Abstract

A nonparametric deconvolution algorithm for recovering the photon time-of-flight distribution (TOFD) from time-resolved (TR) measurements is described. The algorithm combines wavelet denoising and a two-stage deconvolution method based on generalized singular value decomposition and Tikhonov regularization. The efficacy of the algorithm was tested on simulated and experimental TR data and the results show that it can recover the photon TOFD with high fidelity. Combined with the microscopic Beer–Lambert law, the algorithm enables accurate quantification of absorption changes from arbitrary time-of-flight windows, thereby optimizing the depth sensitivity provided by TR measurements.

© 2012 Optical Society of America

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2011

M. Makitalo and A. Foi, IEEE Trans. Image Process 20, 99 (2011).
[CrossRef]

2010

M. Diop, K. M. Tichauer, J. T. Elliott, M. Migueis, T. Y. Lee, and K. St. Lawrence, J. Biomed. Opt. 15, 057004 (2010).
[CrossRef]

2008

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, J. Biomed. Opt. 13, 054019 (2008).
[CrossRef]

2007

P. C. Hansen, Numer. Algorithms 46, 189 (2007).
[CrossRef]

2006

2005

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas, J. Biomed. Opt. 10, 11013 (2005).
[CrossRef]

2004

2003

A. Liebert, H. Wabnitz, D. Grosenick, and R. Macdonald, J. Biomed. Opt. 8, 512 (2003).
[CrossRef]

2001

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, Phys. Med. Biol. 46, 879 (2001).
[CrossRef]

1999

1997

1988

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef]

Bérubé-Lauzière, Y.

G. Bodi and Y. Bérubé-Lauzière, in European Conference on Biomedical Optics (Optical Society of America, 2009), p. 7369.

Boas, D. A.

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, J. Biomed. Opt. 13, 054019 (2008).
[CrossRef]

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas, J. Biomed. Opt. 10, 11013 (2005).
[CrossRef]

Bodi, G.

G. Bodi and Y. Bérubé-Lauzière, in European Conference on Biomedical Optics (Optical Society of America, 2009), p. 7369.

Chabrier, R.

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef]

Contini, D.

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, F. Paglia, and R. Cubeddu, Opt. Express 14, 5418 (2006).
[CrossRef]

H. Wabnitz, A. Liebert, D. Contini, L. Spinelli, and A. Torricelli, in Biomedical Optics (Optical Society of America, 2008), p. BMD9.

Cubeddu, R.

Diop, M.

M. Diop, K. M. Tichauer, J. T. Elliott, M. Migueis, T. Y. Lee, and K. St. Lawrence, J. Biomed. Opt. 15, 057004 (2010).
[CrossRef]

Dogariu, A.

Elliott, J. T.

M. Diop, K. M. Tichauer, J. T. Elliott, M. Migueis, T. Y. Lee, and K. St. Lawrence, J. Biomed. Opt. 15, 057004 (2010).
[CrossRef]

Foi, A.

M. Makitalo and A. Foi, IEEE Trans. Image Process 20, 99 (2011).
[CrossRef]

Franceschini, M. A.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas, J. Biomed. Opt. 10, 11013 (2005).
[CrossRef]

Gagnon, L.

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, J. Biomed. Opt. 13, 054019 (2008).
[CrossRef]

Gauthier, C.

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, J. Biomed. Opt. 13, 054019 (2008).
[CrossRef]

Grosenick, D.

A. Liebert, H. Wabnitz, D. Grosenick, and R. Macdonald, J. Biomed. Opt. 8, 512 (2003).
[CrossRef]

Hansen, P. C.

P. C. Hansen, Numer. Algorithms 46, 189 (2007).
[CrossRef]

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (Society for Industrial and Applied Mathematics, 2010).

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef]

Hoge, R. D.

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, J. Biomed. Opt. 13, 054019 (2008).
[CrossRef]

Kienle, A.

