The University of Montana

Department of Mathematical Sciences

Technical report #8/2006

Total Variation-Penalized Poisson Likelihood
Estimation for Ill-Posed Problems

**J. Bardsley**

Department of Mathematical Sciences

The University of Montana (USA)

**Aaron Luttmann**

Division of Science and Mathematics

Bethany Lutheran College (USA)

**Abstract**

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 likelihood functional 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.

**Keywords:** total variation regularization, ill-posed problems,
maximum likelihood estimation,
image deblurring, nonnegatively constrained minimization

**PACS numbers:** 02.30.Zz, 02.50.-r, 07.05.Pj

**Download Technical Report:** Pdf (238
KB)