Poisson inverse problems

By Anestis Antoniadis, Jérémie Bigot. 2006

50 citations recorded by the source corpus.

In this paper we focus on nonparametric estimators in inverse problems for Poisson processes involving the use of wavelet decompositions. Adopting an adaptive wavelet Galerkin discretization, we find that our method combines the well-known theoretical advantages of wavelet–vaguelette decompositions for inverse problems in terms of optimally adapting to the unknown smoothness of the solution, together with the remarkably simple closed-form expressions of Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann. Statist. 19 (1991) 1347–1369] to the context of log-intensity functions approximated by wavelet series with the use of the Kullback–Leibler distance between two point processes, we also present an asymptotic analysis of convergence rates that justifies our approach. In order to shed some light on the theoretical results obtained and to examine the accuracy of our estimates in finite samples, we illustrate our method by the analysis of some simulated examples.

Subfields: Computer Vision and Pattern Recognition, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/529b63ca8d0543a168c15a8463a26e9ac03f7be3

Citations in the corpus

• Restoration of Poissonian Images Using Alternating Direction Optimization (2010, unknown): “Other wavelet-based approaches to Poissonian inverse problems include [2] and [43] (both based upon the wavelet-vaguelette decomposition [19]), [8] (based upon the vaguelette-wavelet decomposition [1]), and [41] (based upon a cross-validation filter design [40]).”