Abstract

Total variation (TV) is a powerful regularization method that has been widely applied in different imaging applications, but is difficult to apply to diffuse optical tomography (DOT) image reconstruction (inverse problem) due to unstructured discretization of complex geometries, non-linearity of the data fitting and regularization terms, and non-differentiability of the regularization term. We develop several approaches to overcome these difficulties by: i) defining discrete differential operators for TV regularization using both finite element and graph representations; ii) developing an optimization algorithm based on the alternating direction method of multipliers (ADMM) for the non-differentiable and non-linear minimization problem; iii) investigating isotropic and anisotropic variants of TV regularization, and comparing their finite element (FEM)- and graph-based implementations. These approaches are evaluated on experiments on simulated data and real data acquired from a tissue phantom. Our results show that both FEM and graph-based TV regularization is able to accurately reconstruct both sparse and non-sparse distributions without the over-smoothing effect of Tikhonov regularization and the over-sparsifying effect of L1 regularization. The graph representation was found to out-perform the FEM method for low-resolution meshes, and the FEM method was found to be more accurate for high-resolution meshes.

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

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References

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2018 (1)

2017 (3)

J. Tang, B. Han, W. Han, B. Bi, and L. Li, “Mixed total variation and regularization method for optical tomography based on radiative transfer equation,” Comput. Math. Methods Med. 2017, 2953560 (2017).
[Crossref]

J. Duan, W. OC. Ward, L. Sibbett, Z. Pan, and L. Bai, “Introducing diffusion tensor to high order variational model for image reconstruction,” Digit. Signal Process. 69, 323–336 (2017).
[Crossref]

G. González, V. Kolehmainen, and A. Seppänen, “Isotropic and anisotropic total variation regularization in electrical impedance tomography,” Comput. Math. Appl. 74(3), 564–576 (2017).
[Crossref]

2016 (3)

J. Duan, Z. Qiu, W. Lu, G. Wang, Z. Pan, and L. Bai, “An edge-weighted second order variational model for image decomposition,” Digit. Signal Process. 49, 162–181 (2016).
[Crossref]

W. Lu, J. Duan, Z. Qiu, Z. Pan, R. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Methods Appl. Sci. 39, 4208–4233 (2016).
[Crossref]

J. Duan, W. Lu, C. Tench, I. Gottlob, F. Proudlock, and et al., “Denoising optical coherence tomography using second order total generalized variation decomposition,” Biomed. Signal Process. Control 24, 120–127 (2016).
[Crossref]

2015 (1)

J. Duan, Z. Pan, B. Zhang, W. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J. Glob. Optim. 62(4), 853–876 (2015).
[Crossref]

2014 (5)

K. Papafitsoros and C. B. Schönlieb, “A combined first and second order variational approach for image reconstruction,” J. Math. Imaging Vis. 48(2), 308–338 (2014).
[Crossref]

A. T. Eggebrecht, S. L. Ferradal, A. R. Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photon. 8(6), 448–454 (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. Quantum. Electron. 20(2), 74–82 (2014).
[Crossref]

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on ?1 relaxations of cheeger cut and Mumford-Shah-Potts model,” J. Math. Imaging Vis. 49(1), 191–201 (2014).
[Crossref]

X. Wu, A. T. Eggebrecht, S. L. Ferradal, J. P. Culver, and H. Dehghani, “Quantitative evaluation of atlas-based high-density diffuse optical tomography for imaging of the human visual cortex,” Biomed. Opt. Express 5(11), 3882–3900 (2014).
[Crossref] [PubMed]

2013 (3)

E. Merkurjev, T. Kostic, and A. L. Bertozzi, “An mbo scheme on graphs for classification and image processing,” SIAM J. Imaging Sci. 6(4), 1903–1930 (2013).
[Crossref]

J. Duan, Z. Pan, W. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 4(03), 43 (2013).
[Crossref]

Q. Lyu, Z. Lin, Y. She, and C. Zhang, “A comparison of typical Lp minimization algorithms,” Neurocomputing 119, 413–424 (2013).
[Crossref]

2012 (4)

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, and et al., “A quantitative spatial comparison of high-density diffuse optical tomography and FMRI cortical mapping,” Neuroimage 61(4), 1120–1128 (2012).
[Crossref] [PubMed]

A. L. Bertozzi and A. Flenner, “Diffuse interface models on graphs for classification of high dimensional data,” Multiscale Model. Simul. 10(3), 1090–1118 (2012).
[Crossref]

C. B. Shaw and P. K. Yalavarthy, “Effective contrast recovery in rapid dynamic near-infrared diffuse optical tomography using l1-norm-based linear image reconstruction method,” J. Biomed. Opt. 17(8), 0860091 (2012).
[Crossref]

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]

2011 (4)

S. Okawa, Y. Hoshi, and Y. Yamada, “Improvement of image quality of time-domain diffuse optical tomography with lp sparsity regularization,” Biomed. Opt. Express 2(12), 3334–3348 (2011).
[Crossref] [PubMed]

