Abstract

Fluorescence lifetime imaging (FLI) is a popular method for extracting useful information that is otherwise unavailable from a conventional intensity image. Usually, however, it requires expensive equipment, is often limited to either distinctly frequency- or time-domain modalities, and demands calibration measurements and precise knowledge of the illumination signal. Here, we present a generalized time-based, cost-effective method for estimating lifetimes by repurposing a consumer-grade time-of-flight sensor. By developing mathematical theory that unifies time- and frequency-domain approaches, we can interpret a time-based signal as a combination of multiple frequency measurements. We show that we can estimate lifetimes without knowledge of the illumination signal and without any calibration. We experimentally demonstrate this blind, reference-free method using a quantum dot solution and discuss the method’s implementation in FLI applications.

© 2015 Optical Society of America

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References

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2015 (2)

G. Satat, B. Heshmat, C. Barsi, D. Raviv, O. Chen, M. G. Bawendi, and R. Raskar, “Locating and classifying fluorescent tags behind turbid layers using time-resolved inversion,” Nat. Commun. 6, 6796 (2015).
[Crossref]

H. Qiao, J. Lin, Y. Liu, M. B. Hullin, and Q. Dai, “Resolving transient time profile in ToF imaging via log-sum sparse regularization,” Opt. Lett. 40, 918–921 (2015).
[Crossref]

2014 (4)

2013 (2)

A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. Dorrington, and R. Raskar, “Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles,” ACM Trans. Graph. 32, 167 (2013).
[Crossref]

F. Heide, M. B. Hullin, J. Gregson, and W. Heidrich, “Low-budget transient imaging using photonic mixer devices,” ACM Trans. Graph. 32, 1 (2013).

2011 (1)

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (ToF) cameras: a survey,” IEEE Sens. J. 11, 1917–1926 (2011).
[Crossref]

2009 (1)

2008 (4)

A. T. Kumar, S. B. Raymond, B. J. Bacskai, and D. A. Boas, “Comparison of frequency-domain and time-domain fluorescence lifetime tomography,” Opt. Lett. 33, 470–472 (2008).
[Crossref]

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, “The phasor approach to fluorescence lifetime imaging analysis,” Biophys. J. 94, L14–L16 (2008).
[Crossref]

A. T. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 27, 1152–1163 (2008).
[Crossref]

G.-J. Kremers, E. B. Van Munster, J. Goedhart, and T. W. Gadella, “Quantitative lifetime unmixing of multiexponentially decaying fluorophores using single-frequency fluorescence lifetime imaging microscopy,” Biophys. J. 95, 378–389 (2008).
[Crossref]

2007 (2)

2005 (1)

2003 (2)

J. Philip and K. Carlsson, “Theoretical investigation of the signal-to-noise ratio in fluorescence lifetime imaging,” J. Opt. Soc. Am. A 20, 368–379 (2003).
[Crossref]

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368–375 (2003).
[Crossref]

2001 (1)

Q. S. Hanley, V. Subramaniam, D. J. Arndt-Jovin, and T. M. Jovin, “Fluorescence lifetime imaging: multi-point calibration, minimum resolvable differences, and artifact suppression,” Cytometry 43, 248–260 (2001).
[Crossref]

2000 (1)

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, “Multiple frequency fluorescence lifetime imaging microscopy,” J. Microsc. 197, 136–149 (2000).
[Crossref]

1998 (1)

H. He, B. K. Nunnally, L.-C. Li, and L. B. McGown, “On-the-fly fluorescence lifetime detection of dye-labeled DNA primers for multiplex analysis,” Anal. Chem. 70, 3413–3418 (1998).
[Crossref]

1997 (1)

1996 (1)

T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).
[Crossref]

1984 (1)

E. Gratton, M. Limkeman, J. R. Lakowicz, B. P. Maliwal, H. Cherek, and G. Laczko, “Resolution of mixtures of fluorophores using variable-frequency phase and modulation data,” Biophys. J. 46, 479–486 (1984).
[Crossref]

Akbari, N.

Alenya, G.

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (ToF) cameras: a survey,” IEEE Sens. J. 11, 1917–1926 (2011).
[Crossref]

Allen, R.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368–375 (2003).
[Crossref]

Arndt-Jovin, D. J.

Q. S. Hanley, V. Subramaniam, D. J. Arndt-Jovin, and T. M. Jovin, “Fluorescence lifetime imaging: multi-point calibration, minimum resolvable differences, and artifact suppression,” Cytometry 43, 248–260 (2001).
[Crossref]

Bacskai, B. J.

A. T. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 27, 1152–1163 (2008).
[Crossref]

A. T. Kumar, S. B. Raymond, B. J. Bacskai, and D. A. Boas, “Comparison of frequency-domain and time-domain fluorescence lifetime tomography,” Opt. Lett. 33, 470–472 (2008).
[Crossref]

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368–375 (2003).
[Crossref]

Barsi, C.

G. Satat, B. Heshmat, C. Barsi, D. Raviv, O. Chen, M. G. Bawendi, and R. Raskar, “Locating and classifying fluorescent tags behind turbid layers using time-resolved inversion,” Nat. Commun. 6, 6796 (2015).
[Crossref]

A. Bhandari, A. Kadambi, R. Whyte, C. Barsi, M. Feigin, A. Dorrington, and R. Raskar, “Resolving multipath interference in time-of-flight imaging via modulation frequency diversity and sparse regularization,” Opt. Lett. 39, 1705–1708 (2014).
[Crossref]

A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. Dorrington, and R. Raskar, “Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles,” ACM Trans. Graph. 32, 167 (2013).
[Crossref]

Bastiaens, P. I. H.

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, “Multiple frequency fluorescence lifetime imaging microscopy,” J. Microsc. 197, 136–149 (2000).
[Crossref]

Bawendi, M. G.

