October  2019, 13(5): 1113-1137. doi: 10.3934/ipi.2019050

Kullback-Leibler residual and regularization for inverse problems with noisy data and noisy operator

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

Received  March 2019 Revised  May 2019 Published  July 2019

We study the properties of a regularization method for inverse problems with joint Kullback-Leibler data term and regularization when the data and the operator are corrupted by some noise. We show the convergence of the method and we obtain convergence rates for the approximate solution of the inverse problem and for the operator when it is characterized by some kernel, under the assumption that some source conditions are satisfied. Numerical results showing the effect of the noise levels on the reconstructed solution are provided for Spectral Computerized Tomography.

Citation: Bruno Sixou, Cyril Mory. Kullback-Leibler residual and regularization for inverse problems with noisy data and noisy operator. Inverse Problems & Imaging, 2019, 13 (5) : 1113-1137. doi: 10.3934/ipi.2019050
References:
[1]

S. W. Anzengruber and R. Ramlau, Discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001, 17pp. doi: 10.1088/0266-5611/26/2/025001.  Google Scholar

[2]

R. F. Barber, E. Y. Sidky, T. G. Schmidt and X. Pan, An algorithm for constrained one-step of spectral CT data, Physics in Medicine and Biology, 61 (2016), 3784. Google Scholar

[3]

J. M. Bardsley and J. Goldes, Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation, Inverse Problems, 25 (2009), 095005, 18pp. doi: 10.1088/0266-5611/25/9/095005.  Google Scholar

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

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M. Bertero, P. Boccacci, C. Talenti, R. Zanella and L. Zanni, A discrepancy principle for Poisson data, Inverse Problems, 25 (2010), 105004, 20pp. doi: 10.1088/0266-5611/26/10/105004.  Google Scholar

[6]

J. R. Bleyer and R. Ramlau, A double regularization approach for inverse problems with noisy data and inexact operator, Inverse Problems, 29 (2013), 025004, 16pp. doi: 10.1088/0266-5611/29/2/025004.  Google Scholar

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I. R. Bleyer and R. Ramlau, An alternating iterative minimisation algorithm for the double-regularised total least square functional, Inverse Problems, 31 (2015), 075004, 21pp. doi: 10.1088/0266-5611/31/7/075004.  Google Scholar

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M. Burger and S. Osher, Convergence rates of convex variational reglarization, Inverse Problems, 10 (2004), 1411-1422.  doi: 10.1088/0266-5611/20/5/005.  Google Scholar

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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

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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

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

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G. H. GolubP. C. Hansen and D. P. O'Leary, Tikhonov regularization and total least squares, SIAM J. Numer. Anal. Appl., 21 (1999), 185-194.  doi: 10.1137/S0895479897326432.  Google Scholar

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P. C. Hansen and M. Saxild-Hansen, AIR Tools-A MATLAB package of algebraic iterative reconstruction methods, Journal of Computational and Applied Mathematics, 236 (2012), 2067-2178.  doi: 10.1016/j.cam.2011.09.039.  Google Scholar

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B. HofmannB. KaltenbacherC. Poschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.  doi: 10.1088/0266-5611/23/3/009.  Google Scholar

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T. Hohage and F. Werner, Inverse problems with Poisson data: Statistical regularization theory, applications and algorithms, Inverse Problems, 32 (2016), 093001, 56pp. doi: 10.1088/0266-5611/32/9/093001.  Google Scholar

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

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S. LuS. V. Pereverzev and U. Tautenhahn, Regularized total least squares: Computational aspects and error bounds, SIAM J. Numer. Anal. Appl., 31 (2009), 918-941.  doi: 10.1137/070709086.  Google Scholar

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S. Lu and J. Flemming, Convergence rate analysis of Tikhonov regularization for nonlinear ill-posed problems with noisy operators, Inverse Problems, 28 (2012), 104003, 19pp. doi: 10.1088/0266-5611/28/10/104003.  Google Scholar

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K. MechlemS. EhenT. SellererE. BraigD. MunzelF. Pfeibber and P. B. Noel, Joint satistical iterative material image reconstruction for spectral computed tomography using a semi-empirical forward model, IEEE Transactions on Medical Imaging, 37 (2018), 68-80.   Google Scholar

