5,465 research outputs found
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
Mammographic image restoration using maximum entropy deconvolution
An image restoration approach based on a Bayesian maximum entropy method
(MEM) has been applied to a radiological image deconvolution problem, that of
reduction of geometric blurring in magnification mammography. The aim of the
work is to demonstrate an improvement in image spatial resolution in realistic
noisy radiological images with no associated penalty in terms of reduction in
the signal-to-noise ratio perceived by the observer. Images of the TORMAM
mammographic image quality phantom were recorded using the standard
magnification settings of 1.8 magnification/fine focus and also at 1.8
magnification/broad focus and 3.0 magnification/fine focus; the latter two
arrangements would normally give rise to unacceptable geometric blurring.
Measured point-spread functions were used in conjunction with the MEM image
processing to de-blur these images. The results are presented as comparative
images of phantom test features and as observer scores for the raw and
processed images. Visualization of high resolution features and the total image
scores for the test phantom were improved by the application of the MEM
processing. It is argued that this successful demonstration of image
de-blurring in noisy radiological images offers the possibility of weakening
the link between focal spot size and geometric blurring in radiology, thus
opening up new approaches to system optimization.Comment: 18 pages, 10 figure
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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