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June  2018, 12(3): 607-634. doi: 10.3934/ipi.2018026

Morozov principle for Kullback-Leibler residual term and Poisson noise

CREATIS, CNRS UMR 5220, INSERM U1044, INSA de Lyon, Université de Lyon 1, Université de Lyon, 69621, Villeurbanne Cedex, France

Received  June 2017 Revised  November 2017 Published  March 2018

We study the properties of a regularization method for inverse problems corrupted by Poisson noise with Kullback-Leibler divergence as data term. The regularization parameter is chosen according to a Morozov type principle. We show that this method of choice of the parameter is well-defined. This a posteriori choice leads to a convergent regularization method. Convergences rates are obtained for this a posteriori choice of the regularization parameter when some source condition is satisfied.

Citation: Bruno Sixou, Tom Hohweiller, Nicolas Ducros. Morozov principle for Kullback-Leibler residual term and Poisson noise. Inverse Problems & Imaging, 2018, 12 (3) : 607-634. doi: 10.3934/ipi.2018026
References:
[1]

V. AlbaniA. De Cezaro and J. P. Zubelli, On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy, Inverse Problems and Imaging, 10 (2016), 1-25.  doi: 10.3934/ipi.2016.10.1.  Google Scholar

[2]

A. Antoniadis and J. Bigot, Poisson inverse problems, Ann. Stat., 34 (2006), 2132-2158.  doi: 10.1214/009053606000000687.  Google Scholar

[3]

S. Anzengruber and R. Ramlau, Morozov discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001, 17pp.   Google Scholar

[4]

S. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities, Inverse Problems, 27 (2010), 105007, 18pp.   Google Scholar

[5]

A. BanerjeeS. MeruguI. Dhilon and J. Ghosh, Clustering with Bregman divergences, Journal of Machine Learning Research, 6 (2005), 1705-1749.   Google Scholar

[6]

J. M. Bardsley and J. Goldes, Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation, Inverse Problems, 25 (2009), 095005, 18pp.   Google Scholar

[7]

J. M. Bardsley, A theoretical framework for the regularization of Poisson Likelihood estimation problems, Inverse Problems and Imaging, 4 (2010), 11-17.  doi: 10.3934/ipi.2010.4.11.  Google Scholar

[8]

0. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory, Wiley Publishers, 1978.  Google Scholar

[9]

M. BerteroP. BoccacciG. TalentiR. Zanella and L. Zanni, A discrepancy principle for Poisson data, Inverse Problems, 26 (2010), 105004, 20 pp.   Google Scholar

[10]

G. L. BesneraisJ. F. Bercher and G. Demoment, A new look at entropy for solving linear inverse problems, IEEE Transactions on Information Theory, 45 (1999), 1566-1578.  doi: 10.1109/18.771159.  Google Scholar

[11]

T. Bonesky, Morozov's discrepancy principle and Tikhonov-type functionals, Inverse Problems, 25 (2009), 015015, 11pp.   Google Scholar

[12]

J. M. Borwein and A. S. Lewis, Convergences of best entropy estimates, SIAM Journal on Optimization, 1 (1991), 191-205.  doi: 10.1137/0801014.  Google Scholar

[13]

L. M. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming, USSR Computational Mathematics and Mathematical Physics, 7 (1967), 200-217.   Google Scholar

[14]

M. Burger and S. Osher, Convergence rates of convex variational reglarization, Inverse Problems, 20 (2004), 1411-1421.  doi: 10.1088/0266-5611/20/5/005.  Google Scholar

[15]

C. Byrne, Iterative Optimization in Inverse Problems, Taylor and Francis, 2014.  Google Scholar

[16]

C. Byrne and P. Eggermont, EM algorithms, Handbook of Mathematical Methods in Imaging, Vol. 1, 2, 3,305-388, Springer, New York, 2015.  Google Scholar

[17]

L. Cavalier and J. Y. Koo, Poisson intensity estimation for tomographic data using a wavelet shrinkage approach, IEEE Trans. on Information Theory, 48 (2002), 2794-2802.  doi: 10.1109/TIT.2002.802632.  Google Scholar

