Abstract

We present a shape-based approach to three-dimensional image reconstruction from diffuse optical data. Our approach differs from others in the literature in that we jointly reconstruct object and background characterization and localization simultaneously, rather than sequentially process for optical properties and postprocess for edges. The key to the efficiency and robustness of our algorithm is in the model we propose for the optical properties of the background and anomaly: We use a low-order parameterization of the background and another for the interior of the anomaly, and we use an ellipsoid to describe the boundary of the anomaly. This model has the effect of regularizing the inversion problem and provides a natural means of including additional physical properties if they are known a priori. A Gauss-Newton-type algorithm with line search is implemented to solve the underlying nonlinear least-squares problem and thereby determine the coefficients of the parameterizations and the descriptors of the ellipsoid. Numerical results show the effectiveness of this method.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2002

M. Belge, M. Kilmer, E. Miller, “Efficient selection of multiple regularization parameters in a generalized L-curve framework,” Inverse Prob. 18, 1161–1183 (2002).
[CrossRef]

2001

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

V. Ntziachristos, B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Res. 3, 41–46 (2001).
[CrossRef] [PubMed]

D. Boas, T. Gaudette, G. Strangman, X. Cheng, J. Marota, J. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001).
[CrossRef] [PubMed]

V. Ntziachristos, A. Hielscher, A. Yodh, B. Chance, “Diffuse optical tomography of highly heterogeneous media,” IEEE Trans. Med. Imaging 20, 470–478 (2001).
[CrossRef] [PubMed]

2000

M. Kilmer, E. Miller, D. Boas, D. Brooks, “A shape-based reconstruction technique for DPDW data,” Opt. Express 7, 481–491 (2000); http://www.opticsexpress.org .

O. Dorn, E. Miller, C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

V. Kolehmainen, M. Vauhkonen, J. Kaipio, S. R. Arridge, “Recovery of piecewise constant coefficients in optical diffusion tomography,” Opt. Express 7, 468–480 (2000); http://www.opticsexpress.org .

E. Miller, M. Kilmer, C. Rappaport, “A new shape-based method for object localization and characterization from scattered field data,” IEEE Trans. Geosci. Remote Sens. 38, 1682–1696 (2000).
[CrossRef]

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

M. Braunstein, R. Levine, “Three-dimensional tomographic reconstruction of an absorptive perturbation with diffuse photon density waves,” J. Opt. Soc. Am. A 17, 11–20 (2000).
[CrossRef]

1999

M. Schweiger, 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]

J. Ye, K. Webb, C. Bouman, R. Millane, “Optical diffusion tomography by iterative-coordinate-descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400–2412 (1999).
[CrossRef]

H. Jiang, “Optical image reconstruction based on the third-order diffusion equations,” Opt. Exp. 4, 241–246 (1999); http://www.opticsexpress.org .

B. Pogue, T. McBride, J. Prewitt, U. Sterberg, K. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. 38, 2950–2961 (1999).
[CrossRef]

S. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

1995

1994

1980

D. O’Leary, “The block conjugate gradient algorithm and related methods,” Linear Algebr. Appl. 29, 293–322 (1980).
[CrossRef]

Arridge, S.

Arridge, S. R.

V. Kolehmainen, M. Vauhkonen, J. Kaipio, S. R. Arridge, “Recovery of piecewise constant coefficients in optical diffusion tomography,” Opt. Express 7, 468–480 (2000); http://www.opticsexpress.org .

M. Schweiger, 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]

Belge, M.

M. Belge, M. Kilmer, E. Miller, “Efficient selection of multiple regularization parameters in a generalized L-curve framework,” Inverse Prob. 18, 1161–1183 (2002).
[CrossRef]

Benaron, D.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Boas, D.

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

D. Boas, T. Gaudette, G. Strangman, X. Cheng, J. Marota, J. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001).
[CrossRef] [PubMed]

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

M. Kilmer, E. Miller, D. Boas, D. Brooks, “A shape-based reconstruction technique for DPDW data,” Opt. Express 7, 481–491 (2000); http://www.opticsexpress.org .

