Abstract

The problem of reconstruction of a refractive-index distribution (RID) in optical refraction tomography (ORT) with optical path-length difference (OPD) data is solved using two adaptive-estimation-based extended-Kalman-filter (EKF) approaches. First, a basic single-resolution EKF (SR-EKF) is applied to a state variable model describing the tomographic process, to estimate the RID of an optically transparent refracting object from noisy OPD data. The initialization of the biases and covariances corresponding to the state and measurement noise is discussed. The state and measurement noise biases and covariances are adaptively estimated. An EKF is then applied to the wavelet-transformed state variable model to yield a wavelet-based multiresolution EKF (MR-EKF) solution approach. To numerically validate the adaptive EKF approaches, we evaluate them with benchmark studies of standard stationary cases, where comparative results with commonly used efficient deterministic approaches can be obtained. Detailed reconstruction studies for the SR-EKF and two versions of the MR-EKF (with Haar and Daubechies-4 wavelets) compare well with those obtained from a typically used variant of the (deterministic) algebraic reconstruction technique, the average correction per projection method, thus establishing the capability of the EKF for ORT. To the best of our knowledge, the present work contains unique reconstruction studies encompassing the use of EKF for ORT in single-resolution and multiresolution formulations, and also in the use of adaptive estimation of the EKF’s noise covariances.

© 2010 Optical Society of America

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2009 (1)

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]

2007 (1)

A.-P. Tossavainen, M. Vauhkonen, and V. Kolehmainen, “A three-dimensional shape estimation approach for tracking of phase interfaces in sedimentation processes using electrical impedance tomography,” Meas. Sci. Technol. 18, 1413-1424(2007).
[CrossRef]

2006 (2)

2004 (2)

2003 (1)

2001 (1)

Seppanen, M. Vaukhonen, E. Somersalo, and J. P. Kaipio, “State space models in process tomography-approximation of state noise covariance,” Inv. Prob. Eng. 9, 561-585 (2001).

2000 (2)

R. E. Pierson, D. F. Olson, E. Y. Chen, and L. McMackin, “Comparison of reconstruction algorithm performance for optical phase tomography of a heated air flow,” Opt. Eng. 39, 838-846 (2000).
[CrossRef]

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase microscopy,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

1999 (5)

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, “Quantitative visualization of high speed 3D turbulent flow structures using holographic interferometric tomography,” Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

G. Keshava Datta and R. M. Vasu, “Non-interferometric methods of phase estimation for application in optical tomography,” J. Mod. Opt. 46, 1377-1388 (1999).

N. Naik and R. M. Vasu, “An extended Kalman filter based reconstruction approach to curved ray optical tomography,” Proc. SPIE 3729, 153-157 (1999).
[CrossRef]

N. Naik and R. M. Vasu, “A wavelet based multiresolution extended Kalman filter approach to the reconstruction problem of curved ray optical tomography,” Proc. SPIE 3642, 156-165 (1999).
[CrossRef]

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

1998 (2)

1997 (2)

W. Zhu, Y. Deng, Y. Yao, and R. L. Barbour, “A wavelet based multiresolution regularised least squares reconstruction approach to optical tomography,” IEEE Trans. Med. Imaging 16, 210-217 (1997).
[CrossRef]

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlund, “Deterministic edge preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef]

1995 (1)

F. Denis, O. Basset, and G. Gimenez, “Ultrasonic transmission tomography in refracting media: reduction of refraction artifacts by curved ray techniques,” IEEE Trans. Med. Imaging 14, 173-188 (1995).
[CrossRef]

1994 (1)

T. M. Chin and A. J. Mariano, “Wavelet based compression of covariances in Kalman filtering of geophysical flows,” Proc. SPIE 2242, 842-850 (1994).
[CrossRef]

1992 (1)

D. D. Verhoeven, “MART type CT algorithms for the reconstruction of multidirectional interferometric data,” Proc. SPIE 1553, 376-387 (1992).
[CrossRef]

1991 (2)

S. Bahl and J. A. Liburdy, “Three-dimensional image reconstruction using interferometric data from a limited field of view with noise,” Appl. Opt. 30, 4218-4226 (1991).
[CrossRef]

