-field fluorescence tomography with composited epi-illumination of multi-frequency sinusoidal patterns,” Appl. Opt. 56(29), 8283–8290 (2017).
[Crossref] [PubMed]

X. Chen, W. Lin, C. Wang, S. Chen, J. Sheng, B. Zeng, and M. Xu, “In vivo real-time imaging of cutaneous hemoglobin concentration, oxygen saturation, scattering properties, melanin content, and epidermal thickness with visible spatially modulated light,” Biomed. Opt. Express 8(12), 5468–5482 (2017).
[Crossref] [PubMed]

2016 (3)

2015 (4)

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
[Crossref] [PubMed]

M. A. Yücel, C. M. Aasted, M. P. Petkov, D. Borsook, D. A. Boas, and L. Becerra, “Specificity of hemodynamic brain responses to painful stimuli: a functional near-infrared spectroscopy study,” Sci. Rep. 5(1), 9469 (2015).
[Crossref] [PubMed]

Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field fluorescence molecular tomography based on structured light and single-pixel detection,” Opt. Lett. 40(3), 431–434 (2015).
[Crossref] [PubMed]

D. G. Winters, R. A. Bartels, and S. R. Domingue, “Hyperspectral imaging via labeled excitation light and background-free absorption spectroscopy,” Optica 2(11), 929–932 (2015).
[Crossref]

2014 (1)

A. C. Ehlis, S. Schneider, T. Dresler, and A. J. Fallgatter, “Application of functional near-infrared spectroscopy in psychiatry,” Neuroimage 85(Pt 1), 478–488 (2014).
[Crossref] [PubMed]

2013 (2)

S. Dokouzyannis and T. Tziortzios, “High throughput and energy efficient two-dimensional inverse discrete cosine transform architecture,” IET Image Processing 7(5), 533–541 (2013).
[Crossref] [PubMed]

S. S. Welsh, M. P. Edgar, R. Bowman, P. Jonathan, B. Sun, and M. J. Padgett, “Fast full-color computational imaging with single-pixel detectors,” Opt. Express 21(20), 23068–23074 (2013).
[Crossref] [PubMed]

2012 (2)

Y. Wang and S. Zhang, “Comparison of the squared binary, sinusoidal pulse width modulation, and optimal pulse width modulation methods for three-dimensional shape measurement with projector defocusing,” Appl. Opt. 51(7), 861–872 (2012).
[Crossref] [PubMed]

S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
[Crossref] [PubMed]

2010 (5)

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15(1), 016006 (2010).
[Crossref] [PubMed]

R. B. Saager, D. J. Cuccia, and A. J. Durkin, “Determination of optical properties of turbid media spanning visible and near-infrared regimes via spatially modulated quantitative spectroscopy,” J. Biomed. Opt. 15(1), 017012 (2010).
[Crossref] [PubMed]

C. D’Andrea, N. Ducros, A. Bassi, S. Arridge, and G. Valentini, “Fast 3D optical reconstruction in turbid media using spatially modulated light,” Biomed. Opt. Express 1(2), 471–481 (2010).
[Crossref] [PubMed]

A. Mazhar, D. J. Cuccia, S. Gioux, A. J. Durkin, J. V. Frangioni, and B. J. Tromberg, “Structured illumination enhances resolution and contrast in thick tissue fluorescence imaging,” J. Biomed. Opt. 15(1), 010506 (2010).
[Crossref] [PubMed]

H. Akbari, Y. Kosugi, K. Kojima, and N. Tanaka, “Detection and analysis of the intestinal ischemia using visible and invisible hyperspectral imaging,” IEEE Trans. Biomed. Eng. 57(8), 2011–2017 (2010).
[Crossref] [PubMed]

2009 (2)

S. D. Konecky, A. Mazhar, D. Cuccia, A. J. Durkin, J. C. Schotland, and B. J. Tromberg, “Quantitative optical tomography of sub-surface heterogeneities using spatially modulated structured light,” Opt. Express 17(17), 14780–14790 (2009).
[Crossref] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14(2), 024012 (2009).
[Crossref] [PubMed]

2008 (2)

J. M. Masciotti, J. M. Lasker, and A. H. Hielscher, “Digital lock-in detection for discriminating multiple modulation frequencies with high accuracy and computational efficiency,” IEEE Trans. Instrum. Meas. 57(1), 182–189 (2008).
[Crossref]

A. Bassi, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, “Spatial shift of spatially modulated light projected on turbid media,” J. Opt. Soc. Am. A 25(11), 2833–2839 (2008).
[Crossref] [PubMed]

2007 (2)

D.-L. Qin, H.-J. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid media by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[Crossref]

S. V. Panasyuk, S. Yang, D. V. Faller, D. Ngo, R. A. Lew, J. E. Freeman, and A. E. Rogers, “Medical hyperspectral imaging to facilitate residual tumor identification during surgery,” Cancer Biol. Ther. 6(3), 439–446 (2007).
[Crossref] [PubMed]

2006 (2)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Y.-B. Tong, Q.-S. Zhang, and Y.-P. Qi, “Image quality assessing by combining PSNR with SSIM,” J. Image Graphics 11(12), 1758–1763 (2006).

