Sparse Poisson Intensity Reconstruction Algorithms

TL;DR

This paper develops algorithms for reconstructing sparse Poisson intensities from count data, addressing challenges of non-Gaussian noise and high-dimensionality, with applications in photon counting and similar fields.

Contribution

It introduces computational methods using quadratic approximations and multiscale estimation for solving sparse Poisson inverse problems with nonnegativity constraints.

Findings

Effective algorithms for sparse Poisson intensity reconstruction.

Improved accuracy over traditional methods for count data.

Efficient handling of high-dimensional inverse problems.

Abstract

The observations in many applications consist of counts of discrete events, such as photons hitting a dector, which cannot be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise model. As a result, accurate reconstruction of a spatially or temporally distributed phenomenon (f) from Poisson data (y) cannot be accomplished by minimizing a conventional l2-l1 objective function. The problem addressed in this paper is the estimation of f from y in an inverse problem setting, where (a) the number of unknowns may potentially be larger than the number of observations and (b) f admits a sparse approximation in some basis. The optimization formulation considered in this paper uses a negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). This paper describes computational methods for solving the constrained sparse Poisson inverse problem. In particular, the proposed approach incorporates key ideas of using quadratic separable approximations to the objective function at each iteration and computationally efficient partition-based multiscale estimation methods.