G. Beylkin, R. R. Coifman, and V. Rokhlin, “Fast wavelet transforms and numerical algorithms,” Commun. Pure Appl. Math. 44, 141-183 (1991).
[CrossRef]

1990 (1)

S.-Y. Lu and J. G. Berryman, “Inverse scattering, seismic traveltime tomography, and neural networks,” J. Imag. Sci. Technol. 2, 112-118 (1990).
[CrossRef]

1989 (1)

J. G. Berryman, “Fermat's principle and nonlinear traveltime tomography,” Phys. Rev. Lett. 62, 2953-2956 (1989).
[CrossRef]

1988 (1)

1987 (3)

1986 (1)

R. L. Kirlin and A. Moghaddamjoo, “Robust adaptive Kalman filtering for systems with unknown step inputs and non-Gaussian measurement errors,” IEEE Trans. Acoust. Speech Signal Process. 34, 252-263 (1986).
[CrossRef]

1985 (2)

1984 (1)

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81-94 (1984).
[CrossRef]

1982 (2)

S. J. Norton and J. M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imaging 4, 201-233 (1982).
[CrossRef]

A. H. Andersen and A. C. Kak, “Digital ray tracing in two dimensional refractive fields,” J. Acoust. Soc. Am. 72, 1593-1606 (1982).
[CrossRef]

1981 (1)

1980 (1)

R. J. Lytle and K. A. Dines, “Iterative ray tracing between boreholes for underground image reconstructions,” IEEE Trans. Geosci. Remote Sens. 18, 234-240 (1980).
[CrossRef]

1976 (1)

K. A. Myers and B. D. Tapley, “Adaptive sequential estimation with unknown noise statistics,” IEEE Trans. Automat. Contr. 21, 520-523 (1976).
[CrossRef]

Andersen, A. H.

A. H. Andersen, “Ray linking for computed tomography by rebinning of projection data,” J. Acoust. Soc. Am. 81, 1190-1192 (1987).
[CrossRef]

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81-94 (1984).
[CrossRef]

A. H. Andersen and A. C. Kak, “Digital ray tracing in two dimensional refractive fields,” J. Acoust. Soc. Am. 72, 1593-1606 (1982).
[CrossRef]

Arridge, S. R.

Aubert, G.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlund, “Deterministic edge preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef]

Bahl, S.

Banerjee, B.

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]

Barbour, R. L.

W. Zhu, Y. Deng, Y. Yao, and R. L. Barbour, “A wavelet based multiresolution regularised least squares reconstruction approach to optical tomography,” IEEE Trans. Med. Imaging 16, 210-217 (1997).
[CrossRef]

Barlaud, M.

L. Blanc-Feraud, P. Charbonnier, P. Lobel, and M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D-wavelet transform domain,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1994), Vol. 5, pp. 305-308.

Barlund, M.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlund, “Deterministic edge preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef]

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase microscopy,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Basset, O.

F. Denis, O. Basset, and G. Gimenez, “Ultrasonic transmission tomography in refracting media: reduction of refraction artifacts by curved ray techniques,” IEEE Trans. Med. Imaging 14, 173-188 (1995).
[CrossRef]

Berryman, J. G.

S.-Y. Lu and J. G. Berryman, “Inverse scattering, seismic traveltime tomography, and neural networks,” J. Imag. Sci. Technol. 2, 112-118 (1990).
[CrossRef]

J. G. Berryman, “Fermat's principle and nonlinear traveltime tomography,” Phys. Rev. Lett. 62, 2953-2956 (1989).
[CrossRef]

Beylkin, G.

G. Beylkin, R. R. Coifman, and V. Rokhlin, “Fast wavelet transforms and numerical algorithms,” Commun. Pure Appl. Math. 44, 141-183 (1991).
[CrossRef]

Bierman, G. J.

G. J. Bierman, Factorization Methods for Discrete Sequential Estimation (Academic, 1977).

Blanc-Feraud, L.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlund, “Deterministic edge preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef]

L. Blanc-Feraud, P. Charbonnier, P. Lobel, and M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D-wavelet transform domain,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1994), Vol. 5, pp. 305-308.

Bryanston-Cross, P. J.

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, “Quantitative visualization of high speed 3D turbulent flow structures using holographic interferometric tomography,” Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

Byer, R. L.

Cha, S.