[18]

A. Das Gupta, On a Differential Equation and one Step Recursion for Poisson Moments, Purdue University, Technical Report 98-02. Google Scholar

[19]

N. DucrosJ. F. P. AbascalB. SixouS. Rit and F. Peyrin, Regularization of nonlinear decomposition of spectral X-ray projection images, Med.Phys., 44 (2017), 174-187.   Google Scholar

[20]

P. P. B. Eggermont, Maximum entropy regularization for Fredholm integral equations of the first kind, SIAM Journal on Mathematical Analysis, 24 (1993), 1557-1576.  doi: 10.1137/0524088.  Google Scholar

[21]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers: Dordrecht, 1996.  Google Scholar

[22]

F. FavatiG. LottiO. Menchi and F. Menchi, Performance analysis of maximum likekihood methods for regularization problems with nonnegativity constraints, Inverse Problems, 26 (2010), 085013, 18pp.   Google Scholar

[23]

I. M. Gelfand and G. E. Shilov, Generalized Functions, Academic Press, San Diego, 1964.  Google Scholar

[24]

R. M. Gray, Probability, Random Processes and Ergodic Properties, Second edition. Springer, Dordrecht, 2009.  Google Scholar

[25]

G. Grubb, Distributions and Operators, Springer-Verlag, New York, 2009.  Google Scholar

[26]

T. Hohage and F. Werner, Inhomogeneous Poisson intensity estimation via information projections onto wavelet subspaces, Inverse Problems, 32 (2016), 093001, 56pp.   Google Scholar

[27]

K. Ito, Fundations of Stochastic Differential Equations in Infinite Dimensional Spaces, SIAM, 1984.  Google Scholar

[28]

J. Y. Koo and W. C. Kim, Inhomogeneous Poisson intensity estimation via information projections onto wavelet subspaces, Journal of Korean Statistical Society, 31 (2002), 343-357.   Google Scholar

[29]

S. Kullback and R. A. Leibler, On information and sufficiency, Annals of Mathematical Statistics, 22 (1951), 79-86.  doi: 10.1214/aoms/1177729694.  Google Scholar

[30]

Y. Long and J. A. Fessler, Multi-material decomposition using statistical image reconstruction for spectral CT, Trans. Med. Imaging, 33 (2014), 1614-1626.   Google Scholar

[31]

R. D. Nowak and E. D. Kolaczyk, A Bayesian multiscale framework for Poisson inverse problems, IEEE Trans. on Information Theory, 46 (2000), 1811-1825.  doi: 10.1109/18.857793.  Google Scholar

[32]

M. H. Neumann, Absolute regularity and ergodicity of Poisson count processes, Bernoulli, 17 (2011), 1268-1284.  doi: 10.3150/10-BEJ313.  Google Scholar

[33]

J. M. Ollinger and J. A. Fessler, Positon-emission tomography, IEEE Signal Processing Magazine, 14 (1997), 43-55.   Google Scholar

[34]

C. Pöschl, Tikhonov Regularization with General Residual Term, PhD Thesis, Univeristat Innsbruck, 2008. Google Scholar

[35]

E. Resmerita and R. S. Anderssen, Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problem, Mathematical Methods in the Applied sciences, 30 (2007), 1527-1544.  doi: 10.1002/mma.855.  Google Scholar

[36]

E. Resmerita, Regularization of ill-posed inverse problems in Banach spaces: Convergence rates, Inverse Problems, 21 (2005), 1301-1314.  doi: 10.1088/0266-5611/21/4/007.  Google Scholar

[37]

P. Reynaud-Bouret, Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities, Probab. Theory Relat. Fields, 126 (2003), 103-153.  doi: 10.1007/s00440-003-0259-1.  Google Scholar

[38]

R. J. Santos and A. R. De Pierro, A new parameters choice method for ill-posed problem with Poisson data and its application to emission tomographic imaging, International Journal of Tomography and Statistics, 11 (2009), 33-52.   Google Scholar

