Abstract

Fluorescence diffuse optical tomography is a powerful tool for the investigation of molecular events in studies for new therapeutic developments. Here, the stress is put on the mathematical problem of tomography, which can be formulated in terms of an estimation of physical parameters appearing as a set of partial differential equations and solved by the finite element method. This method is well known to be time consuming, and our principal objective is to reduce the model in order to speed up computation. A method based on a wavelet multiresolution technique is presented in detail. A validation study was conducted on synthetic data and experiments.

© 2009 Optical Society of America

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2008 (1)

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

2007 (1)

2006 (2)

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, 61640R1-10 (2006).

2005 (1)

M. N. O. Sadiku, C. M. Akujuobi, and R. C. Garcia, “An introduction to wavelets in electromagnetics,” IEEE Microwave Magazine 6, 63-72 (2005).
[CrossRef]

2004 (2)

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12, 5402-5417 (2004).
[CrossRef]

2003 (4)

R. Roy, A. Godavarty, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical tomography using referenced measurements of heterogeneous media,” IEEE Trans. Med. Imaging 22, 824-836 (2003).
[CrossRef]

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9, 123-128 (2003).
[CrossRef]

F. Fedele, J. P. Laible, and M. J. Eppstein, “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation,” J. Comput. Phys. 187, 597-619 (2003).
[CrossRef]

M. J. Eppstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging 22, 1215-1223 (2003).
[CrossRef]

2002 (1)

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1-N10(2002).
[CrossRef]

2001 (1)

1999 (1)

S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet-based compression and denoising of optical tomography data,” Opt. Commun. 167, 37-46 (1999).
[CrossRef]

1998 (1)

1997 (4)

T. A. Davis and I. S. Duff, “An unsymmetric pattern multifrontal method for sparse LU factorization,” SIAM J. Matrix Anal. Appl. 18, 140-158 (1997).
[CrossRef]

Z. Xiang and Y. Lu, “An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations,” IEEE Trans. Antennas Propag. 45, 1205-1213 (1997).
[CrossRef]

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

F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein, and D. Walnut, “Wavelet-based multiresolution local tomography,” IEEE Trans. Image Process. 10, 1412-1430 (1997).
[CrossRef]

1996 (1)

M. Bathia, W. C. Karl, and A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” IEEE Trans. Med. Imaging 15, 92-101 (1996).
[CrossRef]

1995 (1)

R. L. Wagner and W. C. Chew, “A study of wavelets for the solution of electromagnetic integral equations,” IEEE Trans. Antennas Propag. 43, 802-810 (1995).
[CrossRef]

1994 (1)

1993 (1)

B. Sahiner and A. E. Yagle, “Image reconstruction from projections under wavelet constraints,” IEEE Trans. Signal Process. 41, 3579-3584(1993).
[CrossRef]

1992 (1)

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef]

1989 (1)

S. G. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674-693 (1989).
[CrossRef]

1986 (1)

Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp. 7, 856-869 (1986).
[CrossRef]

1984 (1)

A. Grossmann and J. Morlet, “Decomposition of hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723-736 (1984).
[CrossRef]

Agoshkov, V. I.

G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems (CRC, 1996).

Akujuobi, C. M.

M. N. O. Sadiku, C. M. Akujuobi, and R. C. Garcia, “An introduction to wavelets in electromagnetics,” IEEE Microwave Magazine 6, 63-72 (2005).
[CrossRef]

Arridge, S. R.

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef]

Bangerth, W.

Bansal, P.

B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, 61640R1-10 (2006).

Barbour, R. L.

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

Bathia, M.

M. Bathia, W. C. Karl, and A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” IEEE Trans. Med. Imaging 15, 92-101 (1996).
[CrossRef]

Berenstein, C. A.

F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein, and D. Walnut, “Wavelet-based multiresolution local tomography,” IEEE Trans. Image Process. 10, 1412-1430 (1997).
[CrossRef]

Berger, M.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46, 4896-4906 (2007).
[CrossRef]

Bilgot, A.

A. Bilgot, V. Perrier, and L. Desbat, “Wavelets, local tomography and interventional x-ray imaging,” IEEE, Management International Conference Proceedings M9, 183 (2004).

Bogdanov, A.

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

Boutet, J.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46, 4896-4906 (2007).
[CrossRef]

Burnett, D. S.