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imaging 30(5), 1143–1153 (2011).
[Crossref] [PubMed]

H. Niu, S. Khadka, F. Tian, Z. J. Lin, C. Lu, C. Zhu, and H. Liu, “Resting-state functional connectivity assessed with two diffuse optical tomographic systems,” J. Biomed. Opt. 16(4), 046006 (2011).
[Crossref] [PubMed]

L. Yao and H. Jiang, “Enhancing finite element-based photoacoustic tomography using total variation minimization,” Appl. Op. 50(25), 5031–5041 (2011).
[Crossref]

2010 (4)

I. Daubechies, R. DeVore, M. Fornasier, and C. S. Güntürk, “Iteratively reweighted least squares minimization for sparse recovery,” Commun. Pure Appl. Math. 63(1), 1–38 (2010).
[Crossref]

A. Custo, D. A. Boas, D. Tsuzuki, I. Dan, R. Mesquita, B. Fischl, W. Eric, L. Grimson, and W. Wells, “Anatomical atlas-guided diffuse optical tomography of brain activation,” Neuroimage 49(1), 561–567 (2010).
[Crossref]

H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 1: L1 regularization,” Opt. Express 18(3), 1854–1871 (2010).
[Crossref] [PubMed]

M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt. 49(19), 3741–3747 (2010).
[Crossref] [PubMed]

2009 (4)

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Appl. Opt. 48(10), D137–D143 (2009).
[Crossref] [PubMed]

T. Goldstein and S. Osher, “The split bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2(2), 323–343 (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]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, and et al., “Near infrared optical tomography using NIRFAST: Algorithms for numerical model and image reconstruction,” Int. J. Numer. Method. Biomed. Eng. 25(6), 711–732 (2009).

2008 (2)

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Sim. 7(3), 1005–1028 (2008).
[Crossref]

A. Elmoataz, O. Lezoray, and S. Bougleux, “Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing,” IEEE T. Image Process. 17(7), 1047–1060 (2008).
[Crossref]

2007 (1)

2005 (2)

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical tomography,” Phys. Med. Biol. 50(4), 31–9155 (2005).
[Crossref]

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[Crossref]

2004 (2)

2003 (3)

J. Ashburner and K. J. Friston, “Image segmentation,” Human Brain Function 2003, 2 (2003).

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42(1), 135–145 (2003).
[Crossref] [PubMed]

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

1997 (1)

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: ii. modelling and reconstruction,” Phys. Med. Biol. 42(5), 841 (1997).
[Crossref] [PubMed]

1996 (1)

1995 (1)

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical tomography,” Phys. Med. Biol. 50(4), 31–9155 (2005).
[Crossref]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: ii. modelling and reconstruction,” Phys. Med. Biol. 42(5), 841 (1997).
[Crossref] [PubMed]

S. R. Arridge and M Schweiger, “Photon-measurement density functions. part 2: Finite-element-method calculations,” Appl. Opt. 34(34), 8026–8037 (1995).
[Crossref] [PubMed]

Ashburner, J.

J. Ashburner and K. J. Friston, “Image segmentation,” Human Brain Function 2003, 2 (2003).

Bai, L.

J. Duan, W. OC. Ward, L. Sibbett, Z. Pan, and L. Bai, “Introducing diffusion tensor to high order variational model for image reconstruction,” Digit. Signal Process. 69, 323–336 (2017).
[Crossref]

J. Duan, Z. Qiu, W. Lu, G. Wang, Z. Pan, and L. Bai, “An edge-weighted second order variational model for image decomposition,” Digit. Signal Process. 49, 162–181 (2016).
[Crossref]

W. Lu, J. Duan, Z. Qiu, Z. Pan, R. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Methods Appl. Sci. 39, 4208–4233 (2016).
[Crossref]

Baraniuk, R.

T. Goldstein, C. Studer, and R. Baraniuk, “A field guide to forward-backward splitting with a FASTA implementation,” arXiv preprint arXiv:1411.3406 (2014).

Baritaux, J. C.

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imaging 30(5), 1143–1153 (2011).
[Crossref] [PubMed]

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]

Belli, A.

M. Clancy, A. Belli, D. Davies, S. JE. Lucas, Z. Su, and H. Dehghani, “Comparison of neurological NIRS signals during standing valsalva maneuvers, pre and post vasopressor injection,” In European Conference on Biomedical Optics, 953817 (Optical Society of America, 2015).

Belongie, S.

C. Fowlkes, S. Belongie, F. Chung, and J. Malik, “Spectral grouping using the Nystrom method,” IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 214–225 (2004).
[Crossref] [PubMed]

Bertozzi, A. L.

E. Merkurjev, T. Kostic, and A. L. Bertozzi, “An mbo scheme on graphs for classification and image processing,” SIAM J. Imaging Sci. 6(4), 1903–1930 (2013).
[Crossref]

A. L. Bertozzi and A. Flenner, “Diffuse interface models on graphs for classification of high dimensional data,” Multiscale Model. Simul. 10(3), 1090–1118 (2012).
[Crossref]

Bi, B.

