Abstract

High density diffuse optical tomography has become increasingly important to detect underlying neuronal activities. Conventional methods first estimate the time courses of the changes in the absorption coefficients for all the voxels, and then estimate the hemodynamic response function (HRF). Activation-level maps are extracted at last based on this HRF. However, the error propagation among the successive processes degrades and even misleads the final results. Besides, the computation burden is heavy. To address the above problems, a direct method is proposed in this paper to simultaneously estimate the HRF and the activation-level maps from the boundary fluxes. It is assumed that all the voxels in the same activated brain region share the same HRF but differ in the activation levels, and no prior information is imposed on the specific shape of the HRF. The dynamic simulation and phantom experiments demonstrate that the proposed method outperforms the conventional one in terms of the estimation accuracy and computation speed.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (3)

M. D. Wheelock, J. P. Culver, and A. T. Eggebrecht, “High-density diffuse optical tomography for imaging human brain function,” Rev. Sci. Instrum. 90(5), 051101 (2019).
[Crossref]

B. Wang, T. Pan, Y. Zhang, D. Liu, J. Jiang, H. Zhao, and F. Gao, “A Kalman-based tomographic scheme for directly reconstructing activation levels of brain function,” Opt. Express 27(3), 3229–3246 (2019).
[Crossref]

D. Liu, B. Wang, T. Pan, J. Li, Z. Qin, L. Zhang, Z. Zhou, and F. Gao, “Toward quantitative near infrared brain functional imaging: lock-in photon counting instrumentation combined with tomographic reconstruction,” IEEE Access 7, 86829–86842 (2019).
[Crossref]

2017 (1)

A. Aarabi, V. Osharina, and F. Wallois, “Effect of confounding variables on hemodynamic response function estimation using averaging and deconvolution analysis: An event-related NIRS study,” NeuroImage 155, 25–49 (2017).
[Crossref]

2016 (1)

T. Sato, I. Nambu, K. Takeda, T. Aihara, O. Yamashita, Y. Isogaya, Y. Inoue, Y. Otaka, Y. Wada, M. Kawato, M. A. Sato, and R. Osu, “Reduction of global interference of scalp-hemodynamics in functional near-infrared spectroscopy using short distance probes,” NeuroImage 141, 120–132 (2016).
[Crossref]

2015 (2)

2014 (8)

M. S. Hassanpour, B. R. White, A. T. Eggebrecht, S. L. Ferradal, A. Z. Snyder, and J. P. Culver, “Statistical analysis of high density diffuse optical tomography,” NeuroImage 85(1), 104–116 (2014).
[Crossref]

N. Bazargani and A. Nosratinia, “Joint maximum likelihood estimation of activation and hemodynamic Response Function for fMRI,” Med. Image Anal. 18(5), 711–724 (2014).
[Crossref]

F. Scholkmann, S. Kleiser, A. J. Metz, R. Zimmermann, J. Mata Pavia, U. Wolf, and M. Wolf, “A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology,” NeuroImage 85(1), 6–27 (2014).
[Crossref]

A. Shah and A. K. Seghouane, “An integrated framework for joint HRF and drift estimation and HbO/HbR signal improvement in fNIRS data,” IEEE Trans. Med. Imaging 33(11), 2086–2097 (2014).
[Crossref]

J. R. Goodwin, C. R. Gaudet, and A. J. Berger, “Short-channel functional near-infrared spectroscopy regressions improve when source-detector separation is reduced,” Neurophotonics 1(1), 015002 (2014).
[Crossref]

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref]

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse recovery methods hold promise for diffuse optical tomographic image reconstruction,” Ieee J Sel Top Quant 20(2), 74–82(2014).
[Crossref]

C. Habermehl, J. Steinbrink, K. R. Muller, and S. Haufe, “Optimizing the regularization for image reconstruction of cerebral diffuse optical tomography,” J. Biomed. Opt. 19(9), 096006 (2014).
[Crossref]

2013 (3)

F. Scarpa, S. Brigadoi, S. Cutini, P. Scatturin, M. Zorzi, R. Dell’acqua, and G. Sparacino, “A reference-channel based methodology to improve estimation of event-related hemodynamic response from fNIRS measurements,” NeuroImage 72, 106–119 (2013).
[Crossref]

T. Shimokawa, T. Kosaka, O. Yamashita, N. Hiroe, T. Amita, Y. Inoue, and M. A. Sato, “Extended hierarchical Bayesian diffuse optical tomography for removing scalp artifact,” Biomed. Opt. Express 4(11), 2411–2432 (2013).
[Crossref]

T. Zhang, F. Li, L. Beckes, and J. A. Coan, “A semi-parametric model of the hemodynamic response for multi-subject fMRI data,” NeuroImage 75, 136–145 (2013).
[Crossref]

2012 (2)

A. K. Seghouane and A. Shah, “HRF estimation in fMRI data with an unknown drift matrix by iterative minimization of the Kullback-Leibler divergence,” IEEE Trans. Med. Imaging 31(2), 192–206 (2012).
[Crossref]

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

2011 (2)

V. Fonov, A. C. Evans, K. Botteron, C. R. Almli, R. C. McKinstry, and D. L. Collins, “Unbiased average age-appropriate atlases for pediatric studies,” NeuroImage 54(1), 313–327 (2011).
[Crossref]

L. Gagnon, K. Perdue, D. N. Greve, D. Goldenholz, G. Kaskhedikar, and D. A. Boas, “Improved recovery of the hemodynamic response in diffuse optical imaging using short optode separations and state-space modeling,” NeuroImage 56(3), 1362–1371 (2011).
[Crossref]

2010 (3)

S. Lloyd-Fox, A. Blasi, and C. E. Elwell, “Illuminating the developing brain: the past, present and future of functional near infrared spectroscopy,” Neurosci. Biobehav. Rev. 34(3), 269–284 (2010).
[Crossref]

Q. Q. Fang, “Mesh-based Monte Carlo method using fast ray-tracing in Plucker coordinates,” Biomed. Opt. Express 1(1), 165–175 (2010).
[Crossref]

N. M. Gregg, B. R. White, B. W. Zeff, A. J. Berger, and J. P. Culver, “Brain specificity of diffuse optical imaging: improvements from superficial signal regression and tomography,” Front. Neuroenerg. 2, 14 (2010).

