Abstract

Conventional reconstruction of diffuse optical tomography (DOT) is based on the Tikhonov regularization and the white Gaussian noise assumption. Consequently, the reconstructed DOT images usually have a low spatial resolution. In this work, we have derived a novel quantification method for noise variance based on the linear Rytov approximation of the photon diffusion equation. Specifically, we have implemented this quantification of noise variance to normalize the measurement signals from all source-detector channels along with sparsity regularization to provide high-quality DOT images. Multiple experiments from computer simulations and laboratory phantoms were performed to validate and support the newly developed algorithm. The reconstructed images demonstrate that quantification and normalization of noise variance with sparsity regularization (QNNVSR) is an effective reconstruction approach to greatly enhance the spatial resolution and the shape fidelity for DOT images. Since noise variance can be estimated by our derived expression with relatively limited resources available, this approach is practically useful for many DOT applications.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref] [PubMed]

2014 (1)

F. Tian and H. Liu, “Depth-compensated diffuse optical tomography enhanced by general linear model analysis and an anatomical atlas of human head,” Neuroimage 85(Pt 1), 166–180 (2014).
[Crossref] [PubMed]

2012 (5)

M. Ferrari and V. Quaresima, “A brief review on the history of human functional near-infrared spectroscopy (fNIRS) development and fields of application,” Neuroimage 63(2), 921–935 (2012).
[Crossref] [PubMed]

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] [PubMed]

F. Tian, F. A. Kozel, A. Yennu, P. E. Croarkin, S. M. McClintock, K. S. Mapes, M. M. Husain, and H. Liu, “Test-retest assessment of cortical activation induced by repetitive transcranial magnetic stimulation with brain atlas-guided optical topography,” J. Biomed. Opt. 17(11), 116020 (2012).
[Crossref] [PubMed]

V. C. Kavuri, Z. J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express 3(5), 943–957 (2012).
[Crossref] [PubMed]

A. Jin, B. Yazici, A. Ale, and V. Ntziachristos, “Preconditioning of the fluorescence diffuse optical tomography sensing matrix based on compressive sensing,” Opt. Lett. 37(20), 4326–4328 (2012).
[Crossref] [PubMed]

2011 (3)

B. Khan, P. Chand, and G. Alexandrakis, “Spatiotemporal relations of primary sensorimotor and secondary motor activation patterns mapped by NIR imaging,” Biomed. Opt. Express 2(12), 3367–3386 (2011).
[Crossref] [PubMed]

O. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: noniterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging 30(5), 1129–1142 (2011).
[Crossref] [PubMed]

D. R. Leff, F. Orihuela-Espina, C. E. Elwell, T. Athanasiou, D. T. Delpy, A. W. Darzi, and G. Z. Yang, “Assessment of the cerebral cortex during motor task behaviours in adults: a systematic review of functional near infrared spectroscopy (fNIRS) studies,” Neuroimage 54(4), 2922–2936 (2011).
[Crossref] [PubMed]

2010 (6)

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
[Crossref] [PubMed]

H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed. Opt. 15(4), 046005 (2010).
[Crossref] [PubMed]

M. V. Afonso, J. M. Bioucas-Dias, and M. A. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19(9), 2345–2356 (2010).
[Crossref] [PubMed]

H. Niu, F. Tian, Z. J. Lin, and H. Liu, “Development of a compensation algorithm for accurate depth localization in diffuse optical tomography,” Opt. Lett. 35(3), 429–431 (2010).
[Crossref] [PubMed]

M. Süzen, A. Giannoula, and T. Durduran, “Compressed sensing in diffuse optical tomography,” Opt. Express 18(23), 23676–23690 (2010).
[Crossref] [PubMed]

F. Tian, M. R. Delgado, S. C. Dhamne, B. Khan, G. Alexandrakis, M. I. Romero, L. Smith, D. Reid, N. J. Clegg, and H. Liu, “Quantification of functional near infrared spectroscopy to assess cortical reorganization in children with cerebral palsy,” Opt. Express 18(25), 25973–25986 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (2)

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

M. Lustig, D. Donoho, J. Santos, and J. Pauly, “Compressed Sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[Crossref]

2007 (6)

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE Sel. Top. Sig. Process. 1(4), 606–617 (2007).
[Crossref]

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

A. Douiri, M. Schweiger, J. Riley, and R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18(1), 87–95 (2007).
[Crossref]