Charbonnier, P.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlund, “Deterministic edge preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef]

L. Blanc-Feraud, P. Charbonnier, P. Lobel, and M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D-wavelet transform domain,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1994), Vol. 5, pp. 305-308.

Chen, E. Y.

R. E. Pierson, D. F. Olson, E. Y. Chen, and L. McMackin, “Comparison of reconstruction algorithm performance for optical phase tomography of a heated air flow,” Opt. Eng. 39, 838-846 (2000).
[CrossRef]

Chin, T. M.

T. M. Chin and A. J. Mariano, “Wavelet based compression of covariances in Kalman filtering of geophysical flows,” Proc. SPIE 2242, 842-850 (1994).
[CrossRef]

T. M. Chin and A. J. Mariano, “Kalman filtering of large scale geophysical flows by approximations based on Markov random fields and wavelets,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1995), Vol. 5, pp. 2785-2788.

Coifman, R. R.

G. Beylkin, R. R. Coifman, and V. Rokhlin, “Fast wavelet transforms and numerical algorithms,” Commun. Pure Appl. Math. 44, 141-183 (1991).
[CrossRef]

Daubechies, I.

I. Daubechies, M. Defrise, and C. de Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457(2004).
[CrossRef]

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992).

de Mol, C.

I. Daubechies, M. Defrise, and C. de Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457(2004).
[CrossRef]

Defrise, M.

I. Daubechies, M. Defrise, and C. de Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457(2004).
[CrossRef]

Deng, Y.

W. Zhu, Y. Deng, Y. Yao, and R. L. Barbour, “A wavelet based multiresolution regularised least squares reconstruction approach to optical tomography,” IEEE Trans. Med. Imaging 16, 210-217 (1997).
[CrossRef]

Denis, F.

F. Denis, O. Basset, and G. Gimenez, “Ultrasonic transmission tomography in refracting media: reduction of refraction artifacts by curved ray techniques,” IEEE Trans. Med. Imaging 14, 173-188 (1995).
[CrossRef]

Dines, K. A.

R. J. Lytle and K. A. Dines, “Iterative ray tracing between boreholes for underground image reconstructions,” IEEE Trans. Geosci. Remote Sens. 18, 234-240 (1980).
[CrossRef]

Dougherty, D. E.

Eppstein, M. J.

Faris, G. W.

Gimenez, G.

F. Denis, O. Basset, and G. Gimenez, “Ultrasonic transmission tomography in refracting media: reduction of refraction artifacts by curved ray techniques,” IEEE Trans. Med. Imaging 14, 173-188 (1995).
[CrossRef]

He, A.

Hesselink, L.

Jazwinski, A. H.

A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic, 1970).

Kaipio, J. P.

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

Seppanen, M. Vaukhonen, E. Somersalo, and J. P. Kaipio, “State space models in process tomography-approximation of state noise covariance,” Inv. Prob. Eng. 9, 561-585 (2001).

Kak, A. C.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81-94 (1984).
[CrossRef]

A. H. Andersen and A. C. Kak, “Digital ray tracing in two dimensional refractive fields,” J. Acoust. Soc. Am. 72, 1593-1606 (1982).
[CrossRef]

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Kanjirodan, R.

Keshava Datta, G.

G. Keshava Datta and R. M. Vasu, “Non-interferometric methods of phase estimation for application in optical tomography,” J. Mod. Opt. 46, 1377-1388 (1999).

Kirlin, R. L.

R. L. Kirlin and A. Moghaddamjoo, “Robust adaptive Kalman filtering for systems with unknown step inputs and non-Gaussian measurement errors,” IEEE Trans. Acoust. Speech Signal Process. 34, 252-263 (1986).
[CrossRef]

Kolehmainen, V.

A.-P. Tossavainen, M. Vauhkonen, and V. Kolehmainen, “A three-dimensional shape estimation approach for tracking of phase interfaces in sedimentation processes using electrical impedance tomography,” Meas. Sci. Technol. 18, 1413-1424(2007).
[CrossRef]

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

Koornwinder, T. H.

T. H. Koornwinder, Wavelets: An Elementary Treatment of Theory and Applications (World Scientific, 1993).

Langoju, R.

Liburdy, J. A.

Lin, B.