G. Satat, B. Heshmat, C. Barsi, D. Raviv, O. Chen, M. G. Bawendi, and R. Raskar, “Locating and classifying fluorescent tags behind turbid layers using time-resolved inversion,” Nat. Commun. 6, 6796 (2015).
[Crossref]

Bec, J.

D. R. Yankelevich, D. Ma, J. Liu, Y. Sun, Y. Sun, J. Bec, D. S. Elson, and L. Marcu, “Design and evaluation of a device for fast multispectral time-resolved fluorescence spectroscopy and imaging,” Rev. Sci. Instrum. 85, 034303 (2014).
[Crossref]

Bhandari, A.

A. Bhandari, A. Kadambi, R. Whyte, C. Barsi, M. Feigin, A. Dorrington, and R. Raskar, “Resolving multipath interference in time-of-flight imaging via modulation frequency diversity and sparse regularization,” Opt. Lett. 39, 1705–1708 (2014).
[Crossref]

A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. Dorrington, and R. Raskar, “Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles,” ACM Trans. Graph. 32, 167 (2013).
[Crossref]

A. Bhandari, M. Feigin, S. Izadi, C. Rhemann, M. Schmidt, and R. Raskar, “Resolving multipath interference in Kinect: An inverse problem approach,” in IEEE Sensors (IEEE, 2014), pp. 614–617.

A. Kadambi, A. Bhandari, R. Whyte, A. Dorrington, and R. Raskar, “Demultiplexing illumination via low cost sensing and nanosecond coding,” in IEEE International Conference on Computational Photography (ICCP) (2014).

A. Bhandari, A. Bourquard, S. Izadi, and R. Raskar, “Blind transmitted and reflected image separation using depth diversity and time-of-flight sensors,” in Computational Optical Sensing and Imaging (Optical Society of America, 2015), paper CT4F-2.

A. Bhandari, A. Kadambi, and R. Raskar, “Sparse linear operator identification without sparse regularization? Applications to mixed pixel problem in time-of-flight/range imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2014), pp. 365–369.

Boas, D. A.

A. T. Kumar, S. B. Raymond, B. J. Bacskai, and D. A. Boas, “Comparison of frequency-domain and time-domain fluorescence lifetime tomography,” Opt. Lett. 33, 470–472 (2008).
[Crossref]

A. T. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 27, 1152–1163 (2008).
[Crossref]

Bourquard, A.

A. Bhandari, A. Bourquard, S. Izadi, and R. Raskar, “Blind transmitted and reflected image separation using depth diversity and time-of-flight sensors,” in Computational Optical Sensing and Imaging (Optical Society of America, 2015), paper CT4F-2.

Buckley, B. W.

Caiolfa, V. R.

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, “The phasor approach to fluorescence lifetime imaging analysis,” Biophys. J. 94, L14–L16 (2008).
[Crossref]

Carli, P.

Carlsson, K.

Chan, J. C. K.

Chen, O.

G. Satat, B. Heshmat, C. Barsi, D. Raviv, O. Chen, M. G. Bawendi, and R. Raskar, “Locating and classifying fluorescent tags behind turbid layers using time-resolved inversion,” Nat. Commun. 6, 6796 (2015).
[Crossref]

Cherek, H.

E. Gratton, M. Limkeman, J. R. Lakowicz, B. P. Maliwal, H. Cherek, and G. Laczko, “Resolution of mixtures of fluorophores using variable-frequency phase and modulation data,” Biophys. J. 46, 479–486 (1984).
[Crossref]

Chuang, F. S.

Cicchi, R.

Coleman, T. F.

T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).
[Crossref]

Cree, M. J.

D. Lefloch, R. Nair, F. Lenzen, H. Schäfer, L. Streeter, M. J. Cree, R. Koch, and A. Kolb, “Technical foundation and calibration methods for time-of-flight cameras,” in Time-of-Flight and Depth Imaging: Sensors, Algorithms, and Applications (Springer, 2013), pp. 3–24.

Dai, Q.

Davis, J.

S. Schuon, C. Theobalt, J. Davis, and S. Thrun, “Lidarboost: Depth superresolution for ToF 3D shape scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2009), pp. 343–350.

De Giorgi, V.

Diebold, E. D.

Digman, M. A.

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, “The phasor approach to fluorescence lifetime imaging analysis,” Biophys. J. 94, L14–L16 (2008).
[Crossref]

Dorrington, A.

A. Bhandari, A. Kadambi, R. Whyte, C. Barsi, M. Feigin, A. Dorrington, and R. Raskar, “Resolving multipath interference in time-of-flight imaging via modulation frequency diversity and sparse regularization,” Opt. Lett. 39, 1705–1708 (2014).
[Crossref]

A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. Dorrington, and R. Raskar, “Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles,” ACM Trans. Graph. 32, 167 (2013).
[Crossref]

A. Kadambi, A. Bhandari, R. Whyte, A. Dorrington, and R. Raskar, “Demultiplexing illumination via low cost sensing and nanosecond coding,” in IEEE International Conference on Computational Photography (ICCP) (2014).

Dragulescu-Andrasi, A.

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18, 17–25 (2007).
[Crossref]

Dunn, A. K.

A. T. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 27, 1152–1163 (2008).
[Crossref]

Elson, D. S.

D. R. Yankelevich, D. Ma, J. Liu, Y. Sun, Y. Sun, J. Bec, D. S. Elson, and L. Marcu, “Design and evaluation of a device for fast multispectral time-resolved fluorescence spectroscopy and imaging,” Rev. Sci. Instrum. 85, 034303 (2014).
[Crossref]

Y. Sun, J. Phipps, D. S. Elson, H. Stoy, S. Tinling, J. Meier, B. Poirier, F. S. Chuang, D. G. Farwell, and L. Marcu, “Fluorescence lifetime imaging microscopy: in vivo application to diagnosis of oral carcinoma,” Opt. Lett. 34, 2081–2083 (2009).
[Crossref]

Esposito, A.

