Abstract

Diffuse optical tomography (DOT) aims at recovering three-dimensional images of absorption and scattering parameters inside diffusive body based on small number of transmission measurements at the boundary of the body. This image reconstruction problem is known to be an ill-posed inverse problem, which requires use of prior information for successful reconstruction. We present a shape based method for DOT, where we assume a priori that the unknown body consist of disjoint subdomains with different optical properties. We utilize spherical harmonics expansion to parameterize the reconstruction problem with respect to the subdomain boundaries, and introduce a finite element (FEM) based algorithm that uses a novel 3D mesh subdivision technique to describe the mapping from spherical harmonics coefficients to the 3D absorption and scattering distributions inside a unstructured volumetric FEM mesh. We evaluate the shape based method by reconstructing experimental DOT data, from a cylindrical phantom with one inclusion with high absorption and one with high scattering. The reconstruction was monitored, and we found a 87% reduction in the Hausdorff measure between targets and reconstructed inclusions, 96% success in recovering the location of the centers of the inclusions and 87% success in average in the recovery for the volumes.

© 2009 Optical Society of America

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    [CrossRef]
  6. M. Schweiger and S.R. Arridge, “Optical Tomographic Reconstruction in a Complex Head Model Using a priori Region Boundary Information,” Phys. Med. Biol. 44, 2703–2721 (1999).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  29. S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
    [CrossRef] [PubMed]
  30. A. Björck, “Numerical methods for least square problems,” SIAM, (1996).
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    [CrossRef]
  32. T. Moller and B. Trumbore, Fast, minimum storage ray-triangle intersection,” J. Graphics Tools 2, 21–28 (1997).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. V. Kolehmainen, S.R. Arridge, M. Vauhkonen, and J.P. Kaipio, “Recovery of constant coefficients in optical diffusion tomography,” Opt. Express 7, 468–480 (2000).
    [CrossRef] [PubMed]

2009 (3)

2008 (2)

P. Hiltunen, D. Calvetti, and E. Somersalo, “An adaptive smoothness regularization algorithm for optical tomography,” Opt. Express 16, 19957–19977 (2008).
[CrossRef] [PubMed]

G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef] [PubMed]

2007 (1)

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

2006 (5)

M. Soleimani, W. R. B. Lionheart, and O. Dorn, “Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data,” Inverse Problems in Sc. and Eng. 14, 193–210 (2006).
[CrossRef]

M. Soleimani, O. Dorn, and W. R. B. Lionheart, “A narrow-band level set method applied to EIT in brain for cryosurgery monitoring,” IEEE Trans. Biomed. Eng. 53, 2257–2264 (2006).
[CrossRef] [PubMed]

A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems 22, 1509–1532 (2006).
[CrossRef]

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Problems 22, R67–R131, (2006).
[CrossRef]

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. 21, no. 4, 471–473 (2006).
[CrossRef]

2005 (3)

M. Schweiger, S. Arridge, and I. Nissila, “GaussNewton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[CrossRef] [PubMed]

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–58, (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43(2005).
[CrossRef] [PubMed]

2004 (1)

X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical tomography with physiological and spatial a priori constraints,” Phys. Med. Biol. 49, N155–63, (2004).
[CrossRef] [PubMed]

2003 (3)

B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen, and KD. “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE Sel. Top. Quantum Electron. 9, 199–209, (2003).
[CrossRef]

G. Bal, “Optical tomography for small volume absorbing inclusions,” Inverse Problems,  19, 371–386 (2003).
[CrossRef]

M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. 42, 3129–3144 (2003).
[CrossRef] [PubMed]

2002 (2)

I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila., Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. 73, 3306–3312 (2002).
[CrossRef]

V. Ntziachristos, A. G. Yodh, M. D. Schnall, and B. Chance, “MRI-guided diffuse optical spectroscopy of malignant and benign breast lesions,”  4, 347–354, (2002).