2005 (2)

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30(11), 1354–1356 (2005).
[Crossref] [PubMed]

A. Restelli, R. Abbiati, and A. Geraci, “Digital field programmable gate array-based lock-in amplifier for high performance photon counting applications,” Rev. Sci. Instrum. 76(9), 093112 (2005).
[Crossref]

2002 (2)

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47(1), N1–N10 (2002).
[Crossref] [PubMed]

D. Braun and A. Libchaber, “Computer-based photon-counting lock-in for phase detection at the shot-noise limit,” Opt. Lett. 27(16), 1418–1420 (2002).
[Crossref] [PubMed]

2001 (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Quan Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[Crossref]

1994 (1)

1991 (1)

Aasted, C. M.

M. A. Yücel, C. M. Aasted, M. P. Petkov, D. Borsook, D. A. Boas, and L. Becerra, “Specificity of hemodynamic brain responses to painful stimuli: a functional near-infrared spectroscopy study,” Sci. Rep. 5(1), 9469 (2015).
[Crossref] [PubMed]

Abbiati, R.

A. Restelli, R. Abbiati, and A. Geraci, “Digital field programmable gate array-based lock-in amplifier for high performance photon counting applications,” Rev. Sci. Instrum. 76(9), 093112 (2005).
[Crossref]

Abran, M.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15(1), 016006 (2010).
[Crossref] [PubMed]

Akbari, H.

H. Akbari, Y. Kosugi, K. Kojima, and N. Tanaka, “Detection and analysis of the intestinal ischemia using visible and invisible hyperspectral imaging,” IEEE Trans. Biomed. Eng. 57(8), 2011–2017 (2010).
[Crossref] [PubMed]

Arridge, S.

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[Crossref]

Ayers, F. R.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14(2), 024012 (2009).
[Crossref] [PubMed]

Bartels, R. A.

M. Torabzadeh, I. Y. Park, R. A. Bartels, A. J. Durkin, and B. J. Tromberg, “Compressed single pixel imaging in the spatial frequency domain,” J. Biomed. Opt. 22(3), 030501 (2017).
[Crossref] [PubMed]

D. G. Winters, R. A. Bartels, and S. R. Domingue, “Hyperspectral imaging via labeled excitation light and background-free absorption spectroscopy,” Optica 2(11), 929–932 (2015).
[Crossref]

Bassi, A.

Becerra, L.

M. A. Yücel, C. M. Aasted, M. P. Petkov, D. Borsook, D. A. Boas, and L. Becerra, “Specificity of hemodynamic brain responses to painful stimuli: a functional near-infrared spectroscopy study,” Sci. Rep. 5(1), 9469 (2015).
[Crossref] [PubMed]

Bélanger, S.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15(1), 016006 (2010).
[Crossref] [PubMed]

Bevilacqua, F.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14(2), 024012 (2009).
[Crossref] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30(11), 1354–1356 (2005).
[Crossref] [PubMed]

Boas, D. A.

M. A. Yücel, C. M. Aasted, M. P. Petkov, D. Borsook, D. A. Boas, and L. Becerra, “Specificity of hemodynamic brain responses to painful stimuli: a functional near-infrared spectroscopy study,” Sci. Rep. 5(1), 9469 (2015).
[Crossref] [PubMed]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Quan Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]

Borsook, D.

M. A. Yücel, C. M. Aasted, M. P. Petkov, D. Borsook, D. A. Boas, and L. Becerra, “Specificity of hemodynamic brain responses to painful stimuli: a functional near-infrared spectroscopy study,” Sci. Rep. 5(1), 9469 (2015).
[Crossref] [PubMed]

Bowman, R.

Braun, D.

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Quan Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Cao, Z.-L.

M. Xu, Z.-L. Cao, W.-H. Lin, X.-L. Chen, L.-F. Zheng, and B.-X. Zeng, “Single snapshot multiple frequency modulated imaging of subsurface optical properties of turbid media with structured light,” AIP Adv. 6(12), 125208 (2016).
[Crossref]

Casanova, C.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15(1), 016006 (2010).
[Crossref] [PubMed]

Chance, B.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47(1), N1–N10 (2002).
[Crossref] [PubMed]

Chen, S.