[39]

O. Scherzer, M. Grassmair, H. Grossauer, M. Haltmaier and F. Lenzen, Variational Methods in Imaging, Springer Verlag, New York, 2009.  Google Scholar

[40]

J.P. SchlomkaE. RoesslR. DorscheidS. DillG. MartensT. IstelC. BumerC. HerrmannR. SteadmanG. ZeitlerA. Livne and R. Proska, Experimental feasiblitiy of multi-energy photon-counting k-edge imaging in pre-clinical computed tomography, Physics in Medicine and Biology, 53 (2008), p4031.   Google Scholar

[41]

L. Schwarz, Theorie des Distributions, Hermann, Paris, 1966.  Google Scholar

[42]

Spray tollbox, https://www.creatis.insa-lyon.fr/~ducros/WebPage/spray.html Google Scholar

[43]

J. L. Stark and F. Murtagh, Astronomical Image and Data Analysis, Springer Verlag, New York, 2006. Google Scholar

[44]

A. N. Tikhonov and V. Y. Arsenin, Solutions to Ill-Posed Problems, Winston-Wiley, New York, 1977.  Google Scholar

[45]

H. Wang and P. C. Miller, Scaled heavy-ball acceleration of the Richardson-Lucy algorithm for 3D microscopy image restoration, IEEE Transactions on Image Processing, 23 (2014), 848-854.  doi: 10.1109/TIP.2013.2291324.  Google Scholar

[46]

F. Werner and T. Hohage, Convergence rate in expectation ofr Tikhonov-type regularization of inverse problems with Poisson data, Inverse Problems, 28 (2012), 104004, 15pp.   Google Scholar

[47]

F. Werner, Inverse problems with Poisson data: Tikhonov-type Regularization and Iteratively Regularized Newton Methods, PhD thesis, University of Göttingen, 2012. Google Scholar

[48]

R. ZanellaP. BoccacciL. Zani and M. Bertero, Efficient gradient projection methods for edge-preserving removal of poisson noise, Inverse Problems, 25 (2009), 045010, 24pp.   Google Scholar

show all references

References:
[1]

V. AlbaniA. De Cezaro and J. P. Zubelli, On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy, Inverse Problems and Imaging, 10 (2016), 1-25.  doi: 10.3934/ipi.2016.10.1.  Google Scholar

[2]

A. Antoniadis and J. Bigot, Poisson inverse problems, Ann. Stat., 34 (2006), 2132-2158.  doi: 10.1214/009053606000000687.  Google Scholar

[3]

S. Anzengruber and R. Ramlau, Morozov discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001, 17pp.   Google Scholar

[4]

S. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities, Inverse Problems, 27 (2010), 105007, 18pp.   Google Scholar

[5]

A. BanerjeeS. MeruguI. Dhilon and J. Ghosh, Clustering with Bregman divergences, Journal of Machine Learning Research, 6 (2005), 1705-1749.   Google Scholar

[6]

J. M. Bardsley and J. Goldes, Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation, Inverse Problems, 25 (2009), 095005, 18pp.   Google Scholar

[7]

J. M. Bardsley, A theoretical framework for the regularization of Poisson Likelihood estimation problems, Inverse Problems and Imaging, 4 (2010), 11-17.  doi: 10.3934/ipi.2010.4.11.  Google Scholar

[8]

0. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory, Wiley Publishers, 1978.  Google Scholar

[9]

M. BerteroP. BoccacciG. TalentiR. Zanella and L. Zanni, A discrepancy principle for Poisson data, Inverse Problems, 26 (2010), 105004, 20 pp.   Google Scholar

[10]

G. L. BesneraisJ. F. Bercher and G. Demoment, A new look at entropy for solving linear inverse problems, IEEE Transactions on Information Theory, 45 (1999), 1566-1578.  doi: 10.1109/18.771159.  Google Scholar

[11]

T. Bonesky, Morozov's discrepancy principle and Tikhonov-type functionals, Inverse Problems, 25 (2009), 015015, 11pp.   Google Scholar

