# American Institute of Mathematical Sciences

February  2010, 4(1): 11-17. doi: 10.3934/ipi.2010.4.11

## A theoretical framework for the regularization of Poisson likelihood estimation problems

 1 Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812, United States

Received  October 2008 Revised  October 2009 Published  February 2010

Let $z=Au+\gamma$ be an ill-posed, linear operator equation. Such a model arises, for example, in both astronomical and medical imaging, in which case $\gamma$ corresponds to background, $u$ the unknown true image, $A$ the forward operator, and $z$ the data. Regularized solutions of this equation can be obtained by solving

$R_\alpha(A,z)= arg\min_{u\geq 0} \{T_0(Au;z)+\alpha J(u)\},$

where $T_0(Au;z)$ is the negative-log of the Poisson likelihood functional, and $\alpha>0$ and $J$ are the regularization parameter and functional, respectively. Our goal in this paper is to determine general conditions which guarantee that $R_\alpha$ defines a regularization scheme for $z=Au+\gamma$. Determining the appropriate definition for regularization scheme in this context is important: not only will it serve to unify previous theoretical arguments in this direction, it will provide a framework for future theoretical analyses. To illustrate the latter, we end the paper with an application of the general framework to a case in which an analysis has not been done.

Citation: 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
 [1] Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems & Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1 [2] Johnathan M. Bardsley. An efficient computational method for total variation-penalized Poisson likelihood estimation. Inverse Problems & Imaging, 2008, 2 (2) : 167-185. doi: 10.3934/ipi.2008.2.167 [3] Frank Natterer. Incomplete data problems in wave equation imaging. Inverse Problems & Imaging, 2010, 4 (4) : 685-691. doi: 10.3934/ipi.2010.4.685 [4] T. Varslo, C E Yarman, M. Cheney, B Yazıcı. A variational approach to waveform design for synthetic-aperture imaging. Inverse Problems & Imaging, 2007, 1 (3) : 577-592. doi: 10.3934/ipi.2007.1.577 [5] Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139 [6] Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267 [7] Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215 [8] Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018 [9] Durga Prasad Challa, Anupam Pal Choudhury, Mourad Sini. Mathematical imaging using electric or magnetic nanoparticles as contrast agents. Inverse Problems & Imaging, 2018, 12 (3) : 573-605. doi: 10.3934/ipi.2018025 [10] Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511 [11] Abraão D. C. Nascimento, Leandro C. Rêgo, Raphaela L. B. A. Nascimento. Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation. Inverse Problems & Imaging, 2019, 13 (4) : 787-803. doi: 10.3934/ipi.2019036 [12] Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347 [13] Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343 [14] You-Wei Wen, Raymond Honfu Chan. Using generalized cross validation to select regularization parameter for total variation regularization problems. Inverse Problems & Imaging, 2018, 12 (5) : 1103-1120. doi: 10.3934/ipi.2018046 [15] Avner Friedman. PDE problems arising in mathematical biology. Networks & Heterogeneous Media, 2012, 7 (4) : 691-703. doi: 10.3934/nhm.2012.7.691 [16] Matthew A. Fury. Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space. Conference Publications, 2013, 2013 (special) : 259-272. doi: 10.3934/proc.2013.2013.259 [17] Huan Han. A variational model with fractional-order regularization term arising in registration of diffusion tensor image. Inverse Problems & Imaging, 2018, 12 (6) : 1263-1291. doi: 10.3934/ipi.2018053 [18] Jie Zhang, Shuang Lin, Li-Wei Zhang. A log-exponential regularization method for a mathematical program with general vertical complementarity constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 561-577. doi: 10.3934/jimo.2013.9.561 [19] Tim Hoheisel, Christian Kanzow, Alexandra Schwartz. Improved convergence properties of the Lin-Fukushima-Regularization method for mathematical programs with complementarity constraints. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 49-60. doi: 10.3934/naco.2011.1.49 [20] Liping Pang, Na Xu, Jian Lv. The inexact log-exponential regularization method for mathematical programs with vertical complementarity constraints. Journal of Industrial & Management Optimization, 2019, 15 (1) : 59-79. doi: 10.3934/jimo.2018032

2018 Impact Factor: 1.469