2000 (1)

1999 (4)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, 41—93 (1999).
[CrossRef]

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

V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems 15, 1375–1391 (1999).
[CrossRef]

M. Schweiger and S.R. Arridge, “Optical Tomographic Reconstruction in a Complex Head Model Using a priori Region Boundary Information,” Phys. Med. Biol. 44, 2703–2721 (1999).
[CrossRef] [PubMed]

1998 (3)

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Problems 14, 1107–1130 (1998).
[CrossRef]

D.J. Cedio-Fengya, S. Moskow, and M.S. Vogelius, “Identification of conductivity imperfections of small diameter by bounday measurements. Continuous dependence and computational reconstruction,” Inverse Problems 14, 553–595 (1998).
[CrossRef]

B. W. Pogue and K. D. Paulsen, “High-resolution near-infrared tomographic imaging simulations of the rat cranium by use of a priori magnetic resonance imaging structural information,” Opt. Lett. 23, Is.21, 1716–1718, (1998).
[CrossRef]

1997 (2)

J. Chang, H.L. Graber, P. C. Koo, R. Aronson, S.-L. S. Barbour, and R. L. Barbour, “Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources,” IEEE Trans. 16, 68–77,(1997).

T. Moller and B. Trumbore, Fast, minimum storage ray-triangle intersection,” J. Graphics Tools 2, 21–28 (1997).

1993 (1)

S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

1991 (1)

R. Kopperman, P. Meyer, and R.G. Wilson, “A Jordan surface theorem for three-dimensional digital spaces,” Discrete Comput. Geom. 6, 155–161 (1991).
[CrossRef]

Alvarez, D.

Aronson, R.

J. Chang, H.L. Graber, P. C. Koo, R. Aronson, S.-L. S. Barbour, and R. L. Barbour, “Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources,” IEEE Trans. 16, 68–77,(1997).

Arridge, S.

M. Schweiger, S. Arridge, and I. Nissila, “GaussNewton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge and J.C. Schotland, “Optical Tomography: Forward and Inverse Problems”, Inverse Problems 25, 12, (2009).
[CrossRef]

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. A 26, no. 5, 1277–1290 (2009).
[CrossRef]

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

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. 21, no. 4, 471–473 (2006).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43(2005).
[CrossRef] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, 41—93 (1999).
[CrossRef]

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Translational multimodality optical imaging, Chapter 5: Multimodal Diffuse Optical Tomography: Theory,” Artech Press, 101–124 (2008).

S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” Journal of Physics: Conference Series 125,012001 (2008).

Arridge, S.R

A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems 22, 1509–1532 (2006).
[CrossRef]

Arridge, S.R.

V. Kolehmainen, S.R. Arridge, M. Vauhkonen, and J.P. Kaipio, “Recovery of constant coefficients in optical diffusion tomography,” Opt. Express 7, 468–480 (2000).
[CrossRef] [PubMed]

M. Schweiger and S.R. Arridge, “Optical Tomographic Reconstruction in a Complex Head Model Using a priori Region Boundary Information,” Phys. Med. Biol. 44, 2703–2721 (1999).
[CrossRef] [PubMed]

V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems 15, 1375–1391 (1999).
[CrossRef]

S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Bal, G.

G. Bal, “Optical tomography for small volume absorbing inclusions,” Inverse Problems,  19, 371–386 (2003).
[CrossRef]

Barbaro, A.

Barbour, R. L.

J. Chang, H.L. Graber, P. C. Koo, R. Aronson, S.-L. S. Barbour, and R. L. Barbour, “Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources,” IEEE Trans. 16, 68–77,(1997).

Barbour, S.-L. S.

J. Chang, H.L. Graber, P. C. Koo, R. Aronson, S.-L. S. Barbour, and R. L. Barbour, “Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources,” IEEE Trans. 16, 68–77,(1997).

Beatson, R.

N. Naik, R. Beatson, J. Eriksson, and E. van Houten, “An implicit radial basis function based reconstruction approach to electromagnetic shape tomography,” Inverse Problems25, no. 2 (2009).
[CrossRef]

Björck, A.

