1,698 research outputs found
The WaveD Transform in R: Performs Fast Translation-Invariant Wavelet Deconvolution
This paper provides an introduction to a software package called waved making available all code necessary for reproducing the figures in the recently published articles on the WaveD transform for wavelet deconvolution of noisy signals. The forward WaveD transforms and their inverses can be computed using any wavelet from the Meyer family. The WaveD coefficients can be depicted according to time and resolution in several ways for data analysis. The algorithm which implements the translation invariant WaveD transform takes full advantage of the fast Fourier transform (FFT) and runs in O(n(log n)^2)steps only. The waved package includes functions to perform thresholding and tne resolution tuning according to methods in the literature as well as newly designed visual and statistical tools for assessing WaveD fits. We give a waved tutorial session and review benchmark examples of noisy convolutions to illustrate the non-linear adaptive properties of wavelet deconvolution.
Wavelet Deconvolution in a Periodic Setting with Long-Range Dependent Errors
In this paper, a hard thresholding wavelet estimator is constructed for a
deconvolution model in a periodic setting that has long-range dependent noise.
The estimation paradigm is based on a maxiset method that attains a near
optimal rate of convergence for a variety of L_p loss functions and a wide
variety of Besov spaces in the presence of strong dependence. The effect of
long-range dependence is detrimental to the rate of convergence. The method is
implemented using a modification of the WaveD-package in R and an extensive
numerical study is conducted. The numerical study supplements the theoretical
results and compares the LRD estimator with na\"ively using the standard WaveD
approach
Wavelet Deconvolution in a Periodic Setting Using Cross-Validation
The wavelet deconvolution method WaveD using band-limited wavelets offers both theoretical and computational advantages over traditional compactly supported wavelets. The translation-invariant WaveD with a fast algorithm improves further. The twofold cross-validation method for choosing the threshold parameter and the finest resolution level in WaveD is introduced. The algorithm’s performance is compared with the fixed constant tuning and the default tuning in WaveD
A deconvolution approach to estimation of a common shape in a shifted curves model
This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature
Log-Density Deconvolution by Wavelet Thresholding
This paper proposes a new wavelet-based method for deconvolving a density. The estimator combines the ideas of nonlinear wavelet thresholding with periodised Meyer wavelets and estimation by information projection. It is guaranteed to be in the class of density functions, in particular it is positive everywhere by construction. The asymptotic optimality of the estimator is established in terms of rate of convergence of the Kullback-Leibler discrepancy over Besov classes. Finite sample properties is investigated in detail, and show the excellent empirical performance of the estimator, compared with other recently introduced estimators.deconvolution, wavelet thresholding,adaptive estimation
The WaveD Transform in R: Performs Fast Translation-Invariant Wavelet Deconvolution
This paper provides an introduction to a software package called waved making available all code necessary for reproducing the figures in the recently published articles on the WaveD transform for wavelet deconvolution of noisy signals. The forward WaveD transforms and their inverses can be computed using any wavelet from the Meyer family. The WaveD coefficients can be depicted according to time and resolution in several ways for data analysis. The algorithm which implements the translation invariant WaveD transform takes full advantage of the fast Fourier transform (FFT) and runs in O(n(log n)2)steps only. The waved package includes functions to perform thresholding and tne resolution tuning according to methods in the literature as well as newly designed visual and statistical tools for assessing WaveD fits. We give a waved tutorial session and review benchmark examples of noisy convolutions to illustrate the non-linear adaptive properties of wavelet deconvolution
Recent Progress in Image Deblurring
This paper comprehensively reviews the recent development of image
deblurring, including non-blind/blind, spatially invariant/variant deblurring
techniques. Indeed, these techniques share the same objective of inferring a
latent sharp image from one or several corresponding blurry images, while the
blind deblurring techniques are also required to derive an accurate blur
kernel. Considering the critical role of image restoration in modern imaging
systems to provide high-quality images under complex environments such as
motion, undesirable lighting conditions, and imperfect system components, image
deblurring has attracted growing attention in recent years. From the viewpoint
of how to handle the ill-posedness which is a crucial issue in deblurring
tasks, existing methods can be grouped into five categories: Bayesian inference
framework, variational methods, sparse representation-based methods,
homography-based modeling, and region-based methods. In spite of achieving a
certain level of development, image deblurring, especially the blind case, is
limited in its success by complex application conditions which make the blur
kernel hard to obtain and be spatially variant. We provide a holistic
understanding and deep insight into image deblurring in this review. An
analysis of the empirical evidence for representative methods, practical
issues, as well as a discussion of promising future directions are also
presented.Comment: 53 pages, 17 figure
Pinsker estimators for local helioseismology
A major goal of helioseismology is the three-dimensional reconstruction of
the three velocity components of convective flows in the solar interior from
sets of wave travel-time measurements. For small amplitude flows, the forward
problem is described in good approximation by a large system of convolution
equations. The input observations are highly noisy random vectors with a known
dense covariance matrix. This leads to a large statistical linear inverse
problem.
Whereas for deterministic linear inverse problems several computationally
efficient minimax optimal regularization methods exist, only one
minimax-optimal linear estimator exists for statistical linear inverse
problems: the Pinsker estimator. However, it is often computationally
inefficient because it requires a singular value decomposition of the forward
operator or it is not applicable because of an unknown noise covariance matrix,
so it is rarely used for real-world problems. These limitations do not apply in
helioseismology. We present a simplified proof of the optimality properties of
the Pinsker estimator and show that it yields significantly better
reconstructions than traditional inversion methods used in helioseismology,
i.e.\ Regularized Least Squares (Tikhonov regularization) and SOLA (approximate
inverse) methods.
Moreover, we discuss the incorporation of the mass conservation constraint in
the Pinsker scheme using staggered grids. With this improvement we can
reconstruct not only horizontal, but also vertical velocity components that are
much smaller in amplitude
Anisotropic Denoising in Functional Deconvolution Model with Dimension-free Convergence Rates
In the present paper we consider the problem of estimating a periodic
-dimensional function based on observations from its noisy
convolution. We construct a wavelet estimator of , derive minimax lower
bounds for the -risk when belongs to a Besov ball of mixed smoothness
and demonstrate that the wavelet estimator is adaptive and asymptotically
near-optimal within a logarithmic factor, in a wide range of Besov balls. We
prove in particular that choosing this type of mixed smoothness leads to rates
of convergence which are free of the "curse of dimensionality" and, hence, are
higher than usual convergence rates when is large. The problem studied in
the paper is motivated by seismic inversion which can be reduced to solution of
noisy two-dimensional convolution equations that allow to draw inference on
underground layer structures along the chosen profiles. The common practice in
seismology is to recover layer structures separately for each profile and then
to combine the derived estimates into a two-dimensional function. By studying
the two-dimensional version of the model, we demonstrate that this strategy
usually leads to estimators which are less accurate than the ones obtained as
two-dimensional functional deconvolutions. Indeed, we show that unless the
function is very smooth in the direction of the profiles, very spatially
inhomogeneous along the other direction and the number of profiles is very
limited, the functional deconvolution solution has a much better precision
compared to a combination of solutions of separate convolution equations. A
limited simulation study in the case of confirms theoretical claims of
the paper.Comment: 29 pages, 1 figure, 1 tabl
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