J. Tang, B. Han, W. Han, B. Bi, and L. Li, “Mixed total variation and regularization method for optical tomography based on radiative transfer equation,” Comput. Math. Methods Med. 2017, 2953560 (2017).
[Crossref]

Bischof, H.

T. Pock, D. Cremers, H. Bischof, and A. Chambolle, “An algorithm for minimizing the Mumford-Shah functional,” In Computer Vision, 2009 IEEE 12th International Conference on, 1133–1140. IEEE (2009).
[Crossref]

Boas, D. A.

A. Custo, D. A. Boas, D. Tsuzuki, I. Dan, R. Mesquita, B. Fischl, W. Eric, L. Grimson, and W. Wells, “Anatomical atlas-guided diffuse optical tomography of brain activation,” Neuroimage 49(1), 561–567 (2010).
[Crossref]

D. A. Boas, K. Chen, D. Grebert, and M. A. Franceschini, “Improving the diffuse optical imaging spatial resolution of the cerebral hemodynamic response to brain activation in humans,” Opt. Lett. 29(13), 1506–1508 (2004).
[Crossref] [PubMed]

Bougleux, S.

A. Elmoataz, O. Lezoray, and S. Bougleux, “Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing,” IEEE T. Image Process. 17(7), 1047–1060 (2008).
[Crossref]

Bresson, X.

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W. Lu, J. Duan, Z. Qiu, Z. Pan, R. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Methods Appl. Sci. 39, 4208–4233 (2016).
[Crossref]

J. Duan, Z. Pan, B. Zhang, W. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J. Glob. Optim. 62(4), 853–876 (2015).
[Crossref]

J. Duan, Z. Pan, W. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 4(03), 43 (2013).
[Crossref]

Papafitsoros, K.

K. Papafitsoros and C. B. Schönlieb, “A combined first and second order variational approach for image reconstruction,” J. Math. Imaging Vis. 48(2), 308–338 (2014).
[Crossref]

Paulsen, K. D.

Pock, T.

T. Pock, D. Cremers, H. Bischof, and A. Chambolle, “An algorithm for minimizing the Mumford-Shah functional,” In Computer Vision, 2009 IEEE 12th International Conference on, 1133–1140. IEEE (2009).
[Crossref]

Pogue, B. W.

Poplack, S. P.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42(1), 135–145 (2003).
[Crossref] [PubMed]

Prakash, J.

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. Quantum. Electron. 20(2), 74–82 (2014).
[Crossref]

Proudlock, F.

J. Duan, W. Lu, C. Tench, I. Gottlob, F. Proudlock, and et al., “Denoising optical coherence tomography using second order total generalized variation decomposition,” Biomed. Signal Process. Control 24, 120–127 (2016).
[Crossref]

Qiu, Z.

J. Duan, Z. Qiu, W. Lu, G. Wang, Z. Pan, and L. Bai, “An edge-weighted second order variational model for image decomposition,” Digit. Signal Process. 49, 162–181 (2016).
[Crossref]

W. Lu, J. Duan, Z. Qiu, Z. Pan, R. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Methods Appl. Sci. 39, 4208–4233 (2016).
[Crossref]

Quintard, H.

C. Ichai, H. Quintard, and J. C. Orban, Metabolic disorders and critically ill patients: from pathophysiology to treatment (Springer, 2017).

Reed, W. H.

W. H. Reed and T. R. Hill, “Triangular mesh methods for the neutron transport equation,” Technical report, Los Alamos Scientific Lab., N. Mex.(USA) (1973).

Sanyal, S.

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imaging 30(5), 1143–1153 (2011).
[Crossref] [PubMed]

Scharfetter, H.

Schönlieb, C. B.

K. Papafitsoros and C. B. Schönlieb, “A combined first and second order variational approach for image reconstruction,” J. Math. Imaging Vis. 48(2), 308–338 (2014).
[Crossref]

Schweiger, M

Seppänen, A.

G. González, V. Kolehmainen, and A. Seppänen, “Isotropic and anisotropic total variation regularization in electrical impedance tomography,” Comput. Math. Appl. 74(3), 564–576 (2017).
[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. Quantum. Electron. 20(2), 74–82 (2014).
[Crossref]

C. B. Shaw and P. K. Yalavarthy, “Effective contrast recovery in rapid dynamic near-infrared diffuse optical tomography using l1-norm-based linear image reconstruction method,” J. Biomed. Opt. 17(8), 0860091 (2012).
[Crossref]

She, Y.

Q. Lyu, Z. Lin, Y. She, and C. Zhang, “A comparison of typical Lp minimization algorithms,” Neurocomputing 119, 413–424 (2013).
[Crossref]

Sibbett, L.

