540 research outputs found
Bregman Cost for Non-Gaussian Noise
One of the tasks of the Bayesian inverse problem is to find a good estimate
based on the posterior probability density. The most common point estimators
are the conditional mean (CM) and maximum a posteriori (MAP) estimates, which
correspond to the mean and the mode of the posterior, respectively. From a
theoretical point of view it has been argued that the MAP estimate is only in
an asymptotic sense a Bayes estimator for the uniform cost function, while the
CM estimate is a Bayes estimator for the means squared cost function. Recently,
it has been proven that the MAP estimate is a proper Bayes estimator for the
Bregman cost if the image is corrupted by Gaussian noise. In this work we
extend this result to other noise models with log-concave likelihood density,
by introducing two related Bregman cost functions for which the CM and the MAP
estimates are proper Bayes estimators. Moreover, we also prove that the CM
estimate outperforms the MAP estimate, when the error is measured in a certain
Bregman distance, a result previously unknown also in the case of additive
Gaussian noise
Iterative algorithms for a non-linear inverse problem in atmospheric lidar
We consider the inverse problem of retrieving aerosol extinction coefficients
from Raman lidar measurements. In this problem the unknown and the data are
related through the exponential of a linear operator, the unknown is
non-negative and the data follow the Poisson distribution. Standard methods
work on the log-transformed data and solve the resulting linear inverse
problem, but neglect to take into account the noise statistics. In this study
we show that proper modelling of the noise distribution can improve
substantially the quality of the reconstructed extinction profiles. To achieve
this goal, we consider the non-linear inverse problem with non-negativity
constraint, and propose two iterative algorithms derived using the
Karush-Kuhn-Tucker conditions. We validate the algorithms with synthetic and
experimental data. As expected, the proposed algorithms outperform standard
methods in terms of sensitivity to noise and reliability of the estimated
profile.Comment: 19 pages, 6 figure
Bayesian Activity Estimation and Uncertainty Quantification of Spent Nuclear Fuel Using Passive Gamma Emission Tomography
In this paper, we address the problem of activity estimation in passive gamma emission tomography (PGET) of spent nuclear fuel. Two different noise models are considered and compared, namely, the isotropic Gaussian and the Poisson noise models. The problem is formulated within a Bayesian framework as a linear inverse problem and prior distributions are assigned to the unknown model parameters. In particular, a Bernoulli-truncated Gaussian prior model is considered to promote sparse pin configurations. A Markov chain Monte Carlo (MCMC) method, based on a split and augmented Gibbs sampler, is then used to sample the posterior distribution of the unknown parameters. The proposed algorithm is first validated by simulations conducted using synthetic data, generated using the nominal models. We then consider more realistic data simulated using a bespoke simulator, whose forward model is non-linear and not available analytically. In that case, the linear models used are mis-specified and we analyse their robustness for activity estimation. The results demonstrate superior performance of the proposed approach in estimating the pin activities in different assembly patterns, in addition to being able to quantify their uncertainty measures, in comparison with existing methods
Revisiting maximum-a-posteriori estimation in log-concave models
Maximum-a-posteriori (MAP) estimation is the main Bayesian estimation
methodology in imaging sciences, where high dimensionality is often addressed
by using Bayesian models that are log-concave and whose posterior mode can be
computed efficiently by convex optimisation. Despite its success and wide
adoption, MAP estimation is not theoretically well understood yet. The
prevalent view in the community is that MAP estimation is not proper Bayesian
estimation in a decision-theoretic sense because it does not minimise a
meaningful expected loss function (unlike the minimum mean squared error (MMSE)
estimator that minimises the mean squared loss). This paper addresses this
theoretical gap by presenting a decision-theoretic derivation of MAP estimation
in Bayesian models that are log-concave. A main novelty is that our analysis is
based on differential geometry, and proceeds as follows. First, we use the
underlying convex geometry of the Bayesian model to induce a Riemannian
geometry on the parameter space. We then use differential geometry to identify
the so-called natural or canonical loss function to perform Bayesian point
estimation in that Riemannian manifold. For log-concave models, this canonical
loss is the Bregman divergence associated with the negative log posterior
density. We then show that the MAP estimator is the only Bayesian estimator
that minimises the expected canonical loss, and that the posterior mean or MMSE
estimator minimises the dual canonical loss. We also study the question of MAP
and MSSE estimation performance in large scales and establish a universal bound
on the expected canonical error as a function of dimension, offering new
insights into the good performance observed in convex problems. These results
provide a new understanding of MAP and MMSE estimation in log-concave settings,
and of the multiple roles that convex geometry plays in imaging problems.Comment: Accepted for publication in SIAM Imaging Science
Bayesian Image Restoration for Poisson Corrupted Image using a Latent Variational Method with Gaussian MRF
We treat an image restoration problem with a Poisson noise chan- nel using a
Bayesian framework. The Poisson randomness might be appeared in observation of
low contrast object in the field of imaging. The noise observation is often
hard to treat in a theo- retical analysis. In our formulation, we interpret the
observation through the Poisson noise channel as a likelihood, and evaluate the
bound of it with a Gaussian function using a latent variable method. We then
introduce a Gaussian Markov random field (GMRF) as the prior for the Bayesian
approach, and derive the posterior as a Gaussian distribution. The latent
parameters in the likelihood and the hyperparameter in the GMRF prior could be
treated as hid- den parameters, so that, we propose an algorithm to infer them
in the expectation maximization (EM) framework using loopy belief
propagation(LBP). We confirm the ability of our algorithm in the computer
simulation, and compare it with the results of other im- age restoration
frameworks.Comment: 9 pages, 6 figures, The of this manuscript is submitting to the
Information Processing Society of Japan(IPSJ), Transactions on Mathematical
Modeling and its Applications (TOM
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