Abstract

We recast the reconstruction problem of diffuse optical tomography (DOT) in a pseudo-dynamical framework and develop a method to recover the optical parameters using particle filters, i.e., stochastic filters based on Monte Carlo simulations. In particular, we have implemented two such filters, viz., the bootstrap (BS) filter and the Gaussian-sum (GS) filter and employed them to recover optical absorption coefficient distribution from both numerically simulated and experimentally generated photon fluence data. Using either indicator functions or compactly supported continuous kernels to represent the unknown property distribution within the inhomogeneous inclusions, we have drastically reduced the number of parameters to be recovered and thus brought the overall computation time to within reasonable limits. Even though the GS filter outperformed the BS filter in terms of accuracy of reconstruction, both gave fairly accurate recovery of the height, radius, and location of the inclusions. Since the present filtering algorithms do not use derivatives, we could demonstrate accurate contrast recovery even in the middle of the object where the usual deterministic algorithms perform poorly owing to the poor sensitivity of measurement of the parameters. Consistent with the fact that the DOT recovery, being ill posed, admits multiple solutions, both the filters gave solutions that were verified to be admissible by the closeness of the data computed through them to the data used in the filtering step (either numerically simulated or experimentally generated).

© 2011 Optical Society of America

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  1. S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. USA 100, 12349–12354 (2003).
    [CrossRef] [PubMed]
  2. D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75(2001).
    [CrossRef]
  3. B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, O. K. S., U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
    [PubMed]
  4. B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
    [CrossRef] [PubMed]
  5. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  6. P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34, 2085–2098 (2007).
    [CrossRef] [PubMed]
  7. C. R. Vogel, Computational Methods for Inverse Problems(Academic, 2002).
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  8. P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
    [CrossRef]
  9. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of the Inverse Problem (Academic, 1996).
    [CrossRef]
  10. V. Kolehmainen, S. Prince, S. R. Arridge, and J. P. Kaipo, “State-estimation approach to the nonstationary optical tomography problem,” J. Opt. Soc. Am. A 20, 876–889 (2003).
    [CrossRef]
  11. M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevic-Muraca, “Biomedical optical tomography using dynamic parametrization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38, 2138–2150 (1999).
    [CrossRef]
  12. M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three-dimensional, Baysian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
    [CrossRef] [PubMed]
  13. M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Sevick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imag. 20, 147–163 (2001).
    [CrossRef]
  14. B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
    [CrossRef]
  15. B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. A 465, 1561–1579 (2009).
    [CrossRef]
  16. A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
    [CrossRef]
  17. A. Doucet, N. de Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice (Academic, 2001).
  18. S. Arulampalam, N. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188(2002).
    [CrossRef]
  19. N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEE Proceedings F Radar and Signal Processing (IEEE, 1993), Vol.  140, pp. 107–113.
    [CrossRef]
  20. K. Murphy and S. Russell, “Rao-Blackwellised particle filtering for dynamic Bayesian networks,” in Sequential Monte Carlo Methods in Practice, A.Doucet, N.de Freitas, and N.Gordon, eds. (Academic2001), pp. 499–515.
  21. J. H. Kotecha and P. M. Djuric, “Gaussian sum particle filtering,” IEEE Trans. Signal Process. 51, 2602–2612 (2003).
    [CrossRef]
  22. J. H. Kotecha and P. M. Djuric, “Gaussian sum particle filtering for dynamic state space models,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 3465–3468.
  23. M. Schweiger, S. R. Arridge, and I. Nissila, “Gauss–Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
    [CrossRef] [PubMed]
  24. K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Quart. Appl. Math. 2, 164–168(1944).
  25. D. W. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441(1963).
    [CrossRef]
  26. D. Roy, “Explorations of the phase space linearization method for deterministic and stochastic non-linear dynamical systems,” Nonlinear Dyn. 23, 225–258 (2000).
    [CrossRef]
  27. D. Roy, “A new numeric-analytical principle for nonlinear deterministic and stochastic dynamical systems,” Proc. R. Soc. London Ser. A 457, 539–566 (2001).
    [CrossRef]
  28. G. Kallianpur, Stochastic Filtering Theory (Academic, 1980).
  29. A. D. Zacharopoulos, M. Schweiger, V. Kolehmainen, and S. Arridge, “3D shape based reconstruction of experimental data in diffuse optical tomography,” Opt. Express 17, 18940–18956(2009).
    [CrossRef]
  30. R. Lipster and A. Shiryaev, Statistics of Random Processes(Academic, 2001).
  31. P. Fernhead, “Sequential Monte Carlo methods in filter theory,” Ph.D. thesis (University of Oxford, 1998).
  32. S. Gupta, P. K. Yalavarthy, D. Roy, D. Piao, and R. M. Vasu, “Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography,” Med. Phys. 36, 5559–5567 (2009).
    [CrossRef]
  33. B. Banerjee, D. Roy, and R. M. Vasu, “Efficient implementations of a pseudo-dynamical stochastic filtering strategy for static elastography,” Med. Phys. 36, 3470–3476 (2009).
    [CrossRef] [PubMed]

