Abstract We consider multichannel deconvolution in a periodic setting with long-memory errors under three different scenarios for the convolution operators, i.e., super-smooth, regular-smooth and box-car convolutions. We investigate global performances of linear and hard-thresholded non-linear wavelet estimators for functions over a wide range of Besov spaces and for a variety of loss functions defining the risk. In particular, we obtain upper bounds on convergence rates using the L p -risk ( 1 ≤ p ∞ ) . Contrary to the case where the errors follow independent Brownian motions, it is demonstrated that multichannel deconvolution with errors that follow independent fractional Brownian motions with different Hurst parameters results in a much more involved situation. An extensive finite-sample numerical study is performed to supplement the theoretical findings.
The case where @math and @math in -- refers to the so-called standard deconvolution model which attracted attention of a number of researchers. (Note that the standard deconvolution model is typically ill-posed in the sense of Hadamard: the inversion does not depend continuously on the observed data, i.e., small noise in the convolved signal leads to a significant error in the estimation procedure.) After a rather rapid progress in this problem in late eighties--early nineties, authors turned to adaptive wavelet solutions of the problem that are optimal (in the minimax or the maxiset sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls and for a variety of loss functions defining the risk, and under mild conditions on the blurring function (see, e.g., @cite_7 @cite_6 @cite_26 @cite_2 @cite_11 @cite_21 @cite_29 @cite_8 ).
@cite_26 @cite_7 @cite_8 @cite_29 @cite_21 @cite_6 @cite_2 @cite_11
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Thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded. For signals in a set Θ), sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence. Deconvolutions with kernels having a Fourier transform which vanishes at high frequencies are examples of unstable inverse problems, where a thresholding in a wavelet basis is a suboptimal estimator. A new mirror wavelet basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals. This thresholding estimator is used to restore blurred satellite images. We describe the wavelet–vaguelette decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type—such as numerical differentiation, inversion of Abel-type transforms, certain convolution transforms, and the Radon transform. We propose to solve ill-posed linear inverse problems by nonlinearly "shrinking" the WVD coefficients of the noisy, indirect data. Our approach offers significant advantages over traditional SVD inversion in recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L2 loss, over the entire scale of Besov spaces. The important case of Besov spaces Bσp,q, p < 2, which model spatial inhomogeneity, is included. In comparison, linear procedures— SVD included—cannot attain optimal rates of convergence over such classes in the case p < 2. For example, our methods achieve faster rates of convergence for objects known to lie in the bump algebra or in bounded variation than any linear procedure. We consider the non-parametric estimation of a function that is observed in white noise after convolution with a boxcar, the indicator of an interval ( a; a). In a recent paper Johnstone, Kerkyacharian, Picard and Raimondo (2004) have developed a wavelet deconvolution method (called WaveD) that can be used for " boxcar kernels. For example, WaveD can be tuned to achieve near optimal rates over Besov spaces when a is a Badly Approximable (BA) irrational number. While the set of all BA's contains quadratic irrationals, e.g., a = p 5, it has Lebesgue measure zero. In this paper we derive two tuning scenarios of WaveD that are valid for all" boxcar convolutions (i.e., when a 2 A where A is a full Lebesgue measure set). We propose (i) a tuning inspired from Minimax theory over Besov spaces; (ii) a tuning derived from Maxiset theory providing similar rates as for WaveD in the BA widths setting. Asymptotic theory nds that (i) in the worst case scenario, departures from the BA assumption aect WaveD convergence rates, at most, by log factors; (ii) the Maxiset tuning, which yields smaller thresholds, is superior to the Minimax tuning over a whole range of Besov sub-scales. Our asymptotic results are illustrated in an extensive simulation of boxcar convolution observed in white noise. SUMMARY A wide variety of scientific settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise. Primary interest is in some function f(t) but data are accessible only about some linear transform corrupted by noise. The usual linear methods for such inverse problems do not perform satisfactorily when f(t) is spatially inhomogeneous. One existing nonlinear alternative is the wavelet-vaguelette decomposition method, based on the expansion of the unknown f(t) in wavelet series. In the vaguelette-wavelet decomposition method proposed here, the observed data are expanded directly in wavelet series. The performances of various methods are compared through exact risk calculations, in the context of the estimation of the derivative of a function observed subject to noise. A result is proved demonstrating that, with a suitable universal threshold somewhat larger than that used for standard denoising problems, both the wavelet-based approaches have an ideal spatial adaptivity property. Deconvolution problems are naturally represented in the Fourier domain, whereas thresholding in wavelet bases is known to have broad adaptivity properties. We study a method which combines both fast Fourier and fast wavelet transforms and can recover a blurred function observed in white noise with "O"l"n" log ("n")-super-2r steps. In the periodic setting, the method applies to most deconvolution problems, including certain 'boxcar' kernels, which are important as a model of motion blur, but having poor Fourier characteristics. Asymptotic theory informs the choice of tuning parameters and yields adaptivity properties for the method over a wide class of measures of error and classes of function. The method is tested on simulated light detection and ranging data suggested by underwater remote sensing. Both visual and numerical results show an improvement over competing approaches. Finally, the theory behind our estimation paradigm gives a complete characterization of the 'maxiset' of the method: the set of functions where the method attains a near optimal rate of convergence for a variety of "L"-super-"p" loss functions. Copyright 2004 Royal Statistical Society. Deconvolution of a noisy signal in a periodic band-limited wavelet basis exhibits visual artifacts in the neighbourhood of discontinuities. This phenomenon is similar to that appearing in denoising with compactly-supported wavelet transforms and can be reduced by "cycle spinning" as in Coifman and Donoho [3]. In this paper we present an algorithm which "cycle-spins" a periodic band-limited wavelet estimator over all circulant shifts in O(n(log(n))2) steps. Our approach is based on a mathematical idea and takes full advantage of the Fast Fourier Transform. A particular feature of our algorithm is to bounce from the Fourier domain (where deconvolution is performed) to the wavelet domain (where denoising is performed). For both smooth and boxcar convolutions observed in white noise, we illustrate the visual and numerical performances of our algorithm in an extensive simulation study of the estimator recently proposed by Johnstone, Kerkyacharian, Picard, and Raimondo [8]. All figures presented here are reproducible using the software package.