2009 (2)

J. Markham, B. R. White, B. W. Zeff, and J. P. Culver, “Blind identification of evoked human brain activity with independent component analysis of optical data,” Hum. Brain Mapp. 30(8), 2382–2392 (2009).
[Crossref]

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2(1), 183–202 (2009).
[Crossref]

2008 (1)

R. Casanova, S. Ryali, J. Serences, L. Yang, R. Kraft, P. J. Laurienti, and J. A. Maldjian, “The impact of temporal regularization on estimates of the BOLD hemodynamic response function: a comparative analysis,” NeuroImage 40(4), 1606–1618 (2008).
[Crossref]

2007 (3)

S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, and K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12(6), 062111 (2007).
[Crossref]

M. A. Lindquist and T. D. Wager, “Validity and power in hemodynamic response modeling: a comparison study and a new approach,” Hum. Brain Mapp. 28(8), 764–784 (2007).
[Crossref]

J. Cohen-Adad, S. Chapuisat, J. Doyon, S. Rossignol, J. M. Lina, H. Benali, and F. Lesage, “Activation detection in diffuse optical imaging by means of the general linear model,” Med. Image Anal. 11(6), 616–629 (2007).
[Crossref]

2006 (2)

S. G. Diamond, T. J. Huppert, V. Kolehmainen, M. A. Franceschini, J. P. Kaipio, S. R. Arridge, and D. A. Boas, “Dynamic physiological modeling for functional diffuse optical tomography,” NeuroImage 30(1), 88–101 (2006).
[Crossref]

B. Vojnovic, “Advanced time-correlated single photon counting techniques,” J. Microsc. 222(1), 65–66 (2006).
[Crossref]

2005 (2)

Y. Zhang, D. H. Brooks, and D. A. Boas, “A haemodynamic response function model in spatio-temporal diffuse optical tomography,” Phys. Med. Biol. 50(19), 4625–4644 (2005).
[Crossref]

R. B. Saager and A. J. Berger, “Direct characterization and removal of interfering absorption trends in two-layer turbid media,” J. Opt. Soc. Am. A 22(9), 1874–1882 (2005).
[Crossref]

2004 (2)

F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49(6), 1055–1078 (2004).
[Crossref]

G. Marrelec, P. Ciuciu, M. Pelegrini-Issac, and H. Benali, “Estimation of the hemodynamic response in event-related functional MRI: Bayesian networks as a framework for efficient Bayesian modeling and inference,” IEEE Trans. Med. Imaging 23(8), 959–967 (2004).
[Crossref]

1999 (1)

G. H. Glover, “Deconvolution of impulse response in event-related BOLD fMRI,” NeuroImage 9(4), 416–429 (1999).
[Crossref]

1998 (1)

K. J. Friston, O. Josephs, G. Rees, and R. Turner, “Nonlinear event-related responses in fMRI,” Magn. Reson. Med. 39(1), 41–52 (1998).
[Crossref]

1936 (1)

C. Eckart and G. Young, “The approximation of one matrix by another of lower rank,” Psychometrika 1(3), 211–218 (1936).
[Crossref]

Aarabi, A.

A. Aarabi, V. Osharina, and F. Wallois, “Effect of confounding variables on hemodynamic response function estimation using averaging and deconvolution analysis: An event-related NIRS study,” NeuroImage 155, 25–49 (2017).
[Crossref]

Aihara, T.

T. Sato, I. Nambu, K. Takeda, T. Aihara, O. Yamashita, Y. Isogaya, Y. Inoue, Y. Otaka, Y. Wada, M. Kawato, M. A. Sato, and R. Osu, “Reduction of global interference of scalp-hemodynamics in functional near-infrared spectroscopy using short distance probes,” NeuroImage 141, 120–132 (2016).
[Crossref]

Almli, C. R.

V. Fonov, A. C. Evans, K. Botteron, C. R. Almli, R. C. McKinstry, and D. L. Collins, “Unbiased average age-appropriate atlases for pediatric studies,” NeuroImage 54(1), 313–327 (2011).
[Crossref]

Amita, T.

T. Shimokawa, T. Kosaka, O. Yamashita, N. Hiroe, T. Amita, Y. Inoue, and M. A. Sato, “Extended hierarchical Bayesian diffuse optical tomography for removing scalp artifact,” Biomed. Opt. Express 4(11), 2411–2432 (2013).
[Crossref]

S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, and K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12(6), 062111 (2007).
[Crossref]

Arridge, S. R.

S. G. Diamond, T. J. Huppert, V. Kolehmainen, M. A. Franceschini, J. P. Kaipio, S. R. Arridge, and D. A. Boas, “Dynamic physiological modeling for functional diffuse optical tomography,” NeuroImage 30(1), 88–101 (2006).
[Crossref]

Bazargani, N.

N. Bazargani and A. Nosratinia, “Joint maximum likelihood estimation of activation and hemodynamic Response Function for fMRI,” Med. Image Anal. 18(5), 711–724 (2014).
[Crossref]

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2(1), 183–202 (2009).
[Crossref]

Beckes, L.

