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.
Total variation regularization Ill-posed problems Maximum likelihood estimation Image deblurring Nonnegatively constrained minimization
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)MATHCrossRefGoogle Scholar
Sardy, S., Antoniadis, A., Tseng, P.: Automatic smoothing with wavelets for a wide class of distributions. J. Comput. Graph. Statist. 13(2), 1–23 (2004)MathSciNetGoogle Scholar
Sardy, S., Tseng, P.: On the statistical analysis of smoothing by maximizing dirty Markov random field posterior distributions. J. Amer. Statist. Assoc. 99(465), 191–204(14) (2004)MATHCrossRefMathSciNetGoogle Scholar
Snyder, D.L., Hammoud, A.M., White, R.L.: Image recovery from data acquired with a charge-coupled-device camera. J. Opt. Soc. Amer. A 10, 1014–1023 (1993)CrossRefGoogle Scholar
Snyder, D.L., Helstrom, C.W., Lanterman, A.D., Faisal, M., White, R.L.: Compensation for readout noise in CCD images. J. Opt. Soc. Amer. A 12, 272–283 (1995)CrossRefGoogle Scholar
Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer Academic, Boston (1990)MATHGoogle Scholar