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34(6), 2085–2098 (2007).
[Crossref] [PubMed]

N. Cao and A. Nehorai, “Tumor localization using diffuse optical tomography and linearly constrained minimum variance beamforming,” Opt. Express 15(3), 896–909 (2007).
[Crossref] [PubMed]

N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express 15(21), 13695–13708 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (2)

2004 (3)

X. Song, B. W. Pogue, S. Jiang, M. M. Doyley, H. Dehghani, T. D. Tosteson, and K. D. Paulsen, “Automated region detection based on the contrast-to-noise ratio in near-infrared tomography,” Appl. Opt. 43(5), 1053–1062 (2004).
[Crossref] [PubMed]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004).
[Crossref] [PubMed]

2001 (2)

J. C. Ye, A. Bouman, J. Webb, and P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Image Processing 10(6), 909–922 (2001).
[Crossref]

J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett. 26(10), 701–703 (2001).
[Crossref] [PubMed]

2000 (2)

A. Zourabian, A. Siegel, B. Chance, N. Ramanujan, M. Rode, and D. A. Boas, “Trans-abdominal monitoring of fetal arterial blood oxygenation using pulse oximetry,” J. Biomed. Opt. 5(4), 391–405 (2000).
[Crossref] [PubMed]

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45(4), 1051–1070 (2000).
[Crossref] [PubMed]

1999 (1)

1995 (1)

1988 (1)

M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 215, 183–189 (1988).
[Crossref] [PubMed]

1963 (1)

A. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Soviet Mathematics Doklady 4, 1035–1038 (1963).

Afonso, M. V.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19(9), 2345–2356 (2010).
[Crossref] [PubMed]

Ale, A.

Alexandrakis, G.

Arridge, R.

A. Douiri, M. Schweiger, J. Riley, and R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18(1), 87–95 (2007).
[Crossref]

Arridge, S.

Arridge, S. R.

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25(12), 123010 (2009).
[Crossref]

Athanasiou, T.

D. R. Leff, F. Orihuela-Espina, C. E. Elwell, T. Athanasiou, D. T. Delpy, A. W. Darzi, and G. Z. Yang, “Assessment of the cerebral cortex during motor task behaviours in adults: a systematic review of functional near infrared spectroscopy (fNIRS) studies,” Neuroimage 54(4), 2922–2936 (2011).
[Crossref] [PubMed]

Baker, W. B.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
[Crossref] [PubMed]

Bioucas-Dias, J. M.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19(9), 2345–2356 (2010).
[Crossref] [PubMed]

Boas, D. A.

D. K. Joseph, T. J. Huppert, M. A. Franceschini, and D. A. Boas, “Diffuse optical tomography system to image brain activation with improved spatial resolution and validation with functional magnetic resonance imaging,” Appl. Opt. 45(31), 8142–8151 (2006).
[Crossref] [PubMed]

D. A. Boas and A. M. Dale, “Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of human brain function,” Appl. Opt. 44(10), 1957–1968 (2005).
[Crossref] [PubMed]

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004).
[Crossref] [PubMed]

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45(4), 1051–1070 (2000).
[Crossref] [PubMed]

A. Zourabian, A. Siegel, B. Chance, N. Ramanujan, M. Rode, and D. A. Boas, “Trans-abdominal monitoring of fetal arterial blood oxygenation using pulse oximetry,” J. Biomed. Opt. 5(4), 391–405 (2000).
[Crossref] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20(5), 426–428 (1995).
[Crossref] [PubMed]

Bouman, A.

J. C. Ye, A. Bouman, J. Webb, and P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Image Processing 10(6), 909–922 (2001).
[Crossref]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Boyd, S.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE Sel. Top. Sig. Process. 1(4), 606–617 (2007).
[Crossref]

Bresler, Y.

O. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: noniterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging 30(5), 1129–1142 (2011).
[Crossref] [PubMed]

Brooks, D. H.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45(4), 1051–1070 (2000).
[Crossref] [PubMed]

Candes, E.

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Cao, N.

Chance, B.

Chand, P.

Chen, C.

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] [PubMed]

Choe, R.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
[Crossref] [PubMed]

Clegg, N. J.

Cope, M.

M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 215, 183–189 (1988).
[Crossref] [PubMed]

Croarkin, P. E.

