2,296 research outputs found
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
Projected Newton Method for noise constrained Tikhonov regularization
Tikhonov regularization is a popular approach to obtain a meaningful solution
for ill-conditioned linear least squares problems. A relatively simple way of
choosing a good regularization parameter is given by Morozov's discrepancy
principle. However, most approaches require the solution of the Tikhonov
problem for many different values of the regularization parameter, which is
computationally demanding for large scale problems. We propose a new and
efficient algorithm which simultaneously solves the Tikhonov problem and finds
the corresponding regularization parameter such that the discrepancy principle
is satisfied. We achieve this by formulating the problem as a nonlinear system
of equations and solving this system using a line search method. We obtain a
good search direction by projecting the problem onto a low dimensional Krylov
subspace and computing the Newton direction for the projected problem. This
projected Newton direction, which is significantly less computationally
expensive to calculate than the true Newton direction, is then combined with a
backtracking line search to obtain a globally convergent algorithm, which we
refer to as the Projected Newton method. We prove convergence of the algorithm
and illustrate the improved performance over current state-of-the-art solvers
with some numerical experiments
A GCV based Arnoldi-Tikhonov regularization method
For the solution of linear discrete ill-posed problems, in this paper we
consider the Arnoldi-Tikhonov method coupled with the Generalized Cross
Validation for the computation of the regularization parameter at each
iteration. We study the convergence behavior of the Arnoldi method and its
properties for the approximation of the (generalized) singular values, under
the hypothesis that Picard condition is satisfied. Numerical experiments on
classical test problems and on image restoration are presented
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