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]
  2. 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]
  3. 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]
  4. 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]
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    [CrossRef]
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  24. 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]
  25. P. Hansen, M. Kilmer, R. Kjeldsen, “Exploiting residual information in the regularization of discrete ill-posed problems,” SIAM J. Matrix Anal. Appl.

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

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]

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]

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]

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]

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. 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 .

1999

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]

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

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]

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

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, 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. 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 .

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]

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

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 .

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]

Ntziachristos, V.

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]

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]

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.

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

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

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 .

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).

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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