TY - JOUR
T1 - An analysis of regularization by diffusion for ill-posed Poisson likelihood estimations
AU - Bardsley, Johnathan M.
AU - Laobeul, N'Djekornom
N1 - Funding Information:
J.M. Bardsley was supported by the NSF under grant DMS-0504325. N. Laobeul was supported by the University of Montana (UM) through the Bertha Moon Summer Research Scholarship and the Bryan Family Math Sciences Summer Research Scholarship through the UM Math Department.
PY - 2009/6
Y1 - 2009/6
N2 - The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative-log Poisson likelihood function in place of the least-squares fit-to-data function. However, if the underlying mathematical model is assumed to have the form z= Au+γ, where z is the data and A is a compact operator and γ is the background light intensity, minimizing the negative-log Poisson likelihood function is an ill-posed problem, and hence some form of regularization is required. In previous work, the authors have performed theoretical analyses of two approaches for regularization in this setting: standard Tikhonov and total variation regularization. In this article, we consider a class of regularization functionals defined by differential operators of diffusion type, and our main results constitute a theoretical justification of this approach. However, in order to demonstrate that the approach is effective in practice, we follow our theoretical analysis with a numerical experiment.
AB - The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative-log Poisson likelihood function in place of the least-squares fit-to-data function. However, if the underlying mathematical model is assumed to have the form z= Au+γ, where z is the data and A is a compact operator and γ is the background light intensity, minimizing the negative-log Poisson likelihood function is an ill-posed problem, and hence some form of regularization is required. In previous work, the authors have performed theoretical analyses of two approaches for regularization in this setting: standard Tikhonov and total variation regularization. In this article, we consider a class of regularization functionals defined by differential operators of diffusion type, and our main results constitute a theoretical justification of this approach. However, in order to demonstrate that the approach is effective in practice, we follow our theoretical analysis with a numerical experiment.
KW - Compact operator equations
KW - Ill-posed problems
KW - Optimization
KW - Poisson likelihood estimation
UR - http://www.scopus.com/inward/record.url?scp=70350669488&partnerID=8YFLogxK
U2 - 10.1080/17415970802231594
DO - 10.1080/17415970802231594
M3 - Article
AN - SCOPUS:70350669488
SN - 1741-5977
VL - 17
SP - 537
EP - 550
JO - Inverse Problems in Science and Engineering
JF - Inverse Problems in Science and Engineering
IS - 4
ER -