45,693 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
Task adapted reconstruction for inverse problems
The paper considers the problem of performing a task defined on a model
parameter that is only observed indirectly through noisy data in an ill-posed
inverse problem. A key aspect is to formalize the steps of reconstruction and
task as appropriate estimators (non-randomized decision rules) in statistical
estimation problems. The implementation makes use of (deep) neural networks to
provide a differentiable parametrization of the family of estimators for both
steps. These networks are combined and jointly trained against suitable
supervised training data in order to minimize a joint differentiable loss
function, resulting in an end-to-end task adapted reconstruction method. The
suggested framework is generic, yet adaptable, with a plug-and-play structure
for adjusting both the inverse problem and the task at hand. More precisely,
the data model (forward operator and statistical model of the noise) associated
with the inverse problem is exchangeable, e.g., by using neural network
architecture given by a learned iterative method. Furthermore, any task that is
encodable as a trainable neural network can be used. The approach is
demonstrated on joint tomographic image reconstruction, classification and
joint tomographic image reconstruction segmentation
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