T. Zhang, F. Li, L. Beckes, and J. A. Coan, “A semi-parametric model of the hemodynamic response for multi-subject fMRI data,” NeuroImage 75, 136–145 (2013).
[Crossref]

Benali, H.

J. Cohen-Adad, S. Chapuisat, J. Doyon, S. Rossignol, J. M. Lina, H. Benali, and F. Lesage, “Activation detection in diffuse optical imaging by means of the general linear model,” Med. Image Anal. 11(6), 616–629 (2007).
[Crossref]

G. Marrelec, P. Ciuciu, M. Pelegrini-Issac, and H. Benali, “Estimation of the hemodynamic response in event-related functional MRI: Bayesian networks as a framework for efficient Bayesian modeling and inference,” IEEE Trans. Med. Imaging 23(8), 959–967 (2004).
[Crossref]

Berger, A. J.

J. R. Goodwin, C. R. Gaudet, and A. J. Berger, “Short-channel functional near-infrared spectroscopy regressions improve when source-detector separation is reduced,” Neurophotonics 1(1), 015002 (2014).
[Crossref]

N. M. Gregg, B. R. White, B. W. Zeff, A. J. Berger, and J. P. Culver, “Brain specificity of diffuse optical imaging: improvements from superficial signal regression and tomography,” Front. Neuroenerg. 2, 14 (2010).

R. B. Saager and A. J. Berger, “Direct characterization and removal of interfering absorption trends in two-layer turbid media,” J. Opt. Soc. Am. A 22(9), 1874–1882 (2005).
[Crossref]

Blasi, A.

S. Lloyd-Fox, A. Blasi, and C. E. Elwell, “Illuminating the developing brain: the past, present and future of functional near infrared spectroscopy,” Neurosci. Biobehav. Rev. 34(3), 269–284 (2010).
[Crossref]

Boas, D. A.

L. Gagnon, K. Perdue, D. N. Greve, D. Goldenholz, G. Kaskhedikar, and D. A. Boas, “Improved recovery of the hemodynamic response in diffuse optical imaging using short optode separations and state-space modeling,” NeuroImage 56(3), 1362–1371 (2011).
[Crossref]

S. G. Diamond, T. J. Huppert, V. Kolehmainen, M. A. Franceschini, J. P. Kaipio, S. R. Arridge, and D. A. Boas, “Dynamic physiological modeling for functional diffuse optical tomography,” NeuroImage 30(1), 88–101 (2006).
[Crossref]

Y. Zhang, D. H. Brooks, and D. A. Boas, “A haemodynamic response function model in spatio-temporal diffuse optical tomography,” Phys. Med. Biol. 50(19), 4625–4644 (2005).
[Crossref]

Bonomini, V.

Botteron, K.

V. Fonov, A. C. Evans, K. Botteron, C. R. Almli, R. C. McKinstry, and D. L. Collins, “Unbiased average age-appropriate atlases for pediatric studies,” NeuroImage 54(1), 313–327 (2011).
[Crossref]

Brigadoi, S.

F. Scarpa, S. Brigadoi, S. Cutini, P. Scatturin, M. Zorzi, R. Dell’acqua, and G. Sparacino, “A reference-channel based methodology to improve estimation of event-related hemodynamic response from fNIRS measurements,” NeuroImage 72, 106–119 (2013).
[Crossref]

Brooks, D. H.

Y. Zhang, D. H. Brooks, and D. A. Boas, “A haemodynamic response function model in spatio-temporal diffuse optical tomography,” Phys. Med. Biol. 50(19), 4625–4644 (2005).
[Crossref]

Casanova, R.

R. Casanova, S. Ryali, J. Serences, L. Yang, R. Kraft, P. J. Laurienti, and J. A. Maldjian, “The impact of temporal regularization on estimates of the BOLD hemodynamic response function: a comparative analysis,” NeuroImage 40(4), 1606–1618 (2008).
[Crossref]

Chapuisat, S.

J. Cohen-Adad, S. Chapuisat, J. Doyon, S. Rossignol, J. M. Lina, H. Benali, and F. Lesage, “Activation detection in diffuse optical imaging by means of the general linear model,” Med. Image Anal. 11(6), 616–629 (2007).
[Crossref]

Chen, C.

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref]

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

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Re, R.

Rees, G.

K. J. Friston, O. Josephs, G. Rees, and R. Turner, “Nonlinear event-related responses in fMRI,” Magn. Reson. Med. 39(1), 41–52 (1998).
[Crossref]

Rossignol, S.

J. Cohen-Adad, S. Chapuisat, J. Doyon, S. Rossignol, J. M. Lina, H. Benali, and F. Lesage, “Activation detection in diffuse optical imaging by means of the general linear model,” Med. Image Anal. 11(6), 616–629 (2007).
[Crossref]

Ryali, S.

R. Casanova, S. Ryali, J. Serences, L. Yang, R. Kraft, P. J. Laurienti, and J. A. Maldjian, “The impact of temporal regularization on estimates of the BOLD hemodynamic response function: a comparative analysis,” NeuroImage 40(4), 1606–1618 (2008).
[Crossref]

Saager, R. B.

Sato, M. A.

T. Sato, I. Nambu, K. Takeda, T. Aihara, O. Yamashita, Y. Isogaya, Y. Inoue, Y. Otaka, Y. Wada, M. Kawato, M. A. Sato, and R. Osu, “Reduction of global interference of scalp-hemodynamics in functional near-infrared spectroscopy using short distance probes,” NeuroImage 141, 120–132 (2016).
[Crossref]

T. Shimokawa, T. Kosaka, O. Yamashita, N. Hiroe, T. Amita, Y. Inoue, and M. A. Sato, “Extended hierarchical Bayesian diffuse optical tomography for removing scalp artifact,” Biomed. Opt. Express 4(11), 2411–2432 (2013).
[Crossref]

Sato, T.

