Split operator method for fluorescence diffuse optical tomography using anisotropic diffusion regularisation with prior anatomical information

Teresa Correia; Juan Aguirre; Alejandro Sisniega; Judit Chamorro-Servent; Juan Abascal; Juan J. Vaquero; Manuel Desco; Ville Kolehmainen; Simon Arridge

doi:10.1364/BOE.2.002632

Split operator method for fluorescence diffuse optical tomography using anisotropic diffusion regularisation with prior anatomical information

Teresa Correia, Juan Aguirre, Alejandro Sisniega, Judit Chamorro-Servent, Juan Abascal, Juan J. Vaquero, Manuel Desco, Ville Kolehmainen, and Simon Arridge

Biomedical Optics Express
Vol. 2,
Issue 9,
pp. 2632-2648
(2011)
•https://doi.org/10.1364/BOE.2.002632

Get Citation
Copy Citation Text
Teresa Correia, Juan Aguirre, Alejandro Sisniega, Judit Chamorro-Servent, Juan Abascal, Juan J. Vaquero, Manuel Desco, Ville Kolehmainen, and Simon Arridge, "Split operator method for fluorescence diffuse optical tomography using anisotropic diffusion regularisation with prior anatomical information," Biomed. Opt. Express 2, 2632-2648 (2011)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (3224 KB)
Set citation alerts
Save article

Related Topics
Table of Contents Category
- Image Reconstruction and Inverse Problems
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Inverse problems (100.3190)
- Tomography (170.6960)

About this Article
History
- Original Manuscript: July 7, 2011
- Revised Manuscript: August 14, 2011
- Manuscript Accepted: August 15, 2011
- Published: August 19, 2011

Accessible

Open Access

Abstract

Fluorescence diffuse optical tomography (fDOT) is an imaging modality that provides images of the fluorochrome distribution within the object of study. The image reconstruction problem is ill-posed and highly underdetermined and, therefore, regularisation techniques need to be used. In this paper we use a nonlinear anisotropic diffusion regularisation term that incorporates anatomical prior information. We introduce a split operator method that reduces the nonlinear inverse problem to two simpler problems, allowing fast and efficient solution of the fDOT problem. We tested our method using simulated, phantom and ex-vivo mouse data, and found that it provides reconstructions with better spatial localisation and size of fluorochrome inclusions than using the standard Tikhonov penalty term.

Full Article | PDF Article

OSA Recommended Articles

Feasibility of U-curve method to select the regularization parameter for fluorescence diffuse optical tomography in phantom and small animal studies

Judit Chamorro-Servent, Juan Aguirre, Jorge Ripoll, Juan José Vaquero, and Manuel Desco
Opt. Express 19(12) 11490-11506 (2011)

Subsurface fluorescence molecular tomography with prior information

Wei He, Huangsheng Pu, Guanglei Zhang, Xu Cao, Bin Zhang, Fei Liu, Jianwen Luo, and Jing Bai
Appl. Opt. 53(3) 402-409 (2014)

Enhancement of fluorescence molecular tomography with structural-prior-based diffuse optical tomography: combating optical background uncertainty

Linhui Wu, Huijuan Zhao, Xin Wang, Xi Yi, Weiting Chen, and Feng Gao
Appl. Opt. 53(30) 6970-6982 (2014)

