Wavelet decomposition approaches to statistical inverse problems

Abramovich, Felix; Silverman, B. W.

doi:10.1093/biomet/85.1.115

Cited by 181 publications

(178 citation statements)

References 17 publications

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“…In Abramovitch and Silverman [1], this method was compared with the similar vaguelette-wavelet decomposition. Other wavelet approaches, might be mentioned such as Antoniadis and Bigot [2], Antoniadis & al [3] and especially for the deconvolution problem, Penski & Vidakovic [37], Fan & Koo [17], Kalifa & Mallat [24], Neelamani & al [34].…”

Section: Projection Methodsmentioning

confidence: 99%

Estimation in inverse problems and second-generation wavelets

Kerkyacharian¹,

Picard²

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We consider the problem of recovering a function f , when we receive a blurred (by a linear operator) and noisy version : Yε = Kf + εẆ . We will have as guides 2 famous examples of such inverse problems : the deconvolution and the Wicksell problem. The direct problem (K is the identity) isolates the denoising operation. It can't be solved unless accepting to estimate a smoothed version of f : for instance, if f has an expansion on a basis, this smoothing might correspond to stopping the expansion at some stage m. Then a crucial problem lies in finding an equilibrium for m considering the fact that for m large, the difference between f and its smoothed version is small, whereas the random effect introduces an error which is increasing with m. In the true inverse problem, in addition to denoising we have to 'inverse the operator' K, which operation not only creates the usual difficulties, but also introduces the necessity to control the additional instability due to the inversion of the random noise. Our purpose here is to emphasize the fact that in such a problem, there generally exists a basis which is fully adapted to the problem, where for instance the inversion remains very stable : this is the Singular Value Decomposition basis. On the other hand, the SVD basis might be difficult to determine and manipulate numerically, it also might not be appropriate for the accurate description of the solution with a small number of parameters. Also in many practical situations, the signal provides inhomogeneous regularity, and its local features are especially interesting to recover. In such cases, other bases (in particular localised bases such as wavelet bases) may be much more appropriate to give a good representation of the object at hand. Our approach here will be to produce estimation procedures trying to keep the advantages of localisation without loosing the stability and computability of SVD decompositions. We will especially consider two cases. In the first one (which is the case of the deconvolution example) we show that a fairly simple algorithm (WAVE-VD) using an appropriate thresholding technique performed on a standard wavelet system, enables us to estimate the object with rates which are almost optimal up to logarithm factors for any Lp loss function, and on the whole range of Besov spaces. In the second case (which is the case of the Wicksell example where the SVD bases lies in the range of Jacobi polynomials), we prove that quite a similar algorithm (NEED-VD) can be performed provided replacing the standard wavelet system by a second generation wavelet-type basis : the Needlets. We use here the construction (essentially following the work of Petrushev and co-authors) of a localised frame linked with a prescribed basis (here Jacobi polynomials) using a Littlewood Paley decomposition combined with a cubature formula. Section 5 describes the direct case (K = I). It has its own interest and will act as a guide for understanding the 'true' inverse models for a reader which is unfamiliar with nonpar...

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Section: Projection Methodsmentioning

confidence: 99%

Estimation in inverse problems and second-generation wavelets

Kerkyacharian¹,

Picard²

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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“…To overcome this limitation, Kalifa et al proposed a wavelet packet based method that matches the frequency behavior of certain convolution operators [22]. Additional wavelet-based techniques have been proposed in [16] and [23]- [25].…”

Section: B Historical Perspectivementioning

confidence: 99%

“…In the implementation stage, to deal with boundary effects, some concepts in the literature have centered around the idea of noise shrinkage either before or after the application of the deconvolution procedure (see [14] and [16]). However, to carry out such schemes effectively, one needs a transform that can be implemented in a nonrecursive formulation as is done in this work with the shearlet transform.…”

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confidence: 99%

Shearlet-Based Deconvolution

Patel

Easley

Healy

2009

IEEE Trans. on Image Process.

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Abstract-In this paper, a new type of deconvolution algorithm is proposed that is based on estimating the image from a shearlet decomposition. Shearlets provide a multidirectional and multiscale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. Constructions such as curvelets and contourlets share similar properties, yet their implementations are significantly different from that of shearlets. Taking advantage of unique properties of a new M-channel implementation of the shearlet transform, we develop an algorithm that allows for the approximation inversion operator to be controlled on a multiscale and multidirectional basis. A key improvement over closely related approaches such as ForWaRD is the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation (GCV). Various tests show that this method can perform significantly better than many competitive deconvolution algorithms.

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“…Amongst them, Donoho (1995), Abramovich & Silverman (1998), Jonhstone, Kerkyacharian, Picard & Raimondo (2004) and Chenseau (2008) proposed various wavelet thresholding estimators of the unknown response function in this model that achieve optimal (in the minimax or the maxiset sense), or near-optimal within a logarithmic factor, convergence rates over a wide range of Besov balls and for a range of L p -loss functions defining the risk.…”

Section: Introductionmentioning

confidence: 99%

Minimax convergence rates under the -risk in the functional deconvolution model

Petsa¹,

Sapatinas²

2009

Statistics & Probability Letters

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To cite this version:Athanasia Petsa, Theofanis Sapatinas. Minimax convergence rates under the -risk in the functional deconvolution model. Statistics and Probability Letters, Elsevier, 2009, 79 (13) This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A C C E P T E D M A N U S C R I P T

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Wavelet decomposition approaches to statistical inverse problems

Cited by 181 publications

References 17 publications

Estimation in inverse problems and second-generation wavelets

Estimation in inverse problems and second-generation wavelets

Shearlet-Based Deconvolution

Minimax convergence rates under the -risk in the functional deconvolution model

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