A. Björck, “Numerical methods for least square problems,” SIAM, (1996).

Boas, D.

Boas, D. A.

A. G. Yodh and D. A. Boas, “Functional imaging with diffusing light,” Biomedical photonics handbook(CRC Press, 2003).

Boas, D.A.

G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef] [PubMed]

Boverman, G.

G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef] [PubMed]

Brooks, D.H.

G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef] [PubMed]

Brooksby, B.A.

B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen, and KD. “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE Sel. Top. Quantum Electron. 9, 199–209, (2003).
[CrossRef]

Calvetti, D.

Cedio-Fengya, D.J.

D.J. Cedio-Fengya, S. Moskow, and M.S. Vogelius, “Identification of conductivity imperfections of small diameter by bounday measurements. Continuous dependence and computational reconstruction,” Inverse Problems 14, 553–595 (1998).
[CrossRef]

Chance, B.

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–58, (2005).
[CrossRef] [PubMed]

X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical tomography with physiological and spatial a priori constraints,” Phys. Med. Biol. 49, N155–63, (2004).
[CrossRef] [PubMed]

V. Ntziachristos, A. G. Yodh, M. D. Schnall, and B. Chance, “MRI-guided diffuse optical spectroscopy of malignant and benign breast lesions,”  4, 347–354, (2002).

Chang, J.

J. Chang, H.L. Graber, P. C. Koo, R. Aronson, S.-L. S. Barbour, and R. L. Barbour, “Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources,” IEEE Trans. 16, 68–77,(1997).

Dehghani, H.

B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen, and KD. “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE Sel. Top. Quantum Electron. 9, 199–209, (2003).
[CrossRef]

Delpy, D.T.

S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Dorn, O.

A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems 22, 1509–1532 (2006).
[CrossRef]

M. Soleimani, W. R. B. Lionheart, and O. Dorn, “Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data,” Inverse Problems in Sc. and Eng. 14, 193–210 (2006).
[CrossRef]

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. 21, no. 4, 471–473 (2006).
[CrossRef]

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Problems 22, R67–R131, (2006).
[CrossRef]

M. Soleimani, O. Dorn, and W. R. B. Lionheart, “A narrow-band level set method applied to EIT in brain for cryosurgery monitoring,” IEEE Trans. Biomed. Eng. 53, 2257–2264 (2006).
[CrossRef] [PubMed]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Problems 14, 1107–1130 (1998).
[CrossRef]

S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” Journal of Physics: Conference Series 125,012001 (2008).

Douiri, A.

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

Eriksson, J.

N. Naik, R. Beatson, J. Eriksson, and E. van Houten, “An implicit radial basis function based reconstruction approach to electromagnetic shape tomography,” Inverse Problems25, no. 2 (2009).
[CrossRef]

Fallström, K.

I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila., Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. 73, 3306–3312 (2002).
[CrossRef]

Gibson, A. P.

Graber, H.L.

J. Chang, H.L. Graber, P. C. Koo, R. Aronson, S.-L. S. Barbour, and R. L. Barbour, “Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources,” IEEE Trans. 16, 68–77,(1997).

Guven, M.

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–58, (2005).
[CrossRef] [PubMed]

X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical tomography with physiological and spatial a priori constraints,” Phys. Med. Biol. 49, N155–63, (2004).
[CrossRef] [PubMed]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43(2005).
[CrossRef] [PubMed]

Hiltunen, P.

Hiraoka, M.

S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Intes, X.

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–58, (2005).
[CrossRef] [PubMed]

X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical tomography with physiological and spatial a priori constraints,” Phys. Med. Biol. 49, N155–63, (2004).
[CrossRef] [PubMed]

Isaacson, D.

G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef] [PubMed]

Kaipio, J. P.

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

Kaipio, J.P.

V. Kolehmainen, S.R. Arridge, M. Vauhkonen, and J.P. Kaipio, “Recovery of constant coefficients in optical diffusion tomography,” Opt. Express 7, 468–480 (2000).
[CrossRef] [PubMed]

V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems 15, 1375–1391 (1999).
[CrossRef]

Katila, T.