References

View by:
Article Order
|
Year
|
Author
|
Publication

V. Ntziachristos, “Fluorescence molecular imaging,” Ann. Rev. Biomed. Eng. 8, 1–33 (2006).
[Crossref]
V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13, 195–208 (2003).
[PubMed]
G. Zacharakis, H. Kambara, H. Shih, J. Ripoll, J. Grimm, Y. Saeki, R. Weissleder, and V. Ntziachristos, “Volumetric tomography of fluorescent proteins through small animals in vivo,” Proc. Natl. Acad. Sci. U.S.A. 12, 18252–18257 (2005).
[Crossref]
D. S. Kepshire, S. L. Gibbs-Strauss, J. A. OHara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[Crossref] [PubMed]
A. Martin, J. A. J, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes.” Molec. Imaging 7, 157–167 (2008).
A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15, 6696–6716 (2007).
[Crossref] [PubMed]
V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97, 2767–2772 (2000).
[Crossref] [PubMed]
S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. V. der Mark, M. V. Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: First clinical experience in patients,” Molec. Imaging Biol. 12, 343–248 (2010).
[Crossref]
S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl. 25, 123010 (2009).
[Crossref]
A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12, 5402–5417 (2004).
[Crossref] [PubMed]
A. D. Zacharopoulos, P. Svenmarker, J. Axelsson, M. Schweiger, S. R. Arridge, and S. Andersson-Engels, “A matrix-free algorithm for multiple wavelength fluorescence tomography,” Opt. Express 17, 3025–3035 (2009).
[Crossref] [PubMed]
A. Ale, R. B. Schulz, A. Sarantopoulos, and V. Ntziachristos, “Imaging performance of a hybrid x-ray computed tomography-fluorescence molecular tomography system using priors,” Med. Phys. 37, 1976–1986 (2010).
[Crossref] [PubMed]
S. C. Davis, H. Dehghani, J. Wang, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express 15, 4066–4082 (2007).
[Crossref] [PubMed]
Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt. 48, 1328–1336 (2009).
[Crossref] [PubMed]
C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).
[Crossref]
M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion in fluorescence optical tomography,” IEEE Trans. Biomed. Eng. 57, 2723–2729 (2010).
[Crossref]
M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt. 49, 3741–3747 (2010).
[Crossref] [PubMed]
A. Chambolle and P. Lions, “Image recovery via total variation minimization and related problems,” Num. Math. 76, 167–188 (1997).
[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, 2439–2441 (2005).
[Crossref] [PubMed]
A. Douiri, M. Schweiger, J. Riley, and S. R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. 18, 87–95 (2007).
[Crossref]
S. R. Arridge, V. Kolehmainen, and M. J. Schweiger, “Reconstruction and regularisation in optical tomography,” in “Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT),”(Pisa, Italy, 2007), pp. 1–18.
J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inv. Probl. 15, 713–729 (1999).
[Crossref]
V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26, 893–895 (2001).
[Crossref]
M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters 21, 158–160 (1996).
[Crossref]
A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24, 1377–1386 (2005).
[Crossref] [PubMed]
T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. 35, 763–765 (2010).
[Crossref] [PubMed]
F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal. 29, 182–193 (1992).
[Crossref]
P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990).
[Crossref]
J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Process. 7, 398–410 (1998).
[Crossref]
M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image Process. 7, 421–432 (1998).
[Crossref]
M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[Crossref] [PubMed]
H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Mathematics and its applications (Kluwer Academic Publishers, 2000).
O. Scherzer and C. Groetsch, “Inverse scale space theory for inverse problems,” in “Scale-Space and Morphology in Computer Vision,”, vol. 2106 of Lecture Notes in Computer Science, M. Kerckhove, ed. (Springer, 2006), pp. 317–325.
[Crossref]
B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inv. Probl. 13, 729 (1997).
[Crossref]
M. Hanke, “A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems,” Inv. Probl. 13, 79 (1997).
[Crossref]
P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[Crossref]
T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14, 034044 (2009).
[Crossref] [PubMed]
Q. Fang and D. Boas, “Tetrahedral mesh generation from volumetric binary and gray-scale images,” in “Proceedings of IEEE International Symposium on Biomedical Imaging,”(2009), pp. 1142–1145.
[Crossref]
B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577 (2007).
[Crossref] [PubMed]
G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225 (2005).
[Crossref] [PubMed]
J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Design and development of a co-planar fluorescence and X-ray tomograph,” in “Nuclear Science Symposium Conference Record, 2008. NSS ’08. IEEE,”(2008), pp. 5412–5413.
[Crossref]
D. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. thesis, University of Pennsylvania, Philadelphia (USA) (1996).

2010 (5)

S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. V. der Mark, M. V. Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: First clinical experience in patients,” Molec. Imaging Biol. 12, 343–248 (2010).
[Crossref]

A. Ale, R. B. Schulz, A. Sarantopoulos, and V. Ntziachristos, “Imaging performance of a hybrid x-ray computed tomography-fluorescence molecular tomography system using priors,” Med. Phys. 37, 1976–1986 (2010).
[Crossref] [PubMed]

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion in fluorescence optical tomography,” IEEE Trans. Biomed. Eng. 57, 2723–2729 (2010).
[Crossref]

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. 35, 763–765 (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, 3741–3747 (2010).
[Crossref] [PubMed]

2009 (6)

A. D. Zacharopoulos, P. Svenmarker, J. Axelsson, M. Schweiger, S. R. Arridge, and S. Andersson-Engels, “A matrix-free algorithm for multiple wavelength fluorescence tomography,” Opt. Express 17, 3025–3035 (2009).
[Crossref] [PubMed]

Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt. 48, 1328–1336 (2009).
[Crossref] [PubMed]

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14, 034044 (2009).
[Crossref] [PubMed]

Q. Fang and D. Boas, “Tetrahedral mesh generation from volumetric binary and gray-scale images,” in “Proceedings of IEEE International Symposium on Biomedical Imaging,”(2009), pp. 1142–1145.
[Crossref]