T. Sato, I. Nambu, K. Takeda, T. Aihara, O. Yamashita, Y. Isogaya, Y. Inoue, Y. Otaka, Y. Wada, M. Kawato, M. A. Sato, and R. Osu, “Reduction of global interference of scalp-hemodynamics in functional near-infrared spectroscopy using short distance probes,” NeuroImage 141, 120–132 (2016).
[Crossref]

Scarpa, F.

F. Scarpa, S. Brigadoi, S. Cutini, P. Scatturin, M. Zorzi, R. Dell’acqua, and G. Sparacino, “A reference-channel based methodology to improve estimation of event-related hemodynamic response from fNIRS measurements,” NeuroImage 72, 106–119 (2013).
[Crossref]

Scatturin, P.

F. Scarpa, S. Brigadoi, S. Cutini, P. Scatturin, M. Zorzi, R. Dell’acqua, and G. Sparacino, “A reference-channel based methodology to improve estimation of event-related hemodynamic response from fNIRS measurements,” NeuroImage 72, 106–119 (2013).
[Crossref]

Schmitz, C.

C. Habermehl, C. Schmitz, S. P. Koch, J. Mehnert, and J. Steinbrink, “Investigating hemodynamics in scalp and brain using high-resolution diffuse optical tomography in humans,” BSu2A.2 (2012).

Scholkmann, F.

F. Scholkmann, S. Kleiser, A. J. Metz, R. Zimmermann, J. Mata Pavia, U. Wolf, and M. Wolf, “A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology,” NeuroImage 85(1), 6–27 (2014).
[Crossref]

Seghouane, A. K.

A. Shah and A. K. Seghouane, “An integrated framework for joint HRF and drift estimation and HbO/HbR signal improvement in fNIRS data,” IEEE Trans. Med. Imaging 33(11), 2086–2097 (2014).
[Crossref]

A. K. Seghouane and A. Shah, “HRF estimation in fMRI data with an unknown drift matrix by iterative minimization of the Kullback-Leibler divergence,” IEEE Trans. Med. Imaging 31(2), 192–206 (2012).
[Crossref]

Seiyama, A.

S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, and K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12(6), 062111 (2007).
[Crossref]

Serences, J.

R. Casanova, S. Ryali, J. Serences, L. Yang, R. Kraft, P. J. Laurienti, and J. A. Maldjian, “The impact of temporal regularization on estimates of the BOLD hemodynamic response function: a comparative analysis,” NeuroImage 40(4), 1606–1618 (2008).
[Crossref]

Shah, A.

A. Shah and A. K. Seghouane, “An integrated framework for joint HRF and drift estimation and HbO/HbR signal improvement in fNIRS data,” IEEE Trans. Med. Imaging 33(11), 2086–2097 (2014).
[Crossref]

A. K. Seghouane and A. Shah, “HRF estimation in fMRI data with an unknown drift matrix by iterative minimization of the Kullback-Leibler divergence,” IEEE Trans. Med. Imaging 31(2), 192–206 (2012).
[Crossref]

Shaw, C. B.

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse recovery methods hold promise for diffuse optical tomographic image reconstruction,” Ieee J Sel Top Quant 20(2), 74–82(2014).
[Crossref]

Shimizu, K.

S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, and K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12(6), 062111 (2007).
[Crossref]

Shimokawa, T.

Shynk, J. J.

J. J. Shynk, Probability, Random Variables and Random Processes Theory and Signal Processing Applications (John Wiley & Sons, Inc., 2013), p. 291.

Snyder, A. Z.

M. S. Hassanpour, B. R. White, A. T. Eggebrecht, S. L. Ferradal, A. Z. Snyder, and J. P. Culver, “Statistical analysis of high density diffuse optical tomography,” NeuroImage 85(1), 104–116 (2014).
[Crossref]

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

Sparacino, G.

F. Scarpa, S. Brigadoi, S. Cutini, P. Scatturin, M. Zorzi, R. Dell’acqua, and G. Sparacino, “A reference-channel based methodology to improve estimation of event-related hemodynamic response from fNIRS measurements,” NeuroImage 72, 106–119 (2013).
[Crossref]

Spinelli, L.

Steinbrink, J.

C. Habermehl, J. Steinbrink, K. R. Muller, and S. Haufe, “Optimizing the regularization for image reconstruction of cerebral diffuse optical tomography,” J. Biomed. Opt. 19(9), 096006 (2014).
[Crossref]

C. Habermehl, C. Schmitz, S. P. Koch, J. Mehnert, and J. Steinbrink, “Investigating hemodynamics in scalp and brain using high-resolution diffuse optical tomography in humans,” BSu2A.2 (2012).

Takeda, K.

T. Sato, I. Nambu, K. Takeda, T. Aihara, O. Yamashita, Y. Isogaya, Y. Inoue, Y. Otaka, Y. Wada, M. Kawato, M. A. Sato, and R. Osu, “Reduction of global interference of scalp-hemodynamics in functional near-infrared spectroscopy using short distance probes,” NeuroImage 141, 120–132 (2016).
[Crossref]

Tanikawa, Y.

F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49(6), 1055–1078 (2004).
[Crossref]

Teboulle, M.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2(1), 183–202 (2009).
[Crossref]

Thirion, B.

F. Pedregosa, M. Eickenberg, P. Ciuciu, B. Thirion, and A. Gramfort, “Data-driven HRF estimation for encoding and decoding models,” NeuroImage 104, 209–220 (2015).
[Crossref]

Tian, F.