Bouman, C.

Braunstein, M.

Brooks, D.

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

M. Kilmer, E. Miller, D. Boas, D. Brooks, “A shape-based reconstruction technique for DPDW data,” Opt. Express 7, 481–491 (2000); http://www.opticsexpress.org .

Chance, B.

V. Ntziachristos, A. Hielscher, A. Yodh, B. Chance, “Diffuse optical tomography of highly heterogeneous media,” IEEE Trans. Med. Imaging 20, 470–478 (2001).
[CrossRef] [PubMed]

V. Ntziachristos, B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Res. 3, 41–46 (2001).
[CrossRef] [PubMed]

Cheng, X.

D. Boas, T. Gaudette, G. Strangman, X. Cheng, J. Marota, J. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001).
[CrossRef] [PubMed]

Cheong, W.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Dennis, J.

J. Dennis, R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

DiMarzio, C.

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Dorn, O.

O. Dorn, E. Miller, C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

Feng, T.

Frahn, J.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Gaudette, R.

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Gaudette, T.

D. Boas, T. Gaudette, G. Strangman, X. Cheng, J. Marota, J. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001).
[CrossRef] [PubMed]

Hansen, P.

P. Hansen, M. Kilmer, R. Kjeldsen, “Exploiting residual information in the regularization of discrete ill-posed problems,” SIAM J. Matrix Anal. Appl.

Haskell, R.

Hassani, S.

S. Hassani, Foundations of Mathematical Physics (Allyn and Bacon, Boston, 1991).

Hielscher, A.

V. Ntziachristos, A. Hielscher, A. Yodh, B. Chance, “Diffuse optical tomography of highly heterogeneous media,” IEEE Trans. Med. Imaging 20, 470–478 (2001).
[CrossRef] [PubMed]

Hintz, S.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Hirth, C.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Jiang, H.

H. Jiang, “Optical image reconstruction based on the third-order diffusion equations,” Opt. Exp. 4, 241–246 (1999); http://www.opticsexpress.org .

Kaipio, J.

Kermit, E.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Kilmer, M.

M. Belge, M. Kilmer, E. Miller, “Efficient selection of multiple regularization parameters in a generalized L-curve framework,” Inverse Prob. 18, 1161–1183 (2002).
[CrossRef]

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

M. Kilmer, E. Miller, D. Boas, D. Brooks, “A shape-based reconstruction technique for DPDW data,” Opt. Express 7, 481–491 (2000); http://www.opticsexpress.org .

E. Miller, M. Kilmer, C. Rappaport, “A new shape-based method for object localization and characterization from scattered field data,” IEEE Trans. Geosci. Remote Sens. 38, 1682–1696 (2000).
[CrossRef]

P. Hansen, M. Kilmer, R. Kjeldsen, “Exploiting residual information in the regularization of discrete ill-posed problems,” SIAM J. Matrix Anal. Appl.

Kjeldsen, R.

P. Hansen, M. Kilmer, R. Kjeldsen, “Exploiting residual information in the regularization of discrete ill-posed problems,” SIAM J. Matrix Anal. Appl.

Kleinschmidt, A.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Kolehmainen, V.

Levine, R.

Mandeville, J.

D. Boas, T. Gaudette, G. Strangman, X. Cheng, J. Marota, J. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001).
[CrossRef] [PubMed]

Marota, J.

D. Boas, T. Gaudette, G. Strangman, X. Cheng, J. Marota, J. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001).
[CrossRef] [PubMed]

McAdams, M.

McBride, T.

Millane, R.

Miller, E.

M. Belge, M. Kilmer, E. Miller, “Efficient selection of multiple regularization parameters in a generalized L-curve framework,” Inverse Prob. 18, 1161–1183 (2002).
[CrossRef]

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

M. Kilmer, E. Miller, D. Boas, D. Brooks, “A shape-based reconstruction technique for DPDW data,” Opt. Express 7, 481–491 (2000); http://www.opticsexpress.org .