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

[21]

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

[22]

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

[23]

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

[24]

O. Scherzer, M. Grassmair, H. Grossauer, M. Haltmaier and F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences, 167. Springer, New York, 2009.  Google Scholar

[25]

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Bumer, C. Herrmann, R. Steadman, G. Zeitler, A. 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), 4031. Google Scholar

[26]

U. Tautenhahn, Regularization of linear ill-posed problems with noisy right hand side and noisy operator, SIAM J.Inverse Ill-Posed Problems, 16 (2008), 507-523.  doi: 10.1515/JIIP.2008.027.  Google Scholar

[27]

Y. Wang, A. C. Yagola and C. Yang, Optimization and Regularization for Computational Inverse Problems and Applications, Berlin: Springer; 2010. Google Scholar

[28]

T. WeidingerT. M. BuzugT. FlohrS. Kappler and K. Stierstorfer, Polychromatic iterative satistical material image reconstruction for photon-counting computed tomography, International Journal of Biomedical Imaging, 2 (2016), 1-15.   Google Scholar

[29]

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. doi: 10.1088/0266-5611/28/10/104004.  Google Scholar

[30]

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

show all references

References:
[1]

S. W. Anzengruber and R. Ramlau, Discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001, 17pp. doi: 10.1088/0266-5611/26/2/025001.  Google Scholar

[2]

R. F. Barber, E. Y. Sidky, T. G. Schmidt and X. Pan, An algorithm for constrained one-step of spectral CT data, Physics in Medicine and Biology, 61 (2016), 3784. Google Scholar

[3]

J. M. Bardsley and J. Goldes, Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation, Inverse Problems, 25 (2009), 095005, 18pp. doi: 10.1088/0266-5611/25/9/095005.  Google Scholar

[4]

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

[5]

M. Bertero, P. Boccacci, C. Talenti, R. Zanella and L. Zanni, A discrepancy principle for Poisson data, Inverse Problems, 25 (2010), 105004, 20pp. doi: 10.1088/0266-5611/26/10/105004.  Google Scholar

[6]

J. R. Bleyer and R. Ramlau, A double regularization approach for inverse problems with noisy data and inexact operator, Inverse Problems, 29 (2013), 025004, 16pp. doi: 10.1088/0266-5611/29/2/025004.  Google Scholar

[7]

I. R. Bleyer and R. Ramlau, An alternating iterative minimisation algorithm for the double-regularised total least square functional, Inverse Problems, 31 (2015), 075004, 21pp. doi: 10.1088/0266-5611/31/7/075004.  Google Scholar

[8]

M. Burger and S. Osher, Convergence rates of convex variational reglarization, Inverse Problems, 10 (2004), 1411-1422.  doi: 10.1088/0266-5611/20/5/005.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

G. H. GolubP. C. Hansen and D. P. O'Leary, Tikhonov regularization and total least squares, SIAM J. Numer. Anal. Appl., 21 (1999), 185-194.  doi: 10.1137/S0895479897326432.  Google Scholar

[13]

P. C. Hansen and M. Saxild-Hansen, AIR Tools-A MATLAB package of algebraic iterative reconstruction methods, Journal of Computational and Applied Mathematics, 236 (2012), 2067-2178.  doi: 10.1016/j.cam.2011.09.039.  Google Scholar

[14]

B. HofmannB. KaltenbacherC. Poschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.  doi: 10.1088/0266-5611/23/3/009.  Google Scholar

[15]

T. Hohage and F. Werner, Inverse problems with Poisson data: Statistical regularization theory, applications and algorithms, Inverse Problems, 32 (2016), 093001, 56pp. doi: 10.1088/0266-5611/32/9/093001.  Google Scholar

[16]

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

[17]

S. LuS. V. Pereverzev and U. Tautenhahn, Regularized total least squares: Computational aspects and error bounds, SIAM J. Numer. Anal. Appl., 31 (2009), 918-941.  doi: 10.1137/070709086.  Google Scholar

[18]

S. Lu and J. Flemming, Convergence rate analysis of Tikhonov regularization for nonlinear ill-posed problems with noisy operators, Inverse Problems, 28 (2012), 104003, 19pp. doi: 10.1088/0266-5611/28/10/104003.  Google Scholar