Farwell, D. G.

Feigin, M.

A. Bhandari, A. Kadambi, R. Whyte, C. Barsi, M. Feigin, A. Dorrington, and R. Raskar, “Resolving multipath interference in time-of-flight imaging via modulation frequency diversity and sparse regularization,” Opt. Lett. 39, 1705–1708 (2014).
[Crossref]

A. Bhandari, M. Feigin, S. Izadi, C. Rhemann, M. Schmidt, and R. Raskar, “Resolving multipath interference in Kinect: An inverse problem approach,” in IEEE Sensors (IEEE, 2014), pp. 614–617.

Foix, S.

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (ToF) cameras: a survey,” IEEE Sens. J. 11, 1917–1926 (2011).
[Crossref]

Gadella, T. W.

G.-J. Kremers, E. B. Van Munster, J. Goedhart, and T. W. Gadella, “Quantitative lifetime unmixing of multiexponentially decaying fluorophores using single-frequency fluorescence lifetime imaging microscopy,” Biophys. J. 95, 378–389 (2008).
[Crossref]

Gerritsen, H.

Goedhart, J.

G.-J. Kremers, E. B. Van Munster, J. Goedhart, and T. W. Gadella, “Quantitative lifetime unmixing of multiexponentially decaying fluorophores using single-frequency fluorescence lifetime imaging microscopy,” Biophys. J. 95, 378–389 (2008).
[Crossref]

Golomb, S. W.

S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar (Cambridge University, 2005).

Gong, G.

S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar (Cambridge University, 2005).

Gratton, E.

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, “The phasor approach to fluorescence lifetime imaging analysis,” Biophys. J. 94, L14–L16 (2008).
[Crossref]

E. Gratton, M. Limkeman, J. R. Lakowicz, B. P. Maliwal, H. Cherek, and G. Laczko, “Resolution of mixtures of fluorophores using variable-frequency phase and modulation data,” Biophys. J. 46, 479–486 (1984).
[Crossref]

Gregson, J.

F. Heide, M. B. Hullin, J. Gregson, and W. Heidrich, “Low-budget transient imaging using photonic mixer devices,” ACM Trans. Graph. 32, 1 (2013).

Hanley, Q. S.

Q. S. Hanley, V. Subramaniam, D. J. Arndt-Jovin, and T. M. Jovin, “Fluorescence lifetime imaging: multi-point calibration, minimum resolvable differences, and artifact suppression,” Cytometry 43, 248–260 (2001).
[Crossref]

He, H.

H. He, B. K. Nunnally, L.-C. Li, and L. B. McGown, “On-the-fly fluorescence lifetime detection of dye-labeled DNA primers for multiplex analysis,” Anal. Chem. 70, 3413–3418 (1998).
[Crossref]

Heide, F.

F. Heide, L. Xiao, A. Kolb, M. B. Hullin, and W. Heidrich, “Imaging in scattering media using correlation image sensors and sparse convolutional coding,” Opt. Express 22, 26338–26350 (2014).
[Crossref]

F. Heide, M. B. Hullin, J. Gregson, and W. Heidrich, “Low-budget transient imaging using photonic mixer devices,” ACM Trans. Graph. 32, 1 (2013).

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2014), pp. 3222–3229.

Heidrich, W.

F. Heide, L. Xiao, A. Kolb, M. B. Hullin, and W. Heidrich, “Imaging in scattering media using correlation image sensors and sparse convolutional coding,” Opt. Express 22, 26338–26350 (2014).
[Crossref]

F. Heide, M. B. Hullin, J. Gregson, and W. Heidrich, “Low-budget transient imaging using photonic mixer devices,” ACM Trans. Graph. 32, 1 (2013).

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2014), pp. 3222–3229.

Heshmat, B.

G. Satat, B. Heshmat, C. Barsi, D. Raviv, O. Chen, M. G. Bawendi, and R. Raskar, “Locating and classifying fluorescent tags behind turbid layers using time-resolved inversion,” Nat. Commun. 6, 6796 (2015).
[Crossref]

Hickey, G. A.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368–375 (2003).
[Crossref]

Hullin, M. B.

H. Qiao, J. Lin, Y. Liu, M. B. Hullin, and Q. Dai, “Resolving transient time profile in ToF imaging via log-sum sparse regularization,” Opt. Lett. 40, 918–921 (2015).
[Crossref]

F. Heide, L. Xiao, A. Kolb, M. B. Hullin, and W. Heidrich, “Imaging in scattering media using correlation image sensors and sparse convolutional coding,” Opt. Express 22, 26338–26350 (2014).
[Crossref]

F. Heide, M. B. Hullin, J. Gregson, and W. Heidrich, “Low-budget transient imaging using photonic mixer devices,” ACM Trans. Graph. 32, 1 (2013).

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2014), pp. 3222–3229.

Hyman, B. T.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368–375 (2003).
[Crossref]

Izadi, S.

A. Bhandari, M. Feigin, S. Izadi, C. Rhemann, M. Schmidt, and R. Raskar, “Resolving multipath interference in Kinect: An inverse problem approach,” in IEEE Sensors (IEEE, 2014), pp. 614–617.

A. Bhandari, A. Bourquard, S. Izadi, and R. Raskar, “Blind transmitted and reflected image separation using depth diversity and time-of-flight sensors,” in Computational Optical Sensing and Imaging (Optical Society of America, 2015), paper CT4F-2.

Jalali, B.

Jovin, T. M.

Q. S. Hanley, V. Subramaniam, D. J. Arndt-Jovin, and T. M. Jovin, “Fluorescence lifetime imaging: multi-point calibration, minimum resolvable differences, and artifact suppression,” Cytometry 43, 248–260 (2001).
[Crossref]

Kadambi, A.