D. S. Burnett, Finite Element Analysis: From Concepts to Applications (Addison-Wesley, 1987).

Chance, B.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1-N10(2002).
[CrossRef]

Chew, W. C.

R. L. Wagner and W. C. Chew, “A study of wavelets for the solution of electromagnetic integral equations,” IEEE Trans. Antennas Propag. 43, 802-810 (1995).
[CrossRef]

Coll, J. L.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef]

Culver, J. P.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1-N10(2002).
[CrossRef]

Da Silva, A.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46, 4896-4906 (2007).
[CrossRef]

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992), pp. 167-213.

Davis, T. A.

T. A. Davis and I. S. Duff, “An unsymmetric pattern multifrontal method for sparse LU factorization,” SIAM J. Matrix Anal. Appl. 18, 140-158 (1997).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef]

Desbat, L.

A. Bilgot, V. Perrier, and L. Desbat, “Wavelets, local tomography and interventional x-ray imaging,” IEEE, Management International Conference Proceedings M9, 183 (2004).

Dinten, J. M.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46, 4896-4906 (2007).
[CrossRef]

Duff, I. S.

T. A. Davis and I. S. Duff, “An unsymmetric pattern multifrontal method for sparse LU factorization,” SIAM J. Matrix Anal. Appl. 18, 140-158 (1997).
[CrossRef]

Eppstein, M. J.

F. Fedele, J. P. Laible, and M. J. Eppstein, “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation,” J. Comput. Phys. 187, 597-619 (2003).
[CrossRef]

M. J. Eppstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging 22, 1215-1223 (2003).
[CrossRef]

Fedele, F.

M. J. Eppstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging 22, 1215-1223 (2003).
[CrossRef]

F. Fedele, J. P. Laible, and M. J. Eppstein, “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation,” J. Comput. Phys. 187, 597-619 (2003).
[CrossRef]

Feng, T. C.

Garcia, R. C.

M. N. O. Sadiku, C. M. Akujuobi, and R. C. Garcia, “An introduction to wavelets in electromagnetics,” IEEE Microwave Magazine 6, 63-72 (2005).
[CrossRef]

Godavarty, A.

M. J. Eppstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging 22, 1215-1223 (2003).
[CrossRef]

R. Roy, A. Godavarty, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical tomography using referenced measurements of heterogeneous media,” IEEE Trans. Med. Imaging 22, 824-836 (2003).
[CrossRef]

Graves, E.

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

Grigori, L.

L. Grigori, “A framework for efficient sparse LU factorization in a cluster based platform,” in Proceedings of International Workshop on Cluster Computing (Springer-Verlag, 2001).

Grossmann, A.

A. Grossmann and J. Morlet, “Decomposition of hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723-736 (1984).
[CrossRef]

Haskell, R. C.

Hervé, L.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46, 4896-4906 (2007).
[CrossRef]

Intes, X.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1-N10(2002).
[CrossRef]

Josephson, L.

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

Joshi, A.

Josserand, V.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

Kaipio, J. P.

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987), Chap. 7.

Kanmani, B.

B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, 61640R1-10 (2006).

Karl, W. C.

M. Bathia, W. C. Karl, and A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” IEEE Trans. Med. Imaging 15, 92-101 (1996).
[CrossRef]

Koenig, A.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46, 4896-4906 (2007).
[CrossRef]

Kolehmainen, V.

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

Laible, J.

M. J. Eppstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging 22, 1215-1223 (2003).
[CrossRef]

Laible, J. P.

F. Fedele, J. P. Laible, and M. J. Eppstein, “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation,” J. Comput. Phys. 187, 597-619 (2003).
[CrossRef]

Lascaux, P.

P. Lascaux and R. Theodor, Analyse Numérique Matricielle Appliquée à l'Art de l'Ingénieur, 2nd ed. (Masson, 1998).

Liu, K. J. R.

F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein, and D. Walnut, “Wavelet-based multiresolution local tomography,” IEEE Trans. Image Process. 10, 1412-1430 (1997).
[CrossRef]

Lu, Y.

Z. Xiang and Y. Lu, “An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations,” IEEE Trans. Antennas Propag. 45, 1205-1213 (1997).
[CrossRef]

Mallat, S. G.

S. G. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674-693 (1989).
[CrossRef]

Marchuk, G. I.

G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems (CRC, 1996).

McAdams, M. S.

Morlet, J.

A. Grossmann and J. Morlet, “Decomposition of hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723-736 (1984).
[CrossRef]

Nath, S. K.