J. Duan, W. OC. Ward, L. Sibbett, Z. Pan, and L. Bai, “Introducing diffusion tensor to high order variational model for image reconstruction,” Digit. Signal Process. 69, 323–336 (2017).
[Crossref]

Snyder, A. Z.

A. T. Eggebrecht, S. L. Ferradal, A. R. Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photon. 8(6), 448–454 (2014).
[Crossref]

Soho, S.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Srinivasan, S.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, and et al., “Near infrared optical tomography using NIRFAST: Algorithms for numerical model and image reconstruction,” Int. J. Numer. Method. Biomed. Eng. 25(6), 711–732 (2009).

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Studer, C.

T. Goldstein, C. Studer, and R. Baraniuk, “A field guide to forward-backward splitting with a FASTA implementation,” arXiv preprint arXiv:1411.3406 (2014).

Styles, I. B.

Su, Z.

M. Clancy, A. Belli, D. Davies, S. JE. Lucas, Z. Su, and H. Dehghani, “Comparison of neurological NIRS signals during standing valsalva maneuvers, pre and post vasopressor injection,” In European Conference on Biomedical Optics, 953817 (Optical Society of America, 2015).

Szlam, A.

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on ?1 relaxations of cheeger cut and Mumford-Shah-Potts model,” J. Math. Imaging Vis. 49(1), 191–201 (2014).
[Crossref]

Tai, X. C.

J. Duan, Z. Pan, B. Zhang, W. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J. Glob. Optim. 62(4), 853–876 (2015).
[Crossref]

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on ?1 relaxations of cheeger cut and Mumford-Shah-Potts model,” J. Math. Imaging Vis. 49(1), 191–201 (2014).
[Crossref]

J. Duan, Z. Pan, W. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 4(03), 43 (2013).
[Crossref]

Tang, J.

J. Tang, B. Han, W. Han, B. Bi, and L. Li, “Mixed total variation and regularization method for optical tomography based on radiative transfer equation,” Comput. Math. Methods Med. 2017, 2953560 (2017).
[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]

Tench, C.

J. Duan, W. Lu, C. Tench, I. Gottlob, F. Proudlock, and et al., “Denoising optical coherence tomography using second order total generalized variation decomposition,” Biomed. Signal Process. Control 24, 120–127 (2016).
[Crossref]

Tian, F.

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, S. Khadka, F. Tian, Z. J. Lin, C. Lu, C. Zhu, and H. Liu, “Resting-state functional connectivity assessed with two diffuse optical tomographic systems,” J. Biomed. Opt. 16(4), 046006 (2011).
[Crossref] [PubMed]

Tizzard, A.

Tosteson, T. D.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Tsuzuki, D.

A. Custo, D. A. Boas, D. Tsuzuki, I. Dan, R. Mesquita, B. Fischl, W. Eric, L. Grimson, and W. Wells, “Anatomical atlas-guided diffuse optical tomography of brain activation,” Neuroimage 49(1), 561–567 (2010).
[Crossref]

Unser, M.

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imaging 30(5), 1143–1153 (2011).
[Crossref] [PubMed]

Viehoever, A. R.

A. T. Eggebrecht, S. L. Ferradal, A. R. Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photon. 8(6), 448–454 (2014).
[Crossref]

Vlasov, V. V.

A. B. Konovalov and V. V. Vlasov, “Total variation based reconstruction of scattering inhomogeneities in tissue from time-resolved optical projections,” In PALS 9917: 99170S, International Society for Optics and Photonics (2016).

Wang, G.

J. Duan, Z. Qiu, W. Lu, G. Wang, Z. Pan, and L. Bai, “An edge-weighted second order variational model for image decomposition,” Digit. Signal Process. 49, 162–181 (2016).
[Crossref]

Ward, W. OC.

J. Duan, W. OC. Ward, L. Sibbett, Z. Pan, and L. Bai, “Introducing diffusion tensor to high order variational model for image reconstruction,” Digit. Signal Process. 69, 323–336 (2017).
[Crossref]

Wells, W.

A. Custo, D. A. Boas, D. Tsuzuki, I. Dan, R. Mesquita, B. Fischl, W. Eric, L. Grimson, and W. Wells, “Anatomical atlas-guided diffuse optical tomography of brain activation,” Neuroimage 49(1), 561–567 (2010).
[Crossref]

White, B. R.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, and et al., “A quantitative spatial comparison of high-density diffuse optical tomography and FMRI cortical mapping,” Neuroimage 61(4), 1120–1128 (2012).
[Crossref] [PubMed]

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Appl. Opt. 48(10), D137–D143 (2009).
[Crossref] [PubMed]

Wu, X.

Xu, J.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[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. Quantum. Electron. 20(2), 74–82 (2014).
[Crossref]

C. B. Shaw and P. K. Yalavarthy, “Effective contrast recovery in rapid dynamic near-infrared diffuse optical tomography using l1-norm-based linear image reconstruction method,” J. Biomed. Opt. 17(8), 0860091 (2012).
[Crossref]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, and et al., “Near infrared optical tomography using NIRFAST: Algorithms for numerical model and image reconstruction,” Int. J. Numer. Method. Biomed. Eng. 25(6), 711–732 (2009).