F. Tian, F. A. Kozel, A. Yennu, P. E. Croarkin, S. M. McClintock, K. S. Mapes, M. M. Husain, and H. Liu, “Test-retest assessment of cortical activation induced by repetitive transcranial magnetic stimulation with brain atlas-guided optical topography,” J. Biomed. Opt. 17(11), 116020 (2012).
[Crossref] [PubMed]

Culver, J. P.

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] [PubMed]

J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis,” Opt. Lett. 26(10), 701–703 (2001).
[Crossref] [PubMed]

Dale, A. M.

D. A. Boas and A. M. Dale, “Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of human brain function,” Appl. Opt. 44(10), 1957–1968 (2005).
[Crossref] [PubMed]

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004).
[Crossref] [PubMed]

Darzi, A. W.

D. R. Leff, F. Orihuela-Espina, C. E. Elwell, T. Athanasiou, D. T. Delpy, A. W. Darzi, and G. Z. Yang, “Assessment of the cerebral cortex during motor task behaviours in adults: a systematic review of functional near infrared spectroscopy (fNIRS) studies,” Neuroimage 54(4), 2922–2936 (2011).
[Crossref] [PubMed]

Dehghani, H.

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] [PubMed]

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34(6), 2085–2098 (2007).
[Crossref] [PubMed]

X. Song, B. W. Pogue, S. Jiang, M. M. Doyley, H. Dehghani, T. D. Tosteson, and K. D. Paulsen, “Automated region detection based on the contrast-to-noise ratio in near-infrared tomography,” Appl. Opt. 43(5), 1053–1062 (2004).
[Crossref] [PubMed]

Delgado, M. R.

Delpy, D. T.

D. R. Leff, F. Orihuela-Espina, C. E. Elwell, T. Athanasiou, D. T. Delpy, A. W. Darzi, and G. Z. Yang, “Assessment of the cerebral cortex during motor task behaviours in adults: a systematic review of functional near infrared spectroscopy (fNIRS) studies,” Neuroimage 54(4), 2922–2936 (2011).
[Crossref] [PubMed]

M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 215, 183–189 (1988).
[Crossref] [PubMed]

Dhamne, S.

H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed. Opt. 15(4), 046005 (2010).
[Crossref] [PubMed]

Dhamne, S. C.

DiMarzio, C. A.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45(4), 1051–1070 (2000).
[Crossref] [PubMed]

Donoho, D.

M. Lustig, D. Donoho, J. Santos, and J. Pauly, “Compressed Sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[Crossref]

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Douiri, A.

A. Douiri, M. Schweiger, J. Riley, and R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18(1), 87–95 (2007).
[Crossref]

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Local diffusion regularization method for optical tomography reconstruction by using robust statistics,” Opt. Lett. 30(18), 2439–2441 (2005).
[Crossref] [PubMed]

Doyley, M. M.

Durduran, T.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
[Crossref] [PubMed]

M. Süzen, A. Giannoula, and T. Durduran, “Compressed sensing in diffuse optical tomography,” Opt. Express 18(23), 23676–23690 (2010).
[Crossref] [PubMed]

Eggebrecht, A. T.

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] [PubMed]

Elwell, C. E.

D. R. Leff, F. Orihuela-Espina, C. E. Elwell, T. Athanasiou, D. T. Delpy, A. W. Darzi, and G. Z. Yang, “Assessment of the cerebral cortex during motor task behaviours in adults: a systematic review of functional near infrared spectroscopy (fNIRS) studies,” Neuroimage 54(4), 2922–2936 (2011).
[Crossref] [PubMed]

Ferradal, S. L.

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] [PubMed]

Ferrari, M.

M. Ferrari and V. Quaresima, “A brief review on the history of human functional near-infrared spectroscopy (fNIRS) development and fields of application,” Neuroimage 63(2), 921–935 (2012).
[Crossref] [PubMed]

Figueiredo, M. A.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19(9), 2345–2356 (2010).
[Crossref] [PubMed]

Franceschini, M. A.

D. K. Joseph, T. J. Huppert, M. A. Franceschini, and D. A. Boas, “Diffuse optical tomography system to image brain activation with improved spatial resolution and validation with functional magnetic resonance imaging,” Appl. Opt. 45(31), 8142–8151 (2006).
[Crossref] [PubMed]

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004).
[Crossref] [PubMed]

Gaudette, R. J.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45(4), 1051–1070 (2000).
[Crossref] [PubMed]

Gaudette, T.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45(4), 1051–1070 (2000).
[Crossref] [PubMed]

Giannoula, A.

