Bayesian wavelet denoising: Besov priors and non-Gaussian noises

Leporini, D.; Pesquet, Jean-Christophe

doi:10.1016/s0165-1684(00)00190-0

Cited by 35 publications

(15 citation statements)

References 17 publications

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“…This contradicts the first inequality in (39). Hence {z δ } is bounded in B t 1 (T d ) for any t < s − d and there existsz ∈ B t 1 (T d ) and a subsequence (also denoted by) {z δ } ⊂ B t 1 (T d ) such that z δ converges toz in the weak*-topology.…”

Section: Proofs Of Results In Sectionmentioning

confidence: 92%

Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

Agapiou¹,

Burger²,

Dashti³

et al. 2018

Inverse Problems

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We consider the inverse problem of recovering an unknown functional parameter u in a separable Banach space, from a noisy observation y of its image through a known possibly non-linear ill-posed map G. The data y is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach to the problem and consider Besov space priors (see [36]), which are well-known for their edge-preserving and sparsity-promoting properties and have recently attracted wide attention especially in the medical imaging community.Our key result is to show that in this non-parametric setup the maximum a posteriori (MAP) estimates are characterized by the minimizers of a generalized Onsager-Machlup functional of the posterior. This is done independently for the so-called weak and strong MAP estimates, which as we show coincide in our context. In addition, we prove a form of weak consistency for the MAP estimators in the infinitely informative data limit. Our results are remarkable for two reasons: first, the prior distribution is non-Gaussian and does not meet the smoothness conditions required in previous research on nonparametric MAP estimates. Second, the result analytically justifies existing uses of the MAP estimate in finite but high dimensional discretizations of Bayesian inverse problems with the considered Besov priors.

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Section: Proofs Of Results In Sectionmentioning

confidence: 92%

Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

Agapiou¹,

Burger²,

Dashti³

et al. 2018

Inverse Problems

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show abstract

“…Proof. In order to prove (41) we first observe that by Lemma 10(ii) and Cauchy-Schwarz inequality it is enough to estimate the expectation…”

Section: Quantitative Estimates For Reconstructorsmentioning

confidence: 99%

Discretization-invariant Bayesian inversion and Besov space priors

Lassas¹,

Saksman²,

Siltanen³

2009

Inverse Problems &Amp; Imaging

148

249

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Bayesian solution of an inverse problem for indirect measurement M = AU +E is considered, where U is a function on a domain of R d . Here A is a smoothing linear operator and E is Gaussian white noise. The data is a realization m k of the random variable M k = P k AU + P k E, where P k is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as U n = T n U, where T n is a finite dimensional projection, leading to the computational measurement model M kn = P k AU n + P k E. Bayes formula gives then the posterior distribution π kn (u n | m kn ) ∼ Π n (u n ) exp(− 1 2 m kn − P k Au n 2 2 ) in R d , and the mean u kn := u n π kn (u n | m k ) du n is considered as the reconstruction of U. We discuss a systematic way of choosing prior distributions Π n for all n ≥ n 0 > 0 by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is discretization-invariant in the sense that Π n represent the same a priori information for all n and that the mean u kn converges to a limit estimate as k, n → ∞. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with B 1 11 prior is related to penalizing the ℓ 1 norm of the wavelet coefficients of U.

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“…In the literature, many wavelet decompositions and extensions have been reported offering different features in order to provide sparse image representations. For instance, decompositions onto orthonormal dyadic wavelet bases (Daubechies, 1988) including the Haar transform (Haar, 1910) as a special simple case or decompositions onto biorthogonal dyadic wavelets (Cohen et al, 1992), M-band wavelet representations (Steffen et al, 1993) and wavelet packet representations (Coifman and Wickerhauser, 1992) have been extensively investigated in image denoising (Donoho and Johnstone, 1995;Leporini and Pesquet, 2001;Mueller and Vidakovic, 1999;Heurta, 2005;Daubechies et al, 2004;Daubechies and Teschke, 2005) and deconvolution (Daubechies et al, 2004;Daubechies and Teschke, 2005;Vonesch and Unser, 2008;Chaux et al, 2007). In medical imaging, wavelet decompositions have also been widely used for image denoising (Weaver et al, 1991;Wang and Haomin, 2006;Pizurica et al, 2006), coil sensitivity map estimation and encoding schemes (Lin et al, 2003;Gelman and Wood, 1996;Wendt et al, 1998) in MRI, activation detection in fMRI (Ruttimann et al, 1998;Meyer, 2003;Van De Ville et al, 2004;Van De Ville et al, 2006), tissue characterization in ultrasound imaging (Mojsilovic et al, 1998) and tomographic reconstruction (Olson, 1992).…”

Section: Motivationmentioning

confidence: 99%