I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila., Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. 73, 3306–3312 (2002).
[CrossRef]

KD.,

B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen, and KD. “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE Sel. Top. Quantum Electron. 9, 199–209, (2003).
[CrossRef]

Kilmer, M. E.

Kolehmainen, V.

A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems 22, 1509–1532 (2006).
[CrossRef]

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. 21, no. 4, 471–473 (2006).
[CrossRef]

V. Kolehmainen, S.R. Arridge, M. Vauhkonen, and J.P. Kaipio, “Recovery of constant coefficients in optical diffusion tomography,” Opt. Express 7, 468–480 (2000).
[CrossRef] [PubMed]

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

V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems 15, 1375–1391 (1999).
[CrossRef]

S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” Journal of Physics: Conference Series 125,012001 (2008).

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Translational multimodality optical imaging, Chapter 5: Multimodal Diffuse Optical Tomography: Theory,” Artech Press, 101–124 (2008).

Koo, P. C.

J. Chang, H.L. Graber, P. C. Koo, R. Aronson, S.-L. S. Barbour, and R. L. Barbour, “Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources,” IEEE Trans. 16, 68–77,(1997).

Kopperman, R.

R. Kopperman, P. Meyer, and R.G. Wilson, “A Jordan surface theorem for three-dimensional digital spaces,” Discrete Comput. Geom. 6, 155–161 (1991).
[CrossRef]

Kotilahti, K.

I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila., Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. 73, 3306–3312 (2002).
[CrossRef]

Leahy, R. M.

Lesselier, D.

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Problems 22, R67–R131, (2006).
[CrossRef]

Lionheart, W. R. B.

M. Soleimani, O. Dorn, and W. R. B. Lionheart, “A narrow-band level set method applied to EIT in brain for cryosurgery monitoring,” IEEE Trans. Biomed. Eng. 53, 2257–2264 (2006).
[CrossRef] [PubMed]

M. Soleimani, W. R. B. Lionheart, and O. Dorn, “Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data,” Inverse Problems in Sc. and Eng. 14, 193–210 (2006).
[CrossRef]

Lionheart, W.R.B.

V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems 15, 1375–1391 (1999).
[CrossRef]

Maloux, C.

X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical tomography with physiological and spatial a priori constraints,” Phys. Med. Biol. 49, N155–63, (2004).
[CrossRef] [PubMed]

Medina, P.

Meyer, P.

R. Kopperman, P. Meyer, and R.G. Wilson, “A Jordan surface theorem for three-dimensional digital spaces,” Discrete Comput. Geom. 6, 155–161 (1991).
[CrossRef]

Miller, E. L.

Miller, E.L.

G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef] [PubMed]

Moller, T.

T. Moller and B. Trumbore, Fast, minimum storage ray-triangle intersection,” J. Graphics Tools 2, 21–28 (1997).

Moscoso, M.

Moskow, S.

D.J. Cedio-Fengya, S. Moskow, and M.S. Vogelius, “Identification of conductivity imperfections of small diameter by bounday measurements. Continuous dependence and computational reconstruction,” Inverse Problems 14, 553–595 (1998).
[CrossRef]

Naik, N.

N. Naik, R. Beatson, J. Eriksson, and E. van Houten, “An implicit radial basis function based reconstruction approach to electromagnetic shape tomography,” Inverse Problems25, no. 2 (2009).
[CrossRef]

Nissila, I.

M. Schweiger, S. Arridge, and I. Nissila, “GaussNewton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[CrossRef] [PubMed]

Nissilä, I.

I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila., Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. 73, 3306–3312 (2002).
[CrossRef]

Ntziachristos, V.

V. Ntziachristos, A. G. Yodh, M. D. Schnall, and B. Chance, “MRI-guided diffuse optical spectroscopy of malignant and benign breast lesions,”  4, 347–354, (2002).

Panagiotou, C.