Gorinevsky, D.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE Sel. Top. Sig. Process. 1(4), 606–617 (2007).
[Crossref]

Holboke, M. J.

Huppert, T. J.

Husain, M. M.

F. Tian, F. A. Kozel, A. Yennu, P. E. Croarkin, S. M. McClintock, K. S. Mapes, M. M. Husain, and H. Liu, “Test-retest assessment of cortical activation induced by repetitive transcranial magnetic stimulation with brain atlas-guided optical topography,” J. Biomed. Opt. 17(11), 116020 (2012).
[Crossref] [PubMed]

Jacobs, M.

Jiang, S.

Jin, A.

Joseph, D. K.

Kavuri, V. C.

Khan, B.

Kilmer, M. E.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45(4), 1051–1070 (2000).
[Crossref] [PubMed]

Kim, J. M.

O. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: noniterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging 30(5), 1129–1142 (2011).
[Crossref] [PubMed]

Kim, S. J.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE Sel. Top. Sig. Process. 1(4), 606–617 (2007).
[Crossref]

Koh, K.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE Sel. Top. Sig. Process. 1(4), 606–617 (2007).
[Crossref]

Kozel, F. A.

F. Tian, F. A. Kozel, A. Yennu, P. E. Croarkin, S. M. McClintock, K. S. Mapes, M. M. Husain, and H. Liu, “Test-retest assessment of cortical activation induced by repetitive transcranial magnetic stimulation with brain atlas-guided optical topography,” J. Biomed. Opt. 17(11), 116020 (2012).
[Crossref] [PubMed]

Lee, O.

O. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: noniterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging 30(5), 1129–1142 (2011).
[Crossref] [PubMed]

Leff, D. R.

D. R. Leff, F. Orihuela-Espina, C. E. Elwell, T. Athanasiou, D. T. Delpy, A. W. Darzi, and G. Z. Yang, “Assessment of the cerebral cortex during motor task behaviours in adults: a systematic review of functional near infrared spectroscopy (fNIRS) studies,” Neuroimage 54(4), 2922–2936 (2011).
[Crossref] [PubMed]

Lin, Z. J.

Liu, H.

F. Tian and H. Liu, “Depth-compensated diffuse optical tomography enhanced by general linear model analysis and an anatomical atlas of human head,” Neuroimage 85(Pt 1), 166–180 (2014).
[Crossref] [PubMed]

F. Tian, F. A. Kozel, A. Yennu, P. E. Croarkin, S. M. McClintock, K. S. Mapes, M. M. Husain, and H. Liu, “Test-retest assessment of cortical activation induced by repetitive transcranial magnetic stimulation with brain atlas-guided optical topography,” J. Biomed. Opt. 17(11), 116020 (2012).
[Crossref] [PubMed]

V. C. Kavuri, Z. J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express 3(5), 943–957 (2012).
[Crossref] [PubMed]

H. Niu, F. Tian, Z. J. Lin, and H. Liu, “Development of a compensation algorithm for accurate depth localization in diffuse optical tomography,” Opt. Lett. 35(3), 429–431 (2010).
[Crossref] [PubMed]

F. Tian, M. R. Delgado, S. C. Dhamne, B. Khan, G. Alexandrakis, M. I. Romero, L. Smith, D. Reid, N. J. Clegg, and H. Liu, “Quantification of functional near infrared spectroscopy to assess cortical reorganization in children with cerebral palsy,” Opt. Express 18(25), 25973–25986 (2010).
[Crossref] [PubMed]

H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed. Opt. 15(4), 046005 (2010).
[Crossref] [PubMed]

F. Tian, G. Alexandrakis, and H. Liu, “Optimization of probe geometry for diffuse optical brain imaging based on measurement density and distribution,” Appl. Opt. 48(13), 2496–2504 (2009).
[Crossref] [PubMed]

Lustig, M.

M. Lustig, D. Donoho, J. Santos, and J. Pauly, “Compressed Sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[Crossref]

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE Sel. Top. Sig. Process. 1(4), 606–617 (2007).
[Crossref]

Mapes, K. S.