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express 15(13), 8043–8058 (2007).
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Yamada, Y.

Yao, L.

L. Yao and H. Jiang, “Enhancing finite element-based photoacoustic tomography using total variation minimization,” Appl. Op. 50(25), 5031–5041 (2011).
[Crossref]

Yin, W.

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[Crossref]

Zeff, B. W.

Zhan, Y.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, and et al., “A quantitative spatial comparison of high-density diffuse optical tomography and FMRI cortical mapping,” Neuroimage 61(4), 1120–1128 (2012).
[Crossref] [PubMed]

Zhang, B.

J. Duan, Z. Pan, B. Zhang, W. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J. Glob. Optim. 62(4), 853–876 (2015).
[Crossref]

Zhang, C.

Q. Lyu, Z. Lin, Y. She, and C. Zhang, “A comparison of typical Lp minimization algorithms,” Neurocomputing 119, 413–424 (2013).
[Crossref]

Zhao, H.

Zhu, C.

H. Niu, S. Khadka, F. Tian, Z. J. Lin, C. Lu, C. Zhu, and H. Liu, “Resting-state functional connectivity assessed with two diffuse optical tomographic systems,” J. Biomed. Opt. 16(4), 046006 (2011).
[Crossref] [PubMed]

Appl. Op. (1)

L. Yao and H. Jiang, “Enhancing finite element-based photoacoustic tomography using total variation minimization,” Appl. Op. 50(25), 5031–5041 (2011).
[Crossref]

Appl. Opt. (5)

Biomed. Opt. Express (4)

Biomed. Signal Process. Control (1)

J. Duan, W. Lu, C. Tench, I. Gottlob, F. Proudlock, and et al., “Denoising optical coherence tomography using second order total generalized variation decomposition,” Biomed. Signal Process. Control 24, 120–127 (2016).
[Crossref]

Commun. Pure Appl. Math. (1)

I. Daubechies, R. DeVore, M. Fornasier, and C. S. Güntürk, “Iteratively reweighted least squares minimization for sparse recovery,” Commun. Pure Appl. Math. 63(1), 1–38 (2010).
[Crossref]

Comput. Math. Appl. (1)

G. González, V. Kolehmainen, and A. Seppänen, “Isotropic and anisotropic total variation regularization in electrical impedance tomography,” Comput. Math. Appl. 74(3), 564–576 (2017).
[Crossref]

Comput. Math. Methods Med. (1)

J. Tang, B. Han, W. Han, B. Bi, and L. Li, “Mixed total variation and regularization method for optical tomography based on radiative transfer equation,” Comput. Math. Methods Med. 2017, 2953560 (2017).
[Crossref]

Digit. Signal Process. (2)

J. Duan, Z. Qiu, W. Lu, G. Wang, Z. Pan, and L. Bai, “An edge-weighted second order variational model for image decomposition,” Digit. Signal Process. 49, 162–181 (2016).
[Crossref]

J. Duan, W. OC. Ward, L. Sibbett, Z. Pan, and L. Bai, “Introducing diffusion tensor to high order variational model for image reconstruction,” Digit. Signal Process. 69, 323–336 (2017).
[Crossref]

Human Brain Function (1)

J. Ashburner and K. J. Friston, “Image segmentation,” Human Brain Function 2003, 2 (2003).

IEEE J. Sel. Top. Quantum. Electron. (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. Quantum. Electron. 20(2), 74–82 (2014).
[Crossref]

IEEE T. Image Process. (1)

A. Elmoataz, O. Lezoray, and S. Bougleux, “Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing,” IEEE T. Image Process. 17(7), 1047–1060 (2008).
[Crossref]

IEEE Trans. Med. Imaging (1)

J. C. Baritaux, K. Hassler, M. Bucher, S. Sanyal, and M. Unser, “Sparsity-driven reconstruction for FDOT with anatomical priors,” IEEE Trans. Med. Imaging 30(5), 1143–1153 (2011).
[Crossref] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

C. Fowlkes, S. Belongie, F. Chung, and J. Malik, “Spectral grouping using the Nystrom method,” IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 214–225 (2004).
[Crossref] [PubMed]

Int. J. Numer. Method. Biomed. Eng. (1)

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, and et al., “Near infrared optical tomography using NIRFAST: Algorithms for numerical model and image reconstruction,” Int. J. Numer. Method. Biomed. Eng. 25(6), 711–732 (2009).