2009

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. A 465, 1561–1579 (2009).
[CrossRef]

A. D. Zacharopoulos, M. Schweiger, V. Kolehmainen, and S. Arridge, “3D shape based reconstruction of experimental data in diffuse optical tomography,” Opt. Express 17, 18940–18956(2009).
[CrossRef]

S. Gupta, P. K. Yalavarthy, D. Roy, D. Piao, and R. M. Vasu, “Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography,” Med. Phys. 36, 5559–5567 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “Efficient implementations of a pseudo-dynamical stochastic filtering strategy for static elastography,” Med. Phys. 36, 3470–3476 (2009).
[CrossRef] [PubMed]

2007

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34, 2085–2098 (2007).
[CrossRef] [PubMed]

2005

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

2003

J. H. Kotecha and P. M. Djuric, “Gaussian sum particle filtering,” IEEE Trans. Signal Process. 51, 2602–2612 (2003).
[CrossRef]

V. Kolehmainen, S. Prince, S. R. Arridge, and J. P. Kaipo, “State-estimation approach to the nonstationary optical tomography problem,” J. Opt. Soc. Am. A 20, 876–889 (2003).
[CrossRef]

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

2002

S. Arulampalam, N. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188(2002).
[CrossRef]

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three-dimensional, Baysian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef] [PubMed]

2001

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Sevick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imag. 20, 147–163 (2001).
[CrossRef]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75(2001).
[CrossRef]

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, O. K. S., U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

D. Roy, “A new numeric-analytical principle for nonlinear deterministic and stochastic dynamical systems,” Proc. R. Soc. London Ser. A 457, 539–566 (2001).
[CrossRef]

2000

D. Roy, “Explorations of the phase space linearization method for deterministic and stochastic non-linear dynamical systems,” Nonlinear Dyn. 23, 225–258 (2000).
[CrossRef]

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
[CrossRef] [PubMed]

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
[CrossRef]

1999

1992

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

1963

D. W. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441(1963).
[CrossRef]

1944

K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Quart. Appl. Math. 2, 164–168(1944).

Andrieu, C.

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
[CrossRef]

Arridge, S.

Arridge, S. R.

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

V. Kolehmainen, S. Prince, S. R. Arridge, and J. P. Kaipo, “State-estimation approach to the nonstationary optical tomography problem,” J. Opt. Soc. Am. A 20, 876–889 (2003).
[CrossRef]

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

Arulampalam, S.

S. Arulampalam, N. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188(2002).
[CrossRef]

Banerjee, B.

B. Banerjee, D. Roy, and R. M. Vasu, “Efficient implementations of a pseudo-dynamical stochastic filtering strategy for static elastography,” Med. Phys. 36, 3470–3476 (2009).
[CrossRef] [PubMed]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. A 465, 1561–1579 (2009).
[CrossRef]

Boas, D. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75(2001).
[CrossRef]

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75(2001).
[CrossRef]

Butler, J.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
[CrossRef] [PubMed]

Cerussi, A.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
[CrossRef] [PubMed]

Clapp, T.

S. Arulampalam, N. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188(2002).
[CrossRef]

de Freitas, N.

A. Doucet, N. de Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice (Academic, 2001).

Dehghani, H.

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34, 2085–2098 (2007).
[CrossRef] [PubMed]

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

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75(2001).
[CrossRef]

Djuric, P. M.

J. H. Kotecha and P. M. Djuric, “Gaussian sum particle filtering,” IEEE Trans. Signal Process. 51, 2602–2612 (2003).
[CrossRef]

J. H. Kotecha and P. M. Djuric, “Gaussian sum particle filtering for dynamic state space models,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 3465–3468.

Doucet, A.

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
[CrossRef]

A. Doucet, N. de Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice (Academic, 2001).

Dougherty, D. E.