F. Tian, F. A. Kozel, A. Yennu, P. E. Croarkin, S. M. McClintock, K. S. Mapes, M. M. Husain, and H. Liu, “Test-retest assessment of cortical activation induced by repetitive transcranial magnetic stimulation with brain atlas-guided optical topography,” J. Biomed. Opt. 17(11), 116020 (2012).
[Crossref] [PubMed]

McClintock, S. M.

F. Tian, F. A. Kozel, A. Yennu, P. E. Croarkin, S. M. McClintock, K. S. Mapes, M. M. Husain, and H. Liu, “Test-retest assessment of cortical activation induced by repetitive transcranial magnetic stimulation with brain atlas-guided optical topography,” J. Biomed. Opt. 17(11), 116020 (2012).
[Crossref] [PubMed]

Millane, P.

J. C. Ye, A. Bouman, J. Webb, and P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Image Processing 10(6), 909–922 (2001).
[Crossref]

Miller, E. L.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45(4), 1051–1070 (2000).
[Crossref] [PubMed]

Nehorai, A.

Niu, H.

H. Niu, F. Tian, Z. J. Lin, and H. Liu, “Development of a compensation algorithm for accurate depth localization in diffuse optical tomography,” Opt. Lett. 35(3), 429–431 (2010).
[Crossref] [PubMed]

H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed. Opt. 15(4), 046005 (2010).
[Crossref] [PubMed]

Ntziachristos, V.

O’Leary, M. A.

Orihuela-Espina, F.

D. R. Leff, F. Orihuela-Espina, C. E. Elwell, T. Athanasiou, D. T. Delpy, A. W. Darzi, and G. Z. Yang, “Assessment of the cerebral cortex during motor task behaviours in adults: a systematic review of functional near infrared spectroscopy (fNIRS) studies,” Neuroimage 54(4), 2922–2936 (2011).
[Crossref] [PubMed]

Paulsen, K. D.

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34(6), 2085–2098 (2007).
[Crossref] [PubMed]

X. Song, B. W. Pogue, S. Jiang, M. M. Doyley, H. Dehghani, T. D. Tosteson, and K. D. Paulsen, “Automated region detection based on the contrast-to-noise ratio in near-infrared tomography,” Appl. Opt. 43(5), 1053–1062 (2004).
[Crossref] [PubMed]

Pauly, J.

M. Lustig, D. Donoho, J. Santos, and J. Pauly, “Compressed Sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[Crossref]

Pauly, J. M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Pogue, B. W.

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34(6), 2085–2098 (2007).
[Crossref] [PubMed]

X. Song, B. W. Pogue, S. Jiang, M. M. Doyley, H. Dehghani, T. D. Tosteson, and K. D. Paulsen, “Automated region detection based on the contrast-to-noise ratio in near-infrared tomography,” Appl. Opt. 43(5), 1053–1062 (2004).
[Crossref] [PubMed]

Quaresima, V.

M. Ferrari and V. Quaresima, “A brief review on the history of human functional near-infrared spectroscopy (fNIRS) development and fields of application,” Neuroimage 63(2), 921–935 (2012).
[Crossref] [PubMed]

Ramanujan, N.

A. Zourabian, A. Siegel, B. Chance, N. Ramanujan, M. Rode, and D. A. Boas, “Trans-abdominal monitoring of fetal arterial blood oxygenation using pulse oximetry,” J. Biomed. Opt. 5(4), 391–405 (2000).
[Crossref] [PubMed]

Reid, D.

Reynolds, E. O.

M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 215, 183–189 (1988).
[Crossref] [PubMed]

Riley, J.

A. Douiri, M. Schweiger, J. Riley, and R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18(1), 87–95 (2007).
[Crossref]

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Local diffusion regularization method for optical tomography reconstruction by using robust statistics,” Opt. Lett. 30(18), 2439–2441 (2005).
[Crossref] [PubMed]

Rode, M.

A. Zourabian, A. Siegel, B. Chance, N. Ramanujan, M. Rode, and D. A. Boas, “Trans-abdominal monitoring of fetal arterial blood oxygenation using pulse oximetry,” J. Biomed. Opt. 5(4), 391–405 (2000).
[Crossref] [PubMed]

Romero, M. I.

Santos, J.

M. Lustig, D. Donoho, J. Santos, and J. Pauly, “Compressed Sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[Crossref]

Schotland, J. C.

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25(12), 123010 (2009).
[Crossref]

Schweiger, M.