B. Lin, B. Nguyen, and E. T. Olsen, “Orthogonal wavelets and signal processing,” Signal Processing Methods for Audio, Images and Telecommunication, P. M. Clarkson and H. Stark, eds. (Academic, 1995) pp. 1-69.

Linzer, J. M.

S. J. Norton and J. M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imaging 4, 201-233 (1982).
[CrossRef]

Lira, I. H.

I. H. Lira and C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. 26, 3919-3928(1987).
[CrossRef]

I. H. Lira, “Correcting for refraction effects in holographic interferometry and tomography of transparent objects,” Ph.D. dissertation (University of Michigan, 1987).

Lobel, P.

L. Blanc-Feraud, P. Charbonnier, P. Lobel, and M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D-wavelet transform domain,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1994), Vol. 5, pp. 305-308.

Lu, S.-Y.

S.-Y. Lu and J. G. Berryman, “Inverse scattering, seismic traveltime tomography, and neural networks,” J. Imag. Sci. Technol. 2, 112-118 (1990).
[CrossRef]

Lytle, R. J.

R. J. Lytle and K. A. Dines, “Iterative ray tracing between boreholes for underground image reconstructions,” IEEE Trans. Geosci. Remote Sens. 18, 234-240 (1980).
[CrossRef]

Mandelis, A.

Mariano, A. J.

T. M. Chin and A. J. Mariano, “Wavelet based compression of covariances in Kalman filtering of geophysical flows,” Proc. SPIE 2242, 842-850 (1994).
[CrossRef]

T. M. Chin and A. J. Mariano, “Kalman filtering of large scale geophysical flows by approximations based on Markov random fields and wavelets,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1995), Vol. 5, pp. 2785-2788.

Maybeck, P.

P. Maybeck, Stochastic Models, Estimation and Control (Academic, 1982).

McMackin, L.

R. E. Pierson, D. F. Olson, E. Y. Chen, and L. McMackin, “Comparison of reconstruction algorithm performance for optical phase tomography of a heated air flow,” Opt. Eng. 39, 838-846 (2000).
[CrossRef]

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J. M. Mendel, Lessons in Digital Estimation Theory (Prentice Hall, 1987).

Miller, E. L.

Moghaddamjoo, A.

R. L. Kirlin and A. Moghaddamjoo, “Robust adaptive Kalman filtering for systems with unknown step inputs and non-Gaussian measurement errors,” IEEE Trans. Acoust. Speech Signal Process. 34, 252-263 (1986).
[CrossRef]

Myers, K. A.

K. A. Myers and B. D. Tapley, “Adaptive sequential estimation with unknown noise statistics,” IEEE Trans. Automat. Contr. 21, 520-523 (1976).
[CrossRef]

Naik, N.

N. Naik and R. M. Vasu, “A wavelet based multiresolution extended Kalman filter approach to the reconstruction problem of curved ray optical tomography,” Proc. SPIE 3642, 156-165 (1999).
[CrossRef]

N. Naik and R. M. Vasu, “An extended Kalman filter based reconstruction approach to curved ray optical tomography,” Proc. SPIE 3729, 153-157 (1999).
[CrossRef]

N. Naik, “Studies on the development of models and reconstruction algorithms for optical tomography,” Ph.D. dissertation (Department of Instrumentation, Indian Institute of Science, 2000).

Nguyen, B.

B. Lin, B. Nguyen, and E. T. Olsen, “Orthogonal wavelets and signal processing,” Signal Processing Methods for Audio, Images and Telecommunication, P. M. Clarkson and H. Stark, eds. (Academic, 1995) pp. 1-69.

Nicolaides, L.

Norton, S. J.

S. J. Norton and J. M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imaging 4, 201-233 (1982).
[CrossRef]

Nugent, K. A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase microscopy,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Olsen, E. T.

B. Lin, B. Nguyen, and E. T. Olsen, “Orthogonal wavelets and signal processing,” Signal Processing Methods for Audio, Images and Telecommunication, P. M. Clarkson and H. Stark, eds. (Academic, 1995) pp. 1-69.

Olson, D. F.

R. E. Pierson, D. F. Olson, E. Y. Chen, and L. McMackin, “Comparison of reconstruction algorithm performance for optical phase tomography of a heated air flow,” Opt. Eng. 39, 838-846 (2000).
[CrossRef]

Padmaram, R.

