7,906 research outputs found
Bias-Reduction in Variational Regularization
The aim of this paper is to introduce and study a two-step debiasing method
for variational regularization. After solving the standard variational problem,
the key idea is to add a consecutive debiasing step minimizing the data
fidelity on an appropriate set, the so-called model manifold. The latter is
defined by Bregman distances or infimal convolutions thereof, using the
(uniquely defined) subgradient appearing in the optimality condition of the
variational method. For particular settings, such as anisotropic and
TV-type regularization, previously used debiasing techniques are shown to be
special cases. The proposed approach is however easily applicable to a wider
range of regularizations. The two-step debiasing is shown to be well-defined
and to optimally reduce bias in a certain setting.
In addition to visual and PSNR-based evaluations, different notions of bias
and variance decompositions are investigated in numerical studies. The
improvements offered by the proposed scheme are demonstrated and its
performance is shown to be comparable to optimal results obtained with Bregman
iterations.Comment: Accepted by JMI
CLEAR: Covariant LEAst-square Re-fitting with applications to image restoration
In this paper, we propose a new framework to remove parts of the systematic
errors affecting popular restoration algorithms, with a special focus for image
processing tasks. Generalizing ideas that emerged for regularization,
we develop an approach re-fitting the results of standard methods towards the
input data. Total variation regularizations and non-local means are special
cases of interest. We identify important covariant information that should be
preserved by the re-fitting method, and emphasize the importance of preserving
the Jacobian (w.r.t. the observed signal) of the original estimator. Then, we
provide an approach that has a "twicing" flavor and allows re-fitting the
restored signal by adding back a local affine transformation of the residual
term. We illustrate the benefits of our method on numerical simulations for
image restoration tasks
Generalized SURE for Exponential Families: Applications to Regularization
Stein's unbiased risk estimate (SURE) was proposed by Stein for the
independent, identically distributed (iid) Gaussian model in order to derive
estimates that dominate least-squares (LS). In recent years, the SURE criterion
has been employed in a variety of denoising problems for choosing
regularization parameters that minimize an estimate of the mean-squared error
(MSE). However, its use has been limited to the iid case which precludes many
important applications. In this paper we begin by deriving a SURE counterpart
for general, not necessarily iid distributions from the exponential family.
This enables extending the SURE design technique to a much broader class of
problems. Based on this generalization we suggest a new method for choosing
regularization parameters in penalized LS estimators. We then demonstrate its
superior performance over the conventional generalized cross validation
approach and the discrepancy method in the context of image deblurring and
deconvolution. The SURE technique can also be used to design estimates without
predefining their structure. However, allowing for too many free parameters
impairs the performance of the resulting estimates. To address this inherent
tradeoff we propose a regularized SURE objective. Based on this design
criterion, we derive a wavelet denoising strategy that is similar in sprit to
the standard soft-threshold approach but can lead to improved MSE performance.Comment: to appear in the IEEE Transactions on Signal Processin
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