A. Douiri, M. Schweiger, J. Riley, and R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18(1), 87–95 (2007).
[Crossref]

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Local diffusion regularization method for optical tomography reconstruction by using robust statistics,” Opt. Lett. 30(18), 2439–2441 (2005).
[Crossref] [PubMed]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Siegel, A.

A. Zourabian, A. Siegel, B. Chance, N. Ramanujan, M. Rode, and D. A. Boas, “Trans-abdominal monitoring of fetal arterial blood oxygenation using pulse oximetry,” J. Biomed. Opt. 5(4), 391–405 (2000).
[Crossref] [PubMed]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Smith, L.

Snyder, A. Z.

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] [PubMed]

Song, X.

Süzen, M.

Tian, F.

F. Tian and H. Liu, “Depth-compensated diffuse optical tomography enhanced by general linear model analysis and an anatomical atlas of human head,” Neuroimage 85(Pt 1), 166–180 (2014).
[Crossref] [PubMed]

F. Tian, F. A. Kozel, A. Yennu, P. E. Croarkin, S. M. McClintock, K. S. Mapes, M. M. Husain, and H. Liu, “Test-retest assessment of cortical activation induced by repetitive transcranial magnetic stimulation with brain atlas-guided optical topography,” J. Biomed. Opt. 17(11), 116020 (2012).
[Crossref] [PubMed]

V. C. Kavuri, Z. J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express 3(5), 943–957 (2012).
[Crossref] [PubMed]

F. Tian, M. R. Delgado, S. C. Dhamne, B. Khan, G. Alexandrakis, M. I. Romero, L. Smith, D. Reid, N. J. Clegg, and H. Liu, “Quantification of functional near infrared spectroscopy to assess cortical reorganization in children with cerebral palsy,” Opt. Express 18(25), 25973–25986 (2010).
[Crossref] [PubMed]

H. Niu, F. Tian, Z. J. Lin, and H. Liu, “Development of a compensation algorithm for accurate depth localization in diffuse optical tomography,” Opt. Lett. 35(3), 429–431 (2010).
[Crossref] [PubMed]

H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed. Opt. 15(4), 046005 (2010).
[Crossref] [PubMed]

F. Tian, G. Alexandrakis, and H. Liu, “Optimization of probe geometry for diffuse optical brain imaging based on measurement density and distribution,” Appl. Opt. 48(13), 2496–2504 (2009).
[Crossref] [PubMed]

Tikhonov, A.

A. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Soviet Mathematics Doklady 4, 1035–1038 (1963).

Tosteson, T. D.

van der Zee, P.

M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 215, 183–189 (1988).
[Crossref] [PubMed]

Wakin, M.

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Webb, J.

J. C. Ye, A. Bouman, J. Webb, and P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Image Processing 10(6), 909–922 (2001).
[Crossref]

White, B. R.

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] [PubMed]

Wray, S.

M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 215, 183–189 (1988).
[Crossref] [PubMed]

Wyatt, J.

M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 215, 183–189 (1988).
[Crossref] [PubMed]

Yalavarthy, P. K.

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34(6), 2085–2098 (2007).
[Crossref] [PubMed]

Yang, G. Z.

D. R. Leff, F. Orihuela-Espina, C. E. Elwell, T. Athanasiou, D. T. Delpy, A. W. Darzi, and G. Z. Yang, “Assessment of the cerebral cortex during motor task behaviours in adults: a systematic review of functional near infrared spectroscopy (fNIRS) studies,” Neuroimage 54(4), 2922–2936 (2011).
[Crossref] [PubMed]

Yazici, B.

Ye, J. C.

O. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: noniterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging 30(5), 1129–1142 (2011).
[Crossref] [PubMed]

J. C. Ye, A. Bouman, J. Webb, and P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Image Processing 10(6), 909–922 (2001).
[Crossref]

Yennu, A.

F. Tian, F. A. Kozel, A. Yennu, P. E. Croarkin, S. M. McClintock, K. S. Mapes, M. M. Husain, and H. Liu, “Test-retest assessment of cortical activation induced by repetitive transcranial magnetic stimulation with brain atlas-guided optical topography,” J. Biomed. Opt. 17(11), 116020 (2012).
[Crossref] [PubMed]

Yodh, A.

Yodh, A. G.

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] [PubMed]

Zourabian, A.