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

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. OHara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt. 14, 030501 (2009).
[Crossref] [PubMed]

2008 (1)

A. Martin, J. A. J, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes.” Molec. Imaging 7, 157–167 (2008).

2007 (4)

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

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577 (2007).
[Crossref] [PubMed]

S. C. Davis, H. Dehghani, J. Wang, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express 15, 4066–4082 (2007).
[Crossref] [PubMed]

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15, 6696–6716 (2007).
[Crossref] [PubMed]

2006 (1)

V. Ntziachristos, “Fluorescence molecular imaging,” Ann. Rev. Biomed. Eng. 8, 1–33 (2006).
[Crossref]

2005 (5)

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[Crossref] [PubMed]

G. Zacharakis, H. Kambara, H. Shih, J. Ripoll, J. Grimm, Y. Saeki, R. Weissleder, and V. Ntziachristos, “Volumetric tomography of fluorescent proteins through small animals in vivo,” Proc. Natl. Acad. Sci. U.S.A. 12, 18252–18257 (2005).
[Crossref]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50, 4225 (2005).
[Crossref] [PubMed]

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24, 1377–1386 (2005).
[Crossref] [PubMed]

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

2004 (1)

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12, 5402–5417 (2004).
[Crossref] [PubMed]

2003 (1)

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13, 195–208 (2003).
[PubMed]

2001 (1)

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26, 893–895 (2001).
[Crossref]

2000 (1)

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97, 2767–2772 (2000).
[Crossref] [PubMed]

1999 (1)

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inv. Probl. 15, 713–729 (1999).
[Crossref]

1998 (2)

J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Process. 7, 398–410 (1998).
[Crossref]

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image Process. 7, 421–432 (1998).
[Crossref]

1997 (3)

B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inv. Probl. 13, 729 (1997).
[Crossref]

M. Hanke, “A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems,” Inv. Probl. 13, 79 (1997).
[Crossref]

A. Chambolle and P. Lions, “Image recovery via total variation minimization and related problems,” Num. Math. 76, 167–188 (1997).
[Crossref]

1996 (1)

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters 21, 158–160 (1996).
[Crossref]

1993 (1)

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[Crossref]

1992 (1)

F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal. 29, 182–193 (1992).
[Crossref]

1990 (1)

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990).
[Crossref]

Aguirre, J.

J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Design and development of a co-planar fluorescence and X-ray tomograph,” in “Nuclear Science Symposium Conference Record, 2008. NSS ’08. IEEE,”(2008), pp. 5412–5413.
[Crossref]

Ale, A.

Alexandrakis, G.

Andersson-Engels, S.

Arridge, S.

Arridge, S. R.

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. 35, 763–765 (2010).
[Crossref] [PubMed]

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

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[Crossref] [PubMed]

S. R. Arridge, V. Kolehmainen, and M. J. Schweiger, “Reconstruction and regularisation in optical tomography,” in “Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT),”(Pisa, Italy, 2007), pp. 1–18.

Axelsson, J.

Bakker, L.

Bangerth, W.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12, 5402–5417 (2004).
[Crossref] [PubMed]

Beek, M. V.

Black, M. J.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image Process. 7, 421–432 (1998).
[Crossref]

Boas, D.

D. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. thesis, University of Pennsylvania, Philadelphia (USA) (1996).

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters 21, 158–160 (1996).
[Crossref]

Bremer, C.

Brendel, B.

Catteé, F.

F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal. 29, 182–193 (1992).
[Crossref]

Chambolle, A.

A. Chambolle and P. Lions, “Image recovery via total variation minimization and related problems,” Num. Math. 76, 167–188 (1997).
[Crossref]

Chance, B.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters 21, 158–160 (1996).
[Crossref]

Chatziioannou, A. F.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577 (2007).
[Crossref] [PubMed]

Choe, R.

Clason, C.

Coll, T.

F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal. 29, 182–193 (1992).
[Crossref]

Corlu, A.

Correia, T.

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14, 034044 (2009).
[Crossref] [PubMed]

Davis, S. C.

Dehghani, H.

der Mark, M. V.

Desco, M.

Dogdas, B.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577 (2007).
[Crossref] [PubMed]

Douiri, A.

Durduran, T.

Egger, H.

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion in fluorescence optical tomography,” IEEE Trans. Biomed. Eng. 57, 2723–2729 (2010).
[Crossref]

Elias, S.

Engl, H. W.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Mathematics and its applications (Kluwer Academic Publishers, 2000).

Fang, Q.