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. A 26, no. 5, 1277–1290 (2009).
[CrossRef]

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Translational multimodality optical imaging, Chapter 5: Multimodal Diffuse Optical Tomography: Theory,” Artech Press, 101–124 (2008).

Paulsen, K. D.

Paulsen, K.D.

B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen, and KD. “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE Sel. Top. Quantum Electron. 9, 199–209, (2003).
[CrossRef]

Pogue, B. W.

Pogue, B.W.

B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen, and KD. “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE Sel. Top. Quantum Electron. 9, 199–209, (2003).
[CrossRef]

Qianqian, F.

G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef] [PubMed]

Riley, J.

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

Schnall, M. D.

V. Ntziachristos, A. G. Yodh, M. D. Schnall, and B. Chance, “MRI-guided diffuse optical spectroscopy of malignant and benign breast lesions,”  4, 347–354, (2002).

Schotland, J.C.

S. R. Arridge and J.C. Schotland, “Optical Tomography: Forward and Inverse Problems”, Inverse Problems 25, 12, (2009).
[CrossRef]

Schweiger, M.

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. A 26, no. 5, 1277–1290 (2009).
[CrossRef]

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

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. 21, no. 4, 471–473 (2006).
[CrossRef]

M. Schweiger, S. Arridge, and I. Nissila, “GaussNewton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[CrossRef] [PubMed]

M. Schweiger and S.R. Arridge, “Optical Tomographic Reconstruction in a Complex Head Model Using a priori Region Boundary Information,” Phys. Med. Biol. 44, 2703–2721 (1999).
[CrossRef] [PubMed]

S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Translational multimodality optical imaging, Chapter 5: Multimodal Diffuse Optical Tomography: Theory,” Artech Press, 101–124 (2008).

S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” Journal of Physics: Conference Series 125,012001 (2008).

Sikora, J.

A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems 22, 1509–1532 (2006).
[CrossRef]

Soleimani, M.

M. Soleimani, W. R. B. Lionheart, and O. Dorn, “Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data,” Inverse Problems in Sc. and Eng. 14, 193–210 (2006).
[CrossRef]

M. Soleimani, O. Dorn, and W. R. B. Lionheart, “A narrow-band level set method applied to EIT in brain for cryosurgery monitoring,” IEEE Trans. Biomed. Eng. 53, 2257–2264 (2006).
[CrossRef] [PubMed]

Somayajula, S.

Somersalo, E.

P. Hiltunen, D. Calvetti, and E. Somersalo, “An adaptive smoothness regularization algorithm for optical tomography,” Opt. Express 16, 19957–19977 (2008).
[CrossRef] [PubMed]

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

Trumbore, B.

T. Moller and B. Trumbore, Fast, minimum storage ray-triangle intersection,” J. Graphics Tools 2, 21–28 (1997).

van Houten, E.

N. Naik, R. Beatson, J. Eriksson, and E. van Houten, “An implicit radial basis function based reconstruction approach to electromagnetic shape tomography,” Inverse Problems25, no. 2 (2009).
[CrossRef]

Vauhkonen, M.

V. Kolehmainen, S.R. Arridge, M. Vauhkonen, and J.P. Kaipio, “Recovery of constant coefficients in optical diffusion tomography,” Opt. Express 7, 468–480 (2000).
[CrossRef] [PubMed]

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

V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems 15, 1375–1391 (1999).
[CrossRef]

Vogel, C.R.

C.R. Vogel, “Computational methods for inverse problems,” Frontiers in Applied Mathematics Series, SIAM, 23 (2002).

Vogelius, M.S.

D.J. Cedio-Fengya, S. Moskow, and M.S. Vogelius, “Identification of conductivity imperfections of small diameter by bounday measurements. Continuous dependence and computational reconstruction,” Inverse Problems 14, 553–595 (1998).
[CrossRef]

Wilson, R.G.

R. Kopperman, P. Meyer, and R.G. Wilson, “A Jordan surface theorem for three-dimensional digital spaces,” Discrete Comput. Geom. 6, 155–161 (1991).
[CrossRef]

Yazici, B.