J. Biomed. Opt. (2)

C. B. Shaw and P. K. Yalavarthy, “Effective contrast recovery in rapid dynamic near-infrared diffuse optical tomography using l1-norm-based linear image reconstruction method,” J. Biomed. Opt. 17(8), 0860091 (2012).
[Crossref]

H. Niu, S. Khadka, F. Tian, Z. J. Lin, C. Lu, C. Zhu, and H. Liu, “Resting-state functional connectivity assessed with two diffuse optical tomographic systems,” J. Biomed. Opt. 16(4), 046006 (2011).
[Crossref] [PubMed]

J. Glob. Optim. (1)

J. Duan, Z. Pan, B. Zhang, W. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J. Glob. Optim. 62(4), 853–876 (2015).
[Crossref]

J. Math. Imaging Vis. (2)

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on ?1 relaxations of cheeger cut and Mumford-Shah-Potts model,” J. Math. Imaging Vis. 49(1), 191–201 (2014).
[Crossref]

K. Papafitsoros and C. B. Schönlieb, “A combined first and second order variational approach for image reconstruction,” J. Math. Imaging Vis. 48(2), 308–338 (2014).
[Crossref]

JSIP (1)

J. Duan, Z. Pan, W. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 4(03), 43 (2013).
[Crossref]

Math. Methods Appl. Sci. (1)

W. Lu, J. Duan, Z. Qiu, Z. Pan, R. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Methods Appl. Sci. 39, 4208–4233 (2016).
[Crossref]

Multiscale Model. Sim. (1)

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Sim. 7(3), 1005–1028 (2008).
[Crossref]

Multiscale Model. Simul. (2)

A. L. Bertozzi and A. Flenner, “Diffuse interface models on graphs for classification of high dimensional data,” Multiscale Model. Simul. 10(3), 1090–1118 (2012).
[Crossref]

S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul. 4(2), 460–489 (2005).
[Crossref]

Nat. Photon. (1)

A. T. Eggebrecht, S. L. Ferradal, A. R. Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photon. 8(6), 448–454 (2014).
[Crossref]

Neurocomputing (1)

Q. Lyu, Z. Lin, Y. She, and C. Zhang, “A comparison of typical Lp minimization algorithms,” Neurocomputing 119, 413–424 (2013).
[Crossref]

Neuroimage (2)

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, and et al., “A quantitative spatial comparison of high-density diffuse optical tomography and FMRI cortical mapping,” Neuroimage 61(4), 1120–1128 (2012).
[Crossref] [PubMed]

A. Custo, D. A. Boas, D. Tsuzuki, I. Dan, R. Mesquita, B. Fischl, W. Eric, L. Grimson, and W. Wells, “Anatomical atlas-guided diffuse optical tomography of brain activation,” Neuroimage 49(1), 561–567 (2010).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (2)

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: ii. modelling and reconstruction,” Phys. Med. Biol. 42(5), 841 (1997).
[Crossref] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical tomography,” Phys. Med. Biol. 50(4), 31–9155 (2005).
[Crossref]

Proc. Natl. Acad. Sci. (1)

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

SIAM J. Imaging Sci. (3)

T. Goldstein and S. Osher, “The split bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2(2), 323–343 (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]

E. Merkurjev, T. Kostic, and A. L. Bertozzi, “An mbo scheme on graphs for classification and image processing,” SIAM J. Imaging Sci. 6(4), 1903–1930 (2013).
[Crossref]

Other (7)

W. H. Reed and T. R. Hill, “Triangular mesh methods for the neutron transport equation,” Technical report, Los Alamos Scientific Lab., N. Mex.(USA) (1973).

M. Clancy, A. Belli, D. Davies, S. JE. Lucas, Z. Su, and H. Dehghani, “Comparison of neurological NIRS signals during standing valsalva maneuvers, pre and post vasopressor injection,” In European Conference on Biomedical Optics, 953817 (Optical Society of America, 2015).

C. Ichai, H. Quintard, and J. C. Orban, Metabolic disorders and critically ill patients: from pathophysiology to treatment (Springer, 2017).

A. B. Konovalov and V. V. Vlasov, “Total variation based reconstruction of scattering inhomogeneities in tissue from time-resolved optical projections,” In PALS 9917: 99170S, International Society for Optics and Photonics (2016).

T. Goldstein, C. Studer, and R. Baraniuk, “A field guide to forward-backward splitting with a FASTA implementation,” arXiv preprint arXiv:1411.3406 (2014).

T. Pock, D. Cremers, H. Bischof, and A. Chambolle, “An algorithm for minimizing the Mumford-Shah functional,” In Computer Vision, 2009 IEEE 12th International Conference on, 1133–1140. IEEE (2009).
[Crossref]

D. Lighter, A. Filer, and H. Dehghani, “Multispectral diffuse optical tomography of finger joints,” In European Conference on Biomedical Optics, 104120N (Optical Society of America, 2017).