Chen, W.

Chen, X.

Chen, X.-L.

M. Xu, Z.-L. Cao, W.-H. Lin, X.-L. Chen, L.-F. Zheng, and B.-X. Zeng, “Single snapshot multiple frequency modulated imaging of subsurface optical properties of turbid media with structured light,” AIP Adv. 6(12), 125208 (2016).
[Crossref]

Cuccia, D.

Cuccia, D. J.

A. Mazhar, D. J. Cuccia, S. Gioux, A. J. Durkin, J. V. Frangioni, and B. J. Tromberg, “Structured illumination enhances resolution and contrast in thick tissue fluorescence imaging,” J. Biomed. Opt. 15(1), 010506 (2010).
[Crossref] [PubMed]

R. B. Saager, D. J. Cuccia, and A. J. Durkin, “Determination of optical properties of turbid media spanning visible and near-infrared regimes via spatially modulated quantitative spectroscopy,” J. Biomed. Opt. 15(1), 017012 (2010).
[Crossref] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14(2), 024012 (2009).
[Crossref] [PubMed]

A. Bassi, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, “Spatial shift of spatially modulated light projected on turbid media,” J. Opt. Soc. Am. A 25(11), 2833–2839 (2008).
[Crossref] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30(11), 1354–1356 (2005).
[Crossref] [PubMed]

Culver, J. P.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47(1), N1–N10 (2002).
[Crossref] [PubMed]

D’Andrea, C.

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Quan Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]

Ding, X.-M.

X.-M. Ding, B.-Y. Wang, D.-Y. Liu, Y. Zhang, J. He, H. J. Zhao, and F. Gao, “A three-wavelength multi-channel brain functional imager based on digital lock-in photon counting technique,” Proc. SPIE 10480, 104800S (2018).

Dokouzyannis, S.

S. Dokouzyannis and T. Tziortzios, “High throughput and energy efficient two-dimensional inverse discrete cosine transform architecture,” IET Image Processing 7(5), 533–541 (2013).
[Crossref] [PubMed]

Domingue, S. R.

Dresler, T.

A. C. Ehlis, S. Schneider, T. Dresler, and A. J. Fallgatter, “Application of functional near-infrared spectroscopy in psychiatry,” Neuroimage 85(Pt 1), 478–488 (2014).
[Crossref] [PubMed]

Ducros, N.

Durkin, A. J.

M. Torabzadeh, I. Y. Park, R. A. Bartels, A. J. Durkin, and B. J. Tromberg, “Compressed single pixel imaging in the spatial frequency domain,” J. Biomed. Opt. 22(3), 030501 (2017).
[Crossref] [PubMed]

R. B. Saager, D. J. Cuccia, and A. J. Durkin, “Determination of optical properties of turbid media spanning visible and near-infrared regimes via spatially modulated quantitative spectroscopy,” J. Biomed. Opt. 15(1), 017012 (2010).
[Crossref] [PubMed]

A. Mazhar, D. J. Cuccia, S. Gioux, A. J. Durkin, J. V. Frangioni, and B. J. Tromberg, “Structured illumination enhances resolution and contrast in thick tissue fluorescence imaging,” J. Biomed. Opt. 15(1), 010506 (2010).
[Crossref] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14(2), 024012 (2009).
[Crossref] [PubMed]

S. D. Konecky, A. Mazhar, D. Cuccia, A. J. Durkin, J. C. Schotland, and B. J. Tromberg, “Quantitative optical tomography of sub-surface heterogeneities using spatially modulated structured light,” Opt. Express 17(17), 14780–14790 (2009).
[Crossref] [PubMed]

A. Bassi, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, “Spatial shift of spatially modulated light projected on turbid media,” J. Opt. Soc. Am. A 25(11), 2833–2839 (2008).
[Crossref] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30(11), 1354–1356 (2005).
[Crossref] [PubMed]

Edgar, M. P.

Ehlis, A. C.

A. C. Ehlis, S. Schneider, T. Dresler, and A. J. Fallgatter, “Application of functional near-infrared spectroscopy in psychiatry,” Neuroimage 85(Pt 1), 478–488 (2014).
[Crossref] [PubMed]

Faller, D. V.

S. V. Panasyuk, S. Yang, D. V. Faller, D. Ngo, R. A. Lew, J. E. Freeman, and A. E. Rogers, “Medical hyperspectral imaging to facilitate residual tumor identification during surgery,” Cancer Biol. Ther. 6(3), 439–446 (2007).
[Crossref] [PubMed]

Fallgatter, A. J.