A. Zourabian, A. Siegel, B. Chance, N. Ramanujan, M. Rode, and D. A. Boas, “Trans-abdominal monitoring of fetal arterial blood oxygenation using pulse oximetry,” J. Biomed. Opt. 5(4), 391–405 (2000).
[Crossref] [PubMed]

Adv. Exp. Med. Biol. (1)

M. Cope, D. T. Delpy, E. O. Reynolds, S. Wray, J. Wyatt, and P. van der Zee, “Methods of quantitating cerebral near infrared spectroscopy data,” Adv. Exp. Med. Biol. 215, 183–189 (1988).
[Crossref] [PubMed]

Appl. Opt. (4)

Biomed. Opt. Express (2)

IEEE Image Processing (1)

J. C. Ye, A. Bouman, J. Webb, and P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Image Processing 10(6), 909–922 (2001).
[Crossref]

IEEE Sel. Top. Sig. Process. (1)

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE Sel. Top. Sig. Process. 1(4), 606–617 (2007).
[Crossref]

IEEE Signal Process. Mag. (2)

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

M. Lustig, D. Donoho, J. Santos, and J. Pauly, “Compressed Sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[Crossref]

IEEE Trans. Image Process. (2)

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Figures (10)

Fig. 1
Fig. 1 Noise variance of different S-D pairs from a phantom experiment.
Fig. 2
Fig. 2 Probe array setup for both simulation and phantom experiments. The unit in both x and y directions is cm.
Fig. 3
Fig. 3 Original absorbing object used for simulation experiment; the x and y dimension is in cm. Color bar represents Δμa in cm−1.
Fig. 4
Fig. 4 Comparison of noise variance from all simulated channels. Calculated noise variances by Eq. (17) are represented by y-axis and the actual noise variances computed from simulated 100 measurements are shown in x-axis.
Fig. 5
Fig. 5 Comparison of normalized reconstruction results from the simulation experiment with: (a) Tikhonov method, (b) GLS, (c) SR, and (d) QNNVSR.
Fig. 6
Fig. 6 Calculated SNRs among different channels in the phantom experiment (“+” means outliers). X-axis represents S-D separations, while discrete boxes from Y axis mark the 1st, 2nd, 3rd, … and nth nearest S-D pairs.
Fig. 7
Fig. 7 Absorber used in the second phantom experiment, where the size of the background grid has a total area of 25 mm × 25 mm, with a grid unit of 5 mm in each direction.
Fig. 8
Fig. 8 Normalized reconstructed images of the first phantom experiment (two spherical objects) obtained by using (a) Tikhonov regularization, (b) GLS, (c) SR, and (d) QNNVSR.
Fig. 9
Fig. 9 Recovered amplitudes of absorption changes along y-axis of image plane (see Fig. 2 or Fig. 8) with Tikhonov, GLS, SR, and QNNVSR reconstruction algorithm at x = 0.
Fig. 10
Fig. 10 Normalized images of the second phantom experiment with an L-shape object (see Fig. 7) reconstructed by (a) Tikhonov, (b) GLS, (c)SR, and (d) QNNVSR. The data used for reconstruction were taken from the first to sixth nearest S-D pairs (up to 4.2 cm), having a total number of 188 channels.

Tables (2)

Tables Icon

Table 1 SSIM and CNR values of reconstructed images from numerical experiments with Tikhonov, GLS, SR, and NQNSR

Tables Icon

Table 2 SSIM and CNR values of reconstructed images from phantom experiments with Tikhonov, GLS, SR, and NQNSR

Equations (31)