O. Dorn, E. Miller, C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

E. Miller, M. Kilmer, C. Rappaport, “A new shape-based method for object localization and characterization from scattered field data,” IEEE Trans. Geosci. Remote Sens. 38, 1682–1696 (2000).
[CrossRef]

Ntziachristos, V.

V. Ntziachristos, A. Hielscher, A. Yodh, B. Chance, “Diffuse optical tomography of highly heterogeneous media,” IEEE Trans. Med. Imaging 20, 470–478 (2001).
[CrossRef] [PubMed]

V. Ntziachristos, B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Res. 3, 41–46 (2001).
[CrossRef] [PubMed]

O’Leary, D.

D. O’Leary, “The block conjugate gradient algorithm and related methods,” Linear Algebr. Appl. 29, 293–322 (1980).
[CrossRef]

Obrig, H.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Paulsen, K.

Pogue, B.

Prewitt, J.

Rappaport, C.

E. Miller, M. Kilmer, C. Rappaport, “A new shape-based method for object localization and characterization from scattered field data,” IEEE Trans. Geosci. Remote Sens. 38, 1682–1696 (2000).
[CrossRef]

O. Dorn, E. Miller, C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

Schnabel, R.

J. Dennis, R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Schweiger, M.

M. Schweiger, 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]

Sterberg, U.

Stevenson, D.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Strangman, G.

D. Boas, T. Gaudette, G. Strangman, X. Cheng, J. Marota, J. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001).
[CrossRef] [PubMed]

Svaasand, L.

Tromberg, B.

Tsay, T.

Van Houten, J.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Vauhkonen, M.

Villringer, A.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

Vogel, C.

C. Vogel, Computational Methods for Inverse Problems (SIAM, Philadelphia, Pa., 2002), Chap. 7.
[CrossRef]

Webb, K.

Ye, J.

Yodh, A.

V. Ntziachristos, A. Hielscher, A. Yodh, B. Chance, “Diffuse optical tomography of highly heterogeneous media,” IEEE Trans. Med. Imaging 20, 470–478 (2001).
[CrossRef] [PubMed]

Zhang, Q.

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Appl. Opt.

Breast Cancer Res.

V. Ntziachristos, B. Chance, “Probing physiology and molecular function using optical imaging: applications to breast cancer,” Breast Cancer Res. 3, 41–46 (2001).
[CrossRef] [PubMed]

IEEE Trans. Geosci. Remote Sens.

E. Miller, M. Kilmer, C. Rappaport, “A new shape-based method for object localization and characterization from scattered field data,” IEEE Trans. Geosci. Remote Sens. 38, 1682–1696 (2000).
[CrossRef]

IEEE Trans. Med. Imaging

V. Ntziachristos, A. Hielscher, A. Yodh, B. Chance, “Diffuse optical tomography of highly heterogeneous media,” IEEE Trans. Med. Imaging 20, 470–478 (2001).
[CrossRef] [PubMed]

Inverse Prob.

M. Belge, M. Kilmer, E. Miller, “Efficient selection of multiple regularization parameters in a generalized L-curve framework,” Inverse Prob. 18, 1161–1183 (2002).
[CrossRef]

Inverse Probl.

O. Dorn, E. Miller, C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

S. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

J. Cereb. Blood Flow Metab.

D. Benaron, S. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahn, C. Hirth, H. Obrig, J. Van Houten, E. Kermit, W. Cheong, D. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20, 469–477 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Linear Algebr. Appl.

D. O’Leary, “The block conjugate gradient algorithm and related methods,” Linear Algebr. Appl. 29, 293–322 (1980).
[CrossRef]

Neuroimage

D. Boas, T. Gaudette, G. Strangman, X. Cheng, J. Marota, J. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76–90 (2001).
[CrossRef] [PubMed]

Opt. Exp.

