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February  2013, 7(1): 267-282. doi: 10.3934/ipi.2013.7.267

Absorption and phase retrieval with Tikhonov and joint sparsity regularizations

Bruno Sixou 1, , Valentina Davidoiu 1, , Max Langer 2, and  Francoise Peyrin 2,

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

Received  June 2012 Revised  October 2012 Published  February 2013

The X-ray phase contrast imaging technique relies on the measurement of the Fresnel diffraction intensity patterns associated to a phase shift induced by the object. The simultaneous recovery of the phase and of the absorption is an ill-posed nonlinear inverse problem. In this work, we investigate the resolution of this problem with nonlinear Tikhonov regularization and with a joint sparsity constraint regularization. The regularization functionals are minimized with a Gauss-Newton method and with a fixed point iterative method based on a surrogate functional. The algorithms are evalutated using simulated noisy data. The joint sparsity regularization gives better reconstructions for high noise levels.
Keywords: Tikhonov regularization, phase retrieval, joint sparsity, Nonlinear inverse problems, biomedical imaging., Gauss-Newton mehtod.
Mathematics Subject Classification: Primary: 65J22, 65J20, 65K10; Secondary: 52A4.
Citation: Bruno Sixou, Valentina Davidoiu, Max Langer, Francoise Peyrin. Absorption and phase retrieval with Tikhonov and joint sparsity regularizations. Inverse Problems and Imaging, 2013, 7 (1) : 267-282. doi: 10.3934/ipi.2013.7.267
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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