A. C. Ehlis, S. Schneider, T. Dresler, and A. J. Fallgatter, “Application of functional near-infrared spectroscopy in psychiatry,” Neuroimage 85(Pt 1), 478–488 (2014).
[Crossref] [PubMed]

Feng, T. C.

Frangioni, J. V.

A. Mazhar, D. J. Cuccia, S. Gioux, A. J. Durkin, J. V. Frangioni, and B. J. Tromberg, “Structured illumination enhances resolution and contrast in thick tissue fluorescence imaging,” J. Biomed. Opt. 15(1), 010506 (2010).
[Crossref] [PubMed]

Freeman, J. E.

S. V. Panasyuk, S. Yang, D. V. Faller, D. Ngo, R. A. Lew, J. E. Freeman, and A. E. Rogers, “Medical hyperspectral imaging to facilitate residual tumor identification during surgery,” Cancer Biol. Ther. 6(3), 439–446 (2007).
[Crossref] [PubMed]

Gao, F.

X.-M. Ding, B.-Y. Wang, D.-Y. Liu, Y. Zhang, J. He, H. J. Zhao, and F. Gao, “A three-wavelength multi-channel brain functional imager based on digital lock-in photon counting technique,” Proc. SPIE 10480, 104800S (2018).

T. Li, Z. Qin, W. Chen, H. Zhao, P. Yan, K. Zhao, and F. Gao, “Wide-field fluorescence tomography with composited epi-illumination of multi-frequency sinusoidal patterns,” Appl. Opt. 56(29), 8283–8290 (2017).
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W. Chen, X. Wang, B. Wang, Y. Wang, Y. Zhang, H. Zhao, and F. Gao, “Lock-in-photon-counting-based highly-sensitive and large-dynamic imaging system for continuous-wave diffuse optical tomography,” Biomed. Opt. Express 7(2), 499–511 (2016).
[Crossref] [PubMed]

D.-L. Qin, H.-J. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid media by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[Crossref]

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Quan Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]

Geraci, A.

A. Restelli, R. Abbiati, and A. Geraci, “Digital field programmable gate array-based lock-in amplifier for high performance photon counting applications,” Rev. Sci. Instrum. 76(9), 093112 (2005).
[Crossref]

Gioux, S.

A. Mazhar, D. J. Cuccia, S. Gioux, A. J. Durkin, J. V. Frangioni, and B. J. Tromberg, “Structured illumination enhances resolution and contrast in thick tissue fluorescence imaging,” J. Biomed. Opt. 15(1), 010506 (2010).
[Crossref] [PubMed]

Haskell, R. C.

He, J.

X.-M. Ding, B.-Y. Wang, D.-Y. Liu, Y. Zhang, J. He, H. J. Zhao, and F. Gao, “A three-wavelength multi-channel brain functional imager based on digital lock-in photon counting technique,” Proc. SPIE 10480, 104800S (2018).

Hielscher, A. H.

J. M. Masciotti, J. M. Lasker, and A. H. Hielscher, “Digital lock-in detection for discriminating multiple modulation frequencies with high accuracy and computational efficiency,” IEEE Trans. Instrum. Meas. 57(1), 182–189 (2008).
[Crossref]

Huang, K.

S. Jin, W. Hui, Y. Wang, K. Huang, Q. Shi, C. Ying, D. Liu, Q. Ye, W. Zhou, and J. Tian, “Hyperspectral imaging using the single-pixel Fourier transform technique,” Sci. Rep. 7(1), 45209 (2017).
[Crossref] [PubMed]

Hui, W.

S. Jin, W. Hui, Y. Wang, K. Huang, Q. Shi, C. Ying, D. Liu, Q. Ye, W. Zhou, and J. Tian, “Hyperspectral imaging using the single-pixel Fourier transform technique,” Sci. Rep. 7(1), 45209 (2017).
[Crossref] [PubMed]

Intes, X.

Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field fluorescence molecular tomography based on structured light and single-pixel detection,” Opt. Lett. 40(3), 431–434 (2015).
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S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15(1), 016006 (2010).
[Crossref] [PubMed]

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47(1), N1–N10 (2002).
[Crossref] [PubMed]

Jin, S.

S. Jin, W. Hui, Y. Wang, K. Huang, Q. Shi, C. Ying, D. Liu, Q. Ye, W. Zhou, and J. Tian, “Hyperspectral imaging using the single-pixel Fourier transform technique,” Sci. Rep. 7(1), 45209 (2017).
[Crossref] [PubMed]

Jin, Z.

Jonathan, P.

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Quan Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]

Kojima, K.