A. Bhandari, A. Kadambi, R. Whyte, C. Barsi, M. Feigin, A. Dorrington, and R. Raskar, “Resolving multipath interference in time-of-flight imaging via modulation frequency diversity and sparse regularization,” Opt. Lett. 39, 1705–1708 (2014).
[Crossref]

A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. Dorrington, and R. Raskar, “Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles,” ACM Trans. Graph. 32, 167 (2013).
[Crossref]

A. Kadambi, A. Bhandari, R. Whyte, A. Dorrington, and R. Raskar, “Demultiplexing illumination via low cost sensing and nanosecond coding,” in IEEE International Conference on Computational Photography (ICCP) (2014).

A. Bhandari, A. Kadambi, and R. Raskar, “Sparse linear operator identification without sparse regularization? Applications to mixed pixel problem in time-of-flight/range imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2014), pp. 365–369.

Koch, R.

D. Lefloch, R. Nair, F. Lenzen, H. Schäfer, L. Streeter, M. J. Cree, R. Koch, and A. Kolb, “Technical foundation and calibration methods for time-of-flight cameras,” in Time-of-Flight and Depth Imaging: Sensors, Algorithms, and Applications (Springer, 2013), pp. 3–24.

Kolb, A.

F. Heide, L. Xiao, A. Kolb, M. B. Hullin, and W. Heidrich, “Imaging in scattering media using correlation image sensors and sparse convolutional coding,” Opt. Express 22, 26338–26350 (2014).
[Crossref]

D. Lefloch, R. Nair, F. Lenzen, H. Schäfer, L. Streeter, M. J. Cree, R. Koch, and A. Kolb, “Technical foundation and calibration methods for time-of-flight cameras,” in Time-of-Flight and Depth Imaging: Sensors, Algorithms, and Applications (Springer, 2013), pp. 3–24.

Kremers, G.-J.

G.-J. Kremers, E. B. Van Munster, J. Goedhart, and T. W. Gadella, “Quantitative lifetime unmixing of multiexponentially decaying fluorophores using single-frequency fluorescence lifetime imaging microscopy,” Biophys. J. 95, 378–389 (2008).
[Crossref]

Kumar, A. T.

A. T. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 27, 1152–1163 (2008).
[Crossref]

A. T. Kumar, S. B. Raymond, B. J. Bacskai, and D. A. Boas, “Comparison of frequency-domain and time-domain fluorescence lifetime tomography,” Opt. Lett. 33, 470–472 (2008).
[Crossref]

Kung, S. Y.

S. Y. Kung, Kernel Methods and Machine Learning (Cambridge University, 2014).

Laczko, G.

E. Gratton, M. Limkeman, J. R. Lakowicz, B. P. Maliwal, H. Cherek, and G. Laczko, “Resolution of mixtures of fluorophores using variable-frequency phase and modulation data,” Biophys. J. 46, 479–486 (1984).
[Crossref]

Lakowicz, J.

J. Lakowicz, Principles of Fluorescence Microscopy (Springer, 1982).

Lakowicz, J. R.

E. Gratton, M. Limkeman, J. R. Lakowicz, B. P. Maliwal, H. Cherek, and G. Laczko, “Resolution of mixtures of fluorophores using variable-frequency phase and modulation data,” Biophys. J. 46, 479–486 (1984).
[Crossref]

Lefloch, D.

D. Lefloch, R. Nair, F. Lenzen, H. Schäfer, L. Streeter, M. J. Cree, R. Koch, and A. Kolb, “Technical foundation and calibration methods for time-of-flight cameras,” in Time-of-Flight and Depth Imaging: Sensors, Algorithms, and Applications (Springer, 2013), pp. 3–24.

Lenzen, F.

D. Lefloch, R. Nair, F. Lenzen, H. Schäfer, L. Streeter, M. J. Cree, R. Koch, and A. Kolb, “Technical foundation and calibration methods for time-of-flight cameras,” in Time-of-Flight and Depth Imaging: Sensors, Algorithms, and Applications (Springer, 2013), pp. 3–24.

Li, L.-C.

H. He, B. K. Nunnally, L.-C. Li, and L. B. McGown, “On-the-fly fluorescence lifetime detection of dye-labeled DNA primers for multiplex analysis,” Anal. Chem. 70, 3413–3418 (1998).
[Crossref]

Li, Y.

T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).
[Crossref]

Limkeman, M.

E. Gratton, M. Limkeman, J. R. Lakowicz, B. P. Maliwal, H. Cherek, and G. Laczko, “Resolution of mixtures of fluorophores using variable-frequency phase and modulation data,” Biophys. J. 46, 479–486 (1984).
[Crossref]

Lin, J.

Liu, J.

D. R. Yankelevich, D. Ma, J. Liu, Y. Sun, Y. Sun, J. Bec, D. S. Elson, and L. Marcu, “Design and evaluation of a device for fast multispectral time-resolved fluorescence spectroscopy and imaging,” Rev. Sci. Instrum. 85, 034303 (2014).
[Crossref]

Liu, Y.

Lotti, T.

Lustenberger, F.

Ma, D.

D. R. Yankelevich, D. Ma, J. Liu, Y. Sun, Y. Sun, J. Bec, D. S. Elson, and L. Marcu, “Design and evaluation of a device for fast multispectral time-resolved fluorescence spectroscopy and imaging,” Rev. Sci. Instrum. 85, 034303 (2014).
[Crossref]

Maliwal, B. P.

E. Gratton, M. Limkeman, J. R. Lakowicz, B. P. Maliwal, H. Cherek, and G. Laczko, “Resolution of mixtures of fluorophores using variable-frequency phase and modulation data,” Biophys. J. 46, 479–486 (1984).
[Crossref]

Mao, S.

Marcu, L.