Fels, L.

Freiberger, M.

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion in fluorescence optical tomography,” IEEE Trans. Biomed. Eng. 57, 2723–2729 (2010).
[Crossref]

Garofalakis, A.

Gibbs-Strauss, S. L.

Gibson, A.

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14, 034044 (2009).
[Crossref] [PubMed]

Grimm, J.

Groetsch, C.

O. Scherzer and C. Groetsch, “Inverse scale space theory for inverse problems,” in “Scale-Space and Morphology in Computer Vision,”, vol. 2106 of Lecture Notes in Computer Science, M. Kerckhove, ed. (Springer, 2006), pp. 317–325.
[Crossref]

Gulsen, G.

Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt. 48, 1328–1336 (2009).
[Crossref] [PubMed]

Hanke, M.

M. Hanke, “A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems,” Inv. Probl. 13, 79 (1997).
[Crossref]

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Mathematics and its applications (Kluwer Academic Publishers, 2000).

Hansen, P. C.

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[Crossref]

Hebden, J.

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14, 034044 (2009).
[Crossref] [PubMed]

Heeger, D.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image Process. 7, 421–432 (1998).
[Crossref]

Hutchins, M.

J, J. A.

Jiang, S.

Joshi, A.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12, 5402–5417 (2004).
[Crossref] [PubMed]

Kaipio, J. P.

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inv. Probl. 15, 713–729 (1999).
[Crossref]

Kaltenbacher, B.

B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inv. Probl. 13, 729 (1997).
[Crossref]

Kambara, H.

Kepshire, D. S.

Khayat, M.

Kolehmainen, V.

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inv. Probl. 15, 713–729 (1999).
[Crossref]

Leahy, R. M.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577 (2007).
[Crossref] [PubMed]

Leblond, F.

Lin, Y.

Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt. 48, 1328–1336 (2009).
[Crossref] [PubMed]

Lions, P.

A. Chambolle and P. Lions, “Image recovery via total variation minimization and related problems,” Num. Math. 76, 167–188 (1997).
[Crossref]

F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal. 29, 182–193 (1992).
[Crossref]

Luijten, P.

Mali, W.

Malik, J.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990).
[Crossref]

Mamalaki, C.

Marimont, D. H.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image Process. 7, 421–432 (1998).
[Crossref]

Martin, A.

Meyer, H.

Mincu, N.

Morel, J.

F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal. 29, 182–193 (1992).
[Crossref]

Nachabe, R.

Nalcioglu, O.

Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt. 48, 1328–1336 (2009).
[Crossref] [PubMed]

Neubauer, A.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Mathematics and its applications (Kluwer Academic Publishers, 2000).

Nielsen, T.

Nissilä, I.

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[Crossref] [PubMed]

Ntziachristos, V.

V. Ntziachristos, “Fluorescence molecular imaging,” Ann. Rev. Biomed. Eng. 8, 1–33 (2006).
[Crossref]

O’Leary, D. P.

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[Crossref]

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters 21, 158–160 (1996).
[Crossref]

OHara, J. A.

Paulsen, K. D.

Perona, P.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990).
[Crossref]

Planas, A. M.

Pogue, B. W.

Rannou, F. R.

Riley, J.

Ripoll, J.

Rosen, M. A.

Rudge, T. J.

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. 35, 763–765 (2010).
[Crossref] [PubMed]

Saeki, Y.

Sapiro, G.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image Process. 7, 421–432 (1998).
[Crossref]

Sarantopoulos, A.

Sarasa-Renedo, A.

Scharfetter, H.

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion in fluorescence optical tomography,” IEEE Trans. Biomed. Eng. 57, 2723–2729 (2010).
[Crossref]

Scherzer, O.

Schnall, M.

Schnall, M. D.

Schotland, J. C.

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

Schulz, R. B.

Schweiger, M.

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14, 034044 (2009).
[Crossref] [PubMed]

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[Crossref] [PubMed]

Schweiger, M. J.

Sevick-Muraca, E. M.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12, 5402–5417 (2004).
[Crossref] [PubMed]

Shih, H.

Sisniega, A.

Soloviev, V. Y.

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. 35, 763–765 (2010).
[Crossref] [PubMed]

Somersalo, E.

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inv. Probl. 15, 713–729 (1999).
[Crossref]

Soubret, A.

Srinivasan, S.

Stout, D.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577 (2007).
[Crossref] [PubMed]

Svenmarker, P.

ter Haar Romeny, B. M.

J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Process. 7, 398–410 (1998).
[Crossref]

Tsoukatou, D.

van de Ven, S.

van der Voort, M.

Vaquero, J. J.