Lawrence, K. St.

M. Diop, K. M. Tichauer, J. T. Elliott, M. Migueis, T. Y. Lee, and K. St. Lawrence, J. Biomed. Opt. 15, 057004 (2010).
[CrossRef]

Lee, T. Y.

M. Diop, K. M. Tichauer, J. T. Elliott, M. Migueis, T. Y. Lee, and K. St. Lawrence, J. Biomed. Opt. 15, 057004 (2010).
[CrossRef]

Lesage, F.

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, J. Biomed. Opt. 13, 054019 (2008).
[CrossRef]

Liebert, A.

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Moller, R. Macdonald, A. Villringer, and H. Rinneberg, Appl. Opt. 43, 3037 (2004).
[CrossRef]

A. Liebert, H. Wabnitz, D. Grosenick, and R. Macdonald, J. Biomed. Opt. 8, 512 (2003).
[CrossRef]

H. Wabnitz, A. Liebert, D. Contini, L. Spinelli, and A. Torricelli, in Biomedical Optics (Optical Society of America, 2008), p. BMD9.

Macdonald, R.

Makitalo, M.

M. Makitalo and A. Foi, IEEE Trans. Image Process 20, 99 (2011).
[CrossRef]

Migueis, M.

M. Diop, K. M. Tichauer, J. T. Elliott, M. Migueis, T. Y. Lee, and K. St. Lawrence, J. Biomed. Opt. 15, 057004 (2010).
[CrossRef]

Moller, M.

Montcel, B.

Obrig, H.

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Moller, R. Macdonald, A. Villringer, and H. Rinneberg, Appl. Opt. 43, 3037 (2004).
[CrossRef]

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, Phys. Med. Biol. 46, 879 (2001).
[CrossRef]

Paglia, F.

Patterson, M. S.

Pifferi, A.

Pine, D. J.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef]

Popescu, G.

Poulet, P.

Rinneberg, H.

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Moller, R. Macdonald, A. Villringer, and H. Rinneberg, Appl. Opt. 43, 3037 (2004).
[CrossRef]

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, Phys. Med. Biol. 46, 879 (2001).
[CrossRef]

Selb, J.

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, J. Biomed. Opt. 13, 054019 (2008).
[CrossRef]

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas, J. Biomed. Opt. 10, 11013 (2005).
[CrossRef]

Sorensen, A. G.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas, J. Biomed. Opt. 10, 11013 (2005).
[CrossRef]

Spinelli, L.

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, F. Paglia, and R. Cubeddu, Opt. Express 14, 5418 (2006).
[CrossRef]

H. Wabnitz, A. Liebert, D. Contini, L. Spinelli, and A. Torricelli, in Biomedical Optics (Optical Society of America, 2008), p. BMD9.

Steinbrink, J.

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Moller, R. Macdonald, A. Villringer, and H. Rinneberg, Appl. Opt. 43, 3037 (2004).
[CrossRef]

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, Phys. Med. Biol. 46, 879 (2001).
[CrossRef]

Stott, J. J.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas, J. Biomed. Opt. 10, 11013 (2005).
[CrossRef]

Tichauer, K. M.

M. Diop, K. M. Tichauer, J. T. Elliott, M. Migueis, T. Y. Lee, and K. St. Lawrence, J. Biomed. Opt. 15, 057004 (2010).
[CrossRef]

Torricelli, A.

D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, F. Paglia, and R. Cubeddu, Opt. Express 14, 5418 (2006).
[CrossRef]

H. Wabnitz, A. Liebert, D. Contini, L. Spinelli, and A. Torricelli, in Biomedical Optics (Optical Society of America, 2008), p. BMD9.

Villringer, A.

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Moller, R. Macdonald, A. Villringer, and H. Rinneberg, Appl. Opt. 43, 3037 (2004).
[CrossRef]

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, Phys. Med. Biol. 46, 879 (2001).
[CrossRef]

Wabnitz, H.