Paganin, D.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase microscopy,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Patnaik, L. M.

Pierson, R. E.

R. E. Pierson, D. F. Olson, E. Y. Chen, and L. McMackin, “Comparison of reconstruction algorithm performance for optical phase tomography of a heated air flow,” Opt. Eng. 39, 838-846 (2000).
[CrossRef]

Prince, S.

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase microscopy,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Rokhlin, V.

G. Beylkin, R. R. Coifman, and V. Rokhlin, “Fast wavelet transforms and numerical algorithms,” Commun. Pure Appl. Math. 44, 141-183 (1991).
[CrossRef]

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]

Sarkar, A. K.

A. K. Sarkar, New perspectives in state and parameter estimation of flight vehicles for off-line and real time applications, Ph.D. dissertation (Department of Aerospace Engineering, Indian Institute of Science, 2004).

Seppanen,

Seppanen, M. Vaukhonen, E. Somersalo, and J. P. Kaipio, “State space models in process tomography-approximation of state noise covariance,” Inv. Prob. Eng. 9, 561-585 (2001).

Sevick-Muraca, E. M.

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Snyder, R.

Somersalo, E.

Seppanen, M. Vaukhonen, E. Somersalo, and J. P. Kaipio, “State space models in process tomography-approximation of state noise covariance,” Inv. Prob. Eng. 9, 561-585 (2001).

Song, Y.

Tapley, B. D.

K. A. Myers and B. D. Tapley, “Adaptive sequential estimation with unknown noise statistics,” IEEE Trans. Automat. Contr. 21, 520-523 (1976).
[CrossRef]

Thayyullathil, H.

Timmerman, B. H.

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, “Quantitative visualization of high speed 3D turbulent flow structures using holographic interferometric tomography,” Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

Tossavainen, A.-P.

A.-P. Tossavainen, M. Vauhkonen, and V. Kolehmainen, “A three-dimensional shape estimation approach for tracking of phase interfaces in sedimentation processes using electrical impedance tomography,” Meas. Sci. Technol. 18, 1413-1424(2007).
[CrossRef]

Troy, T. L.

Vasu, R. M.

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]

H. Thayyullathil, R. M. Vasu, and R. Kanjirodan, “Quantitative flow visualization in supersonic jets through tomographic inversion of wavefronts estimated through shadow casting,” Appl. Opt. 45, 5010-5019 (2006).
[CrossRef]

H. Thayyullathil, R. Langoju, R. Padmaram, R. M. Vasu, R. Kanjirodan, and L. M. Patnaik, “Three-dimensional optical tomographic imaging of supersonic jets through inversion of phase data obtained through the transport of intensity equation,” Appl. Opt. 43, 4133-4141 (2004).
[CrossRef]

G. Keshava Datta and R. M. Vasu, “Non-interferometric methods of phase estimation for application in optical tomography,” J. Mod. Opt. 46, 1377-1388 (1999).

N. Naik and R. M. Vasu, “A wavelet based multiresolution extended Kalman filter approach to the reconstruction problem of curved ray optical tomography,” Proc. SPIE 3642, 156-165 (1999).
[CrossRef]

N. Naik and R. M. Vasu, “An extended Kalman filter based reconstruction approach to curved ray optical tomography,” Proc. SPIE 3729, 153-157 (1999).
[CrossRef]

Vauhkonen, M.

A.-P. Tossavainen, M. Vauhkonen, and V. Kolehmainen, “A three-dimensional shape estimation approach for tracking of phase interfaces in sedimentation processes using electrical impedance tomography,” Meas. Sci. Technol. 18, 1413-1424(2007).
[CrossRef]

Vaukhonen, M.

Seppanen, M. Vaukhonen, E. Somersalo, and J. P. Kaipio, “State space models in process tomography-approximation of state noise covariance,” Inv. Prob. Eng. 9, 561-585 (2001).

Verhoeven, D. D.

D. D. Verhoeven, “MART type CT algorithms for the reconstruction of multidirectional interferometric data,” Proc. SPIE 1553, 376-387 (1992).
[CrossRef]

Vest, C. M.

Watt, D. W.

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, “Quantitative visualization of high speed 3D turbulent flow structures using holographic interferometric tomography,” Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

Yao, Y.