H. Akbari, Y. Kosugi, K. Kojima, and N. Tanaka, “Detection and analysis of the intestinal ischemia using visible and invisible hyperspectral imaging,” IEEE Trans. Biomed. Eng. 57(8), 2011–2017 (2010).
[Crossref] [PubMed]

Kolste, K.

S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
[Crossref] [PubMed]

Konecky, S. D.

S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
[Crossref] [PubMed]

S. D. Konecky, A. Mazhar, D. Cuccia, A. J. Durkin, J. C. Schotland, and B. J. Tromberg, “Quantitative optical tomography of sub-surface heterogeneities using spatially modulated structured light,” Opt. Express 17(17), 14780–14790 (2009).
[Crossref] [PubMed]

Kosugi, Y.

H. Akbari, Y. Kosugi, K. Kojima, and N. Tanaka, “Detection and analysis of the intestinal ischemia using visible and invisible hyperspectral imaging,” IEEE Trans. Biomed. Eng. 57(8), 2011–2017 (2010).
[Crossref] [PubMed]

Lasker, J. M.

J. M. Masciotti, J. M. Lasker, and A. H. Hielscher, “Digital lock-in detection for discriminating multiple modulation frequencies with high accuracy and computational efficiency,” IEEE Trans. Instrum. Meas. 57(1), 182–189 (2008).
[Crossref]

Leblond, F.

S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
[Crossref] [PubMed]

Lesage, F.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15(1), 016006 (2010).
[Crossref] [PubMed]

Lew, R. A.

S. V. Panasyuk, S. Yang, D. V. Faller, D. Ngo, R. A. Lew, J. E. Freeman, and A. E. Rogers, “Medical hyperspectral imaging to facilitate residual tumor identification during surgery,” Cancer Biol. Ther. 6(3), 439–446 (2007).
[Crossref] [PubMed]

Li, T.

Libchaber, A.

Lin, W.

Lin, W.-H.

M. Xu, Z.-L. Cao, W.-H. Lin, X.-L. Chen, L.-F. Zheng, and B.-X. Zeng, “Single snapshot multiple frequency modulated imaging of subsurface optical properties of turbid media with structured light,” AIP Adv. 6(12), 125208 (2016).
[Crossref]

Liu, B.-L.

B.-L. Liu, Z.-H. Yang, and L.-A. Wu, “Coloured computational imaging with single-pixel detectors based on a 2D discrete cosine transform,” J. Mod. Opt. 64(3), 259–264 (2017).
[Crossref]

Liu, D.

S. Jin, W. Hui, Y. Wang, K. Huang, Q. Shi, C. Ying, D. Liu, Q. Ye, W. Zhou, and J. Tian, “Hyperspectral imaging using the single-pixel Fourier transform technique,” Sci. Rep. 7(1), 45209 (2017).
[Crossref] [PubMed]

Liu, D.-Y.

X.-M. Ding, B.-Y. Wang, D.-Y. Liu, Y. Zhang, J. He, H. J. Zhao, and F. Gao, “A three-wavelength multi-channel brain functional imager based on digital lock-in photon counting technique,” Proc. SPIE 10480, 104800S (2018).

Lu, R.-F.

Y.-Z. Lu and R.-F. Lu, “Using composite sinusoidal patterns in structured-illumination reflectance imaging (SIRI) for enhanced detection of apple bruise,” J. Food Eng. 199, 54–64 (2017).
[Crossref]

Lu, Y.-Z.

Y.-Z. Lu and R.-F. Lu, “Using composite sinusoidal patterns in structured-illumination reflectance imaging (SIRI) for enhanced detection of apple bruise,” J. Food Eng. 199, 54–64 (2017).
[Crossref]

Ma, H.

Ma, X.

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
[Crossref] [PubMed]

Masciotti, J. M.

J. M. Masciotti, J. M. Lasker, and A. H. Hielscher, “Digital lock-in detection for discriminating multiple modulation frequencies with high accuracy and computational efficiency,” IEEE Trans. Instrum. Meas. 57(1), 182–189 (2008).
[Crossref]

Mazhar, A.

A. Mazhar, D. J. Cuccia, S. Gioux, A. J. Durkin, J. V. Frangioni, and B. J. Tromberg, “Structured illumination enhances resolution and contrast in thick tissue fluorescence imaging,” J. Biomed. Opt. 15(1), 010506 (2010).
[Crossref] [PubMed]

S. D. Konecky, A. Mazhar, D. Cuccia, A. J. Durkin, J. C. Schotland, and B. J. Tromberg, “Quantitative optical tomography of sub-surface heterogeneities using spatially modulated structured light,” Opt. Express 17(17), 14780–14790 (2009).
[Crossref] [PubMed]

McAdams, M. S.