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Moller, R. Macdonald, A. Villringer, and H. Rinneberg, Appl. Opt. 43, 3037 (2004).
[CrossRef]

A. Liebert, H. Wabnitz, D. Grosenick, and R. Macdonald, J. Biomed. Opt. 8, 512 (2003).
[CrossRef]

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, Phys. Med. Biol. 46, 879 (2001).
[CrossRef]

H. Wabnitz, A. Liebert, D. Contini, L. Spinelli, and A. Torricelli, in Biomedical Optics (Optical Society of America, 2008), p. BMD9.

Weitz, D. A.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef]

Appl. Opt.

IEEE Trans. Image Process

M. Makitalo and A. Foi, IEEE Trans. Image Process 20, 99 (2011).
[CrossRef]

J. Biomed. Opt.

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas, J. Biomed. Opt. 10, 11013 (2005).
[CrossRef]

A. Liebert, H. Wabnitz, D. Grosenick, and R. Macdonald, J. Biomed. Opt. 8, 512 (2003).
[CrossRef]

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, J. Biomed. Opt. 13, 054019 (2008).
[CrossRef]

M. Diop, K. M. Tichauer, J. T. Elliott, M. Migueis, T. Y. Lee, and K. St. Lawrence, J. Biomed. Opt. 15, 057004 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Numer. Algorithms

P. C. Hansen, Numer. Algorithms 46, 189 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, Phys. Med. Biol. 46, 879 (2001).
[CrossRef]

Phys. Rev. Lett.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988).
[CrossRef]

Other

“MathWorks,” http://www.mathworks.com/ .

H. Wabnitz, A. Liebert, D. Contini, L. Spinelli, and A. Torricelli, in Biomedical Optics (Optical Society of America, 2008), p. BMD9.

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (Society for Industrial and Applied Mathematics, 2010).

G. Bodi and Y. Bérubé-Lauzière, in European Conference on Biomedical Optics (Optical Society of America, 2009), p. 7369.

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Figures (3)

Fig. 1.
Fig. 1.

(a) The dashed grey curve is a theoretical TOFD generated using the diffusion model with μa=0.02mm1, μs=0.80mm1, and ρ=20mm. Poisson noise was added to the convolution of the theoretical TOFD with the IRF (black curve) to generate the noisy TPSF [red curve in (a)]. The denoised IRF and TPSF (green and blue curves, respectively) show that the wavelet denoising substantially reduces the noise in the curves with negligible distortion of their overall shapes. The dashed red curve at the bottom of (a) is the residual between the theoretical and denoised TPSF. The small magnitude of this residual further confirms the quality of the denoising. The TOFDs recovered from the first and second stage of the deconvolution are shown in (b) and (c), respectively.

Fig. 2.
Fig. 2.

(a) Reflectance (TPSF) measured on a piglet head at a source detector distance of 27 mm; the TPSF was acquired at a count rate of 800 kHz for 0.4 s. The curve in (b) is the recovered TOFD from the deconvolution of the IRF [in Fig. 1(a)] and TPSF.

Fig. 3.
Fig. 3.

Absorption changes obtained using the late part [corresponding to 1.00–1.25 ns in Figs. 1(b) and 1(c)] of the recovered TOFDs from the one-stage algorithm and after the two-stage deconvolution, and using the TPSFs directly [from 2.25 to 2.50 ns in Fig. 1(a)]. The dashed grey line is the true Δμa and it is clear that the two-stage deconvolution provides the most accurate estimate of Δμa. The large error in the absorption change obtained without deconvolution is due to the confounding effects of the IRF. A total of 10 simulations were conducted and the error bars represent standard deviations.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

TPSF=IRF*TOFD,
TPSF(mΔt)=n=0m1IRF([mn]Δt)TOFD(nΔt)Δt,
minx(Axb22+λ2WLx22),

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