[19]

K. MechlemS. EhenT. SellererE. BraigD. MunzelF. Pfeibber and P. B. Noel, Joint satistical iterative material image reconstruction for spectral computed tomography using a semi-empirical forward model, IEEE Transactions on Medical Imaging, 37 (2018), 68-80.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

O. Scherzer, M. Grassmair, H. Grossauer, M. Haltmaier and F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences, 167. Springer, New York, 2009.  Google Scholar

[25]

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Bumer, C. Herrmann, R. Steadman, G. Zeitler, A. 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), 4031. Google Scholar

[26]

U. Tautenhahn, Regularization of linear ill-posed problems with noisy right hand side and noisy operator, SIAM J.Inverse Ill-Posed Problems, 16 (2008), 507-523.  doi: 10.1515/JIIP.2008.027.  Google Scholar

[27]

Y. Wang, A. C. Yagola and C. Yang, Optimization and Regularization for Computational Inverse Problems and Applications, Berlin: Springer; 2010. Google Scholar

[28]

T. WeidingerT. M. BuzugT. FlohrS. Kappler and K. Stierstorfer, Polychromatic iterative satistical material image reconstruction for photon-counting computed tomography, International Journal of Biomedical Imaging, 2 (2016), 1-15.   Google Scholar

[29]

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. doi: 10.1088/0266-5611/28/10/104004.  Google Scholar

[30]

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

Figure 1.  Ground truth maps (a) iodine (b) gadolinium (c) water
Figure 2.  Detector response functions $ d_i(E) $ for the five energy bins
Figure 3.  Evolution of the data term for different noise levels. Bold line ($ \delta^{'} = 0.1 , \delta_F' = 0 $), dashed line ($ \delta' = 0.1 , \delta_F' = 0.1 $), thin line ($ \delta' = 0.1 , \delta_F' = 0.2 $)
Figure 4.  Evolution of the iodine relative reconstruction error. Bold line ($ \delta' = 0.1 , \delta_F' = 0 $), dashed line ($ \delta' = 0.1 , \delta_F' = 0.1 $), thin line ($ \delta' = 0.1 , \delta_F' = 0.2 $)
Figure 5.  Evolution of the gadolinium relative reconstruction error. Bold line ($ \delta^{'} = 0.1 , \delta_F' = 0 $), dashed line ($ \delta^{'} = 0.1 , \delta_F' = 0.1 $), thin line ($ \delta^{'} = 0.1 , \delta_F' = 0.2 $)
Figure 6.  Evolution of the water relative reconstruction error. Bold line ($ \delta' = 0.1 , \delta_F' = 0 $), dashed line ($ \delta = 0.1 , \delta_F' = 0.1 $), thin line ($ \delta' = 0.1 , \delta_F' = 0.2 $)
Figure 7.  Reconstruction maps for the noise levels (a) iodine (b) gadolinium (c) water ($ \delta^{'} = 0.1, \delta_F = 0.1 $)
Figure 8.  Reconstruction maps for the noise levels (a) iodine (b) gadolinium (c) water ($ \delta^{'} = 0.1 , \delta_F = 0.2 $)
Figure 9.  Evolution of the water reconstruction error. Bold line ($ \delta^{'} = 0.1 , \delta_F = 0 $), thin line ($ \delta^{'} = 0.1 , \delta_F' = 0.1 $), dashed line: solution obtained with alternate minimization
Figure 10.  Evolution of the iodine reconstruction error. Bold line ($ \delta^{'} = 0.1 , \delta_F = 0 $), thin line ($ \delta^{'} = 0.1 , \delta_F' = 0.1 $), dashed line: solution obtained with alternate minimization
Figure 11.  Evolution of the gadolinium relative reconstruction error. Bold line ($ \delta^{'} = 0.1 , \delta_F = 0 $), thin line ($ \delta^{'} = 0.05 , \delta_F' = 0.1 $), dashed line: solution obtained with alternate minimization
Figure 12.  Reconstruction maps for (a) iodine (b) gadolinium (c) water obtained with the iterative algorithm starting form ($ \delta' = 0.1 , \delta_F = 0.1 $
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