D. R. Yankelevich, D. Ma, J. Liu, Y. Sun, Y. Sun, J. Bec, D. S. Elson, and L. Marcu, “Design and evaluation of a device for fast multispectral time-resolved fluorescence spectroscopy and imaging,” Rev. Sci. Instrum. 85, 034303 (2014).
[Crossref]

Y. Sun, J. Phipps, D. S. Elson, H. Stoy, S. Tinling, J. Meier, B. Poirier, F. S. Chuang, D. G. Farwell, and L. Marcu, “Fluorescence lifetime imaging microscopy: in vivo application to diagnosis of oral carcinoma,” Opt. Lett. 34, 2081–2083 (2009).
[Crossref]

Massi, D.

McGown, L. B.

H. He, B. K. Nunnally, L.-C. Li, and L. B. McGown, “On-the-fly fluorescence lifetime detection of dye-labeled DNA primers for multiplex analysis,” Anal. Chem. 70, 3413–3418 (1998).
[Crossref]

Meier, J.

Nair, R.

D. Lefloch, R. Nair, F. Lenzen, H. Schäfer, L. Streeter, M. J. Cree, R. Koch, and A. Kolb, “Technical foundation and calibration methods for time-of-flight cameras,” in Time-of-Flight and Depth Imaging: Sensors, Algorithms, and Applications (Springer, 2013), pp. 3–24.

Nunnally, B. K.

H. He, B. K. Nunnally, L.-C. Li, and L. B. McGown, “On-the-fly fluorescence lifetime detection of dye-labeled DNA primers for multiplex analysis,” Anal. Chem. 70, 3413–3418 (1998).
[Crossref]

Oggier, T.

Pavone, F.

Philip, J.

Phipps, J.

Poirier, B.

Qiao, H.

Rao, J.

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18, 17–25 (2007).
[Crossref]

Raskar, R.

G. Satat, B. Heshmat, C. Barsi, D. Raviv, O. Chen, M. G. Bawendi, and R. Raskar, “Locating and classifying fluorescent tags behind turbid layers using time-resolved inversion,” Nat. Commun. 6, 6796 (2015).
[Crossref]

A. Bhandari, A. Kadambi, R. Whyte, C. Barsi, M. Feigin, A. Dorrington, and R. Raskar, “Resolving multipath interference in time-of-flight imaging via modulation frequency diversity and sparse regularization,” Opt. Lett. 39, 1705–1708 (2014).
[Crossref]

A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. Dorrington, and R. Raskar, “Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles,” ACM Trans. Graph. 32, 167 (2013).
[Crossref]

A. Bhandari, M. Feigin, S. Izadi, C. Rhemann, M. Schmidt, and R. Raskar, “Resolving multipath interference in Kinect: An inverse problem approach,” in IEEE Sensors (IEEE, 2014), pp. 614–617.

A. Kadambi, A. Bhandari, R. Whyte, A. Dorrington, and R. Raskar, “Demultiplexing illumination via low cost sensing and nanosecond coding,” in IEEE International Conference on Computational Photography (ICCP) (2014).

A. Bhandari, A. Bourquard, S. Izadi, and R. Raskar, “Blind transmitted and reflected image separation using depth diversity and time-of-flight sensors,” in Computational Optical Sensing and Imaging (Optical Society of America, 2015), paper CT4F-2.

A. Bhandari, A. Kadambi, and R. Raskar, “Sparse linear operator identification without sparse regularization? Applications to mixed pixel problem in time-of-flight/range imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2014), pp. 365–369.

Raviv, D.

G. Satat, B. Heshmat, C. Barsi, D. Raviv, O. Chen, M. G. Bawendi, and R. Raskar, “Locating and classifying fluorescent tags behind turbid layers using time-resolved inversion,” Nat. Commun. 6, 6796 (2015).
[Crossref]

Raymond, S. B.

A. T. Kumar, S. B. Raymond, B. J. Bacskai, and D. A. Boas, “Comparison of frequency-domain and time-domain fluorescence lifetime tomography,” Opt. Lett. 33, 470–472 (2008).
[Crossref]

A. T. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 27, 1152–1163 (2008).
[Crossref]

Rhemann, C.

A. Bhandari, M. Feigin, S. Izadi, C. Rhemann, M. Schmidt, and R. Raskar, “Resolving multipath interference in Kinect: An inverse problem approach,” in IEEE Sensors (IEEE, 2014), pp. 614–617.

Satat, G.

G. Satat, B. Heshmat, C. Barsi, D. Raviv, O. Chen, M. G. Bawendi, and R. Raskar, “Locating and classifying fluorescent tags behind turbid layers using time-resolved inversion,” Nat. Commun. 6, 6796 (2015).
[Crossref]

Schäfer, H.

D. Lefloch, R. Nair, F. Lenzen, H. Schäfer, L. Streeter, M. J. Cree, R. Koch, and A. Kolb, “Technical foundation and calibration methods for time-of-flight cameras,” in Time-of-Flight and Depth Imaging: Sensors, Algorithms, and Applications (Springer, 2013), pp. 3–24.

Schmidt, M.

A. Bhandari, M. Feigin, S. Izadi, C. Rhemann, M. Schmidt, and R. Raskar, “Resolving multipath interference in Kinect: An inverse problem approach,” in IEEE Sensors (IEEE, 2014), pp. 614–617.

Schuon, S.

S. Schuon, C. Theobalt, J. Davis, and S. Thrun, “Lidarboost: Depth superresolution for ToF 3D shape scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2009), pp. 343–350.

Sestini, S.

Skoch, J.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368–375 (2003).
[Crossref]

Squire, A.

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, “Multiple frequency fluorescence lifetime imaging microscopy,” J. Microsc. 197, 136–149 (2000).
[Crossref]

Stoy, H.

Streeter, L.

A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. Dorrington, and R. Raskar, “Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles,” ACM Trans. Graph. 32, 167 (2013).
[Crossref]

D. Lefloch, R. Nair, F. Lenzen, H. Schäfer, L. Streeter, M. J. Cree, R. Koch, and A. Kolb, “Technical foundation and calibration methods for time-of-flight cameras,” in Time-of-Flight and Depth Imaging: Sensors, Algorithms, and Applications (Springer, 2013), pp. 3–24.