S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet-based compression and denoising of optical tomography data,” Opt. Commun. 167, 37-46 (1999).
[CrossRef]

Ntziachristos, V.

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9, 123-128 (2003).
[CrossRef]

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1-N10(2002).
[CrossRef]

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26, 893-895 (2001).
[CrossRef]

Pandit, M.

S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet-based compression and denoising of optical tomography data,” Opt. Commun. 167, 37-46 (1999).
[CrossRef]

Peltié, P.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46, 4896-4906 (2007).
[CrossRef]

Perrier, V.

A. Bilgot, V. Perrier, and L. Desbat, “Wavelets, local tomography and interventional x-ray imaging,” IEEE, Management International Conference Proceedings M9, 183 (2004).

Rashid-Farrokhi, F.

F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein, and D. Walnut, “Wavelet-based multiresolution local tomography,” IEEE Trans. Image Process. 10, 1412-1430 (1997).
[CrossRef]

Ripoll, J.

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

Rizo, P.

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J. M. Dinten, P. Peltié, and P. Rizo, “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt. 46, 4896-4906 (2007).
[CrossRef]

Roy, R.

R. Roy, A. Godavarty, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical tomography using referenced measurements of heterogeneous media,” IEEE Trans. Med. Imaging 22, 824-836 (2003).
[CrossRef]

Saad, Y.

Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp. 7, 856-869 (1986).
[CrossRef]

Sadiku, M. N. O.

M. N. O. Sadiku, C. M. Akujuobi, and R. C. Garcia, “An introduction to wavelets in electromagnetics,” IEEE Microwave Magazine 6, 63-72 (2005).
[CrossRef]

Sahiner, B.

B. Sahiner and A. E. Yagle, “Image reconstruction from projections under wavelet constraints,” IEEE Trans. Signal Process. 41, 3579-3584(1993).
[CrossRef]

Schellenberger, E. A.

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

Schultz, M. H.

Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp. 7, 856-869 (1986).
[CrossRef]

Schweiger, M.

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

Sevick-Muraca, E. M.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12, 5402-5417 (2004).
[CrossRef]

R. Roy, A. Godavarty, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical tomography using referenced measurements of heterogeneous media,” IEEE Trans. Med. Imaging 22, 824-836 (2003).
[CrossRef]

M. J. Eppstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging 22, 1215-1223 (2003).
[CrossRef]

Shutyaev, V. P.

G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems (CRC, 1996).

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987), Chap. 7.

Somersalo, E.

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

Svaasand, L. O.

Tarvainen, T.

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

Theodor, R.

P. Lascaux and R. Theodor, Analyse Numérique Matricielle Appliquée à l'Art de l'Ingénieur, 2nd ed. (Masson, 1998).

Tromberg, B. J.

Tsay, T. T.

Vasu, R. M.

B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, 61640R1-10 (2006).

S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet-based compression and denoising of optical tomography data,” Opt. Commun. 167, 37-46 (1999).
[CrossRef]

Vauhkonen, M.

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

Wagner, R. L.

R. L. Wagner and W. C. Chew, “A study of wavelets for the solution of electromagnetic integral equations,” IEEE Trans. Antennas Propag. 43, 802-810 (1995).
[CrossRef]

Walnut, D.

F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein, and D. Walnut, “Wavelet-based multiresolution local tomography,” IEEE Trans. Image Process. 10, 1412-1430 (1997).
[CrossRef]

Wang, Y.

W. Zhu, Y. Wang, and J. Zhang, “Total least squares reconstruction with wavelets for optical tomography,” J. Opt. Soc. Am. A 15, 2639-2650 (1998).
[CrossRef]

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

Weissleder, R.

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9, 123-128 (2003).
[CrossRef]

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26, 893-895 (2001).
[CrossRef]

Willsky, A. S.

M. Bathia, W. C. Karl, and A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” IEEE Trans. Med. Imaging 15, 92-101 (1996).
[CrossRef]

Xiang, Z.

Z. Xiang and Y. Lu, “An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations,” IEEE Trans. Antennas Propag. 45, 1205-1213 (1997).
[CrossRef]

Yagle, A. E.

B. Sahiner and A. E. Yagle, “Image reconstruction from projections under wavelet constraints,” IEEE Trans. Signal Process. 41, 3579-3584(1993).
[CrossRef]

Yao, Y.

