# American Institute of Mathematical Sciences

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
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##### References:
Ground truth maps (a) iodine (b) gadolinium (c) water
Detector response functions $d_i(E)$ for the five energy bins
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 ($\delta' = 0.1 , \delta_F' = 0.2$)
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 ($\delta' = 0.1 , \delta_F' = 0.2$)
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 ($\delta^{'} = 0.1 , \delta_F' = 0.2$)
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 ($\delta' = 0.1 , \delta_F' = 0.2$)
Reconstruction maps for the noise levels (a) iodine (b) gadolinium (c) water ($\delta^{'} = 0.1, \delta_F = 0.1$)
Reconstruction maps for the noise levels (a) iodine (b) gadolinium (c) water ($\delta^{'} = 0.1 , \delta_F = 0.2$)
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
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
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
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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