Subramaniam, V.

Q. S. Hanley, V. Subramaniam, D. J. Arndt-Jovin, and T. M. Jovin, “Fluorescence lifetime imaging: multi-point calibration, minimum resolvable differences, and artifact suppression,” Cytometry 43, 248–260 (2001).
[Crossref]

Sun, Y.

D. R. Yankelevich, D. Ma, J. Liu, Y. Sun, Y. Sun, J. Bec, D. S. Elson, and L. Marcu, “Design and evaluation of a device for fast multispectral time-resolved fluorescence spectroscopy and imaging,” Rev. Sci. Instrum. 85, 034303 (2014).
[Crossref]

D. R. Yankelevich, D. Ma, J. Liu, Y. Sun, Y. Sun, J. Bec, D. S. Elson, and L. Marcu, “Design and evaluation of a device for fast multispectral time-resolved fluorescence spectroscopy and imaging,” Rev. Sci. Instrum. 85, 034303 (2014).
[Crossref]

Y. Sun, J. Phipps, D. S. Elson, H. Stoy, S. Tinling, J. Meier, B. Poirier, F. S. Chuang, D. G. Farwell, and L. Marcu, “Fluorescence lifetime imaging microscopy: in vivo application to diagnosis of oral carcinoma,” Opt. Lett. 34, 2081–2083 (2009).
[Crossref]

Theobalt, C.

S. Schuon, C. Theobalt, J. Davis, and S. Thrun, “Lidarboost: Depth superresolution for ToF 3D shape scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2009), pp. 343–350.

Thrun, S.

S. Schuon, C. Theobalt, J. Davis, and S. Thrun, “Lidarboost: Depth superresolution for ToF 3D shape scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2009), pp. 343–350.

Tinling, S.

Torras, C.

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (ToF) cameras: a survey,” IEEE Sens. J. 11, 1917–1926 (2011).
[Crossref]

Van Munster, E. B.

G.-J. Kremers, E. B. Van Munster, J. Goedhart, and T. W. Gadella, “Quantitative lifetime unmixing of multiexponentially decaying fluorophores using single-frequency fluorescence lifetime imaging microscopy,” Biophys. J. 95, 378–389 (2008).
[Crossref]

Verveer, P. J.

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, “Multiple frequency fluorescence lifetime imaging microscopy,” J. Microsc. 197, 136–149 (2000).
[Crossref]

Whyte, R.

A. Bhandari, A. Kadambi, R. Whyte, C. Barsi, M. Feigin, A. Dorrington, and R. Raskar, “Resolving multipath interference in time-of-flight imaging via modulation frequency diversity and sparse regularization,” Opt. Lett. 39, 1705–1708 (2014).
[Crossref]

A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. Dorrington, and R. Raskar, “Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles,” ACM Trans. Graph. 32, 167 (2013).
[Crossref]

A. Kadambi, A. Bhandari, R. Whyte, A. Dorrington, and R. Raskar, “Demultiplexing illumination via low cost sensing and nanosecond coding,” in IEEE International Conference on Computational Photography (ICCP) (2014).

Wouters, F.

Xiao, L.

F. Heide, L. Xiao, A. Kolb, M. B. Hullin, and W. Heidrich, “Imaging in scattering media using correlation image sensors and sparse convolutional coding,” Opt. Express 22, 26338–26350 (2014).
[Crossref]

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2014), pp. 3222–3229.

Yamaguchi, I.

Yankelevich, D. R.

D. R. Yankelevich, D. Ma, J. Liu, Y. Sun, Y. Sun, J. Bec, D. S. Elson, and L. Marcu, “Design and evaluation of a device for fast multispectral time-resolved fluorescence spectroscopy and imaging,” Rev. Sci. Instrum. 85, 034303 (2014).
[Crossref]

Yao, H.

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18, 17–25 (2007).
[Crossref]

Zamai, M.

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, “The phasor approach to fluorescence lifetime imaging analysis,” Biophys. J. 94, L14–L16 (2008).
[Crossref]

Zhang, T.

ACM Trans. Graph. (2)

A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. Dorrington, and R. Raskar, “Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles,” ACM Trans. Graph. 32, 167 (2013).
[Crossref]

F. Heide, M. B. Hullin, J. Gregson, and W. Heidrich, “Low-budget transient imaging using photonic mixer devices,” ACM Trans. Graph. 32, 1 (2013).

Anal. Chem. (1)

H. He, B. K. Nunnally, L.-C. Li, and L. B. McGown, “On-the-fly fluorescence lifetime detection of dye-labeled DNA primers for multiplex analysis,” Anal. Chem. 70, 3413–3418 (1998).
[Crossref]

Biomed. Opt. Express (1)

Biophys. J. (3)

E. Gratton, M. Limkeman, J. R. Lakowicz, B. P. Maliwal, H. Cherek, and G. Laczko, “Resolution of mixtures of fluorophores using variable-frequency phase and modulation data,” Biophys. J. 46, 479–486 (1984).
[Crossref]

G.-J. Kremers, E. B. Van Munster, J. Goedhart, and T. W. Gadella, “Quantitative lifetime unmixing of multiexponentially decaying fluorophores using single-frequency fluorescence lifetime imaging microscopy,” Biophys. J. 95, 378–389 (2008).
[Crossref]

M. A. Digman, V. R. Caiolfa, M. Zamai, and E. Gratton, “The phasor approach to fluorescence lifetime imaging analysis,” Biophys. J. 94, L14–L16 (2008).
[Crossref]

Curr. Opin. Biotechnol. (1)

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18, 17–25 (2007).
[Crossref]

Cytometry (1)

Q. S. Hanley, V. Subramaniam, D. J. Arndt-Jovin, and T. M. Jovin, “Fluorescence lifetime imaging: multi-point calibration, minimum resolvable differences, and artifact suppression,” Cytometry 43, 248–260 (2001).
[Crossref]