W. Zhu, Y. Deng, Y. Yao, and R. L. Barbour, “A wavelet based multiresolution regularised least squares reconstruction approach to optical tomography,” IEEE Trans. Med. Imaging 16, 210-217 (1997).
[CrossRef]

Zhang, B.

Zhu, W.

W. Zhu, Y. Deng, Y. Yao, and R. L. Barbour, “A wavelet based multiresolution regularised least squares reconstruction approach to optical tomography,” IEEE Trans. Med. Imaging 16, 210-217 (1997).
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H. Thayyullathil, R. Langoju, R. Padmaram, R. M. Vasu, R. Kanjirodan, and L. M. Patnaik, “Three-dimensional optical tomographic imaging of supersonic jets through inversion of phase data obtained through the transport of intensity equation,” Appl. Opt. 43, 4133-4141 (2004).
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H. Thayyullathil, R. M. Vasu, and R. Kanjirodan, “Quantitative flow visualization in supersonic jets through tomographic inversion of wavefronts estimated through shadow casting,” Appl. Opt. 45, 5010-5019 (2006).
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I. Daubechies, M. Defrise, and C. de Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413-1457(2004).
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M. J. Eppstein and D. E. Dougherty, “Optimal 3-D traveltime tomography,” Geophysics 63, 1053-1061 (1998).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

R. L. Kirlin and A. Moghaddamjoo, “Robust adaptive Kalman filtering for systems with unknown step inputs and non-Gaussian measurement errors,” IEEE Trans. Acoust. Speech Signal Process. 34, 252-263 (1986).
[CrossRef]

IEEE Trans. Automat. Contr. (1)

K. A. Myers and B. D. Tapley, “Adaptive sequential estimation with unknown noise statistics,” IEEE Trans. Automat. Contr. 21, 520-523 (1976).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

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[CrossRef]

IEEE Trans. Med. Imaging (2)

W. Zhu, Y. Deng, Y. Yao, and R. L. Barbour, “A wavelet based multiresolution regularised least squares reconstruction approach to optical tomography,” IEEE Trans. Med. Imaging 16, 210-217 (1997).
[CrossRef]

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[CrossRef]

Inv. Prob. Eng. (1)

Seppanen, M. Vaukhonen, E. Somersalo, and J. P. Kaipio, “State space models in process tomography-approximation of state noise covariance,” Inv. Prob. Eng. 9, 561-585 (2001).

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[CrossRef]

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[CrossRef]

J. Mod. Opt. (1)

G. Keshava Datta and R. M. Vasu, “Non-interferometric methods of phase estimation for application in optical tomography,” J. Mod. Opt. 46, 1377-1388 (1999).

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

A.-P. Tossavainen, M. Vauhkonen, and V. Kolehmainen, “A three-dimensional shape estimation approach for tracking of phase interfaces in sedimentation processes using electrical impedance tomography,” Meas. Sci. Technol. 18, 1413-1424(2007).
[CrossRef]

Opt. Commun. (1)

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, “Quantitative phase microscopy,” Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Opt. Eng. (1)

R. E. Pierson, D. F. Olson, E. Y. Chen, and L. McMackin, “Comparison of reconstruction algorithm performance for optical phase tomography of a heated air flow,” Opt. Eng. 39, 838-846 (2000).
[CrossRef]

Opt. Laser Technol. (1)

B. H. Timmerman, D. W. Watt, and P. J. Bryanston-Cross, “Quantitative visualization of high speed 3D turbulent flow structures using holographic interferometric tomography,” Opt. Laser Technol. 31, 53-65 (1999).
[CrossRef]

Opt. Lett. (1)

Phys. Med. Biol. (1)

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]

Phys. Rev. Lett. (1)

J. G. Berryman, “Fermat's principle and nonlinear traveltime tomography,” Phys. Rev. Lett. 62, 2953-2956 (1989).
[CrossRef]

Proc. SPIE (4)

T. M. Chin and A. J. Mariano, “Wavelet based compression of covariances in Kalman filtering of geophysical flows,” Proc. SPIE 2242, 842-850 (1994).
[CrossRef]

N. Naik and R. M. Vasu, “An extended Kalman filter based reconstruction approach to curved ray optical tomography,” Proc. SPIE 3729, 153-157 (1999).
[CrossRef]

