Inverse problems with poisson noise: Primal and primal-dual splitting

Abstract

In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by Poisson noise. A proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms. Piecing together the data fidelity and the prior terms, the solution to the inverse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding minimizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental results on deconvolution and comparison to prior methods are also reported.

Scaling Techniques for $\epsilon$-Subgradient Methods

“…Corollary 2.1 Let {x (k) } k∈N a sequence satisfying (24) such that the assumptions of Theorem 2.1 are satisfied. Then, {x (k) } k∈N converges to a solution of problem (1).…”

“…A critical point for the implementation of the methods ( 6) and ( 24) is how to select the sequence {α k }; a practical strategy to obtain good performances is still an open problem, since they are in general quite sensitive to this choice [15]. Borrowing the ideas of [17] and [33], in this section we describe a level algorithm that allows to adaptively compute a stepsize α k of the form (5) in the iteration (24). In our scheme we introduce the use of the ǫ-subgradient of f at the current iterate (instead of the subgradient) and a variable metric.…”

“…In particular, we provide an especially tailored implementation of SPDHG for the Total Variation (TV) restoration of images corrupted by Poisson noise. This problem is related to several applications such as astronomical imaging, electronic microscopy, single particle emission computed tomography (SPECT) and positron emission tomography (PET), and a variety of specialized methods have been proposed for its solution (see [4,14,24,25,28,47] and references therein). For this special case of SPDHG, we devise an effective strategy to choose the scaling matrix, showing that significant improvements on the practical convergence speed can be obtained.…”

Faster PET reconstruction with non-smooth priors by randomization and preconditioning

“…Only a few optimization algorithms are capable of combining non-smooth priors and the Poisson noise model, e.g. [7,10,19,[33][34][35][36][37][38][39][40] and most of these are not applicable to solve all regularization models mentioned above.…”

“…PDHG has been used in numerous imaging studies on multiple imaging modalities, including PET, see e.g. [11,27,30,31,36,[41][42][43][44]. While this algorithm is flexible enough to solve a variety of non-smooth optimization problems, in every iteration both the projection and the backprojection have to be applied for all projection bins.…”