IEEE Sens. J. (1)

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (ToF) cameras: a survey,” IEEE Sens. J. 11, 1917–1926 (2011).
[Crossref]

IEEE Trans. Med. Imaging (1)

A. T. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 27, 1152–1163 (2008).
[Crossref]

J. Biomed. Opt. (1)

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368–375 (2003).
[Crossref]

J. Microsc. (1)

A. Squire, P. J. Verveer, and P. I. H. Bastiaens, “Multiple frequency fluorescence lifetime imaging microscopy,” J. Microsc. 197, 136–149 (2000).
[Crossref]

J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

G. Satat, B. Heshmat, C. Barsi, D. Raviv, O. Chen, M. G. Bawendi, and R. Raskar, “Locating and classifying fluorescent tags behind turbid layers using time-resolved inversion,” Nat. Commun. 6, 6796 (2015).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Rev. Sci. Instrum. (1)

D. R. Yankelevich, D. Ma, J. Liu, Y. Sun, Y. Sun, J. Bec, D. S. Elson, and L. Marcu, “Design and evaluation of a device for fast multispectral time-resolved fluorescence spectroscopy and imaging,” Rev. Sci. Instrum. 85, 034303 (2014).
[Crossref]

SIAM J. Optim. (1)

T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996).
[Crossref]

Other (11)

A. Bhandari, A. Bourquard, S. Izadi, and R. Raskar, “Blind transmitted and reflected image separation using depth diversity and time-of-flight sensors,” in Computational Optical Sensing and Imaging (Optical Society of America, 2015), paper CT4F-2.

S. Schuon, C. Theobalt, J. Davis, and S. Thrun, “Lidarboost: Depth superresolution for ToF 3D shape scanning,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2009), pp. 343–350.

D. Lefloch, R. Nair, F. Lenzen, H. Schäfer, L. Streeter, M. J. Cree, R. Koch, and A. Kolb, “Technical foundation and calibration methods for time-of-flight cameras,” in Time-of-Flight and Depth Imaging: Sensors, Algorithms, and Applications (Springer, 2013), pp. 3–24.

S. Y. Kung, Kernel Methods and Machine Learning (Cambridge University, 2014).

S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar (Cambridge University, 2005).

PMD Technologies, “pmd PhotonICs 19k–S3: Specs and reference design,” (2014).

J. Lakowicz, Principles of Fluorescence Microscopy (Springer, 1982).

A. Bhandari, M. Feigin, S. Izadi, C. Rhemann, M. Schmidt, and R. Raskar, “Resolving multipath interference in Kinect: An inverse problem approach,” in IEEE Sensors (IEEE, 2014), pp. 614–617.

F. Heide, L. Xiao, W. Heidrich, and M. B. Hullin, “Diffuse mirrors: 3D reconstruction from diffuse indirect illumination using inexpensive time-of-flight sensors,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2014), pp. 3222–3229.

A. Kadambi, A. Bhandari, R. Whyte, A. Dorrington, and R. Raskar, “Demultiplexing illumination via low cost sensing and nanosecond coding,” in IEEE International Conference on Computational Photography (ICCP) (2014).

A. Bhandari, A. Kadambi, and R. Raskar, “Sparse linear operator identification without sparse regularization? Applications to mixed pixel problem in time-of-flight/range imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2014), pp. 365–369.

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Figures (6)

Fig. 1.
Fig. 1.

Experimental cross-correlated probing function [Eq. (8)] (blue line) and its Fourier series approximation (red dashed–dotted line) [Eq. (9)], with N0=30. Since the time-domain signal Cp,p¯ lasts for Δ=309.902ns, the corresponding fundamental Fourier harmonic is ω0/2π=3.2268MHz. Consequently, Fourier coefficient N0=30 corresponds to 96.80MHz. Inset: Fourier series coefficients ϕ^n,n=N0,,N0.

Fig. 2.
Fig. 2.

ToF depth imaging and its link with FLI. For the SRF h(t,t)=ρδ(tt2d/c), (a) and (b) compare TD- and FD-ToF principles. (a) Probing and reflected signal for TD-ToF. The time delay is proportional to distance d. (b) Probing and reflected signal for FD-ToF. The phase is proportional to distance d and modulation frequency ω0. For the SRF h(t,t)=hDepth(t,t)+hSample(t,t) in Eq. (13), (c) and (d) compare the TD-ToF and FD-ToF methods for FLI. (c) Probing and reflected signal for TD-ToF FLI. The time delay is proportional to distance d and the waveform shape is linked with lifetime τ. (d) Probing and reflected signal for FD-ToF FLI. The phase is proportional to distance d, modulation frequency ω0, and lifetime τ. (e) Experimental setup for FLI estimation via ToF.

Fig. 3.
Fig. 3.

TD-ToF FLI measurements. (a) Time profile of measurements mk=m(kTs) and Ts=7.8120ps based on K=3968 samples. These measurements result from convolving cross correlation of the probing function with the SRF [Eq. (17)]. (b) Phase measurements computed using h^=V+m with N0=15 and Δ=309.9ns. We estimate the distance d˜ and lifetime τ˜ parameters using nonlinear least squares fitting, which results in the fitted curve. The fitted result is shown by the red solid line. The estimated phase contribution [Eq. (20)] due to distance θd˜ and lifetime θτ˜ is also plotted.

Fig. 4.
Fig. 4.