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Quan Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]

Moes, C. J. M.

Ngo, D.

S. V. Panasyuk, S. Yang, D. V. Faller, D. Ngo, R. A. Lew, J. E. Freeman, and A. E. Rogers, “Medical hyperspectral imaging to facilitate residual tumor identification during surgery,” Cancer Biol. Ther. 6(3), 439–446 (2007).
[Crossref] [PubMed]

Ntziachristos, V.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47(1), N1–N10 (2002).
[Crossref] [PubMed]

Owen, C. M.

S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
[Crossref] [PubMed]

Padgett, M. J.

Panasyuk, S. V.

S. V. Panasyuk, S. Yang, D. V. Faller, D. Ngo, R. A. Lew, J. E. Freeman, and A. E. Rogers, “Medical hyperspectral imaging to facilitate residual tumor identification during surgery,” Cancer Biol. Ther. 6(3), 439–446 (2007).
[Crossref] [PubMed]

Park, I. Y.

M. Torabzadeh, I. Y. Park, R. A. Bartels, A. J. Durkin, and B. J. Tromberg, “Compressed single pixel imaging in the spatial frequency domain,” J. Biomed. Opt. 22(3), 030501 (2017).
[Crossref] [PubMed]

Paulsen, K. D.

S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
[Crossref] [PubMed]

Petkov, M. P.

M. A. Yücel, C. M. Aasted, M. P. Petkov, D. Borsook, D. A. Boas, and L. Becerra, “Specificity of hemodynamic brain responses to painful stimuli: a functional near-infrared spectroscopy study,” Sci. Rep. 5(1), 9469 (2015).
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Pian, Q.

Prahl, S. A.

Qi, Y.-P.

Y.-B. Tong, Q.-S. Zhang, and Y.-P. Qi, “Image quality assessing by combining PSNR with SSIM,” J. Image Graphics 11(12), 1758–1763 (2006).

Qin, D.-L.

D.-L. Qin, H.-J. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid media by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[Crossref]

Qin, Z.

Quan Zhang,

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Quan Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]

Restelli, A.

A. Restelli, R. Abbiati, and A. Geraci, “Digital field programmable gate array-based lock-in amplifier for high performance photon counting applications,” Rev. Sci. Instrum. 76(9), 093112 (2005).
[Crossref]

Rice, T.

S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
[Crossref] [PubMed]

Roberts, D. W.

S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
[Crossref] [PubMed]

Rogers, A. E.

S. V. Panasyuk, S. Yang, D. V. Faller, D. Ngo, R. A. Lew, J. E. Freeman, and A. E. Rogers, “Medical hyperspectral imaging to facilitate residual tumor identification during surgery,” Cancer Biol. Ther. 6(3), 439–446 (2007).
[Crossref] [PubMed]

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Saager, R. B.

R. B. Saager, D. J. Cuccia, and A. J. Durkin, “Determination of optical properties of turbid media spanning visible and near-infrared regimes via spatially modulated quantitative spectroscopy,” J. Biomed. Opt. 15(1), 017012 (2010).
[Crossref] [PubMed]

Schneider, S.

A. C. Ehlis, S. Schneider, T. Dresler, and A. J. Fallgatter, “Application of functional near-infrared spectroscopy in psychiatry,” Neuroimage 85(Pt 1), 478–488 (2014).
[Crossref] [PubMed]

Schotland, J. C.

Sheng, J.

Shi, Q.

S. Jin, W. Hui, Y. Wang, K. Huang, Q. Shi, C. Ying, D. Liu, Q. Ye, W. Zhou, and J. Tian, “Hyperspectral imaging using the single-pixel Fourier transform technique,” Sci. Rep. 7(1), 45209 (2017).
[Crossref] [PubMed]

Sun, B.

Svaasand, L. O.

Tanaka, N.

H. Akbari, Y. Kosugi, K. Kojima, and N. Tanaka, “Detection and analysis of the intestinal ischemia using visible and invisible hyperspectral imaging,” IEEE Trans. Biomed. Eng. 57(8), 2011–2017 (2010).
[Crossref] [PubMed]

Tanikawa, Y.

D.-L. Qin, H.-J. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid media by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[Crossref]

Tao, T.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Tian, J.

S. Jin, W. Hui, Y. Wang, K. Huang, Q. Shi, C. Ying, D. Liu, Q. Ye, W. Zhou, and J. Tian, “Hyperspectral imaging using the single-pixel Fourier transform technique,” Sci. Rep. 7(1), 45209 (2017).
[Crossref] [PubMed]

Tong, Y.-B.

