Generalized cross validation for wavelet thresholding

Jansen, Maarten; Malfait, Maurits; Bultheel, Adhemar

doi:10.1016/s0165-1684(97)83621-3

Cited by 157 publications

(87 citation statements)

References 15 publications

(9 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Just as in the case of wavelets, to obtain results similar to those in [50], it is not necessary for the shearlet coefficients to be uncorrelated at any moment; however, it is necessary that the noise be second order stationary [49]. If the noise process is stationary, then using the multiscale and multidirectional structure of shearlets, we obtain the following lemma.…”

Section: Generalized Cross Validation (Gcv) For Shearlet Threshomentioning

confidence: 93%

Shearlet-Based Deconvolution

Patel

Easley

Healy

2009

IEEE Trans. on Image Process.

View full text Add to dashboard Cite

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.

show abstract

Section: Generalized Cross Validation (Gcv) For Shearlet Threshomentioning

confidence: 93%

Shearlet-Based Deconvolution

Patel

Easley

Healy

2009

IEEE Trans. on Image Process.

View full text Add to dashboard Cite

show abstract

“…The GCV method has also been extended in other directions, including to non-Gaussian data [57] and to wavelet thresholding [80]. It can also be applied to iterative regularization methods, in particular the conjugate gradient method [67,71], Krylov methods [87], the ART method [126] and the iteratively regularized Gauss-Newton method [59,142].…”

Section: Generalized Cross-validationmentioning

confidence: 99%

Comparingparameter choice methods for regularization of ill-posed problems

Bauer

Lukas

2011

Mathematics and Computers in Simulation

248

111

View full text Add to dashboard Cite

In the literature on regularization, many different parameter choice methods have been proposed in both deterministic and stochastic settings. However, based on the available information, it is not always easy to know how well a particular method will perform in a given situation and how it compares to other methods. This paper reviews most of the existing parameter choice methods, and evaluates and compares them in a large simulation study for spectral cut-off and Tikhonov regularization. The test cases cover a wide range of linear inverse problems with both white and colored stochastic noise. The results show some marked differences between the methods, in particular, in their stability with respect to the noise and its type. We conclude with a table of properties of the methods and a summary of the simulation results, from which we identify the best methods.

show abstract

“…One may use Akaike's information criterion (AIC) [1], the Bayesian information criterion (BIC) [19], Mallows's C p , [14], or its outlier robust version, [17], cross validation, [15], and its generalized version [10]. An extensive overview of most of the existing model selection techniques is given in [5].…”

Section: The Sparse Penalized Modelmentioning

confidence: 99%