H. Jiang, “Optical image reconstruction based on the third-order diffusion equations,” Opt. Exp. 4, 241–246 (1999); http://www.opticsexpress.org .

Opt. Express

Phys. Med. Biol.

M. Schweiger, 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]

Signal Process. Mag.

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Other

S. Hassani, Foundations of Mathematical Physics (Allyn and Bacon, Boston, 1991).

J. Dennis, R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

C. Vogel, Computational Methods for Inverse Problems (SIAM, Philadelphia, Pa., 2002), Chap. 7.
[CrossRef]

D. O’Leary, Matlab translation of MINPACK subroutine cvsrch; http://www.cs.umd.edu/oleary/m607/cvsrch.m .

D. Boas, D. Brooks, R. Gaudette, T. Gaudette, E. Miller, Q. Zhang, Photon Migration Imaging (PMI) Toolbox, freely available at http://www.nmr.mgh.harvard.edu/DOT/resources/toolbox.htm .

P. Hansen, M. Kilmer, R. Kjeldsen, “Exploiting residual information in the regularization of discrete ill-posed problems,” SIAM J. Matrix Anal. Appl.

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Fig. 1
Fig. 1

Sparsity plots, moving down from the top of the region of interest (where height is 1 cm), depict the location of true and reconstructed anomalies for example 1. The plus marks indicate the location of the true anomaly. The black dots indicate voxels missed by the reconstructed anomaly as the reconstructed anomaly was slightly too small. The horizontal axis is increasing x from left to right but the tick marks give the matrix column index. The vertical axis is decreasing y from top to bottom, and the tick marks indicate the matrix row index.

Fig. 2
Fig. 2

True image of γ slices moving down from surface of the region of interest, examples 2A and 2B. The colormap is truncated to show the background variation (units are in inverse centimeters), so the anomaly with a value of 0.15 cm-1 appears as a bright white spot.

Fig. 3
Fig. 3

Sparsity plots depict the reconstruction error in shape for example 2A. The black plus marks indicate points that were inside the reconstructed anomaly that were not inside the true anomaly. The black dots indicate points that were inside the true anomaly that were not inside the reconstruction. The horizontal axis is increasing x from left to right but the tick marks give the matrix column index. The vertical axis is decreasing y from top to bottom, and the tick marks indicate the matrix row index.

Fig. 4
Fig. 4

Reconstruction with matched background basis functions in example 2B; slices are shown moving away from the top surface. The colormap truncation is the same as was used to display the true image (units are in inverse centimeters), so the reconstructed anomaly with a value of 0.168 cm-1 appears as a bright white spot.

Fig. 5
Fig. 5

Sparsity plots depict the difference between the location of true and reconstructed anomalies for example 3. The black dots indicate which voxels are in the true anomaly that are missed by the reconstruction whereas the plus marks indicate voxels in the reconstruction that are not in the anomaly. Clearly the reconstruction has the anomaly shifted slightly to the right in x. The horizontal axis is increasing x from left to right but the tick marks give the matrix column index. The vertical axis is decreasing y from top to bottom, and the tick marks indicate the matrix row index.

Fig. 6
Fig. 6

Sparsity plots depict the location of true and reconstructed anomalies for example 4. The black dots indicate the true anomaly, and the centers of the black circles show areas of the reconstructed anomaly that lie outside the true anomaly (in other words, the reconstruction is represented by the dots plus the circles). The horizontal axis is increasing x from left to right but the tick marks give the matrix column index. The vertical axis is decreasing y from top to bottom, and the tick marks indicate the matrix row index.

Fig. 7
Fig. 7

True image of γ used in examples 5A and 5B (units are in inverse centimeters). The colormap is truncated to show background detail, so the anomaly with value 0.15 cm-1 appears in bright white.

Fig. 8
Fig. 8

Cross-sectional display of locations of reconstructed ellipsoid (dots and plus marks) versus true ellipsoid (plus marks) for example 5A. The oversized reconstruction is the result of our not appropriately accounting for a lumpy background.