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Figures (11)

Fig. 1
Fig. 1 Modeling a complex geometry using finite element (left) and graph (right) representations.
Fig. 2
Fig. 2 (a)–(c): Discretized computational domain of the three experimental samples; (d): Detailed mesh composition of 2D geometry in finite element and graph representation respectively; (e): Detailed mesh composition of 3D geometry in finite element and graph representation respectively.
Fig. 3
Fig. 3 (a)–(c): Reconstruction on the 2D mesh with low spatial resolution. (d)–(f): Reconstruction on the 2D mesh with high spatial resolution. (a) and (d): 2D reconstruction mesh with sixteen co-located sources and detectors. (b) and (e) give the original target distributions. First row in (c) and (f) represents the results using A-FETV on 0%, 1%, 2% and 3% noisy data while the second row shows the results using A-GTV.
Fig. 4
Fig. 4 (a)–(c): Reconstruction on the 2D mesh with low spatial resolution; (d)–(f): Reconstruction on the 2D mesh with high spatial resolution. (a) and (d): 2D reconstruction mesh with sixteen co-located sources and detectors. (b) and (e) give the original target distributions. First row in (c) and (f) represents the results using I-FETV on 0%, 1%, 2% and 3% noisy data while the second row shows the results by I-GTV.
Fig. 5
Fig. 5 1D cross section of images recovered in Fig. 4. First column corresponds to Fig. 4 (c) where the spatial resolution of the reconstruction mesh is lower. Second column corresponds to Fig. 4 (f) where the spatial resolution of the reconstruction mesh is higher. Top to bottom row: 0%, 1%, 2% and 3% added Gaussian noise.
Fig. 6
Fig. 6 Evaluation metrics comparing the performance of different methods at four different noise levels. Top to bottom row: localization error index; average contrast index; PSNR index and relative recovered volume. Left column corresponds to the reconstructions in Fig. 4 (c) where the reconsturction mesh resolution is low. Right column corresponds to Fig. 4 (f) where the reconsturction mesh resolution is relatively high.
Fig. 7
Fig. 7 First column: distribution of the imaging array with 158 sources (red dots) and 166 detectors (white dots) and the positions of the two simultaneous simulated anomalies. Second to final column: Ground truth and reconstructions by Tikhonov, I-FETV and I-GTV.
Fig. 8
Fig. 8 2D slices of the reconstructions of the absorption coefficient changes on the forehead anomaly (first row in Fig. 7). The ground truth areas are highlighted in white ellipses.
Fig. 9
Fig. 9 2D slices of the reconstructions of the absorption coefficient changes on the back-head anomaly (second row in Fig. 7). The ground truth areas are highlighted in white ellipses.
Fig. 10
Fig. 10 Evaluation metrics comparing the performance of different methods on a 3D head model. The left column represents the reconstruction of the forehead anomaly (first row in Fig. 7), while the right column gives the reconstruction of the back-head anomaly (second row in Fig. 7).
Fig. 11
Fig. 11 (a): Distribution of sources and detectors. (b): Illustration of the overall distribution of three slices. (c): Ground truth and reconstruction results with different methods. From top to bottom: ground truth; results with Tikhonov regularization; results with I-FETV regularization and results with I-GTV regularization.

Tables (6)

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Table 1 Four TV-regularized minimization problems obtained by applying different TV regularizations to Eq. (15). A-FETV, I-FETV, A-GTV and I-GTV respectively represent anisotropic finite element total variation, isotropic finite element total variation, anisotropic graph total variation and isotropic graph total variation.

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Algorithm 1: ADMM-based algorithm for A-FETV and I-FETV.

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Algorithm 2: ADMM-based algorithm for I-GTV and A-GTV.

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Algorithm 3: Algorithm for minimizing the TV-associated inverse problem.

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Table 2 Head tissue optical property for each of five layers.

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Table 3 Evaluation of different methods for reconstruction on a tissue-simulating phantom.

Equations (39)