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

Yessayan, D.

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

Yodh, A.

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1-N10(2002).
[CrossRef]

Yuster, R.

R. Yuster and U. Zwick, “Fast sparse matrix multiplication,” in Proceedings of the 12th Annual European Symposium on Algorithms (Springer-Verlag, 2004).

Zhang, C.

M. J. Eppstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging 22, 1215-1223 (2003).
[CrossRef]

Zhang, J.

Zhu, W.

W. Zhu, Y. Wang, and J. Zhang, “Total least squares reconstruction with wavelets for optical tomography,” J. Opt. Soc. Am. A 15, 2639-2650 (1998).
[CrossRef]

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

Zwick, U.

R. Yuster and U. Zwick, “Fast sparse matrix multiplication,” in Proceedings of the 12th Annual European Symposium on Algorithms (Springer-Verlag, 2004).

Appl. Opt. (1)

IEEE Microwave Magazine (1)

M. N. O. Sadiku, C. M. Akujuobi, and R. C. Garcia, “An introduction to wavelets in electromagnetics,” IEEE Microwave Magazine 6, 63-72 (2005).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. L. Wagner and W. C. Chew, “A study of wavelets for the solution of electromagnetic integral equations,” IEEE Trans. Antennas Propag. 43, 802-810 (1995).
[CrossRef]

Z. Xiang and Y. Lu, “An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations,” IEEE Trans. Antennas Propag. 45, 1205-1213 (1997).
[CrossRef]

IEEE Trans. Image Process. (1)

F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein, and D. Walnut, “Wavelet-based multiresolution local tomography,” IEEE Trans. Image Process. 10, 1412-1430 (1997).
[CrossRef]

IEEE Trans. Med. Imaging (4)

R. Roy, A. Godavarty, and E. M. Sevick-Muraca, “Fluorescence-enhanced optical tomography using referenced measurements of heterogeneous media,” IEEE Trans. Med. Imaging 22, 824-836 (2003).
[CrossRef]

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

M. J. Eppstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E. M. Sevick-Muraca, “A comparison of exact and approximate adjoint sensitivities in fluorescence tomography,” IEEE Trans. Med. Imaging 22, 1215-1223 (2003).
[CrossRef]

M. Bathia, W. C. Karl, and A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” IEEE Trans. Med. Imaging 15, 92-101 (1996).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (1)

S. G. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674-693 (1989).
[CrossRef]

IEEE Trans. Signal Process. (1)

B. Sahiner and A. E. Yagle, “Image reconstruction from projections under wavelet constraints,” IEEE Trans. Signal Process. 41, 3579-3584(1993).
[CrossRef]

Inverse Probl. (1)

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inverse Probl. 22, 175-195 (2006).
[CrossRef]

J. Biomed. Opt. (1)

A. Koenig, L. Hervé, V. Josserand, M. Berger, J. Boutet, A. Da Silva, J. M. Dinten, P. Peltié, J. L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
[CrossRef]

J. Comput. Phys. (1)

F. Fedele, J. P. Laible, and M. J. Eppstein, “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation,” J. Comput. Phys. 187, 597-619 (2003).
[CrossRef]

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

Nat. Med. (1)

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9, 123-128 (2003).
[CrossRef]

Opt. Commun. (1)

S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet-based compression and denoising of optical tomography data,” Opt. Commun. 167, 37-46 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (2)

X. Intes, V. Ntziachristos, J. P. Culver, A. Yodh, and B. Chance, “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol. 47, N1-N10(2002).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

V. Ntziachristos, E. A. Schellenberger, J. Ripoll, D. Yessayan, E. Graves, A. Bogdanov, L. Josephson, and R. Weissleder, “Visualization of antitumor treatment by means of fluorescence molecular tomography with an annexin V-Cy5.5 conjugate,” Proc. Natl. Acad. Sci. USA 101, 12294-12299(2004).
[CrossRef]

Proc. SPIE (1)

B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, 61640R1-10 (2006).

SIAM J. Math. Anal. (1)

A. Grossmann and J. Morlet, “Decomposition of hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723-736 (1984).
[CrossRef]

SIAM J. Matrix Anal. Appl. (1)

T. A. Davis and I. S. Duff, “An unsymmetric pattern multifrontal method for sparse LU factorization,” SIAM J. Matrix Anal. Appl. 18, 140-158 (1997).
[CrossRef]

SIAM J. Sci. Stat. Comp. (1)

Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp. 7, 856-869 (1986).
[CrossRef]

Other (10)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1987), Chap. 7.