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref]

Torricelli, A.

Tsuneishi, S.

S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, and K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12(6), 062111 (2007).
[Crossref]

Turner, R.

K. J. Friston, O. Josephs, G. Rees, and R. Turner, “Nonlinear event-related responses in fMRI,” Magn. Reson. Med. 39(1), 41–52 (1998).
[Crossref]

Vojnovic, B.

B. Vojnovic, “Advanced time-correlated single photon counting techniques,” J. Microsc. 222(1), 65–66 (2006).
[Crossref]

Wada, Y.

T. Sato, I. Nambu, K. Takeda, T. Aihara, O. Yamashita, Y. Isogaya, Y. Inoue, Y. Otaka, Y. Wada, M. Kawato, M. A. Sato, and R. Osu, “Reduction of global interference of scalp-hemodynamics in functional near-infrared spectroscopy using short distance probes,” NeuroImage 141, 120–132 (2016).
[Crossref]

Wager, T. D.

M. A. Lindquist and T. D. Wager, “Validity and power in hemodynamic response modeling: a comparison study and a new approach,” Hum. Brain Mapp. 28(8), 764–784 (2007).
[Crossref]

Wallois, F.

A. Aarabi, V. Osharina, and F. Wallois, “Effect of confounding variables on hemodynamic response function estimation using averaging and deconvolution analysis: An event-related NIRS study,” NeuroImage 155, 25–49 (2017).
[Crossref]

Wang, B.

D. Liu, B. Wang, T. Pan, J. Li, Z. Qin, L. Zhang, Z. Zhou, and F. Gao, “Toward quantitative near infrared brain functional imaging: lock-in photon counting instrumentation combined with tomographic reconstruction,” IEEE Access 7, 86829–86842 (2019).
[Crossref]

B. Wang, T. Pan, Y. Zhang, D. Liu, J. Jiang, H. Zhao, and F. Gao, “A Kalman-based tomographic scheme for directly reconstructing activation levels of brain function,” Opt. Express 27(3), 3229–3246 (2019).
[Crossref]

Wheelock, M. D.

M. D. Wheelock, J. P. Culver, and A. T. Eggebrecht, “High-density diffuse optical tomography for imaging human brain function,” Rev. Sci. Instrum. 90(5), 051101 (2019).
[Crossref]

White, B. R.

M. S. Hassanpour, B. R. White, A. T. Eggebrecht, S. L. Ferradal, A. Z. Snyder, and J. P. Culver, “Statistical analysis of high density diffuse optical tomography,” NeuroImage 85(1), 104–116 (2014).
[Crossref]

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

N. M. Gregg, B. R. White, B. W. Zeff, A. J. Berger, and J. P. Culver, “Brain specificity of diffuse optical imaging: improvements from superficial signal regression and tomography,” Front. Neuroenerg. 2, 14 (2010).

J. Markham, B. R. White, B. W. Zeff, and J. P. Culver, “Blind identification of evoked human brain activity with independent component analysis of optical data,” Hum. Brain Mapp. 30(8), 2382–2392 (2009).
[Crossref]

Wolf, M.

F. Scholkmann, S. Kleiser, A. J. Metz, R. Zimmermann, J. Mata Pavia, U. Wolf, and M. Wolf, “A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology,” NeuroImage 85(1), 6–27 (2014).
[Crossref]

Wolf, U.

F. Scholkmann, S. Kleiser, A. J. Metz, R. Zimmermann, J. Mata Pavia, U. Wolf, and M. Wolf, “A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology,” NeuroImage 85(1), 6–27 (2014).
[Crossref]

Yalavarthy, P. K.

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse recovery methods hold promise for diffuse optical tomographic image reconstruction,” Ieee J Sel Top Quant 20(2), 74–82(2014).
[Crossref]

Yamada, Y.

F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49(6), 1055–1078 (2004).
[Crossref]

Yamashita, O.

T. Sato, I. Nambu, K. Takeda, T. Aihara, O. Yamashita, Y. Isogaya, Y. Inoue, Y. Otaka, Y. Wada, M. Kawato, M. A. Sato, and R. Osu, “Reduction of global interference of scalp-hemodynamics in functional near-infrared spectroscopy using short distance probes,” NeuroImage 141, 120–132 (2016).
[Crossref]

T. Shimokawa, T. Kosaka, O. Yamashita, N. Hiroe, T. Amita, Y. Inoue, and M. A. Sato, “Extended hierarchical Bayesian diffuse optical tomography for removing scalp artifact,” Biomed. Opt. Express 4(11), 2411–2432 (2013).
[Crossref]

Yang, L.

R. Casanova, S. Ryali, J. Serences, L. Yang, R. Kraft, P. J. Laurienti, and J. A. Maldjian, “The impact of temporal regularization on estimates of the BOLD hemodynamic response function: a comparative analysis,” NeuroImage 40(4), 1606–1618 (2008).
[Crossref]

Young, G.

C. Eckart and G. Young, “The approximation of one matrix by another of lower rank,” Psychometrika 1(3), 211–218 (1936).
[Crossref]

Zeff, B. W.

N. M. Gregg, B. R. White, B. W. Zeff, A. J. Berger, and J. P. Culver, “Brain specificity of diffuse optical imaging: improvements from superficial signal regression and tomography,” Front. Neuroenerg. 2, 14 (2010).

J. Markham, B. R. White, B. W. Zeff, and J. P. Culver, “Blind identification of evoked human brain activity with independent component analysis of optical data,” Hum. Brain Mapp. 30(8), 2382–2392 (2009).
[Crossref]

Zhan, Y.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

Zhang, L.

