1,512 research outputs found
Hierarchical Bayesian sparse image reconstruction with application to MRFM
This paper presents a hierarchical Bayesian model to reconstruct sparse
images when the observations are obtained from linear transformations and
corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is
well suited to such naturally sparse image applications as it seamlessly
accounts for properties such as sparsity and positivity of the image via
appropriate Bayes priors. We propose a prior that is based on a weighted
mixture of a positive exponential distribution and a mass at zero. The prior
has hyperparameters that are tuned automatically by marginalization over the
hierarchical Bayesian model. To overcome the complexity of the posterior
distribution, a Gibbs sampling strategy is proposed. The Gibbs samples can be
used to estimate the image to be recovered, e.g. by maximizing the estimated
posterior distribution. In our fully Bayesian approach the posteriors of all
the parameters are available. Thus our algorithm provides more information than
other previously proposed sparse reconstruction methods that only give a point
estimate. The performance of our hierarchical Bayesian sparse reconstruction
method is illustrated on synthetic and real data collected from a tobacco virus
sample using a prototype MRFM instrument.Comment: v2: final version; IEEE Trans. Image Processing, 200
A variational Bayesian method for inverse problems with impulsive noise
We propose a novel numerical method for solving inverse problems subject to
impulsive noises which possibly contain a large number of outliers. The
approach is of Bayesian type, and it exploits a heavy-tailed t distribution for
data noise to achieve robustness with respect to outliers. A hierarchical model
with all hyper-parameters automatically determined from the given data is
described. An algorithm of variational type by minimizing the Kullback-Leibler
divergence between the true posteriori distribution and a separable
approximation is developed. The numerical method is illustrated on several one-
and two-dimensional linear and nonlinear inverse problems arising from heat
conduction, including estimating boundary temperature, heat flux and heat
transfer coefficient. The results show its robustness to outliers and the fast
and steady convergence of the algorithm.Comment: 20 pages, to appear in J. Comput. Phy
Regularization and Bayesian Learning in Dynamical Systems: Past, Present and Future
Regularization and Bayesian methods for system identification have been
repopularized in the recent years, and proved to be competitive w.r.t.
classical parametric approaches. In this paper we shall make an attempt to
illustrate how the use of regularization in system identification has evolved
over the years, starting from the early contributions both in the Automatic
Control as well as Econometrics and Statistics literature. In particular we
shall discuss some fundamental issues such as compound estimation problems and
exchangeability which play and important role in regularization and Bayesian
approaches, as also illustrated in early publications in Statistics. The
historical and foundational issues will be given more emphasis (and space), at
the expense of the more recent developments which are only briefly discussed.
The main reason for such a choice is that, while the recent literature is
readily available, and surveys have already been published on the subject, in
the author's opinion a clear link with past work had not been completely
clarified.Comment: Plenary Presentation at the IFAC SYSID 2015. Submitted to Annual
Reviews in Contro
Sparsity-Promoting Bayesian Dynamic Linear Models
Sparsity-promoting priors have become increasingly popular over recent years
due to an increased number of regression and classification applications
involving a large number of predictors. In time series applications where
observations are collected over time, it is often unrealistic to assume that
the underlying sparsity pattern is fixed. We propose here an original class of
flexible Bayesian linear models for dynamic sparsity modelling. The proposed
class of models expands upon the existing Bayesian literature on sparse
regression using generalized multivariate hyperbolic distributions. The
properties of the models are explored through both analytic results and
simulation studies. We demonstrate the model on a financial application where
it is shown that it accurately represents the patterns seen in the analysis of
stock and derivative data, and is able to detect major events by filtering an
artificial portfolio of assets
Blind deconvolution of sparse pulse sequences under a minimum distance constraint: a partially collapsed Gibbs sampler method
For blind deconvolution of an unknown sparse sequence convolved with an unknown pulse, a powerful Bayesian method employs the Gibbs sampler in combination with a BernoulliâGaussian prior modeling sparsity. In this paper, we extend this method by introducing a minimum distance constraint for the pulses in the sequence. This is physically relevant in applications including layer detection, medical imaging, seismology, and multipath parameter estimation. We propose a Bayesian method for blind deconvolution that is based on a modified BernoulliâGaussian prior including a minimum distance constraint factor. The core of our method is a partially collapsed Gibbs sampler (PCGS) that tolerates and even exploits the strong local dependencies introduced by the minimum distance constraint. Simulation results demonstrate significant performance gains compared to a recently proposed PCGS. The main advantages of the minimum distance constraint are a substantial reduction of computational complexity and of the number of spurious components in the deconvolution result
Adaptive Langevin Sampler for Separation of t-Distribution Modelled Astrophysical Maps
We propose to model the image differentials of astrophysical source maps by
Student's t-distribution and to use them in the Bayesian source separation
method as priors. We introduce an efficient Markov Chain Monte Carlo (MCMC)
sampling scheme to unmix the astrophysical sources and describe the derivation
details. In this scheme, we use the Langevin stochastic equation for
transitions, which enables parallel drawing of random samples from the
posterior, and reduces the computation time significantly (by two orders of
magnitude). In addition, Student's t-distribution parameters are updated
throughout the iterations. The results on astrophysical source separation are
assessed with two performance criteria defined in the pixel and the frequency
domains.Comment: 12 pages, 6 figure
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