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Sevick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imag. 20, 147–163 (2001).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevic-Muraca, “Biomedical optical tomography using dynamic parametrization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38, 2138–2150 (1999).
[CrossRef]

Engl, H. W.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of the Inverse Problem (Academic, 1996).
[CrossRef]

Eppstein, M. J.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three-dimensional, Baysian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef] [PubMed]

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Sevick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imag. 20, 147–163 (2001).
[CrossRef]

M. J. Eppstein, D. E. Dougherty, T. L. Troy, and E. M. Sevic-Muraca, “Biomedical optical tomography using dynamic parametrization and Bayesian conditioning on photon migration measurements,” Appl. Opt. 38, 2138–2150 (1999).
[CrossRef]

Espinoza, J.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
[CrossRef] [PubMed]

Fernhead, P.

P. Fernhead, “Sequential Monte Carlo methods in filter theory,” Ph.D. thesis (University of Oxford, 1998).

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75(2001).
[CrossRef]

Gibson, J. J.

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

Godavarty, A.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three-dimensional, Baysian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef] [PubMed]

Godsill, S.

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
[CrossRef]

Gordon, N.

S. Arulampalam, N. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188(2002).
[CrossRef]

A. Doucet, N. de Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice (Academic, 2001).

Gordon, N. J.

N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEE Proceedings F Radar and Signal Processing (IEEE, 1993), Vol.  140, pp. 107–113.
[CrossRef]

Gupta, S.

S. Gupta, P. K. Yalavarthy, D. Roy, D. Piao, and R. M. Vasu, “Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography,” Med. Phys. 36, 5559–5567 (2009).
[CrossRef]

Hanke, M.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of the Inverse Problem (Academic, 1996).
[CrossRef]

Hansen, P. C.

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Hawrysz, D. J.

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three-dimensional, Baysian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef] [PubMed]

M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Sevick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imag. 20, 147–163 (2001).
[CrossRef]

Jiang, S.

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

Kaipo, J. P.

Kallianpur, G.

G. Kallianpur, Stochastic Filtering Theory (Academic, 1980).

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75(2001).
[CrossRef]

Kogel, C.

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

Kolehmainen, V.

Kotecha, J. H.

J. H. Kotecha and P. M. Djuric, “Gaussian sum particle filtering,” IEEE Trans. Signal Process. 51, 2602–2612 (2003).
[CrossRef]

J. H. Kotecha and P. M. Djuric, “Gaussian sum particle filtering for dynamic state space models,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 3465–3468.

Lanning, R.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
[CrossRef] [PubMed]

Levenberg, K.

K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Quart. Appl. Math. 2, 164–168(1944).

Lipster, R.

R. Lipster and A. Shiryaev, Statistics of Random Processes(Academic, 2001).

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441(1963).
[CrossRef]

Maskell, N.

S. Arulampalam, N. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188(2002).
[CrossRef]

McBride, T. O.

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, O. K. S., U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75(2001).
[CrossRef]

Murphy, K.

K. Murphy and S. Russell, “Rao-Blackwellised particle filtering for dynamic Bayesian networks,” in Sequential Monte Carlo Methods in Practice, A.Doucet, N.de Freitas, and N.Gordon, eds. (Academic2001), pp. 499–515.

Neubauer, A.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of the Inverse Problem (Academic, 1996).
[CrossRef]

Nissila, I.

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

Osterberg, U. L.

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, O. K. S., U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Paulsen, K. D.

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34, 2085–2098 (2007).
[CrossRef] [PubMed]

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

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, O. K. S., U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Pham, T.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
[CrossRef] [PubMed]

Piao, D.

S. Gupta, P. K. Yalavarthy, D. Roy, D. Piao, and R. M. Vasu, “Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography,” Med. Phys. 36, 5559–5567 (2009).
[CrossRef]

Pogue, B. W.

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34, 2085–2098 (2007).
[CrossRef] [PubMed]

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

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, O. K. S., U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Poplack, S. P.

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

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, O. K. S., U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Prince, S.

Roy, D.

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “Efficient implementations of a pseudo-dynamical stochastic filtering strategy for static elastography,” Med. Phys. 36, 3470–3476 (2009).
[CrossRef] [PubMed]

S. Gupta, P. K. Yalavarthy, D. Roy, D. Piao, and R. M. Vasu, “Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography,” Med. Phys. 36, 5559–5567 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. A 465, 1561–1579 (2009).
[CrossRef]

D. Roy, “A new numeric-analytical principle for nonlinear deterministic and stochastic dynamical systems,” Proc. R. Soc. London Ser. A 457, 539–566 (2001).
[CrossRef]

D. Roy, “Explorations of the phase space linearization method for deterministic and stochastic non-linear dynamical systems,” Nonlinear Dyn. 23, 225–258 (2000).
[CrossRef]

Russell, S.