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–58, (2005).
[CrossRef] [PubMed]

X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical tomography with physiological and spatial a priori constraints,” Phys. Med. Biol. 49, N155–63, (2004).
[CrossRef] [PubMed]

Yodh, A. G.

V. Ntziachristos, A. G. Yodh, M. D. Schnall, and B. Chance, “MRI-guided diffuse optical spectroscopy of malignant and benign breast lesions,”  4, 347–354, (2002).

A. G. Yodh and D. A. Boas, “Functional imaging with diffusing light,” Biomedical photonics handbook(CRC Press, 2003).

Zacharopoulos, A.

A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems 22, 1509–1532 (2006).
[CrossRef]

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. 21, no. 4, 471–473 (2006).
[CrossRef]

S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” Journal of Physics: Conference Series 125,012001 (2008).

Appl. Opt. (1)

Discrete Comput. Geom. (1)

R. Kopperman, P. Meyer, and R.G. Wilson, “A Jordan surface theorem for three-dimensional digital spaces,” Discrete Comput. Geom. 6, 155–161 (1991).
[CrossRef]

IEEE Sel. Top. Quantum Electron. (1)

B.A. Brooksby, H. Dehghani, B.W. Pogue, K.D. Paulsen, and KD. “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE Sel. Top. Quantum Electron. 9, 199–209, (2003).
[CrossRef]

IEEE Trans. (1)

J. Chang, H.L. Graber, P. C. Koo, R. Aronson, S.-L. S. Barbour, and R. L. Barbour, “Optical Imaging of Anatomical Maps Derived from Magnetic Resonance Images Using Time-Independent Optical Sources,” IEEE Trans. 16, 68–77,(1997).

IEEE Trans. Biomed. Eng. (1)

M. Soleimani, O. Dorn, and W. R. B. Lionheart, “A narrow-band level set method applied to EIT in brain for cryosurgery monitoring,” IEEE Trans. Biomed. Eng. 53, 2257–2264 (2006).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (1)

G. Boverman, E.L. Miller, D.H. Brooks, D. Isaacson, F. Qianqian, and D.A. Boas, “Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography,” IEEE Trans. Med. Imaging 27, 752–765 (2008).
[CrossRef] [PubMed]

Inverse Problems (9)

A. Zacharopoulos, S.R Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, “Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrisation and a boundary element method,” Inverse Problems 22, 1509–1532 (2006).
[CrossRef]

V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inverse Problems 15, 1375–1391 (1999).
[CrossRef]

D.J. Cedio-Fengya, S. Moskow, and M.S. Vogelius, “Identification of conductivity imperfections of small diameter by bounday measurements. Continuous dependence and computational reconstruction,” Inverse Problems 14, 553–595 (1998).
[CrossRef]

G. Bal, “Optical tomography for small volume absorbing inclusions,” Inverse Problems,  19, 371–386 (2003).
[CrossRef]

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

S. R. Arridge and J.C. Schotland, “Optical Tomography: Forward and Inverse Problems”, Inverse Problems 25, 12, (2009).
[CrossRef]

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Problems 22, R67–R131, (2006).
[CrossRef]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Problems 14, 1107–1130 (1998).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15, 41—93 (1999).
[CrossRef]

Inverse Problems in Sc. and Eng. (1)

M. Soleimani, W. R. B. Lionheart, and O. Dorn, “Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data,” Inverse Problems in Sc. and Eng. 14, 193–210 (2006).
[CrossRef]

J. Graphics Tools (1)

T. Moller and B. Trumbore, Fast, minimum storage ray-triangle intersection,” J. Graphics Tools 2, 21–28 (1997).

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

Meas. Sci. Tech. (1)

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

Med. Phys. (1)

S.R. Arridge, M. Schweiger, M. Hiraoka, and D.T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (2)

Phys. Med. Biol. (5)

M. Schweiger, S. Arridge, and I. Nissila, “GaussNewton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[CrossRef] [PubMed]