D. Liu, B. Wang, T. Pan, J. Li, Z. Qin, L. Zhang, Z. Zhou, and F. Gao, “Toward quantitative near infrared brain functional imaging: lock-in photon counting instrumentation combined with tomographic reconstruction,” IEEE Access 7, 86829–86842 (2019).
[Crossref]

Zhang, T.

T. Zhang, F. Li, L. Beckes, and J. A. Coan, “A semi-parametric model of the hemodynamic response for multi-subject fMRI data,” NeuroImage 75, 136–145 (2013).
[Crossref]

Zhang, Y.

B. Wang, T. Pan, Y. Zhang, D. Liu, J. Jiang, H. Zhao, and F. Gao, “A Kalman-based tomographic scheme for directly reconstructing activation levels of brain function,” Opt. Express 27(3), 3229–3246 (2019).
[Crossref]

Y. Zhang, D. H. Brooks, and D. A. Boas, “A haemodynamic response function model in spatio-temporal diffuse optical tomography,” Phys. Med. Biol. 50(19), 4625–4644 (2005).
[Crossref]

Zhao, H.

B. Wang, T. Pan, Y. Zhang, D. Liu, J. Jiang, H. Zhao, and F. Gao, “A Kalman-based tomographic scheme for directly reconstructing activation levels of brain function,” Opt. Express 27(3), 3229–3246 (2019).
[Crossref]

F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49(6), 1055–1078 (2004).
[Crossref]

D. Qin, Z. Ma, F. Gao, and H. Zhao, “Determination of optical properties in turbid medium based on time-resolved determination,” Fifth International Conference on Photonics and Imaging in Biology and Medicine (2007).

Zhou, Z.

D. Liu, B. Wang, T. Pan, J. Li, Z. Qin, L. Zhang, Z. Zhou, and F. Gao, “Toward quantitative near infrared brain functional imaging: lock-in photon counting instrumentation combined with tomographic reconstruction,” IEEE Access 7, 86829–86842 (2019).
[Crossref]

Zimmermann, R.

F. Scholkmann, S. Kleiser, A. J. Metz, R. Zimmermann, J. Mata Pavia, U. Wolf, and M. Wolf, “A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology,” NeuroImage 85(1), 6–27 (2014).
[Crossref]

Zorzi, M.

F. Scarpa, S. Brigadoi, S. Cutini, P. Scatturin, M. Zorzi, R. Dell’acqua, and G. Sparacino, “A reference-channel based methodology to improve estimation of event-related hemodynamic response from fNIRS measurements,” NeuroImage 72, 106–119 (2013).
[Crossref]

Zucchelli, L.

Biomed. Opt. Express (3)

Front. Neuroenerg. (1)

N. M. Gregg, B. R. White, B. W. Zeff, A. J. Berger, and J. P. Culver, “Brain specificity of diffuse optical imaging: improvements from superficial signal regression and tomography,” Front. Neuroenerg. 2, 14 (2010).

Hum. Brain Mapp. (2)

J. Markham, B. R. White, B. W. Zeff, and J. P. Culver, “Blind identification of evoked human brain activity with independent component analysis of optical data,” Hum. Brain Mapp. 30(8), 2382–2392 (2009).
[Crossref]

M. A. Lindquist and T. D. Wager, “Validity and power in hemodynamic response modeling: a comparison study and a new approach,” Hum. Brain Mapp. 28(8), 764–784 (2007).
[Crossref]

IEEE Access (1)

D. Liu, B. Wang, T. Pan, J. Li, Z. Qin, L. Zhang, Z. Zhou, and F. Gao, “Toward quantitative near infrared brain functional imaging: lock-in photon counting instrumentation combined with tomographic reconstruction,” IEEE Access 7, 86829–86842 (2019).
[Crossref]

Ieee J Sel Top Quant (1)

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse recovery methods hold promise for diffuse optical tomographic image reconstruction,” Ieee J Sel Top Quant 20(2), 74–82(2014).
[Crossref]

IEEE Trans. Med. Imaging (4)

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref]

G. Marrelec, P. Ciuciu, M. Pelegrini-Issac, and H. Benali, “Estimation of the hemodynamic response in event-related functional MRI: Bayesian networks as a framework for efficient Bayesian modeling and inference,” IEEE Trans. Med. Imaging 23(8), 959–967 (2004).
[Crossref]

A. Shah and A. K. Seghouane, “An integrated framework for joint HRF and drift estimation and HbO/HbR signal improvement in fNIRS data,” IEEE Trans. Med. Imaging 33(11), 2086–2097 (2014).
[Crossref]

A. K. Seghouane and A. Shah, “HRF estimation in fMRI data with an unknown drift matrix by iterative minimization of the Kullback-Leibler divergence,” IEEE Trans. Med. Imaging 31(2), 192–206 (2012).
[Crossref]

J. Biomed. Opt. (2)

S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, and K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12(6), 062111 (2007).
[Crossref]

C. Habermehl, J. Steinbrink, K. R. Muller, and S. Haufe, “Optimizing the regularization for image reconstruction of cerebral diffuse optical tomography,” J. Biomed. Opt. 19(9), 096006 (2014).
[Crossref]

J. Microsc. (1)

B. Vojnovic, “Advanced time-correlated single photon counting techniques,” J. Microsc. 222(1), 65–66 (2006).
[Crossref]

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

Magn. Reson. Med. (1)

K. J. Friston, O. Josephs, G. Rees, and R. Turner, “Nonlinear event-related responses in fMRI,” Magn. Reson. Med. 39(1), 41–52 (1998).
[Crossref]

Med. Image Anal. (2)