L. Grigori, “A framework for efficient sparse LU factorization in a cluster based platform,” in Proceedings of International Workshop on Cluster Computing (Springer-Verlag, 2001).

R. Yuster and U. Zwick, “Fast sparse matrix multiplication,” in Proceedings of the 12th Annual European Symposium on Algorithms (Springer-Verlag, 2004).

P. Lascaux and R. Theodor, Analyse Numérique Matricielle Appliquée à l'Art de l'Ingénieur, 2nd ed. (Masson, 1998).

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992), pp. 167-213.

http://www.comsol.com/.

D. S. Burnett, Finite Element Analysis: From Concepts to Applications (Addison-Wesley, 1987).

Y. Desaubies, A. Tarantola, and J. Zinn-Justin, eds., Oceanographic and Geophysical Tomography (North-Holland, 1990).

G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems (CRC, 1996).

A. Bilgot, V. Perrier, and L. Desbat, “Wavelets, local tomography and interventional x-ray imaging,” IEEE, Management International Conference Proceedings M9, 183 (2004).

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Figures (12)

Fig. 1
Fig. 1

Schema of the algorithm and introduction of the wavelet decomposition stage.

Fig. 2
Fig. 2

Haar’s scaling (Φ) and wavelet functions (Ψ).

Fig. 3
Fig. 3

Studied object and the nodes of the finite element mesh: (b) the defined horizontal layers (very coarse example), (c), (d) the classified nodes in each layer (stars in layer 1, squares in layer 2, triangles in layer 3, crosses in layer 4, and circles in layer 5), (e) optimal classification of the nodes in a given layer, (f) example of a stiffness matrix: (top) a sample of the original matrix, (bottom) a sample of the classified matrix (dark-colored pixels, high values; gray-colored pixels, zeros).

Fig. 4
Fig. 4

(a) Schema of the parallelepipedal object: at the bottom, two fluorescent cylinders; at the top, the highly absorbing zone; crossed points, positions of the source; dotted points, detectors. (b)–(h) Results after reconstruction (dark gray, 90% of the maximum of the reconstructed fluorescence signal; light gray, 50%), (b) without model reduction in the FEM equations, 2 14 nodes in the FEM mesh, (c) without model reduction in the FEM equations, 2 13 nodes in the FEM mesh, (d) without model reduction in the FEM equations, 2 12 nodes in the FEM mesh, (e) without model reduction in the FEM equations, 2 11 nodes in the FEM mesh, (f) with model reduction at the first approximation level ( 2 13 nodes), (g) with model reduction at the second approximation level ( 2 12 nodes), (h) with model reduction at the third approximation level ( 2 11 nodes).

Fig. 5
Fig. 5

Comparison of normalized results in the cross sections Z = 5 mm and Y = 10 mm : (a) with model reduction at first approximation level, (b) with model reduction at second approximation level, (c) with model reduction at third approximation level, (d) magnification of (b). The functions represent the intensity of reconstructed fluorescence along the axis X. Discontinuous thin curve, theoretical fluorescence; dotted line, fluorescence after modeling with 2 14 nodes in the FEM, without wavelets; full curve, fluorescence after modeling with 2 13 (a), 2 12 (b), or 2 11 (c) nodes in the FEM, without wavelets; crossed curve, fluorescence after modeling with 2 14 nodes in the FEM, reduced to 2 13 (a), 2 12 (b), or 2 11 (c) nodes by wavelets.

Fig. 6
Fig. 6

(a) Schema of the semicylindrical object: at the bottom, two fluorescent cylinders; crossed points, positions of the source; dotted points, detectors. (b)–(h) Results after reconstruction (dark gray, 90% of the maximum of the reconstructed fluorescence signal; light gray, 50%), (b) without model reduction in the FEM equations, 2 13 nodes in the FEM mesh, (c) without model reduction in the FEM equations, 2 12 nodes in the FEM mesh, (d) without model reduction in the FEM equations, 2 11 nodes in the FEM mesh, (e) without model reduction in the FEM equations, 2 10 nodes in the FEM mesh, (f) with model reduction at the first approximation level ( 2 12 nodes), (g) with model reduction at the second approximation level ( 2 11 nodes), (h) with model reduction at the third approximation level ( 2 10 nodes).