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D( r )Φ( r,ω )+c μ a ( r )Φ( r,ω )+jωΦ( r,ω )=cS( r,ω ),
1 2.3 ln Φ 0 ( r d , r s ) Φ( r d , r s ) =ΔOD=ODO D 0 =[log( I s I )log( I s I 0 )]=Δ μ a L.
Ax=b
x ^ =arg min x ( Axb 2 2 +λ x 2 2 )= A T (A A T +λI) 1 b,
x ^ =arg max x ( lnp (b|x)λ x 1 )
x ^ =arg min x ( Axb 2 2 +λ F(x) 1 ),
x ^ =arg min x ( Axb 2 2 +λ x 1 ).
x ^ =arg min x ( DAxDb 2 2 +λ D x x 2 2 )
x ^ =arg min x ( DAxDb 2 2 +λ x 1 )
b=ln Φ 0 Φ ,
b ˜ =ln Φ 0 + w 1 Φ+ w 2 =ln Φ 0 Φ +ln 1+ w 1 / Φ 0 1+ w 2 /Φ =b+ln 1+ w 1 / Φ 0 1+ w 2 /Φ .
w=ln 1+ w 1 / Φ 0 1+ w 2 /Φ =ln(1+ w 1 Φ 0 )ln(1+ w 2 Φ ).
w(i)=ln(1+ w 1 (i) ϕ 0 (i) )ln(1+ w 2 (i) ϕ(i) ),
w( i ) = ln ( 1+a ) ln ( 1+b )
s w( i ) 2 =E[ w 2 ( i ) ] ( E[ w( i ) ] ) 2 =E{ [ ln( 1+a ) ln( 1+b ) ] 2 } { E[ ln( 1+a ) ln( 1+b ) ] } 2 =E[ ln 2 ( 1+a ) ] +E[ ln 2 ( 1+b ) ] { E[ ln ( 1+a ) ] } 2 { E[ ln( 1+b ) ] } 2 =Var[ ln( 1 +a ) ] + Var[ ln( 1 +b ) ].
g( x ) = ln ( 1 +x ),
g( x )= k=1 (1) k+1 x k k , for 1<x<1.
E[g( x )]= k=1 (1) k+1 E[ x k ] k .
E[ g( x ) ]= k is even ( 1 ) k+1 k σ x k ( k1 )!!= k=1 ( 2k1 )!! 2k σ x 2k ,
E[ g 2 ( x ) ]=E[ ( i=1 (1) i+1 x i i )( j=1 (1) j+1 x j j ) ] = Σ i=1 Σ j=1 ( 1 ) i+j ij E[ x i+j ]= i j ( i+j1 )!! ij σ x i+j ,
E[ g( x ) ]( 1 2 σ x 2 + 3 4 σ x 4 + 5 2 σ x 6 +O( σ x 8 ) ).
E 2 [ g( x ) ] ( 1 2 σ x 2 + 3 4 σ x 4 + 5 2 σ x 6 +O( σ x 8 ) ) 2 1 4 σ x 4 + 3 4 σ x 6 +O( σ x 8 ).
E[ g 2 ( x ) ] σ x 2 + 11 4 σ x 4 + 137 12 σ x 6 +O( σ x 8 )
Var [ g( x ) ]=E[ g 2 ( x ) ] E 2 [g(x)]  σ x 2 + 5 2 σ x 4 + 32 3 σ x 6 +O( σ x 8 ).
E[ w( i ) ]=E{ ln[ 1+ w 1 ( i ) ϕ 0 ( i ) ] }E{ ln[ 1+ w 2 ( i ) ϕ( i ) ] } { 1 2 [ σ 1 2 ϕ 0 2 ( i ) σ 2 2 ϕ 2 ( i ) ]+ 3 4 [ σ 1 4 ϕ 0 4 ( i ) σ 2 4 ϕ 4 ( i ) ]+ 5 2 [ σ 1 6 ϕ 0 6 (i) σ 2 6 ϕ 6 (i) ]+O( σ 8 ) }0
σ w(i) 2 =Var[w( i )][ σ 1 2 ϕ 0 2 ( i ) + σ 2 2 ϕ 2 ( i ) ]+ 5 2 [ σ 1 4 ϕ 0 4 ( i ) + σ 2 4 ϕ 4 ( i ) ]+ 32 3 [ σ 1 6 ϕ 0 6 ( i ) + σ 2 6 ϕ 6 ( i ) ]+O( σ 8 ).
σ w(i) 2 σ w 2 [ 1 ϕ 0 2 ( i ) + 1 ϕ 2 ( i ) ]+ 5 2 σ w 4 [ 1 ϕ 0 4 ( i ) + 1 ϕ 4 ( i ) ]+ 32 3 σ w 6 [ 1 ϕ 0 6 ( i ) + 1 ϕ 6 ( i ) ]+O( σ 8 ).
D= ( 1 σ w(1) 1 σ w(n) ),
SSIM( x,y )=( 2 μ x μ y μ x 2 + μ y 2 )( 2 σ xy σ x 2 + σ y 2 )
CNR= μ ROI ¯ μ BKG ¯ c 1 σ μ ROI 2 + c 2 σ μ BKG 2 ,
c i = Number of pixels in ROI/BKG Total number of pixels in the entire image , i = 1,2.

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