May  2007, 1(2): 399-422. doi: 10.3934/ipi.2007.1.399

Model distortions in Bayesian MAP reconstruction

1. 

CMLA, ENS Cachan, CNRS, PRES UniverSud, 61 Av. President Wilson, F-94230 Cachan, France

Received  December 2006 Published  April 2007

The Bayesian approach and especially the maximum a posteriori (MAP) estimator is most widely used to solve various problems in signal and image processing, such as denoising and deblurring, zooming, and reconstruction. The reason is that it provides a coherent statistical framework to combine observed (noisy) data with prior information on the unknown signal or image which is optimal in a precise statistical sense. This paper presents an objective critical analysis of the MAP approach. It shows that the MAP solutions substantially deviate from both the data-acquisition model and the prior model that underly the MAP estimator. This is explained theoretically using several analytical properties of the MAP solutions and is illustrated using examples and experiments. It follows that the MAP approach is not relevant in the applications where the data-observation and the prior models are accurate. The construction of solutions (estimators) that respect simultaneously two such models remains an open question.
Citation: Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems & Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399
[1]

Alina Toma, Bruno Sixou, Françoise Peyrin. Iterative choice of the optimal regularization parameter in TV image restoration. Inverse Problems & Imaging, 2015, 9 (4) : 1171-1191. doi: 10.3934/ipi.2015.9.1171

[2]

Bartomeu Coll, Joan Duran, Catalina Sbert. Half-linear regularization for nonconvex image restoration models. Inverse Problems & Imaging, 2015, 9 (2) : 337-370. doi: 10.3934/ipi.2015.9.337

[3]

Johnathan M. Bardsley. A theoretical framework for the regularization of Poisson likelihood estimation problems. Inverse Problems & Imaging, 2010, 4 (1) : 11-17. doi: 10.3934/ipi.2010.4.11

[4]

Kokum R. De Silva, Shigetoshi Eda, Suzanne Lenhart. Modeling environmental transmission of MAP infection in dairy cows. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1001-1017. doi: 10.3934/mbe.2017052

[5]

Alexandre M. Bayen, Hélène Frankowska, Jean-Patrick Lebacque, Benedetto Piccoli, H. Michael Zhang. Special issue on Mathematics of Traffic Flow Modeling, Estimation and Control. Networks & Heterogeneous Media, 2013, 8 (3) : i-ii. doi: 10.3934/nhm.2013.8.3i

[6]

W. Y. Tan, L.-J. Zhang, C.W. Chen. Stochastic modeling of carcinogenesis: State space models and estimation of parameters. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 297-322. doi: 10.3934/dcdsb.2004.4.297

[7]

Hui Huang, Eldad Haber, Lior Horesh. Optimal estimation of $\ell_1$-regularization prior from a regularized empirical Bayesian risk standpoint. Inverse Problems & Imaging, 2012, 6 (3) : 447-464. doi: 10.3934/ipi.2012.6.447

[8]

Rafail Krichevskii and Vladimir Potapov. Compression and restoration of square integrable functions. Electronic Research Announcements, 1996, 2: 42-49.

[9]

Len Margolin, Catherine Plesko. Discrete regularization. Evolution Equations & Control Theory, 2019, 8 (1) : 117-137. doi: 10.3934/eect.2019007

[10]

Nahid Banihashemi, C. Yalçın Kaya. Inexact restoration and adaptive mesh refinement for optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 521-542. doi: 10.3934/jimo.2014.10.521

[11]

Amir Averbuch, Pekka Neittaanmäki, Valery Zheludev. Periodic spline-based frames for image restoration. Inverse Problems & Imaging, 2015, 9 (3) : 661-707. doi: 10.3934/ipi.2015.9.661

[12]

Nicolas Lermé, François Malgouyres, Dominique Hamoir, Emmanuelle Thouin. Bayesian image restoration for mosaic active imaging. Inverse Problems & Imaging, 2014, 8 (3) : 733-760. doi: 10.3934/ipi.2014.8.733

[13]

Kamil Rajdl, Petr Lansky. Fano factor estimation. Mathematical Biosciences & Engineering, 2014, 11 (1) : 105-123. doi: 10.3934/mbe.2014.11.105

[14]

Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems & Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1

[15]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006

[16]

Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403

[17]

Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255

[18]

Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1

[19]

John Erik Fornæss, Brendan Weickert. A quantized henon map. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 723-740. doi: 10.3934/dcds.2000.6.723

[20]

Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]