TY - JOUR

T1 - Total variation-penalized Poisson likelihood estimation for ill-posed problems

AU - Bardsley, Johnathan M.

AU - Luttman, Aaron

N1 - Funding Information:
The first author’s work was supported by the NSF under grant DMS-0504325, by the University of Montana through its International Exchange Program, and by the University of Helsinki, which hosted the him during the 2006-07 academic year.

PY - 2009/10

Y1 - 2009/10

N2 - The noise contained in data measured by imaging instruments is often primarily of Poisson type. This motivates, in many cases, the use of the Poisson negative-log likelihood function in place of the ubiquitous least squares data fidelity when solving image deblurring problems. We assume that the underlying blurring operator is compact, so that, as in the least squares case, the resulting minimization problem is ill-posed and must be regularized. In this paper, we focus on total variation regularization and show that the problem of computing the minimizer of the resulting total variation-penalized Poisson likelihood functional is well-posed. We then prove that, as the errors in the data and in the blurring operator tend to zero, the resulting minimizers converge to the minimizer of the exact likelihood function. Finally, the practical effectiveness of the approach is demonstrated on synthetically generated data, and a nonnegatively constrained, projected quasi-Newton method is introduced.

AB - The noise contained in data measured by imaging instruments is often primarily of Poisson type. This motivates, in many cases, the use of the Poisson negative-log likelihood function in place of the ubiquitous least squares data fidelity when solving image deblurring problems. We assume that the underlying blurring operator is compact, so that, as in the least squares case, the resulting minimization problem is ill-posed and must be regularized. In this paper, we focus on total variation regularization and show that the problem of computing the minimizer of the resulting total variation-penalized Poisson likelihood functional is well-posed. We then prove that, as the errors in the data and in the blurring operator tend to zero, the resulting minimizers converge to the minimizer of the exact likelihood function. Finally, the practical effectiveness of the approach is demonstrated on synthetically generated data, and a nonnegatively constrained, projected quasi-Newton method is introduced.

KW - Ill-posed problems

KW - Image deblurring

KW - Maximum likelihood estimation

KW - Nonnegatively constrained minimization

KW - Total variation regularization

UR - http://www.scopus.com/inward/record.url?scp=67349157591&partnerID=8YFLogxK

U2 - 10.1007/s10444-008-9081-8

DO - 10.1007/s10444-008-9081-8

M3 - Article

AN - SCOPUS:67349157591

SN - 1019-7168

VL - 31

SP - 35

EP - 59

JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

IS - 1-3

ER -