Vauhkonen, M.

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inv. Probl. 15, 713–729 (1999).
[Crossref]

Viergever, M. A.

J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Process. 7, 398–410 (1998).
[Crossref]

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).
[Crossref]

Wang, J.

Weickert, J.

J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Process. 7, 398–410 (1998).
[Crossref]

Weissleder, R.

Wiethoff, A.

Yan, H.

Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt. 48, 1328–1336 (2009).
[Crossref] [PubMed]

Yodh, A. G.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters 21, 158–160 (1996).
[Crossref]

Zacharakis, G.

Zacharopoulos, A. D.

Ann. Rev. Biomed. Eng. (1)

V. Ntziachristos, “Fluorescence molecular imaging,” Ann. Rev. Biomed. Eng. 8, 1–33 (2006).
[Crossref]

Appl. Opt. (2)

Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt. 48, 1328–1336 (2009).
[Crossref] [PubMed]

Eur. Radiol. (1)

IEEE Trans. Biomed. Eng. (1)

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion in fluorescence optical tomography,” IEEE Trans. Biomed. Eng. 57, 2723–2729 (2010).
[Crossref]

IEEE Trans. Image Process. (2)

J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Process. 7, 398–410 (1998).
[Crossref]

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image Process. 7, 421–432 (1998).
[Crossref]

IEEE Trans. Med. Imaging (1)

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

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990).
[Crossref]

Inv. Probl. (4)

B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inv. Probl. 13, 729 (1997).
[Crossref]

M. Hanke, “A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems,” Inv. Probl. 13, 79 (1997).
[Crossref]

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inv. Probl. 15, 713–729 (1999).
[Crossref]

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

J. Biomed. Opt. (2)

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14, 034044 (2009).
[Crossref] [PubMed]

Meas. Sci. Technol. (1)

Med. Phys. (1)

Molec. Imaging (1)

Molec. Imaging Biol. (1)

Num. Math. (1)

A. Chambolle and P. Lions, “Image recovery via total variation minimization and related problems,” Num. Math. 76, 167–188 (1997).
[Crossref]

Opt. Express (4)

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12, 5402–5417 (2004).
[Crossref] [PubMed]

Opt. Lett. (3)

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. 35, 763–765 (2010).
[Crossref] [PubMed]

Optics Letters (1)

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters 21, 158–160 (1996).
[Crossref]

Phys. Med. Biol. (3)

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[Crossref] [PubMed]

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577 (2007).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (2)

Proceedings of IEEE International Symposium on Biomedical Imaging (1)

SIAM J. Num. Anal. (1)

F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal. 29, 182–193 (1992).
[Crossref]

SIAM J. Sci. Comput. (1)

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[Crossref]

Other (6)

D. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. thesis, University of Pennsylvania, Philadelphia (USA) (1996).

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Mathematics and its applications (Kluwer Academic Publishers, 2000).

C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1 Reconstructions of h (mm⁻¹) from simulated data (image dimensions 32.5 mm × 88 mm) and PSNR values (dB): (a) the target, (b) reconstruction with Tikhonov prior, (c) reconstruction with anatomical prior and total variation, (d) reconstruction with anatomical prior and Perona-Malik function, (e) reconstruction with anatomical prior and Perona-Malik 2 function, (f) reconstruction with anatomical prior and Huber function, (g) reconstruction with anatomical prior and Tukey function, and (h) reconstruction with anatomical prior and exceedance function.

Download Full Size | PPT Slide | PDF

Fig. 2 Weighted Perona-Malik prior (a) at the first iteration, where only the anatomical edges are visible, and (b) at the last iteration, where both anatomical and fluorescence target edges can be seen.

Download Full Size | PPT Slide | PDF

Fig. 3 Normalised profile plots across the fluorescent target: (a) lateral direction and (b) longitudinal direction.

Download Full Size | PPT Slide | PDF

Fig. 4 Fluorochrome concentrations (μM) reconstructed from phantom data using : (a) Tikhonov prior, (b) structural prior and total variation, (c) structural prior and Perona-Malik function, (d) structural prior and Perona-Malik 2 function, and (e) structural prior and exceedance function. Images have dimensions 150 mm × 150 mm.

Download Full Size | PPT Slide | PDF

Fig. 5 Fluorochrome concentrations (μM) reconstructed from mouse data using : (a) Tikhonov prior, (b) structural prior and Perona-Malik 2. Images have dimensions 32 mm × 32 mm.