[12]

J. M. Borwein and A. S. Lewis, Convergences of best entropy estimates, SIAM Journal on Optimization, 1 (1991), 191-205.  doi: 10.1137/0801014.  Google Scholar

[13]

L. M. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming, USSR Computational Mathematics and Mathematical Physics, 7 (1967), 200-217.   Google Scholar

[14]

M. Burger and S. Osher, Convergence rates of convex variational reglarization, Inverse Problems, 20 (2004), 1411-1421.  doi: 10.1088/0266-5611/20/5/005.  Google Scholar

[15]

C. Byrne, Iterative Optimization in Inverse Problems, Taylor and Francis, 2014.  Google Scholar

[16]

C. Byrne and P. Eggermont, EM algorithms, Handbook of Mathematical Methods in Imaging, Vol. 1, 2, 3,305-388, Springer, New York, 2015.  Google Scholar

[17]

L. Cavalier and J. Y. Koo, Poisson intensity estimation for tomographic data using a wavelet shrinkage approach, IEEE Trans. on Information Theory, 48 (2002), 2794-2802.  doi: 10.1109/TIT.2002.802632.  Google Scholar

[18]

A. Das Gupta, On a Differential Equation and one Step Recursion for Poisson Moments, Purdue University, Technical Report 98-02. Google Scholar

[19]

N. DucrosJ. F. P. AbascalB. SixouS. Rit and F. Peyrin, Regularization of nonlinear decomposition of spectral X-ray projection images, Med.Phys., 44 (2017), 174-187.   Google Scholar

[20]

P. P. B. Eggermont, Maximum entropy regularization for Fredholm integral equations of the first kind, SIAM Journal on Mathematical Analysis, 24 (1993), 1557-1576.  doi: 10.1137/0524088.  Google Scholar

[21]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers: Dordrecht, 1996.  Google Scholar

[22]

F. FavatiG. LottiO. Menchi and F. Menchi, Performance analysis of maximum likekihood methods for regularization problems with nonnegativity constraints, Inverse Problems, 26 (2010), 085013, 18pp.   Google Scholar

[23]

I. M. Gelfand and G. E. Shilov, Generalized Functions, Academic Press, San Diego, 1964.  Google Scholar

[24]

R. M. Gray, Probability, Random Processes and Ergodic Properties, Second edition. Springer, Dordrecht, 2009.  Google Scholar

[25]

G. Grubb, Distributions and Operators, Springer-Verlag, New York, 2009.  Google Scholar

[26]

T. Hohage and F. Werner, Inhomogeneous Poisson intensity estimation via information projections onto wavelet subspaces, Inverse Problems, 32 (2016), 093001, 56pp.   Google Scholar

[27]

K. Ito, Fundations of Stochastic Differential Equations in Infinite Dimensional Spaces, SIAM, 1984.  Google Scholar

[28]

J. Y. Koo and W. C. Kim, Inhomogeneous Poisson intensity estimation via information projections onto wavelet subspaces, Journal of Korean Statistical Society, 31 (2002), 343-357.   Google Scholar

[29]

S. Kullback and R. A. Leibler, On information and sufficiency, Annals of Mathematical Statistics, 22 (1951), 79-86.  doi: 10.1214/aoms/1177729694.  Google Scholar

[30]

Y. Long and J. A. Fessler, Multi-material decomposition using statistical image reconstruction for spectral CT, Trans. Med. Imaging, 33 (2014), 1614-1626.   Google Scholar

[31]

R. D. Nowak and E. D. Kolaczyk, A Bayesian multiscale framework for Poisson inverse problems, IEEE Trans. on Information Theory, 46 (2000), 1811-1825.  doi: 10.1109/18.857793.  Google Scholar

[32]

M. H. Neumann, Absolute regularity and ergodicity of Poisson count processes, Bernoulli, 17 (2011), 1268-1284.  doi: 10.3150/10-BEJ313.  Google Scholar

[33]