Y.-B. Tong, Q.-S. Zhang, and Y.-P. Qi, “Image quality assessing by combining PSNR with SSIM,” J. Image Graphics 11(12), 1758–1763 (2006).

Torabzadeh, M.

M. Torabzadeh, I. Y. Park, R. A. Bartels, A. J. Durkin, and B. J. Tromberg, “Compressed single pixel imaging in the spatial frequency domain,” J. Biomed. Opt. 22(3), 030501 (2017).
[Crossref] [PubMed]

Tromberg, B. J.

M. Torabzadeh, I. Y. Park, R. A. Bartels, A. J. Durkin, and B. J. Tromberg, “Compressed single pixel imaging in the spatial frequency domain,” J. Biomed. Opt. 22(3), 030501 (2017).
[Crossref] [PubMed]

S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
[Crossref] [PubMed]

A. Mazhar, D. J. Cuccia, S. Gioux, A. J. Durkin, J. V. Frangioni, and B. J. Tromberg, “Structured illumination enhances resolution and contrast in thick tissue fluorescence imaging,” J. Biomed. Opt. 15(1), 010506 (2010).
[Crossref] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14(2), 024012 (2009).
[Crossref] [PubMed]

S. D. Konecky, A. Mazhar, D. Cuccia, A. J. Durkin, J. C. Schotland, and B. J. Tromberg, “Quantitative optical tomography of sub-surface heterogeneities using spatially modulated structured light,” Opt. Express 17(17), 14780–14790 (2009).
[Crossref] [PubMed]

A. Bassi, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, “Spatial shift of spatially modulated light projected on turbid media,” J. Opt. Soc. Am. A 25(11), 2833–2839 (2008).
[Crossref] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30(11), 1354–1356 (2005).
[Crossref] [PubMed]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11(10), 2727–2741 (1994).
[Crossref] [PubMed]

Tsay, T. T.

Tziortzios, T.

S. Dokouzyannis and T. Tziortzios, “High throughput and energy efficient two-dimensional inverse discrete cosine transform architecture,” IET Image Processing 7(5), 533–541 (2013).
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Valdés, P. A.

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S. D. Konecky, C. M. Owen, T. Rice, P. A. Valdés, K. Kolste, B. C. Wilson, F. Leblond, D. W. Roberts, K. D. Paulsen, and B. J. Tromberg, “Spatial frequency domain tomography of protoporphyrin IX fluorescence in preclinical glioma models,” J. Biomed. Opt. 17(5), 056008 (2012).
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M. A. Yücel, C. M. Aasted, M. P. Petkov, D. Borsook, D. A. Boas, and L. Becerra, “Specificity of hemodynamic brain responses to painful stimuli: a functional near-infrared spectroscopy study,” Sci. Rep. 5(1), 9469 (2015).
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Figures (11)

Fig. 1
Fig. 1 (a) schematic of the multi-wavelength single-pixel SFD imaging system based on lock-in photon counting detection. D1-D3: LED drivers; (b) configuration of the lock-in photon counting module.
Fig. 2
Fig. 2 Diagram of in-phase and quadrature multiple-period accumulations multiple-period accumulations based on the RWC strategy with an integration time of T.
Fig. 3
Fig. 3 Schematic of the digital PSD block based on the RWC strategy
Fig. 4
Fig. 4 Distribution of the spatial frequency pairs of the 306 sampling patterns.
Fig. 5
Fig. 5 Frequency-decoding results for the detection linearity assessment for the wavelengths of (a) 455 nm, (b) 532 nm and (c) 660 nm.
Fig. 6
Fig. 6 Frequency-decoding results for the stability assessment of the lock-in photon counting detection.
Fig. 7
Fig. 7 Workflow of SFD-DOT reconstruction based on the recovered spatially modulated reflectance images: (a) sketch of the tissue phantom, (b) spatially modulated reflectance images for the three wavelengths recovered with the DCT-SPI scheme, (c) demodulated images of the DC (left) and AC (right) amplitudes for the three wavelengths, (d) horizontal cross-sections at z = 4 mm (left) and vertical cross-sections at y = 20 mm (right), from the SFD-DOT reconstructed tomographic images of the absorption coefficients for the three wavelengths. Dashed circles and rectangles indicate the true locations and sizes of the target.
Fig. 8
Fig. 8 x line-profiles of horizontal cross-sections at z = 4 mm (left) and z line-profiles of vertical cross-sections at y = 20 mm (right), from the SFD-DOT reconstructed tomographic images of the absorption coefficients of the tissue phantom for the wavelengths of (a) 455 nm, (b) 532 nm and (c) 660 nm.
Fig. 9
Fig. 9 SFD-DOT reconstructions with 130 sampling patterns with different acquisition times of 130 s, 65 s, and 32.5 s: (a) horizontal cross-sections at z = 4 mm (top) and vertical cross-sections at y = 20 mm (bottom); (b) corresponding x line-profiles (left) and z line-profiles (right). Dashed circles and rectangles indicate the true locations and sizes of the target.
Fig. 10
Fig. 10 Comparison of DCT- and CS-SPIs under the same experimental conditions: (a) spatially modulated reflectance images recovered with DCT-SPI (left) and CS-SPI (right) schemes; (b) corresponding x line-profiles.
Fig. 11
Fig. 11 Amplitudes of fluence-rate v.s. depth calculated for a semi-infinite geometry under oblique illumination and normal illumination of a sinusoidal pattern, respectively.

