3,019 research outputs found
Exact Distribution of Edge-Preserving MAP Estimators for Linear Signal Models with Gaussian Measurement Noise
We derive the exact statistical distribution of maximum a posteriori (MAP) estimators having edge-preserving nonGaussian priors. Such estimators have been widely advocated for image restoration and reconstruction problems. Previous investigations of these image recovery methods have been primarily empirical; the distribution we derive enables theoretical analysis. The signal model is linear with Gaussian measurement noise. We assume that the energy function of the prior distribution is chosen to ensure a unimodal posterior distribution (for which convexity of the energy function is sufficient), and that the energy function satisfies a uniform Lipschitz regularity condition. The regularity conditions are sufficiently general to encompass popular priors such as the generalized Gaussian Markov random field prior and the Huber prior, even though those priors are not everywhere twice continuously differentiable.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85999/1/Fessler81.pd
Fast Gibbs sampling for high-dimensional Bayesian inversion
Solving ill-posed inverse problems by Bayesian inference has recently
attracted considerable attention. Compared to deterministic approaches, the
probabilistic representation of the solution by the posterior distribution can
be exploited to explore and quantify its uncertainties. In applications where
the inverse solution is subject to further analysis procedures, this can be a
significant advantage. Alongside theoretical progress, various new
computational techniques allow to sample very high dimensional posterior
distributions: In [Lucka2012], a Markov chain Monte Carlo (MCMC) posterior
sampler was developed for linear inverse problems with -type priors. In
this article, we extend this single component Gibbs-type sampler to a wide
range of priors used in Bayesian inversion, such as general priors
with additional hard constraints. Besides a fast computation of the
conditional, single component densities in an explicit, parameterized form, a
fast, robust and exact sampling from these one-dimensional densities is key to
obtain an efficient algorithm. We demonstrate that a generalization of slice
sampling can utilize their specific structure for this task and illustrate the
performance of the resulting slice-within-Gibbs samplers by different computed
examples. These new samplers allow us to perform sample-based Bayesian
inference in high-dimensional scenarios with certain priors for the first time,
including the inversion of computed tomography (CT) data with the popular
isotropic total variation (TV) prior.Comment: submitted to "Inverse Problems
Discretization and Bayesian modeling in inverse problems and imaging
In this thesis the Bayesian modeling and discretization are studied in inverse problems related to imaging. The treatise consists of four articles which focus on the phenomena that appear when more detailed data or a priori information become available. Novel Bayesian methods for solving ill-posed signal processing problems in edge-preserving manner are introduced and analysed. Furthermore, modeling photographs in image processing problems is studied and a novel model is presented
Unsupervised bayesian convex deconvolution based on a field with an explicit partition function
This paper proposes a non-Gaussian Markov field with a special feature: an
explicit partition function. To the best of our knowledge, this is an original
contribution. Moreover, the explicit expression of the partition function
enables the development of an unsupervised edge-preserving convex deconvolution
method. The method is fully Bayesian, and produces an estimate in the sense of
the posterior mean, numerically calculated by means of a Monte-Carlo Markov
Chain technique. The approach is particularly effective and the computational
practicability of the method is shown on a simple simulated example
Performance bounds for expander-based compressed sensing in Poisson noise
This paper provides performance bounds for compressed sensing in the presence
of Poisson noise using expander graphs. The Poisson noise model is appropriate
for a variety of applications, including low-light imaging and digital
streaming, where the signal-independent and/or bounded noise models used in the
compressed sensing literature are no longer applicable. In this paper, we
develop a novel sensing paradigm based on expander graphs and propose a MAP
algorithm for recovering sparse or compressible signals from Poisson
observations. The geometry of the expander graphs and the positivity of the
corresponding sensing matrices play a crucial role in establishing the bounds
on the signal reconstruction error of the proposed algorithm. We support our
results with experimental demonstrations of reconstructing average packet
arrival rates and instantaneous packet counts at a router in a communication
network, where the arrivals of packets in each flow follow a Poisson process.Comment: revised version; accepted to IEEE Transactions on Signal Processin
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