J. M. Ollinger and J. A. Fessler, Positon-emission tomography, IEEE Signal Processing Magazine, 14 (1997), 43-55.   Google Scholar

[34]

C. Pöschl, Tikhonov Regularization with General Residual Term, PhD Thesis, Univeristat Innsbruck, 2008. Google Scholar

[35]

E. Resmerita and R. S. Anderssen, Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problem, Mathematical Methods in the Applied sciences, 30 (2007), 1527-1544.  doi: 10.1002/mma.855.  Google Scholar

[36]

E. Resmerita, Regularization of ill-posed inverse problems in Banach spaces: Convergence rates, Inverse Problems, 21 (2005), 1301-1314.  doi: 10.1088/0266-5611/21/4/007.  Google Scholar

[37]

P. Reynaud-Bouret, Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities, Probab. Theory Relat. Fields, 126 (2003), 103-153.  doi: 10.1007/s00440-003-0259-1.  Google Scholar

[38]

R. J. Santos and A. R. De Pierro, A new parameters choice method for ill-posed problem with Poisson data and its application to emission tomographic imaging, International Journal of Tomography and Statistics, 11 (2009), 33-52.   Google Scholar

[39]

O. Scherzer, M. Grassmair, H. Grossauer, M. Haltmaier and F. Lenzen, Variational Methods in Imaging, Springer Verlag, New York, 2009.  Google Scholar

[40]

J.P. SchlomkaE. RoesslR. DorscheidS. DillG. MartensT. IstelC. BumerC. HerrmannR. SteadmanG. ZeitlerA. Livne and R. Proska, Experimental feasiblitiy of multi-energy photon-counting k-edge imaging in pre-clinical computed tomography, Physics in Medicine and Biology, 53 (2008), p4031.   Google Scholar

[41]

L. Schwarz, Theorie des Distributions, Hermann, Paris, 1966.  Google Scholar

[42]

Spray tollbox, https://www.creatis.insa-lyon.fr/~ducros/WebPage/spray.html Google Scholar

[43]

J. L. Stark and F. Murtagh, Astronomical Image and Data Analysis, Springer Verlag, New York, 2006. Google Scholar

[44]

A. N. Tikhonov and V. Y. Arsenin, Solutions to Ill-Posed Problems, Winston-Wiley, New York, 1977.  Google Scholar

[45]

H. Wang and P. C. Miller, Scaled heavy-ball acceleration of the Richardson-Lucy algorithm for 3D microscopy image restoration, IEEE Transactions on Image Processing, 23 (2014), 848-854.  doi: 10.1109/TIP.2013.2291324.  Google Scholar

[46]

F. Werner and T. Hohage, Convergence rate in expectation ofr Tikhonov-type regularization of inverse problems with Poisson data, Inverse Problems, 28 (2012), 104004, 15pp.   Google Scholar

[47]

F. Werner, Inverse problems with Poisson data: Tikhonov-type Regularization and Iteratively Regularized Newton Methods, PhD thesis, University of Göttingen, 2012. Google Scholar

[48]

R. ZanellaP. BoccacciL. Zani and M. Bertero, Efficient gradient projection methods for edge-preserving removal of poisson noise, Inverse Problems, 25 (2009), 045010, 24pp.   Google Scholar

Figure 1.  Evolution of the Kullback-Leibler functional for $I = 3 $ and $P = 6820$. The value $m/2$ is displayed for comparison
Figure 2.  Maps of the reconstructed bone and soft tissues projected mass for $I = 3 $ and $P = 6820$
Figure 3.  Evolution of the Kullback-Leibler functional for $I = 10 $ and $P = 1737984$.The value $m/2$ is displayed for comparison
Figure 4.  Maps of the reconstructed bone and soft tissues projected mass for $I = 10 $ and $P = 1737984$
Figure 5.  Evolution of the Kullback-Leibler functional for $I = 3 $ and $P = 108624$. The value $m/2$ is displayed for comparison
Figure 6.  Maps of the reconstructed bone and soft tissues projected mass for $I = 3$ and $P = 108624$
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