Tables (3)

Tables Icon

Table 1 Calculated CIs for the three wavelengths

Tables Icon

Table 2 Results of the QC for comparing the performances of the proposed single-pixel SFD imaging system and the conventional camera-based SFD imaging system.

Tables Icon

Table 3 Results of the QR for comparing the performances of the proposed single-pixel SFD imaging system and the conventional camera-based SFD imaging system.

Equations (15)

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S (t) (l) 4 R (l) π n=1 cos[(2n1)2π f (l) t+ θ (l) ],
X (l) = 0 T S (l) (t) H (l) (t) dt{ R (l) (12 θ (l) /π),0< θ (l) <π R (l) (2 θ (l) /π3),π< θ (l) <2π ,
Y (l) = 0 T S (l) (t) Q (l) (t) dt{ R (l) (2 θ (l) /π),0< θ (l) <π/2 R (l) (2 θ (l) /π2),π/2< θ (l) <3π/2 R (l) (2 θ (l) /π4),3π< θ (l) <2π ,
R (l) | X (l) |+| Y (l) |,
R (l) (u,v)= Ω I (l) (x,y)P(x,y;u,v) dxdy,
P(x,y;u,v) 2 MN C u C v cos (2x/0.625+1)uπ 2M cos (2y/0.625+1)vπ 2N ,
R (l) (u,v)= R + (l) (u,v) R (l) (u,v),
I (l) (x,y)= u=0 8 v=0 16 P(x,y;u,v)[ R + (l) (u,v) R (l) (u,v)] ,
I (l) (ρ)= I DC (l) (ρ)+ I AC (l) (ρ)cos(2π f x x),
d 2 d z 2 Φ 0 (l) (z) μ eff (l) Φ 0 (l) (z)= q 0 (l) (z) κ (l)2 ,
Γ 0 (l) ( f x ,ρ)=A Φ 0 (l) ( f x ,r)| z 0 + ,
Γ (l) ( f x ,ρ)= Γ 0 (l) ( f x ,ρ)exp[ Ω r δ μ a (l) (r) G 0 (l) ( f x ,r,ρ) Φ 0 (l) ( f x ,r)dr Φ 0 (l) ( f x ,ρ) ],
G 0 (l) ( f x ,r,ρ)= 1 4π κ (l) [ exp( μ eff (l) ) r 1 (l) r 1 (l) exp( μ eff (l) ) r 2 (l) r 2 (l) ],
B=Wδ μ a
W=[ G 0 (l) ( f 0 , r 1 , ρ 1 ) V 1 Φ 0 (l) ( f 0 , r 1 )/ Γ 0 (l) ( f 0 , ρ 1 ),.., G 0 (l) ( f 0 , r E , ρ 1 ) V E Φ 0 (l) ( f 0 , r E )/ Γ 0 (l) ( f 0 , ρ 1 ) G 0 (l) ( f 0 , r 1 , ρ K ) V 1 Φ 0 (l) ( f 0 , r 1 )/ Γ 0 (l) ( f 0 , ρ K ),.., G 0 (l) ( f 0 , r E , ρ K ) V E Φ 0 (l) ( f 0 , r E )/ Γ 0 (l) ( f 0 , ρ K ) G 0 (l) ( f x , r 1 , ρ 1 ) V 1 Φ 0 (l) ( f x , r 1 )/ Γ 0 (l) ( f x , ρ 1 ),.., G 0 (l) ( f x , r E , ρ 1 ) V E Φ 0 (l) ( f x , r E )/ Γ 0 (l) ( f x , ρ 1 ) G 0 (l) ( f x , r 1 , ρ K ) V 1 Φ 0 (l) ( f x , r 1 )/ Γ 0 (l) ( f x , ρ K ),.., G 0 (l) ( f x , r E , ρ K ) V E Φ 0 (l) ( f x , r E )/ Γ 0 (l) ( f x , ρ K ) ],

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