K. Murphy and S. Russell, “Rao-Blackwellised particle filtering for dynamic Bayesian networks,” in Sequential Monte Carlo Methods in Practice, A.Doucet, N.de Freitas, and N.Gordon, eds. (Academic2001), pp. 499–515.

S., O. K.

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, O. K. S., U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
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N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEE Proceedings F Radar and Signal Processing (IEEE, 1993), Vol.  140, pp. 107–113.
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A. D. Zacharopoulos, M. Schweiger, V. Kolehmainen, and S. Arridge, “3D shape based reconstruction of experimental data in diffuse optical tomography,” Opt. Express 17, 18940–18956(2009).
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M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Sevick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imag. 20, 147–163 (2001).
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B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
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N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEE Proceedings F Radar and Signal Processing (IEEE, 1993), Vol.  140, pp. 107–113.
[CrossRef]

Soho, S.

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

Srinivasan, S.

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

Svaasand, L.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
[CrossRef] [PubMed]

Tosteson, T. D.

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

Tromberg, B. J.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
[CrossRef] [PubMed]

Troy, T. L.

Vasu, R. M.

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. A 465, 1561–1579 (2009).
[CrossRef]

S. Gupta, P. K. Yalavarthy, D. Roy, D. Piao, and R. M. Vasu, “Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography,” Med. Phys. 36, 5559–5567 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “Efficient implementations of a pseudo-dynamical stochastic filtering strategy for static elastography,” Med. Phys. 36, 3470–3476 (2009).
[CrossRef] [PubMed]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[CrossRef]

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C. R. Vogel, Computational Methods for Inverse Problems(Academic, 2002).
[CrossRef]

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B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, O. K. S., U. L. Osterberg, and K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[PubMed]

Yalavarthy, P. K.

S. Gupta, P. K. Yalavarthy, D. Roy, D. Piao, and R. M. Vasu, “Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography,” Med. Phys. 36, 5559–5567 (2009).
[CrossRef]

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34, 2085–2098 (2007).
[CrossRef] [PubMed]

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Zhang, Q.

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[CrossRef]

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M. J. Eppstein, D. E. Dougherty, D. J. Hawrysz, and E. M. Sevick-Muraca, “Three-dimensional Bayesian optical image reconstruction with domain decomposition,” IEEE Trans. Med. Imag. 20, 147–163 (2001).
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S. Gupta, P. K. Yalavarthy, D. Roy, D. Piao, and R. M. Vasu, “Singular value decomposition based computationally efficient algorithm for rapid dynamic near-infrared diffuse optical tomography,” Med. Phys. 36, 5559–5567 (2009).
[CrossRef]

B. Banerjee, D. Roy, and R. M. Vasu, “Efficient implementations of a pseudo-dynamical stochastic filtering strategy for static elastography,” Med. Phys. 36, 3470–3476 (2009).
[CrossRef] [PubMed]

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. 34, 2085–2098 (2007).
[CrossRef] [PubMed]

Neoplasia

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivocharacterization of breast tumors using photon migration spectroscopy,” Neoplasia 2, 26–40 (2000).
[CrossRef] [PubMed]

Nonlinear Dyn.

D. Roy, “Explorations of the phase space linearization method for deterministic and stochastic non-linear dynamical systems,” Nonlinear Dyn. 23, 225–258 (2000).
[CrossRef]

Opt. Express

Phys. Med. Biol.

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

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[CrossRef]

Proc. Natl. Acad. Sci. USA

M. J. Eppstein, D. J. Hawrysz, A. Godavarty, and E. M. Sevick-Muraca, “Three-dimensional, Baysian image reconstruction from sparse and noisy data sets: near-infrared fluorescence tomography,” Proc. Natl. Acad. Sci. USA 99, 9619–9624 (2002).
[CrossRef] [PubMed]

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

Proc. R. Soc. A

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. A 465, 1561–1579 (2009).
[CrossRef]

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N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in IEE Proceedings F Radar and Signal Processing (IEEE, 1993), Vol.  140, pp. 107–113.
[CrossRef]

K. Murphy and S. Russell, “Rao-Blackwellised particle filtering for dynamic Bayesian networks,” in Sequential Monte Carlo Methods in Practice, A.Doucet, N.de Freitas, and N.Gordon, eds. (Academic2001), pp. 499–515.