N. Naik and R. M. Vasu, “A wavelet based multiresolution extended Kalman filter approach to the reconstruction problem of curved ray optical tomography,” Proc. SPIE 3642, 156-165 (1999).
[CrossRef]

D. D. Verhoeven, “MART type CT algorithms for the reconstruction of multidirectional interferometric data,” Proc. SPIE 1553, 376-387 (1992).
[CrossRef]

Ultrason. Imaging (2)

S. J. Norton and J. M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imaging 4, 201-233 (1982).
[CrossRef]

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81-94 (1984).
[CrossRef]

Other (13)

N. Naik, “Studies on the development of models and reconstruction algorithms for optical tomography,” Ph.D. dissertation (Department of Instrumentation, Indian Institute of Science, 2000).

I. H. Lira, “Correcting for refraction effects in holographic interferometry and tomography of transparent objects,” Ph.D. dissertation (University of Michigan, 1987).

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992).

A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic, 1970).

A. K. Sarkar, New perspectives in state and parameter estimation of flight vehicles for off-line and real time applications, Ph.D. dissertation (Department of Aerospace Engineering, Indian Institute of Science, 2004).

P. Maybeck, Stochastic Models, Estimation and Control (Academic, 1982).

L. Blanc-Feraud, P. Charbonnier, P. Lobel, and M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D-wavelet transform domain,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1994), Vol. 5, pp. 305-308.

T. M. Chin and A. J. Mariano, “Kalman filtering of large scale geophysical flows by approximations based on Markov random fields and wavelets,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1995), Vol. 5, pp. 2785-2788.

J. M. Mendel, Lessons in Digital Estimation Theory (Prentice Hall, 1987).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

G. J. Bierman, Factorization Methods for Discrete Sequential Estimation (Academic, 1977).

T. H. Koornwinder, Wavelets: An Elementary Treatment of Theory and Applications (World Scientific, 1993).

B. Lin, B. Nguyen, and E. T. Olsen, “Orthogonal wavelets and signal processing,” Signal Processing Methods for Audio, Images and Telecommunication, P. M. Clarkson and H. Stark, eds. (Academic, 1995) pp. 1-69.

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Figures (9)

Fig. 1
Fig. 1

(a) Surface mesh profile of phantom P 1 . (b) Surface mesh profile of phantom P 2 .

Fig. 2
Fig. 2

Reconstructed surface mesh profiles of phantom P 1 obtained for the dataset P 1 D 1 after the second iteration by (a) the Haar MR-EKF, (b) the Db-4 MR-EKF, (c) the SR-EKF, and (d) the ACP algorithm.

Fig. 3
Fig. 3

Reconstructed profiles of the x = 0 plane of phantom P 1 obtained from the EKFs (dashed curves), as compared to ACP (dashed-dotted curves) algorithm and the actual profile (solid curves) for the dataset P 1 D 1 after the second iteration by (a) the Haar MR-EKF, (b) the Db-4 MR-EKF, and (c) the SR-EKF algorithm.

Fig. 4
Fig. 4

Reconstructed surface mesh profiles of the phantom P 1 obtained for the dataset P 1 D 2 after the second iteration by (a) the Haar MR-EKF, (b) the Db-4 MR-EKF, (c) the SR-EKF, and (d) the ACP algorithm.

Fig. 5
Fig. 5

Reconstructed profiles of the x = 0 plane of phantom P 1 obtained from the EKFs (dashed curves), as compared to ACP (dashed-dotted curves) algorithm and the actual profile (solid curves) for the dataset P 1 D 2 after the second iteration by (a) the Haar MR-EKF, (b) the Db-4 MR-EKF, and (c) the SR-EKF algorithm.

Fig. 6
Fig. 6

Reconstructed surface mesh profiles of the phantom P 2 obtained for the dataset P 2 D 1 after the second iteration by (a) the Haar MR-EKF, (b) the Db-4 MR-EKF, (c) the SR-EKF, and (d) the ACP algorithm.

Fig. 7
Fig. 7

Reconstructed profiles of the x = 0 plane of phantom P 2 obtained from the EKFs (dashed curves), as compared to ACP (dashed-dotted curves) algorithm and the actual profile (solid curves) for the dataset P 2 D 1 after the second iteration by (a) the Haar MR-EKF, (b) the Db-4 MR-EKF, and (c) the SR-EKF algorithm.