M. Schweiger and S.R. Arridge, “Optical Tomographic Reconstruction in a Complex Head Model Using a priori Region Boundary Information,” Phys. Med. Biol. 44, 2703–2721 (1999).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43(2005).
[CrossRef] [PubMed]

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–58, (2005).
[CrossRef] [PubMed]

X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical tomography with physiological and spatial a priori constraints,” Phys. Med. Biol. 49, N155–63, (2004).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila., Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. 73, 3306–3312 (2002).
[CrossRef]

Other (7)

A. Björck, “Numerical methods for least square problems,” SIAM, (1996).

C.R. Vogel, “Computational methods for inverse problems,” Frontiers in Applied Mathematics Series, SIAM, 23 (2002).

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Translational multimodality optical imaging, Chapter 5: Multimodal Diffuse Optical Tomography: Theory,” Artech Press, 101–124 (2008).

A. G. Yodh and D. A. Boas, “Functional imaging with diffusing light,” Biomedical photonics handbook(CRC Press, 2003).

V. Ntziachristos, A. G. Yodh, M. D. Schnall, and B. Chance, “MRI-guided diffuse optical spectroscopy of malignant and benign breast lesions,”  4, 347–354, (2002).

N. Naik, R. Beatson, J. Eriksson, and E. van Houten, “An implicit radial basis function based reconstruction approach to electromagnetic shape tomography,” Inverse Problems25, no. 2 (2009).
[CrossRef]

S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” Journal of Physics: Conference Series 125,012001 (2008).

Supplementary Material (3)

» Media 1: MOV (1185 KB)     
» Media 2: MOV (1243 KB)     
» Media 3: MOV (1173 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Demonstration of the tetrahedral element subdivision scheme used for the two different cases of surface-tetrahedron intersection. (a) In the case when one node denoted (+) is on the one side of the surface while the three others denoted by (-) on the other partition created by the surface. Having found the three intersection points along the adjacent edges of the node, denoted by (x) we divide the tetraherdon into eight subelements shown exploded in (c). (b) In this case, two nodes lay on each side of the surface dividing the tetrahedron into two pentahedral prisms, one on each side of the surface. Having found the four intersection points kv 1kv 4, denoted (x) along the edges connecting each the nodes on the one side (+) of the surface to those on the other (-), we cut the element into six tetrahedra subelements shown exploded in (d).

Fig. 2.
Fig. 2.

The experimental setup used. The sources are denoted by “x” and detectors are de- noted “o”. We also have drawn the positions of the two blobs the one on the left with double the absorption of the background and the one on theright that has double the scattering.

Fig. 3.
Fig. 3.

Comparison between the target, the reconstruction with the pixel based and the shape based method in vertical cross sections along the x-z plane (a) The real position of the µ a target in a slice along the y=0 plane and (b) the real position of the µ′s target in a slice along the y=-12.3. (c) The y=0 slice of the recovered µ a image from the use of a pixel based method as described in [34].(d) The y=-12.3 slice of the recovered µ s image from the use of a pixel based method as described in [34]. We notice that the pixel based reconstruction is successful but the contrast is smoothed out due to the regularisation used and the homogeneous background contains artifacts.(e) A view along the y-axis of the recovered µ a object from the shape based reconstruction in solid color and the initial guess for the object as an outline. (f) A view along the y-axis of the recovered µs object from the shape based reconstruction in solid color and the initial guess for the object as an outline. We notice that the recovered shapes and locations of the object are very close to the expected ones.

Fig. 4.
Fig. 4.

Comparison of the view along the central x-y plane between target, the reconstruction with the pixel based and the shape based method As in figure 3 for horizontal slices along the z=0 plane, the real positions of the targets in (a) and (b) and the result from the pixel based method in (c) and (d), and a view along the z-axis (e) and (f) for the shape based method.

Fig. 5.
Fig. 5.

Metrics used to assess the quality of the reconstruction, plotted for the iterations of the shape evolution. (a) The normalized residual ‖g-A(γn )‖2 2/|g-A(γ 1)‖2 2. (b) the volume ratios between targets and reconstructed shapes. (c) the Hausdorff measure for the mismatch between the surfaces of the shapes and (d) the distance between the center of the target objects and the reconstructed ones.

