SUBMITTED TO IEEE TRANSACTIONS ON IMAGE PROCESSING 1 This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction ALgorithms – Theory and Practice

Abstract—The observations in many applications consist of counts of discrete events, such as photons hitting a detector, 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 effectively accomplished by minimizing a conventional penalized least-squares 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. The optimization formulation considered in this paper uses a penalized negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). In particular, the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to `1 norms of coefficient vectors, total variation seminorms, and partition-based multiscale estimation methods. Index Terms—Photon-limited imaging, Poisson noise, wavelets, convex optimization, sparse approximation, compressed sensing, multiscale, total variation I