Fig. 9
Fig. 9

Reconstructed perturbation (in units of inverse centimeters) for example 5B. The reconstructed contrast is 0.13 cm-1. The colormap truncation is the same as that used in Fig. 7.

Fig. 10
Fig. 10

Cross-sectional slices moving down from the surface for the true solution in Subsection 4.C with s c = 0.005 and w = -1. The colormap is truncated to show background detail, so the perturbation value in the anomaly, 0.15 cm-1, appears in bright white.

Fig. 11
Fig. 11

Cross-sectional slices moving down from the surface for the true solution in Subsection 4.C for s c = 0.005 and w = -2.

Fig. 12
Fig. 12

Cross-sectional slices moving down from the surface for the reconstructed solution in Subsection 4.C for s c = 0.005 and w = -2 obtained with the correct B 2 in the reconstruction. The reconstructed α value is 0.16, and the colormap truncation is the same as used in Fig. 11.

Fig. 13
Fig. 13

Example 8A. Cross-sectional plots moving down from the surface comparing the shape and location of the true object with the reconstructed shape. The true anomaly is marked with black dots, and the reconstruction is overlaid by plus marks. The horizontal axis is increasing x from left to right but the tick marks give the matrix column index. The vertical axis is decreasing y from top to bottom, and the tick marks indicate the matrix row index.

Fig. 14
Fig. 14

Example 8B. Cross-sectional plots moving down from the surface comparing the shape and location of the true object with the reconstructed shape. The true anomaly is indicated with black dots, and the reconstruction is overlaid with plus marks. The horizontal axis is increasing x from left to right but the tick marks give the matrix column index. The vertical axis is decreasing y from top to bottom, and the tick marks indicate the matrix row index.

Tables (5)

Tables Icon

Table 1 Setup for Spherical Inversion Examples

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Table 2 Spherical Reconstruction Results

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Table 3 Setup for Ellipsoid Experiments

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Table 4 Ellipsoid Resultsa

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Table 5 Results for Reconstruction with PWC Basis for Various Lumpy Backgroundsa

Equations (23)

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ϕtot,sr; ω=ϕinc,sr; ω+Ω Gr, r˜; ωϕtot,sr˜; ωγr˜; ωdr˜.
ϕtot,srpϕinc,srp+Δ rkΩ Grp, rkϕtotrkγrk,
In-G1DIAGγϕtot,s1=ϕinc,s1,
ϕtot,s2=ϕinc,s2+G2DIAGγϕtot,s1.
ys=ϕtot,s2-ϕinc,s2=G2DIAGγϕtot,s1=G2DIAGϕtot,s1γ hsγ.
minγWy-hγ22,
minγWy-hγ22+λΘγ,
Sr=1ranomaly0otherwise.
γr=SrBar1×Na α Na×1+1-SrBbr1×Nb β Nb×1.
γ=SB1α+I-SB2β,
B1Rn×Na, B2Rn×Nb, αRNa, βRNb.
D-1UTr-c221.
U=cosθ1sinθ10-sinθ1cosθ10001cosθ20sinθ2010-sinθ20cosθ2×cosθ3sinθ30-sinθ3cosθ30001.
minα,β,c,d,θ Wy-hα, β, c, d, θ22.
minα,β,c,d Wy-hα, β, c, d22,
minα,β,c,d,θ12 εTε.
JTJs=-JTε,
εsp=W hsp=WG2pDIAGϕtot,s1γ=WG2DIAGγϕtot,s1p+DIAGϕtot,s1γp.
In-G1DIAGγϕtot,s1p=G1DIAGϕtot,s1γp.
AX=B, X, B are n×k,
Bbr=sin3x+1, cos8ysin2y+1,sin5z+1,
Bbr=sinωx+1, sinζy+1, sinκz+1,
n1=ny2+1, n2=nx2+1, n3=nz2+1.

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