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[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Parameters estimated through the BS and GS filters for an absorbing object with one inclusion. At the start of recursion, we assume there are two inclusions. (a) h, (b) r, and (c) the Euclidean distance to the center of the recovered inclusion.

Fig. 2
Fig. 2

Recovered μ a images. (a) Gray-level images and (b) cross-sectional plots through the center of the inclusion.

Fig. 3
Fig. 3

Parameters estimated through PF algorithm for a phantom with two inhomogeneities, (a) h, (b) r, and (c) Euclidean distance.

Fig. 4
Fig. 4

Reconstruction of absorption coefficient using (a) BS and (b) GSPF for a phantom with two inhomogeneities.

Fig. 5
Fig. 5

Comparison of simulated reference data and the data computed with the estimates by BS and GSPF.

Fig. 6
Fig. 6

Recovered gray-level images from noisy fluence data using (a) BS filter, (b) GN method. The percentage noise in data is 3.

Fig. 7
Fig. 7

Recovered gray-level images from noisy fluence data using (a) BS filter (b) GN method. The percentage noise in data is 5.

Fig. 8
Fig. 8

Cross section of the phantom with a central inhomogeneity estimated via BS and GSPF along with GN reconstruction and reference.

Fig. 9
Fig. 9

Reconstruction of μ a using (b) GN, (c) GSPF, and (d) BS schemes for an object with a central inclusion; (a) is the reference.

Fig. 10
Fig. 10

Recovered (a) h, (b) r, and (c) the Euclidean distance, estimated through BS, GSPF1, and GSPF2 filters from the experimental data.

Fig. 11
Fig. 11

Reconstructed gray-level image from experimental data using the BS.

Fig. 12
Fig. 12

Reconstructed gray-level image from experimental data using the GSPF1.

Fig. 13
Fig. 13

Reconstructed gray-level image from experimental data using the GSPF2.

Fig. 14
Fig. 14

Comparison of experimental data with computed data using the estimates obtained by BS, GSPF1, and GSPF2 filters.

Fig. 15
Fig. 15

Comparison of experimental data with computed data using the estimates obtained by the GSPF filter for which the number of nodes in the FE discretization is 800.

Tables (1)

Tables Icon

Table 1 Comparison of Integrated Photon Loss

Equations (35)

