Johnathan M. Bardsley

Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation

In image processing applications, image intensity is often measured via the counting of incident photons emitted by the object of interest. In such cases, image data noise is accurately modeled by a Poisson distribution. This motivates the use of Poisson maximum likelihood estimation for image reconstruction. However, when the underlying model equation is ill-posed, regularization is needed. Regularized Poisson likelihood estimation has been studied extensively by the authors, though a problem of high importance remains: the choice of the regularization parameter. We will present three statistically motivated methods for choosing the regularization parameter, and numerical examples will be presented to illustrate their effectiveness.

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

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.

Covariance-Preconditioned Iterative Methods for Nonnegatively Constrained Astronomical Imaging

We consider the problem of solving ill-conditioned linear systems

Randomize-Then-Optimize: A Method for Sampling from Posterior Distributions in Nonlinear Inverse Problems

A\bfx=\bfb

Least-squares methods with Poissonian noise: Analysis and comparison with the Richardson-Lucy algorithm

subject to the nonnegativity constraint

Tikhonov regularized Poisson likelihood estimation: theoretical justification and a computational method

\bfx\geq\bfzero

Analysis of the Gibbs Sampler for Hierarchical Inverse Problems

, and in which the vector

Wavefront Reconstruction Methods for Adaptive Optics Systems on Ground-Based Telescopes

\bfb

An analysis of regularization by diffusion for ill-posed Poisson likelihood estimations

is a realization of a random vector

An Iterative Method for Edge-Preserving MAP Estimation When Data-Noise Is Poisson

\hat{\bfb}