Download Full Size | PPT Slide | PDF

Tables (2)

Table 1 Edge-Preserving Functions

View Table | View all tables in this article

Algorithm 1 Two-Step Image Reconstruction Algorithm

View Table | View all tables in this article

Equations (39)

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

[- \nabla \cdot κ (r, λ_{e}) \nabla + μ_{a} (r, λ_{e})] U (r, λ_{e}) = q (r, λ_{e}), r \in Ω

[- \nabla \cdot κ (r, λ_{f}) \nabla + μ_{a} (r, λ_{f})] U (r, λ_{f}) = U (r, λ_{e}) h (r, λ_{f}), r \in Ω

U (r, λ) + 2 ζ κ (r, λ) \frac{\partial U (r, λ)}{\partial n} = 0, r \in \partial Ω

\begin{matrix} U (r_{s}, r_{d}, λ_{e}) & = & \int_{Ω} q (r, λ_{e}) G (r_{s}, r_{d}, λ_{e}) d r, \\ y_{e} & = & Θ_{e} (r_{s}, r_{d}) U (r_{s}, r_{d}, λ_{e}), \end{matrix}

\begin{matrix} U (r_{s}, r_{d}, λ_{f}) & = & \int_{Ω} h (r, λ_{f}) U (r_{s}, r, λ_{e}) G (r, r_{d}, λ_{f}) d r, \\ y_{f} & = & Θ_{f} (r_{s}, r_{d}, λ_{f}) U (r_{s}, r_{d}, λ_{f}), \end{matrix}

\frac{y_{f}}{y_{e}} = \frac{U (r_{s}, r_{d}, λ_{f})}{U (r_{s}, r_{d}, λ_{e})}

= \frac{1}{U (r_{s}, r_{d}, λ_{e})} \int_{Ω} h (r, λ_{f}) U (r_{s}, r, λ_{e}) G (r, r_{d}, λ_{f}) d r,

\frac{y_{f}}{y_{e}} = \frac{P (\partial Ω \to Σ) U (r_{s}, r_{d}, λ_{e})}{P (\partial Ω \to Σ) U (r_{s}, r_{d}, λ_{e})},

\frac{y_{f}}{y_{e}} = \frac{1}{U (r_{s}, r_{d}, λ_{e})} \sum_{i = 1}^{N} h (r_{i}, λ_{f}) U (r_{s}, r_{i}, λ_{e}) G (r_{i}, r_{d}, λ_{f}) v .

\hat{y} = \frac{y_{f}}{y_{e}} = J h,

y_{i} = \sum_{k}^{M_{x} \times M_{y}} c_{i k} ϕ_{i k}

\tilde{y} = \tilde{J} h .

\frac{\partial h}{\partial t} = \nabla \cdot [g (| \nabla h |) \nabla h],

\nabla \cdot [\frac{ψ' (| \nabla h |)}{| \nabla h |} \nabla h] = ψ' (| \nabla h |) \nabla \cdot (\frac{\nabla h}{| \nabla h |}) + Δ (ψ' (| \nabla h |)) \cdot \frac{\nabla h}{| \nabla h |} .

\nabla \cdot [\frac{ψ' (| \nabla h |)}{| \nabla h |} \nabla h] = \frac{ψ' (| \nabla h |)}{| \nabla h |} (h_{ξ_{1} ξ_{1}} + h_{ξ_{2} ξ_{2}}) + ψ ″ (| \nabla h |) h_{η η},

\nabla . [\frac{ψ' (| \nabla h |)}{| \nabla h |} \nabla h] = h_{ξ_{1} ξ_{1}} + h_{ξ_{2} ξ_{2}} + h_{η η} .

\frac{h_{i}^{k + 1} - h_{i}^{k}}{Δ t} = \sum_{j \in ℵ (i)} \frac{g_{j}^{k} + g_{i}^{k}}{2 d^{2}} (h_{j}^{k} - h_{i}^{k}),

\frac{h_{i}^{k + 1} - h_{i}^{k}}{Δ t} = \sum_{l = 1}^{m} ℒ_{1} (h^{k}) h^{k}

h^{k + 1} = (I + Δ t \sum_{l = 1}^{m} L_{l} (h^{k})) h^{k},

ℓ_{i j} (h^{k}) = {\begin{array}{l} \frac{g_{j}^{k} + g_{i}^{k}}{2 d x^{2}} & j \in ℵ (i) \\ - Σ_{j \in ℵ (i)} \frac{g_{j}^{k} + g_{i}^{k}}{2 d x^{2}} & j = i \\ 0 & e l s e \end{array}

h^{k + 1} = \frac{1}{m} \sum_{l = 1}^{m} {(I - m Δ t ℒ_{1} (h^{k}))}^{- 1} h^{k},

Minimise E (h) = \frac{1}{2} {| | \tilde{y} - \tilde{J} h | |}^{2} + α Ψ (h),

Ψ (h) = \int_{Ω} ψ (| \nabla h |) d Ω .

