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Quantitative photoacoustic tomography with variable index of refraction
Absorption and phase retrieval with Tikhonov and joint sparsity regularizations
1. | CREATIS, CNRS UMR 5220, Inserm U630, INSA Lyon, Université Lyon 1, F-69621 Villeurbanne Cedex, France, France |
2. | European Synchrotron Radiation Facility, 6 rue Jules Horowitz, F-38043, Grenoble Cedex, France, France |
References:
[1] |
S. Bayat, L. Apostol, E. Boller, T. Brochard and F. Peyrin, In vivo imaging of bone micro-architecture in mice with 3D synchrotron radiation micro-tomography, Nucl. Instrum. Methods. Phys. Res., 548 (2005), 247-252. |
[2] |
M. Born and E. Wolf, "Principles of Optics," Cambridge University Press, 1997. |
[3] |
J. H.Bramble, A. Cohen and W. Dahmen, "Multiscale Problems and Methods in Numerical Simulations," Lectures given at the C.I.M.E Summer School held in Martina Franca, September 9-15, 2001, Lecture Notes in Mathematics, 1825, Springer-Verlag, Berlin, 2003. |
[4] |
E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509.
doi: 10.1109/TIT.2005.862083. |
[5] |
C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault and F. Peyrin, Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microarchitecture of human femoral heads, Med. Phys., 33 (2006), 3568-3577. |
[6] |
P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay and M. Schlenker, Phase objects in synchrotron radiation hard X-ray imaging, J. Phys. D, 29 (1996), 133-146. |
[7] |
I. Daubechies, M. Fornasier and I. Loris, Accelerated projected gradient method for linear inverse problems with sparsity constraints, J. Fourier Anal. Appl., 14 (2008), 764-792.
doi: 10.1007/s00041-008-9039-8. |
[8] |
V. Davidoiu, B. Sixou, M. Langer and F. Peyrin, Nonlinear iterative phase retrieval based on Fréchet derivative, Optic Express, 23 (2011), 22809-22819. |
[9] |
G. R. Davis and S. L. Wong, X-ray microtomography of bones and teeth, Physiol. Meas., 17 (1996), 121-146. |
[10] |
M. Defrise, I. Daubechies and C. De Mol, An iterative thresholding algorithm for linear inverse problems with sparsity constraint, Commun. Pure. Appl. Math., 57 (2004), 1413-1457.
doi: 10.1002/cpa.20042. |
[11] |
V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography, Inverse Problems, 15 (1999), 931-960.
doi: 10.1088/0266-5611/15/4/307. |
[12] |
D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306.
doi: 10.1109/TIT.2006.871582. |
[13] |
H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
doi: 10.1007/978-94-009-1740-8. |
[14] |
H. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540. |
[15] |
M. Fornasier and H. Rauhut, Recovery algorithms for vector-valued data with joint sparsity constraints, SIAM J. Numer. Anal., 46 (2008), 577-613.
doi: 10.1137/0606668909. |
[16] |
J. P. Guigay, M. Langer, R. Boistel and P. Cloetens, A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region, Opt. Lett., 32 (2007), 1617-1629. |
[17] |
T. E. Gureyev, Composite techniques for phase retrieval in the Fresnel region, Opt. Commun., 220 (2003), 49-58. |
[18] |
T. E. Gureyev and K. A. Nugent, Phase retrieval with the transport of intensity equation: Orthogonal series solution for non uniform illumination, Opt. Commun., 13 (1996), 1670-1682. |
[19] |
B. Han and Z. Shen, Dual wavelet frames and Riesz bases in Sobolev spaces, Constructive Approximation, 29 (2009), 369-406.
doi: 10.1007/s00365-008-9027-x. |
[20] |
M. Langer, P. Cloetens and F. Peyrin, Regularization of phase retrieval with phase attenuation duality prior for 3-D holotomography, IEEE Trans. Image Process, 19 (2010), 2425-2436.
doi: 10.1109/TIP.2010.2048608. |
[21] |
M. Langer, P. Cloetens, J. P. Guigay and F. Peyrin, Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography, Medical Physics, 35 (2008), 4556-4565. |
[22] |
A. Momose, T. Takeda, Y. Tai, A. Yoneyama and K. Hirano, Phase-contrast tomographic imaging using an X-ray interferometer, J. Synchrotron. Rad., 5 (1998), 309-314. |
[23] |
R. D. Nowak, S. J. Wright and M. A. T. Figueiredo, Sparse reconstruction by separable approximation, IEEE Trans. Sig. Proc., 57 (2009), 2479-2493.