Fig. 6.
Fig. 6.

Reconstructions from experimental data, using different initial positions, as presented in the media files attached to the paper. The first row of the graph consists of the initial guesses in yellow and the experimental targets, red for absorption and blue for scattering displayed in Fig. 6(a) (media 1) Fig. 6(b) (media 2) and Fig. 6(c) (media 3). In the second row we have the reconstruction that resulted from those different initial guesses respectively, with the targets recovered successfully in each case.

Equations (38)

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

D(r)·Φ(r,ω)+μa(r)Φ(r,ω)+iωcΦ(r,ω)=q (r,ω)
D=13(μa+μs),
Φ(r,ω)+2ζD(r)Φ(r,ω)ν=0 , r Ω
y(r)=MΦ=D (r)Φ(r;ω)ν,rΩ
A:𝒫𝓩,A(p)=y, p 𝒫 and y 𝓩
(K(D)+C(μa)+12αE+iωB) Φ=q
Kij=ΩD(r)υi(r)·υj(r)dr
Cij=Ωμa(r)υi(r)υj(r)dr
Eij=Ωυi(r)υj(r)dS
Bij=1cΩυi(r)υj(r)dr
Φ=(K(D)+C(μa)+12αE+iωB)1q
rΓ={xΓ(ϑ,φ)=Σl=0WΣm=llCl,x,mY˜lm(ϑ,φ),yΓ(ϑ,φ)=Σl=0WΣm=llCl,y,mY˜lm(ϑ,φ),zΓ(ϑ,φ)=Σl=0WΣm=llCl,z,mY˜lm(ϑ,φ),
Y˜lm(ϑ,φ) {Re[Ylm](ϑ,φ),whenm0,Im[Ylm](ϑ,φ),whenm>0,
γ={γk}={Cl,x,m,Cl,y,m,Cl,z,m},withl=1 , W ,m= l , l
μa==0Lμa,χ (r),D==0LDχ(r)
Kij = =0Lsupp(υiυj)ΩD(r)υi(r)·υj(r)dr
Cij = =0Lsupp(υiυj)Ωμa,(r)υi(r)·υj(r)dr
I (Γ)={HnHnΩ,andHnΩØ}
(T2νkν)×(T2νT1ν)=0
kν = (xνyνzν) = (l=0Wm=llCl,x,mY˜lm(ϑkν,φkν)l=0Wm=llCl,y,mY˜lm(ϑkν,φkν)l=0Wm=llCl,z,mY˜lm(ϑkν,φkν)).
Ξ (γ)=gA(γ,μa,D)22 = gA(p)22
find γ* so that : Ξ (γ*)=mingA(γ*,μa,D)22
γn+1 = γn + (JnTJn+)1 JnT (gA(γn,μa,D)).
J = A(γ,μa,D)γ ,
S (γ,μa,D) K (γ,D)+C(γ,μa)+12αE+B
S Φγ = Sγ Φ
Φγ = S1 Sγ Φ .
J = M [Φγ] = M S1 Sγ Φ = Φ* Sγ Φ
(Sγk)ij = (δμa)p,pΓγkυp·υp+(δD)p,pΓγkυp·υp
Jsd,k = (δμa)ΦΓ*(d)GkΦΓ(s)(δD)ΦΓ*(d)FkΦΓ(s)
Gppk = Γγkφp(r)·φp(r)
Fppk = Γγkφp(r)·φp(r)
Φ (Ss)=l=14=Φ(l)υl(Ss)
g˜ = (Re(ln(g))Im(ln(g)))
J˜ = (Re[diag(1A(γ,μa,D))J(γ)]Im[diag(1A(γ,μa,D))J(γ)])
Ξ˜ (γ)=gmeasA(γ,μa,D)gref+A(pref)22
gA(γn)22/|gA(γ1)22
C(Γptarger,Γpn)=|ΓpctargerΓpcn|

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