\nabla E (h) = {\tilde{J}}^{T} (\tilde{J} h - \tilde{y}) + α ℒ (h) h,

ℒ (h) = - \nabla \cdot [g (| \nabla h |) \nabla] .

Ψ_{W} (h) = \int_{Ω} W (| \nabla x_{r e f} |) ψ (| \nabla h |) d Ω,

ℒ_{W} (h) = \nabla \cdot [W (| \nabla x_{r e f} |) g (| \nabla h |) \nabla] .

h^{k + 1} = h^{k} + τ {({\tilde{J}}^{T} \tilde{J} + α Ψ " (h^{k}))}^{- 1} {\tilde{J}}^{T} ((\tilde{y} - \tilde{J} h^{k}) - α Ψ' (h^{k})),

L^{T} L = ℒ (h), Γ = {[ℒ (h)]}^{- 1} = L^{- 1} {(L^{- 1})}^{T} .

\overset{⌣}{J} = \tilde{J} L^{- 1}, \overset{⌣}{h} = L h,

Minimise E (\overset{⌣}{h}) = \frac{1}{2} {| | \tilde{y} - \overset{⌣}{J} \overset{⌣}{h} | |}^{2} + α {| | \overset{⌣}{h} | |}^{2} .

{\overset{⌣}{h}}^{k + 1} = {\overset{⌣}{h}}^{k} - {\overset{⌣}{J}}^{T} (\tilde{y} + \overset{⌣}{J} {\overset{⌣}{h}}^{k})

\Rightarrow h^{k + 1} = h^{k} + Γ {\tilde{J}}^{T} (\tilde{y} - \tilde{J} h^{k}) .

h^{k + 1 / 2} = h^{k} + {\tilde{J}}^{T} (\tilde{y} - \tilde{J} h^{k})

ℒ (h^{k + 1}) h^{k + 1} = h^{k + 1 / 2}

h^{k + 1 / 2} = h^{k} + {\tilde{J}}^{T} B (\tilde{y} - \tilde{J} h^{k}),

B = {(\tilde{J} {\tilde{J}}^{T} + λ I)}^{- 1},

P S N R = 10 {log}_{10} \frac{max (h_{t r u e})}{M S E},

M S E = \frac{Σ_{i = 1}^{X} Σ_{j = 1}^{Y} Σ_{k = 1}^{Z} {(h_{t r u e} - h_{r e c o n})}^{2}}{X \times Y \times Z} .

Perona- Malik (Cauchy or Lorentzian) [30]	$ψ (\| \nabla h \|) = \frac{T^{2}}{2} log (1 + {(\frac{\| \nabla h \|}{T})}^{2})$ $g (\| \nabla h \|) = \frac{1}{1 + {(\frac{\| \nabla h \|}{T})}^{2}}$
Perona- Malik 2 (Welsh) [30]	$ψ (\| \nabla h \|) = \frac{T^{2}}{2} [1 - exp (- {(\frac{\| \nabla h \|}{T})}^{2})]$ $g (\| \nabla h \|) = exp (- {(\frac{\| \nabla h \|}{T})}^{2})$
Total variation approximation [21]	$ψ (\| \nabla h \|) = T \sqrt{{\| \nabla h \|}^{2} + T^{2}} - T$ $g (\| \nabla h \|) = \frac{T}{\sqrt{{\| \nabla h \|}^{2} + T^{2}}}$
Huber [21, 30]	$ψ (\| \nabla h \|) = {\begin{matrix} T \| \nabla h \| - \frac{T^{2}}{2} & \| \nabla h \| > T \\ \frac{{\| \nabla h \|}^{2}}{2} & \| \nabla h \| \leq T \end{matrix}$ $g (\| \nabla h \|) = {\begin{array}{l} \frac{T}{\sqrt{{\| \nabla h \|}^{2} + T^{2}}} & \| \nabla h \| > T \\ 1 & \| \nabla h \| \leq T \end{array}$
Tukey [30]	$ψ (\| \nabla h \|) = {\begin{array}{l} \frac{T^{2}}{6} [1 - {[1 - {(\frac{\| \nabla h \|}{T})}^{2}]}^{3}] & \| \nabla h \| < T \\ \frac{T^{2}}{6} & \| \nabla h \| \geq T \end{array}$ $g (\| \nabla h \|) = {\begin{array}{l} {[1 - {(\frac{\| \nabla h \|}{T})}^{2}]}^{2} & \| \nabla h \| < T \\ 0 & \| \nabla h \| \geq T \end{array}$
Exceedance	$ψ (\| \nabla h \|) = \frac{\| \nabla h \|^{2}}{2} (1 - P (\| \nabla h \| \leq X))$ g(\|∇h\|) = 1 – P(\|∇h\| ≤ X) = P(\|∇h\| > X)