J. Cohen-Adad, S. Chapuisat, J. Doyon, S. Rossignol, J. M. Lina, H. Benali, and F. Lesage, “Activation detection in diffuse optical imaging by means of the general linear model,” Med. Image Anal. 11(6), 616–629 (2007).
[Crossref]

N. Bazargani and A. Nosratinia, “Joint maximum likelihood estimation of activation and hemodynamic Response Function for fMRI,” Med. Image Anal. 18(5), 711–724 (2014).
[Crossref]

NeuroImage (13)

G. H. Glover, “Deconvolution of impulse response in event-related BOLD fMRI,” NeuroImage 9(4), 416–429 (1999).
[Crossref]

M. S. Hassanpour, B. R. White, A. T. Eggebrecht, S. L. Ferradal, A. Z. Snyder, and J. P. Culver, “Statistical analysis of high density diffuse optical tomography,” NeuroImage 85(1), 104–116 (2014).
[Crossref]

A. Aarabi, V. Osharina, and F. Wallois, “Effect of confounding variables on hemodynamic response function estimation using averaging and deconvolution analysis: An event-related NIRS study,” NeuroImage 155, 25–49 (2017).
[Crossref]

S. G. Diamond, T. J. Huppert, V. Kolehmainen, M. A. Franceschini, J. P. Kaipio, S. R. Arridge, and D. A. Boas, “Dynamic physiological modeling for functional diffuse optical tomography,” NeuroImage 30(1), 88–101 (2006).
[Crossref]

L. Gagnon, K. Perdue, D. N. Greve, D. Goldenholz, G. Kaskhedikar, and D. A. Boas, “Improved recovery of the hemodynamic response in diffuse optical imaging using short optode separations and state-space modeling,” NeuroImage 56(3), 1362–1371 (2011).
[Crossref]

R. Casanova, S. Ryali, J. Serences, L. Yang, R. Kraft, P. J. Laurienti, and J. A. Maldjian, “The impact of temporal regularization on estimates of the BOLD hemodynamic response function: a comparative analysis,” NeuroImage 40(4), 1606–1618 (2008).
[Crossref]

F. Pedregosa, M. Eickenberg, P. Ciuciu, B. Thirion, and A. Gramfort, “Data-driven HRF estimation for encoding and decoding models,” NeuroImage 104, 209–220 (2015).
[Crossref]

F. Scholkmann, S. Kleiser, A. J. Metz, R. Zimmermann, J. Mata Pavia, U. Wolf, and M. Wolf, “A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology,” NeuroImage 85(1), 6–27 (2014).
[Crossref]

T. Zhang, F. Li, L. Beckes, and J. A. Coan, “A semi-parametric model of the hemodynamic response for multi-subject fMRI data,” NeuroImage 75, 136–145 (2013).
[Crossref]

F. Scarpa, S. Brigadoi, S. Cutini, P. Scatturin, M. Zorzi, R. Dell’acqua, and G. Sparacino, “A reference-channel based methodology to improve estimation of event-related hemodynamic response from fNIRS measurements,” NeuroImage 72, 106–119 (2013).
[Crossref]

T. Sato, I. Nambu, K. Takeda, T. Aihara, O. Yamashita, Y. Isogaya, Y. Inoue, Y. Otaka, Y. Wada, M. Kawato, M. A. Sato, and R. Osu, “Reduction of global interference of scalp-hemodynamics in functional near-infrared spectroscopy using short distance probes,” NeuroImage 141, 120–132 (2016).
[Crossref]

V. Fonov, A. C. Evans, K. Botteron, C. R. Almli, R. C. McKinstry, and D. L. Collins, “Unbiased average age-appropriate atlases for pediatric studies,” NeuroImage 54(1), 313–327 (2011).
[Crossref]

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

Neurophotonics (1)

J. R. Goodwin, C. R. Gaudet, and A. J. Berger, “Short-channel functional near-infrared spectroscopy regressions improve when source-detector separation is reduced,” Neurophotonics 1(1), 015002 (2014).
[Crossref]

Neurosci. Biobehav. Rev. (1)

S. Lloyd-Fox, A. Blasi, and C. E. Elwell, “Illuminating the developing brain: the past, present and future of functional near infrared spectroscopy,” Neurosci. Biobehav. Rev. 34(3), 269–284 (2010).
[Crossref]

Opt. Express (1)

Phys. Med. Biol. (2)

Y. Zhang, D. H. Brooks, and D. A. Boas, “A haemodynamic response function model in spatio-temporal diffuse optical tomography,” Phys. Med. Biol. 50(19), 4625–4644 (2005).
[Crossref]

F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49(6), 1055–1078 (2004).
[Crossref]

Psychometrika (1)

C. Eckart and G. Young, “The approximation of one matrix by another of lower rank,” Psychometrika 1(3), 211–218 (1936).
[Crossref]

Rev. Sci. Instrum. (1)

M. D. Wheelock, J. P. Culver, and A. T. Eggebrecht, “High-density diffuse optical tomography for imaging human brain function,” Rev. Sci. Instrum. 90(5), 051101 (2019).
[Crossref]

SIAM J. Imaging Sci. (1)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2(1), 183–202 (2009).
[Crossref]

Other (8)

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R. A. Horn and C. R. Johnson, Matrix Analysis, second ed. (Cambridge University Press, 2012).

D. Qin, Z. Ma, F. Gao, and H. Zhao, “Determination of optical properties in turbid medium based on time-resolved determination,” Fifth International Conference on Photonics and Imaging in Biology and Medicine (2007).

W. Chen, “Hemodynamic Response Function Modeling,” (University of North Carolina, 2012).

J. J. Shynk, Probability, Random Variables and Random Processes Theory and Signal Processing Applications (John Wiley & Sons, Inc., 2013), p. 291.