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

· ( κ ( x ) φ ( x , ω ) ) + ( μ a ( x ) + j ω c ) φ ( x , ω ) = q 0 ( r 0 ) , x Ω R q ,
φ ( m , ω ) + 2 A κ ( m ) ( φ ( m , ω ) / n ^ ) = 0.
min μ 1 2 ϕ ( m ) ϕ ( μ ) 2 + λ 2 L ( μ ) 2 ,
K ( μ ) Φ = q .
g ( μ ) = J ( μ ) T ( ϕ ( m ) ϕ ( μ ) ) λ L ( μ ) = 0 ,
Δ μ i + 1 = ( J T ( μ i ) J ( μ i ) + λ L ( μ i ) + F ( μ i ) ) 1 ( J T ( μ i ) ( ϕ ( m ) ϕ ( μ i ) ) λ L ( μ i ) ) .
Δ μ i + 1 = ( J T ( μ i ) J ( μ i ) + τ i + 1 I ) 1 ( J T ( μ i ) ( ϕ ( m ) ϕ ( μ i ) ) ) .
μ ˙ + M ( μ * ) ( μ ( t ) μ * ) + V ( μ * ) = 0 ,
μ ( t + Δ t ) = exp ( [ M ( μ * ) ] ( Δ t ) ) μ ( t ) + t t + Δ t exp ( [ M ( μ * ) ] ( t + Δ t s ) ) f d s ,
d μ a ( t ) = d ξ ( t ) ,
d z ( t ) = D K 1 ( μ a ) q d t + d ϑ ( t ) .
f ( μ p , x ) = μ ¯ a + k = 1 P H k I Ω k ( x ) , H k 0 and Ω k Ω     for each   k .
f ( μ p , x ) = μ ¯ a + k = 1 P H k φ k ( x ) .
d π t ( μ p ) = [ π t ( μ p ( D K 1 ( μ p ) q ) T ) π t ( μ p ) π t ( D K 1 ( μ p ) q ) T ] d ϑ ¯ t ,
μ i + 1 = μ i + σ μ ζ i ,
Z i + 1 = DK ( μ ) 1 q + ϑ i + 1 ,
q ( μ i + 1 / Z 1 : i ) = p ( μ i / Z 1 : i ) χ ( μ i + 1 / μ i ) d μ i .
p ( μ i + 1 / Z 1 : i + 1 ) = g ( Z i + 1 / μ i + 1 ) q ( μ i + 1 / Z 1 : i ) g ( Z i + 1 / μ i + 1 ) q ( μ i + 1 / Z 1 : i ) d μ i + 1 ,
g ( Z i + 1 / μ i + 1 ) = p ϑ , i + 1 ( Z i + 1 DK ( μ a ) 1 q ) .
w ˜ i + 1 ( u ) g ( Z i + 1 / μ i + 1 * ( u ) ) ,
P ( d μ i + 1 / Z 1 : i + 1 ) P ( N ) ( d μ i + 1 / Z 1 : i + 1 ) = ( 1 / N ) u = 1 N δ μ i + 1 ( u ) ( d μ i + 1 ) .
P ( d μ i + 1 / Z 1 : i + 1 ) P ( N ) ( d μ i + 1 / Z 1 : i + 1 ) = g ( Z i + 1 / μ i + 1 ) Q ( N ) ( d μ i + 1 / Z 1 : i ) G ( Z i + 1 / μ i + 1 ) Q ( N ) ( d μ i + 1 / Z 1 : i ) = u = 1 N w i + 1 ( u ) δ μ i + 1 * ( u ) d ( μ i + 1 ) .
P N ( d μ i + 1 / Z 1 : i + 1 ) = ( 1 / N ) u = 1 N δ μ i + 1 ( u ) ( d μ i + 1 ) .
μ ^ i + 1 = 1 N u = 1 N μ i + 1 ( u ) .
p ( μ i / Z 1 : i ) l = 1 N G w i l ( μ i ; m i l , Σ i l ) ,
q ( μ i + 1 / Z 1 : i ) = χ ( μ i + 1 / μ i ) p ( μ i / Z 1 : i ) d μ i = ( μ i + 1 ; μ i , Σ ζ i + 1 ) l = 1 N G w i l ( μ i ; m i l , Σ i l ) d μ i = l = 1 N G w i l ( μ i + 1 ; μ i , Σ ζ i + 1 ) ( μ i ; m i l , Σ i l ) d μ i ,
w ¯ i + 1 , l = w i l l = 1 N G w i l .
q ( μ i + 1 / Z 1 : i ) = l = 1 N G w ¯ i + 1 , l ( μ i + 1 ; m ¯ i + 1 , l , Σ ¯ i + 1 , l ) .
p ( μ i + 1 / Z 1 : i + 1 ) = g ( Z 1 : i + 1 / μ i + 1 ) q ( μ i + 1 / Z 1 : i + 1 ) g ( Z 1 : i + 1 / μ i + 1 ) q ( μ i + 1 / Z 1 : i + 1 ) d μ i + 1 g ( Z i + 1 / μ i + 1 ) q ( μ i + 1 / Z 1 : i ) g ( Z i + 1 / μ i + 1 ) l = 1 N G w ¯ i + 1 , l ( μ i + 1 ; m ¯ i + 1 , l , Σ ¯ i + 1 , l ) ,
m i + 1 , l = u = 1 M ψ i + 1 , l ( u ) μ ˜ i + 1 , l ( u ) u = 1 M ψ i + 1 , l ( u ) ,
Σ i + 1 , l = u = 1 M ψ i + 1 , l ( u ) ( μ ˜ i + 1 , l ( u ) m i + 1 , l ) ( μ ˜ i + 1 , l ( u ) m i + 1 , l ) T u = 1 M ψ i + 1 , l ( u ) .
w ˜ i + 1 , l = w ¯ i l u = 1 M ψ i + 1 , l ( u ) l = 1 N G u = 1 M ψ i + 1 , l ( u ) , l = 1 , , N G ,
w i + 1 , l = w ˜ i + 1 , l l = 1 N G w ˜ i + 1 , l .
p ( μ i + 1 / Z 1 : i + 1 ) = l = 1 N G w i + 1 , l ( μ i + 1 ; m i + 1 , l , Σ i + 1 , l ) .
μ ^ i + 1 = l = 1 N G w i + 1 , l m i + 1 , l .

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