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μ * = arg min μ { 1 2 Φ M ( μ ) 2 2 + λ ( μ ) } ,
U = i = 1 N μ i φ i .
φ 1 ( x , y ) = a 1 x + b 1 y + c 1 φ 2 ( x , y ) = a 2 x + b 2 y + c 2 : Ω , φ 3 ( x , y ) = a 3 x + b 3 y + c 3
Ω ( | x U | + | y U | ) d x d y = D x μ 1 + D y μ 1 .
Ω ( x U ) 2 + ( y U ) 2 d x d y = i = 1 M | ( D x μ ) i | 2 + | ( D y μ ) i | 2 .
Ω | x U | d x d y = i = 1 M T i | x U | d x d y = i = 1 M T i | j = 1 N μ j x φ j | d x d y = i = 1 M A T i | a i , 1 μ i , 1 + a i , 2 μ i , 2 + a i , 3 μ i , 3 | = i = 1 M | ( D x μ ) i | = D x μ 1 ,
w μ i ( μ j μ i ) w i j : V .
div w v i j = 1 N ( ν i j ν j i ) w i j : V ,
Δ w μ i 1 2 div w ( w μ i ) = j = 1 N ( μ j μ i ) w i j : V ,
i = 1 N j = 1 N | ( μ j μ i ) w i j | ,
i = 1 N j = 1 N ( μ j μ i ) 2 w i j .
i = 1 N j 𝒩 i | ( μ j μ i ) w i j | ,
i = 1 N j 𝒩 i ( μ j μ i ) 2 w i j ,
( μ ) ( μ k 1 ) + J k 1 ( μ μ k 1 ) ,
δ μ k = arg min δ μ { 1 2 J k 1 δ μ δ Φ k 1 2 2 + λ ( δ μ ) } ,
δ μ n , ν x n , ν y n = arg min δ μ , ν x , ν y { 1 2 J δ μ δ Φ 2 2 + λ ν x 1 + λ ν y 1 + θ 2 ν x D x ( δ μ ) b x n 1 2 2 + θ 2 ν y D y ( δ μ ) b y n 1 2 2 } ,
δ μ n = arg min δ μ { 1 2 J δ μ δ Φ 2 2 + θ 2 ν x n 1 D x ( δ μ ) b x n 1 2 2 + θ 2 ν y n 1 D y ( δ μ ) b y n 1 2 2 } ,
( ( J T J + θ ( D x T D x + D y T D y ) ) ) δ μ n = J T δ Φ θ D x T ( b x n 1 ν x n 1 ) θ D y T ( b y n 1 ν y n 1 ) .
ν x n , ν y n = arg min ν x , ν y { λ ν x 1 + λ ν y 1 + θ 2 ν x D x ( δ μ n ) b x n 1 2 2 + θ 2 ν y D y ( δ μ n ) b y n 1 2 2 } .
ν x n = max ( | D x ( δ μ n ) + b x n 1 | λ θ , 0 ) D x ( δ μ n ) + b x n 1 | D x ( δ μ n ) + b x n 1 | ν y n = max ( | D y ( δ μ n ) + b y n 1 | λ θ , 0 ) D y ( δ μ n ) + b y n 1 | D y ( δ μ n ) + b y n 1 | ,
b x n = b x n 1 + D x ( δ μ n ) ν x n b y n = b y n 1 + D y ( δ μ n ) ν y n .
δ μ n , ν x n , ν y n = arg min δ μ , ν x , ν y { 1 2 J δ μ δ Φ 2 2 + λ ( ν x , ν y ) 2 + θ 2 ν x D x ( δ μ ) b x n 1 2 2 + θ 2 ν y D y ( δ μ ) b y n 1 2 2 } ,
( ν x , ν y ) 2 = i = 1 M | ( ν x ) i | 2 + | ( ν y ) i | 2 ,
ν x n = max ( s n λ θ , 0 ) D x ( δ μ n ) + b x n 1 s n ν y n = max ( s n λ θ , 0 ) D y ( δ μ n ) + b y n 1 s n ,
δ μ n , ν n = arg min δ μ , ν { 1 2 J δ μ δ Φ 2 2 + λ i = 1 N ν i 1 + θ 2 i = 1 N ν i w ( δ μ i ) b i n 1 2 2 } .
δ μ n = arg min δ μ { 1 2 J δ μ δ Φ 2 2 + θ 2 i = 1 N ν i n 1 w ( δ μ i ) b i n 1 2 2 } ,
( J T J δ μ J T δ Φ ) i + θ div w ( ν i n 1 w ( δ μ i ) b i n 1 ) = 0 , i = 1 , , N .
( J T J θ L ) δ μ = J T δ Φ θ g n 1 .
L i , j = { j 𝒩 i w i j if i = j w i j otherwise .
ν n = arg min ν { λ i = 1 N ν i 1 + θ 2 i = 1 N ν i w ( δ μ i n ) b i n 1 2 2 } ,
ν i j n = max ( | w i j ( δ μ j n δ μ i n ) + b i j n 1 | λ θ , 0 ) w i j ( δ μ j n δ μ i n ) + b i j n 1 | w i j ( δ μ j n δ μ i n ) + b i j n 1 | ,
b i j n = b i j n 1 + w i j ( δ μ j n δ μ i n ) ν i j n .
δ μ n , ν n = arg min δ μ , ν { 1 2 J δ μ δ Φ 2 2 + λ i = 1 N ν i 2 + θ 2 i = 1 N ν i w ( δ μ i ) b i n 1 2 2 } .
ν n = arg min ν { λ i = 1 N ν i 2 + θ 2 i = 1 N ν i w ( δ μ i n ) b i n 1 2 2 } ,
ν i j n = max ( j 𝒩 i ( w i j ( δ μ j n δ μ i n ) + b i j n 1 ) 2 λ θ , 0 ) w i j ( δ μ j n δ μ i n ) + b i j n 1 j 𝒩 i ( w i j ( δ μ j n δ μ i n ) + b i j n 1 ) 2 ,
Localization error = X s X r 2 .
Average contrast = ( i = 1 N r μ i / N r ) / μ ˜ ,
PSNR = 10 log 10 ( MAX μ 2 / MSE ) .
V RRV = V r / V s × 100 % ,

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