L. H. Nguyen and K.-S. Hong, “Investigation of the Hemodynamic Response in Near Infrared Spectroscopy Data Analysis,” 2010 Second International Conference on Knowledge and Systems Engineering, 28–32 (2010).
[Crossref]

D. A. M. J. H. B. Mapping, “Optimal experimental design for event-related fMRI,” Hum Brain Mapping8 (1999).

C. Habermehl, C. Schmitz, S. P. Koch, J. Mehnert, and J. Steinbrink, “Investigating hemodynamics in scalp and brain using high-resolution diffuse optical tomography in humans,” BSu2A.2 (2012).

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Figures (13)

Fig. 1.
Fig. 1. Layered head structure: (a) vertical plane at Z=95 mm; (b) coronal plane at X=40 mm. Different colors denote different tissue types.
Fig. 2.
Fig. 2. Brain atlas: (a) right view; (b) back view with assignment of NN-1, NN-2 and NN-3. The green color denotes the activated brain region.
Fig. 3.
Fig. 3. Normalized time courses: (a) ${h_{HbO}}(k)$ and ${h_{HbR}}(k)$; (b) The stimulus and its convolution with ${h_{HbO}}(k)$ and ${h_{HbR}}(k)$; (c) Changes of ${\mu _a}$ in the activated brain region at the wavelengths of 760 nm and 830 nm; (d) The scalp interference signal.
Fig. 4.
Fig. 4. Normalized time courses in the case of global scalp interference: (a) singular value curve; the (b) first and (c) second right singular vector of M.
Fig. 5.
Fig. 5. Normalized HRFs of HbO and HbR concentrations in the case of global scalp interference
Fig. 6.
Fig. 6. Coronal planes at X=40 mm of the normalized activation-level maps in the case of global interference: (a) 760 nm, SFIR and (b) 760 nm, JDE; (c) 830 nm, SFIR and (d) 830 nm, JDE; (e) True.
Fig. 7.
Fig. 7. Normalized time courses in the case of local scalp interference: (a) singular value curve; the (b) first and (c) second right singular vector of M.
Fig. 8.
Fig. 8. Normalized HRFs of HbO and HbR concentrations in the case of local scalp interference
Fig. 9.
Fig. 9. Coronal planes at X=40 mm of the normalized activation-level maps in the case of local interference: (a) 760 nm, SFIR and (b) 760 nm, JDE; (c) 830 nm, SFIR and (d) 830 nm, JDE; (e) True.
Fig. 10.
Fig. 10. Schematic of the dynamic phantoms and the HD-DOT detecting equipment. Cup A, B and C are filled with the target solution, background solution, and waste solution, respectively. Switch 1 and 2 are used to select a cup from which the solution is pumped into the cylinder hole.
Fig. 11.
Fig. 11. Normalized time courses of (a) stimulus, HRF and convolution results and (b) HRF estimated by SFIR and JDE.
Fig. 12.
Fig. 12. Results of the phantom experiments: normalized activation-level maps estimated by (a) SFIR and (b) JDE; The black dotted circles denote the true locations of the activate regions; (c) Horizonal profiles passing through the peak pixel.
Fig. 13.
Fig. 13. Results of the phantom experiments: (a) Normalized activation-level maps estimated by the sparsity regularized JDE; (b) Normalized profiles along the horizonal line passing through the peak pixel.

Tables (8)

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Table 1. Concentrations of HbO and HbR for different tissue types.

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Table 2. Optical properties for different tissue types.

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Table 3. RMSEs of the normalized HRFs of HbO and HbR concentrations estimated by SFIR and JDE in the case of global scalp interference

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Table 4. Metrics of the normalized activation-level maps estimated by SFIR and JDE in the case of global scalp interference

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Table 5. RMSEs of the normalized HRFs of HbO and HbR concentrations estimated by SFIR and JDE in the case of local scalp interference

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Table 6. Metrics of the normalized activation-level maps estimated by SFIR and JDE in the case of local scalp interference

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Table 7. Operation process of the dynamic phantom equipment.

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Table 8. Metrics of the normalized HRF and activation-level maps estimated by SFIR and JDE in the dynamic phantom experiments.

Equations (14)

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M = J X ,
X = β ( S h ) T + α f T + E ,
M = J β ( S h ) T + J α f T + E 1 ,
M i = 1 2 ξ i u i v i T ,
J β ( S h ) T = ξ 2 u 2 v 2 T .
min h | | S h ξ 2 v 2 | | 2 2 + λ | | C h | | 2 2 ,
C = [ 2  1 0   0  1 - 2 1 0   0   0  0 1 - 2 1  0   0 1 - 2 ] .
R M S E h = ( h e s t h t r ) T ( h e s t h t r ) / h t r T h t r .
R M S E β = ( β e s t β t r ) T ( β e s t β t r ) / β t r T β t r ,
C N R = | m e a n ( β a t d ) m e a n ( β b k ) | w var ( β a t d ) + ( 1 w ) var ( β b k ) ,
h H b O ( k ) = ψ 1 [ Γ n ( k , τ 1 , ρ 1 ) ψ 2 Γ n ( k , τ 2 , ρ 2 ) ] ,
M ξ 1 u 1 v 1 T .
E ( X Y ) = x y p X , Y ( x , y ) d x d y = x y p X , ( x ) p Y ( y ) d x d y = x p X ( x ) ( y p Y ( y ) d y ) d x = ( y p Y ( y ) d y ) ( x p X ( x ) d x ) = E ( X ) E ( Y )
r a n g e ( A ) = { y R m : y = A x f o r s o m e x R n }