FD-ToF FLI measurements of a 61×66 pixel patch of a 120×160 pixel sensor image. The size of the scene is approximately 2.3in2. (a) Multifrequency measurements of τ=32ns quantum-dot-based fluorescent sample. We show phase and amplitude images, that is, {z(10f0),z(20f0),z(30f0),z(40f0)} (in radians, [0,2π]) and {|z(10f0)|,|z(20f0)|,|z(30f0)|,|z(40f0)|} (in decibels), respectively. The base modulation frequency for the experiment is f0=ω0/2π=1MHz. The phase at the background pixel is recorded to be z(xb,yb)(40f0)=4.1625rad, which amounts to a depth of 2.48 m, which is consistent with the experimental setup. At the location of the fluorescent sample, we recorded a higher phase value z(xf,yf)(40f0)=5.5822, which is attributed to the fluorescence phenomenon. (b) Multifrequency raw phase measurements {z(x,y)(kf0)}k=1K=40 for four pixels. The measurements confirm with the theoretical hypothesis of Eq. (20), as well as the fitted phase obtained by Eq. (35). The estimated phase contribution [Eq. (20)] due to distance θd˜ and lifetime θτ˜ is also plotted.

Fig. 5.
Fig. 5.

Estimation accuracy of τ1=4ns and τ2=32ns. (a) We plot 2000 estimated values of lifetime as a function of SNR ranging from 0 to 60 dB. As the SNR increases, the estimates cluster around the oracle estimates of τ=4ns and τ=32ns, respectively. (b) We plot the MSEτ on log scale as a function of SNR. After 15 dB, we note a consistent linear relationship [Eq. (36)] between SNR and the log(MSEτ).

Fig. 6.
Fig. 6.

Measured phase as a function of the modulation frequency. We plot the measured phase corresponding to a 120 pixel cross section of the 160×120 ToF sensor image for modulation frequencies f={10,20,30,40}MHz. The first 50 phase measurements are due to the background pixel, which is at a depth d=2.5m. The dashed line marks the average phase value over the first 50 pixels. The average phase value for each modulation frequency leads to a distance estimate: d={2.56,2.52,2.47,2.48} in meters.

Tables (4)

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Table 1. TD- and FD-ToF Depth Imaging

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Table 2. TD- and FD-ToF FLI

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Table 3. TD-ToF FLI (τ=32ns and d=1.05m)

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Table 4. FD-ToF FLI (τ=32ns and d=2.5m)

Equations (49)

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r(t)=p(t)h(t,t)dt.
m(t)=p¯(t)r(tt)dt,
h(t,t)=hSI(tt),
r(t)=(p*hSI)(t)=p(t)hSI(tt)h(t,t)dt.
m(t)=(p*p¯*hSI)(t)(ϕ*hSI)(t),
h(t,t)=ρδ(tt2d/c),
m(t)=ρϕ(t)(tt0)ρ(p*p¯)(tt0).
t˜0=maxtm(t)=maxtϕ(tt0),t0=2dc.
p(t)=1Δn=p^nejnω0t,ω0=2π/Δ,
p^n=0Δp(t)ejnω0tdt,
ϕ(t)=(p*p¯)(t)=n=ϕ^nejω0nt.
ϕ^n=p^np^n*=|p^n|2.
Cp,p¯(t)1Δ|n|N0ϕ^nejnω0t,
p(t)=1+p0cos(ωt),
r(t)=ρp(tt0)=ρ(1+p0cos(ω(tt0))).
m(t)=limB12BB+Bp¯(t)r(tt)dt
=ρ(1+p022cos(ω(t+t0))),
z=(m0m2)+j(m3m1).
ρ˜=|z|p02andd˜=c2ωz.
h(t,t)=hDepth(t,t)+hSample(t,t),
hDepth(t,t)=ρδ(tt2d/c),
hSample(t,t)=μe(tt2d/cτ)Π(tt2d/c).
m(t)=(4)(p*p¯*hSI)(t)(9)(Cp,p¯*hSI)(t)Cp,p¯*(δ(t2d/c)*(ρδ(t)+μetτΠ(t))).
m(t)1Δ|n|N0(ϕ^nh^n)ejω0nt,
h^(ω)=h(t)ejωtdt(Fourier Transform)
=ρejω(2dc)+μτ1+jωτejω(2dc).
|h^(ω)|=(ρ+μτ)2+(ωρτ)21+(ωτ)2,h^(ω)=tan1(ωμτ2ρ+μτ+ρ(ωτ)2)2dcω.
h^(ω)=θτ(ω)θd(ω),
θτ(ω)=tan1(ωτ2τ+ρ(1+(ωτ)2))andθd(ω)=2dω/c.
m=VDϕ^h^,
mTD-FLI(t)=μe(tt0τ)Π(tt0).
mTD-ToF-FLI(t)=(17)1Δ|n|N0(ϕ^nh^n)ejω0nt,
MLS0101110110001111100110100100001,
h^obs=V+m=(θτ(2πn/Δ)+θd(2πn/Δ)),
argmind,τn=N0n=+N0|h^(nω0)h^obs(nω0+β)+α|2,
MSEν=1Nn=0N1|ν˜nν|2.
h^˜(nω0)=(20)tan1(nω0τ˜)2nω0d˜c,
h^obs(nω0)=h^(nω0)+εn.
SNR=20(logh^logh^obsh^),
r(t)=(3)(p*hSI)=|h^(0)|+|h^(ω)|p0cos(ωt+h^(ω)).
m(t)=(10)|h^(0)|+|h^(ω)|p022cos(ωth^(ω)).
|h^(ω0)||z(ω0)|/p02Amplitude Estimateandh^(ω0)(20)z(ω0)Phase Estimate,
h(t)=k=0K1ρkδ(t2dk/c)Fourierh^(ω)=k=0K1ρkejω(2dkc).
{z(kf0)}k=1K,k=1,2,,40.
argmind,τk=1K=40|z(kω0)+θτ(kω0)+θd(kω0)|2.
{d˜,τ˜}={2.4961m,31.024ns}.
f0(1)=0.10N(1)=401,f0(2)=0.25N(2)=161,f0(3)=0.50N(3)=81,f0(4)=1.00N(4)=41.
h^obs(k)(nω0)=(30)h^(nω0)+εn,n=0,,N(k)1,
log(MSEτ)logK0λlog(SNR).

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