Fig. 7
Fig. 7

Comparison of normalized results in the cross sections Z = 12 mm and Y = 12 mm : (a) with model reduction at the first approximation level, (b) with model reduction at the second approximation level, (c) with model reduction at the third approximation level, (d) magnification of (b). The functions represent the intensity of reconstructed fluorescence along the axis, X. Discontinuous thin curve, theoretical fluorescence; dotted curve, fluorescence after modeling with 2 13 nodes in the FEM, without wavelets; full curve, fluorescence after modeling with 2 12 (a), 2 11 (b), or 2 10 (c) nodes in the FEM, without wavelets; crossed curve, fluorescence after modeling with 2 13 nodes in the FEM, reduced to 2 12 (a), 2 11 (b), or 2 10 (c) nodes by wavelets.

Fig. 8
Fig. 8

Photograph (left) and schema (right) of the phantom. Four holes have been made, but only two (filled points in the figure) are filled with fluorophores.

Fig. 9
Fig. 9

Schema of the experimental setup.

Fig. 10
Fig. 10

(a) Schema of the object: the two cylinders are the fluorescent regions; crossed points, positions of the source; dotted points, detectors. (b)–(h) Results obtained on the experiment after reconstruction (dark gray, 90% of the maximum of the reconstructed fluorescence signal; light gray, 70%), (b) without model reduction in the FEM equations, 2 13 nodes in the FEM mesh, (c) without model reduction in the FEM equations, 2 12 nodes in the FEM mesh, (d) without model reduction in the FEM equations, 2 11 nodes in the FEM mesh, (e) without model reduction in the FEM equations, 2 10 nodes in the FEM mesh, (f) with model reduction at the first approximation level ( 2 12 nodes), (g) with model reduction at the second approximation level ( 2 11 nodes), (h) with model reduction at the third approximation level ( 2 10 nodes).

Fig. 11
Fig. 11

Logarithmic graph of the operations number as a function of the size of the stiffness matrix: points, LU factorization; triangles, GMRES; squares, matrices multiplications; stars, total number of operations.

Fig. 12
Fig. 12

Comparison of eigenvalues: dashed curve, 200 highest eigenvalues of the original stiffness matrix A; full curve, 100 highest eigenvalues of the approximation matrix A a ; crossed curve, 100 highest eigenvalues of the diagonal details matrix A d ; dotted curve, 100 highest eigenvalues of the vertical/horizontal details matrix A v / A h .

Tables (2)

Tables Icon

Table 1 Number of Flops: Each Number Is an Average of the Calculated Flops for Three Positions of the Source and for Three Detectors

Tables Icon

Table 2 Ratio of Number of Flops in the Equation Solving, Between the Original Case and the Case of Wavelet Projection

Equations (15)

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

{ · ( D ( r ) u x ( r , r s ) ) + μ a ( r ) u x ( r , r s ) = q ( r s ) · ( D ( r ' ) u m ( r ' , r , r s ) ) + μ a ( r ' ) u m ( r ' , r , r s ) = β ( r ) u x ( r , r s ) .
{ n · ( D u x ( r , r s ) ) + ζ x u x ( r , r s ) = 0 r Ω n · ( D u m ( r ' , r , r s ) ) + ζ m u m ( r ' , r , r s ) = 0 r ' Ω ,
u m obs ( r d , r s ) = Ω u m ( r d , r , r s ) d r = Ω g m ( r , r d ) β ( r ) u x ( r , r s ) d r ,
{ · ( D g m ( r , r d ) ) + μ a g m ( r , r d ) = q ( r d ) n · ( D g m ( r , r d ) ) + ζ m g m ( r , r d ) = 0 r Ω ,
· ( D u ) + μ a u = q .
A u = q ,
u m = W β ,
M A M T M u = M q A u = q .
A a u a = q a .
M A M T = ( A a A v A h A d ) ,
M A M T Mu = Mq A u = q ( A a A v A h A d ) u = q ( A a A v A h A d ) ( u a u d ) = ( q a q d ) ,
( A a A v A h A d ) ( u a u d ) = ( q a q d ) { ( A a u a + A v u d ) = q a ( A h u a + A d u d ) = q d .
A = M A M T = ( a + b 2 0 a b 2 0 0 c + d 2 0 c d 2 a b 2 0 a + b 2 0 0 c d 2 0 c + d 2 ) = ( A a A v A h A d ) .
{ A a u a = q a A a u d = q d .
A a u a = q a .

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