Fig. 8
Fig. 8

Reconstructed surface mesh profiles of the phantom P 2 obtained for the dataset P 2 D 2 after the second iteration by (a) the Haar MR-EKF, (b) the Db-4 MR-EKF, (c) the SR-EKF, and (d) the ACP algorithm.

Fig. 9
Fig. 9

Reconstructed profiles of the x = 0 plane of phantom P 2 obtained from the EKFs (dashed curves), as compared to ACP (dashed-dotted curves) algorithm and the actual profile (solid curves) for the dataset P 2 D 2 after the second iteration by (a) the Haar MR-EKF, (b) the Db-4 MR-EKF, and (c) the SR-EKF algorithm.

Tables (2)

Tables Icon

Table 1 Average Errors of Reconstructions of 1000 ( f ( x , y ) f amb ) a

Tables Icon

Table 2 Adaptive Extended-Kalman-Filter Approach

Equations (29)

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g ( θ , i ) = g 1 ( θ , i ) + v ( θ , i ) ,
g 1 ( θ , i ) Ray ( θ , i ) f ( x , y ) d s f amb L ,
d d s ( f d r d s ) = f ,
f ˙ ( θ ) = b [ f ( θ ) , u ( θ ) , θ ] + w ( θ ) ,
b [ f ( θ ) , u ( θ ) , θ ] = 0 ,
g ( θ ) = h [ f ( θ ) , r ( θ ) , θ ] + v ( θ ) , θ = θ i , i = 1 , 2 , ,
h [ f , r , θ ] = A [ f , θ ] f + r .
f ˜ ˙ ( θ ) = w ˜ ( θ ) ,
g ˜ ( θ ) = h ˜ [ f ˜ ( θ ) , r ˜ ( θ ) , θ ] + v ˜ ( θ ) , θ = θ i , i = 1 , 2 , ,
h ˜ [ f ˜ , r ˜ , θ ] = A ˜ [ f ˜ , θ ] f ˜ + r ˜ ,
r j s = g j A j f ^ 0 ,
cov ( r j s ) = A j P ^ 0 A j + R j ,
C ^ r pred = Σ j = 1 j = L r Γ j L r + R ^ 0 ,
Γ j = A j P ^ 0 A j , R ^ 0 = Σ j = 1 j = L r R j L r .
C ^ r derived = Σ j = 1 j = L r ( r j s r ^ 0 ) ( r j s r ^ 0 ) L r 1 ,
r ^ 0 = Σ j = 1 j = L r r j L r .
R ^ 0 = C ^ r meas Σ j = 1 j = L r Γ j L r .
f ( x , y ) = f amb 0.01 f amb [ exp ( x 2 + ( y 0.1 ) 2 0.09 ) + exp ( x 2 + ( y + 0.5 ) 2 0.04 ) ] .
f ( x , y ) = f amb 0.01 f amb exp ( x 2 + y 2 0.18 ) ,
error av = i = 1 N ( | f ^ ( i ) f ( i ) | ) N f min ,
{ h 0 = h 1 = 1 2 } , { g k = ( 1 ) k h 1 k , k = 0 , 1 } ,
{ h 0 = 1 + 3 2 } , h 1 = 3 + 3 2 , h 2 = 3 3 2 , h 3 = 1 3 2 } , { g k = ( 1 ) k h 1 k , k = 0 , 1 , 2 , 3 } .
f ( x ) = k = 0 N 1 f k ϕ p , k ( x ) ,
( H N M f ) l = k Z h k 2 l f k ,
( G N M f ) l = k Z g k 2 l f k ,
N = ( h 0 h 2 M 1 h 0 h 2 M 1 h 2 M 2 h 2 M 1 h 0 h 2 M 3 h 2 h 2 M 1 h 0 h 1 ) .
f ˜ ( j ) = W 1 ( j ) f = ( N / 2 j 1 G N / 2 2 N / 2 G N / 2 N G N ) ,
F ˜ ( 1 ) = W 1 ( 1 ) F W 1 ( 1 ) .
f ˜ ( j ) = W 2 ( 1 ) f ,

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