α⁰ = λ_lc trace(JJ^T)
u = 0
	ɛ = tolerance value
while \|\|ỹ – J̃h^k\|\|² < ɛ ∨ k < N_k do
h^k^+1/2 = h^k + J̃^T (J̃J̃^T + λI)⁻¹ (ỹ – J̃h^k)
	AOS -Thomas algorithm
for i = 1, ⋯ ,n do
for l = 1, ⋯ , m do
v = (I + mΔtℒ_l(h^k^+1/2))⁻¹ h^k^+1/2
u = u + v
end for
$h^{k + 1 / 2} = \frac{u}{m}$
end for
h^k⁺¹ = h^k^+1/2
if \|\|ỹ – J̃h^k^+1/2\|\|² < \|\|ỹ – J̃h^k\|\|² then
λ^k⁺¹ = λ^k – λ^k * p_λ
else
λ^k⁺¹ = λ^k +λ^k * p_λ
end if
end while

Perona- Malik (Cauchy or Lorentzian) [30]	$ψ (\| \nabla h \|) = \frac{T^{2}}{2} log (1 + {(\frac{\| \nabla h \|}{T})}^{2})$ $g (\| \nabla h \|) = \frac{1}{1 + {(\frac{\| \nabla h \|}{T})}^{2}}$
Perona- Malik 2 (Welsh) [30]	$ψ (\| \nabla h \|) = \frac{T^{2}}{2} [1 - exp (- {(\frac{\| \nabla h \|}{T})}^{2})]$ $g (\| \nabla h \|) = exp (- {(\frac{\| \nabla h \|}{T})}^{2})$
Total variation approximation [21]	$ψ (\| \nabla h \|) = T \sqrt{{\| \nabla h \|}^{2} + T^{2}} - T$ $g (\| \nabla h \|) = \frac{T}{\sqrt{{\| \nabla h \|}^{2} + T^{2}}}$
Huber [21, 30]	$ψ (\| \nabla h \|) = {\begin{matrix} T \| \nabla h \| - \frac{T^{2}}{2} & \| \nabla h \| > T \\ \frac{{\| \nabla h \|}^{2}}{2} & \| \nabla h \| \leq T \end{matrix}$ $g (\| \nabla h \|) = {\begin{array}{l} \frac{T}{\sqrt{{\| \nabla h \|}^{2} + T^{2}}} & \| \nabla h \| > T \\ 1 & \| \nabla h \| \leq T \end{array}$
Tukey [30]	$ψ (\| \nabla h \|) = {\begin{array}{l} \frac{T^{2}}{6} [1 - {[1 - {(\frac{\| \nabla h \|}{T})}^{2}]}^{3}] & \| \nabla h \| < T \\ \frac{T^{2}}{6} & \| \nabla h \| \geq T \end{array}$ $g (\| \nabla h \|) = {\begin{array}{l} {[1 - {(\frac{\| \nabla h \|}{T})}^{2}]}^{2} & \| \nabla h \| < T \\ 0 & \| \nabla h \| \geq T \end{array}$
Exceedance	$ψ (\| \nabla h \|) = \frac{\| \nabla h \|^{2}}{2} (1 - P (\| \nabla h \| \leq X))$ g(\|∇h\|) = 1 – P(\|∇h\| ≤ X) = P(\|∇h\| > X)

α⁰ = λ_lc trace(JJ^T)
u = 0
	ɛ = tolerance value
while \|\|ỹ – J̃h^k\|\|² < ɛ ∨ k < N_k do
h^k^+1/2 = h^k + J̃^T (J̃J̃^T + λI)⁻¹ (ỹ – J̃h^k)
	AOS -Thomas algorithm
for i = 1, ⋯ ,n do
for l = 1, ⋯ , m do
v = (I + mΔtℒ_l(h^k^+1/2))⁻¹ h^k^+1/2
u = u + v
end for
$h^{k + 1 / 2} = \frac{u}{m}$
end for
h^k⁺¹ = h^k^+1/2
if \|\|ỹ – J̃h^k^+1/2\|\|² < \|\|ỹ – J̃h^k\|\|² then
λ^k⁺¹ = λ^k – λ^k * p_λ
else
λ^k⁺¹ = λ^k +λ^k * p_λ
end if
end while