343 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
A CURE for noisy magnetic resonance images: Chi-square unbiased risk estimation
In this article we derive an unbiased expression for the expected
mean-squared error associated with continuously differentiable estimators of
the noncentrality parameter of a chi-square random variable. We then consider
the task of denoising squared-magnitude magnetic resonance image data, which
are well modeled as independent noncentral chi-square random variables on two
degrees of freedom. We consider two broad classes of linearly parameterized
shrinkage estimators that can be optimized using our risk estimate, one in the
general context of undecimated filterbank transforms, and another in the
specific case of the unnormalized Haar wavelet transform. The resultant
algorithms are computationally tractable and improve upon state-of-the-art
methods for both simulated and actual magnetic resonance image data.Comment: 30 double-spaced pages, 11 figures; submitted for publicatio
A proximal iteration for deconvolving Poisson noisy images using sparse representations
We propose an image deconvolution algorithm when the data is contaminated by
Poisson noise. The image to restore is assumed to be sparsely represented in a
dictionary of waveforms such as the wavelet or curvelet transforms. Our key
contributions are: First, we handle the Poisson noise properly by using the
Anscombe variance stabilizing transform leading to a {\it non-linear}
degradation equation with additive Gaussian noise. Second, the deconvolution
problem is formulated as the minimization of a convex functional with a
data-fidelity term reflecting the noise properties, and a non-smooth
sparsity-promoting penalties over the image representation coefficients (e.g.
-norm). Third, a fast iterative backward-forward splitting algorithm is
proposed to solve the minimization problem. We derive existence and uniqueness
conditions of the solution, and establish convergence of the iterative
algorithm. Finally, a GCV-based model selection procedure is proposed to
objectively select the regularization parameter. Experimental results are
carried out to show the striking benefits gained from taking into account the
Poisson statistics of the noise. These results also suggest that using
sparse-domain regularization may be tractable in many deconvolution
applications with Poisson noise such as astronomy and microscopy
Poisson inverse problems
In this paper we focus on nonparametric estimators in inverse problems for
Poisson processes involving the use of wavelet decompositions. Adopting an
adaptive wavelet Galerkin discretization, we find that our method combines the
well-known theoretical advantages of wavelet--vaguelette decompositions for
inverse problems in terms of optimally adapting to the unknown smoothness of
the solution, together with the remarkably simple closed-form expressions of
Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann.
Statist. 19 (1991) 1347--1369] to the context of log-intensity functions
approximated by wavelet series with the use of the Kullback--Leibler distance
between two point processes, we also present an asymptotic analysis of
convergence rates that justifies our approach. In order to shed some light on
the theoretical results obtained and to examine the accuracy of our estimates
in finite samples, we illustrate our method by the analysis of some simulated
examples.Comment: Published at http://dx.doi.org/10.1214/009053606000000687 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Image Denoising in Mixed Poisson-Gaussian Noise
We propose a general methodology (PURE-LET) to design and optimize a wide class of transform-domain thresholding algorithms for denoising images corrupted by mixed Poisson-Gaussian noise. We express the denoising process as a linear expansion of thresholds (LET) that we optimize by relying on a purely data-adaptive unbiased estimate of the mean-squared error (MSE), derived in a non-Bayesian framework (PURE: Poisson-Gaussian unbiased risk estimate). We provide a practical approximation of this theoretical MSE estimate for the tractable optimization of arbitrary transform-domain thresholding. We then propose a pointwise estimator for undecimated filterbank transforms, which consists of subband-adaptive thresholding functions with signal-dependent thresholds that are globally optimized in the image domain. We finally demonstrate the potential of the proposed approach through extensive comparisons with state-of-the-art techniques that are specifically tailored to the estimation of Poisson intensities. We also present denoising results obtained on real images of low-count fluorescence microscopy
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