doi: 10.1109/TSP.2009.2016892. |
[24] |
K. A. Nugent, Coherent mehtods in the X-rays science, Advances in Physics, 59 (2010), 1-99. |
[25] |
S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel and G. Boivin, Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography, Med. Phys., 29 (2002), 2672-2681. |
[26] |
D. M. Paganin, "Coherent X-Ray Optics," Oxford University Press, New York, 2006. |
[27] |
R. Ramlau, Morozov's discrepancy principle for Tikhonov-regularization of nonlinear operators, J. Num. Funct. Anal. Opt., 23 (2002), 147-172.
doi: 10.1081/NFA-120003676. |
[28] |
M. Salome, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel and P. Spanne, A synchrotron radiation microtomography system for the analysis of trabecular bone samples, Med. Phys., 26 (1999), 2194-2204. |
[29] |
O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging," Applied Mathematical Sciences, 167, Springer, New York, 2009. |
[30] |
G. Teschke and C. Borries, Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints, Inverse Problems, 26 (2010), 025007, 23 pp.
doi: 10.1088/0266-5611/26/2/025007. |
[31] |
G. Teschke and R. Ramlau, An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector-valued regimes and an application to color image impainting, Inverse Problems, 23 (2007), 1851-1870.
doi: 10.1088/0266-5611/23/5/005. |
[32] |
R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints, Numer. Math., 104 (2006), 177-203.
doi: 10.1007/s00211-006-0016-3. |
[33] |
J. Tropp, Algorithm for simultaneous sparse approximation. Part II: Convex relaxation, IEEE Transactions on Signal Processing, 86 (2006), 589-602. |
[34] |
T. Weikamp, C. David, O. Bunk, J. Bruder, P. Cloetens and F. Pfeiffer, X-ray phase radiography and tomography of soft tissue using grating interferometry, Eur. J. Radiol., 68 (2008), S13-S17. |
[35] |
S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, Phase contrast imaging using polychromatic X-rays, Nature, 384 (1996), 335-338. |
[36] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators," Springer-Verlag, New York, 1990. |
show all references
References:
[1] |
S. Bayat, L. Apostol, E. Boller, T. Brochard and F. Peyrin, In vivo imaging of bone micro-architecture in mice with 3D synchrotron radiation micro-tomography, Nucl. Instrum. Methods. Phys. Res., 548 (2005), 247-252. |
[2] |
M. Born and E. Wolf, "Principles of Optics," Cambridge University Press, 1997. |
[3] |
J. H.Bramble, A. Cohen and W. Dahmen, "Multiscale Problems and Methods in Numerical Simulations," Lectures given at the C.I.M.E Summer School held in Martina Franca, September 9-15, 2001, Lecture Notes in Mathematics, 1825, Springer-Verlag, Berlin, 2003. |
[4] |
E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509.
doi: 10.1109/TIT.2005.862083. |
[5] |
C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault and F. Peyrin, Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microarchitecture of human femoral heads, Med. Phys., 33 (2006), 3568-3577. |
[6] |
P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay and M. Schlenker, Phase objects in synchrotron radiation hard X-ray imaging, J. Phys. D, 29 (1996), 133-146. |
[7] |
I. Daubechies, M. Fornasier and I. Loris, Accelerated projected gradient method for linear inverse problems with sparsity constraints, J. Fourier Anal. Appl., 14 (2008), 764-792.
doi: 10.1007/s00041-008-9039-8. |
[8] |
V. Davidoiu, B. Sixou, M. Langer and F. Peyrin, Nonlinear iterative phase retrieval based on Fréchet derivative, Optic Express, 23 (2011), 22809-22819. |
[9] |
G. R. Davis and S. L. Wong, X-ray microtomography of bones and teeth, Physiol. Meas., 17 (1996), 121-146. |
[10] |
M. Defrise, I. Daubechies and C. De Mol, An iterative thresholding algorithm for linear inverse problems with sparsity constraint, Commun. Pure. Appl. Math., 57 (2004), 1413-1457.
doi: 10.1002/cpa.20042. |
[11] |
V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography, Inverse Problems, 15 (1999), 931-960.
doi: 10.1088/0266-5611/15/4/307. |
[12] |
D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306.
doi: 10.1109/TIT.2006.871582. |
[13] |
H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
doi: 10.1007/978-94-009-1740-8. |
[14] |
H. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540. |
[15] |
M. Fornasier and H. Rauhut, Recovery algorithms for vector-valued data with joint sparsity constraints, SIAM J. Numer. Anal., 46 (2008), 577-613.
doi: 10.1137/0606668909. |
[16] |
J. P. Guigay, M. Langer, R. Boistel and P. Cloetens, A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region, Opt. Lett., 32 (2007), 1617-1629. |
[17] |
T. E. Gureyev, Composite techniques for phase retrieval in the Fresnel region, Opt. Commun., 220 (2003), 49-58. |
[18] |
T. E. Gureyev and K. A. Nugent, Phase retrieval with the transport of intensity equation: Orthogonal series solution for non uniform illumination, Opt. Commun., 13 (1996), 1670-1682. |
[19] |
B. Han and Z. Shen, Dual wavelet frames and Riesz bases in Sobolev spaces, Constructive Approximation, 29 (2009), 369-406.
doi: 10.1007/s00365-008-9027-x. |
[20] |
M. Langer, P. Cloetens and F. Peyrin, Regularization of phase retrieval with phase attenuation duality prior for 3-D holotomography, IEEE Trans. Image Process, 19 (2010), 2425-2436.
doi: 10.1109/TIP.2010.2048608. |
[21] |
M. Langer, P. Cloetens, J. P. Guigay and F. Peyrin, Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography, Medical Physics, 35 (2008), 4556-4565. |
[22] |
A. Momose, T. Takeda, Y. Tai, A. Yoneyama and K. Hirano, Phase-contrast tomographic imaging using an X-ray interferometer, J. Synchrotron. Rad., 5 (1998), 309-314. |
[23] |
R. D. Nowak, S. J. Wright and M. A. T. Figueiredo, Sparse reconstruction by separable approximation, IEEE Trans. Sig. Proc., 57 (2009), 2479-2493.
doi: 10.1109/TSP.2009.2016892. |
[24] |
K. A. Nugent, Coherent mehtods in the X-rays science, Advances in Physics, 59 (2010), 1-99. |
[25] |
S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel and G. Boivin, Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography, Med. Phys., 29 (2002), 2672-2681. |
[26] |
D. M. Paganin, "Coherent X-Ray Optics," Oxford University Press, New York, 2006. |
[27] |
R. Ramlau, Morozov's discrepancy principle for Tikhonov-regularization of nonlinear operators, J. Num. Funct. Anal. Opt., 23 (2002), 147-172.
doi: 10.1081/NFA-120003676. |
[28] |
M. Salome, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel and P. Spanne, A synchrotron radiation microtomography system for the analysis of trabecular bone samples, Med. Phys., 26 (1999), 2194-2204. |
[29] |
O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging," Applied Mathematical Sciences, 167, Springer, New York, 2009. |
[30] |
G. Teschke and C. Borries, Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints, Inverse Problems, 26 (2010), 025007, 23 pp.
doi: 10.1088/0266-5611/26/2/025007. |
[31] |
G. Teschke and R. Ramlau, An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector-valued regimes and an application to color image impainting, Inverse Problems, 23 (2007), 1851-1870.
doi: 10.1088/0266-5611/23/5/005. |
[32] |
R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints, Numer. Math., 104 (2006), 177-203.
doi: 10.1007/s00211-006-0016-3. |
[33] |
J. Tropp, Algorithm for simultaneous sparse approximation. Part II: Convex relaxation, IEEE Transactions on Signal Processing, 86 (2006), 589-602. |
[34] |
T. Weikamp, C. David, O. Bunk, J. Bruder, P. Cloetens and F. Pfeiffer, X-ray phase radiography and tomography of soft tissue using grating interferometry, Eur. J. Radiol., 68 (2008), S13-S17. |
[35] |
S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, Phase contrast imaging using polychromatic X-rays, Nature, 384 (1996), 335-338. |
[36] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators," Springer-Verlag, New York, 1990. |
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