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Optimal spatial adaptation for patchbased image denoising
 IEEE Trans. Image Process
, 2006
"... Abstract—A novel adaptive and patchbased approach is proposed for image denoising and representation. The method is based on a pointwise selection of small image patches of fixed size in the variable neighborhood of each pixel. Our contribution is to associate with each pixel the weighted sum of da ..."
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Cited by 68 (10 self)
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Abstract—A novel adaptive and patchbased approach is proposed for image denoising and representation. The method is based on a pointwise selection of small image patches of fixed size in the variable neighborhood of each pixel. Our contribution is to associate with each pixel the weighted sum of data points within an adaptive neighborhood, in a manner that it balances the accuracy of approximation and the stochastic error, at each spatial position. This method is general and can be applied under the assumption that there exists repetitive patterns in a local neighborhood of a point. By introducing spatial adaptivity, we extend the work earlier described by Buades et al. which can be considered as an extension of bilateral filtering to image patches. Finally, we propose a nearly parameterfree algorithm for image denoising. The method is applied to both artificially corrupted (white Gaussian noise) and real images and the performance is very close to, and in some cases even surpasses, that of the already published denoising methods. I.
On spatial adaptive estimation of nonparametric regression
 Math. Meth. Statistics
, 1997
"... The paper is devoted to developing spatial adaptive estimates for restoring functions from noisy observations. We show that the traditional least square (piecewise polynomial) estimate equipped with adaptively adjusted window possesses simultaneously many attractive adaptive properties, namely, 1) i ..."
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Cited by 57 (4 self)
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The paper is devoted to developing spatial adaptive estimates for restoring functions from noisy observations. We show that the traditional least square (piecewise polynomial) estimate equipped with adaptively adjusted window possesses simultaneously many attractive adaptive properties, namely, 1) it is near– optimal within ln n–factor for estimating a function (or its derivative) at a single point; 2) it is spatial adaptive in the sense that its quality is close to that one which could be achieved if smoothness of the underlying function was known in advance; 3) it is optimal in order (in the case of “strong ” accuracy measure) or near–optimal within ln n–factor (in the case of “weak ” accuracy measure) for estimating whole function (or its derivative) over wide range of the classes and global loss functions. We demonstrate that the “spatial adaptive abilities ” of our estimate are, in a sense, the best possible. Besides this, our adaptive estimate is computationally efficient and demonstrates reasonable practical behavior. 1
Adaptive hypothesis testing using wavelets
 Annals of Statistics
, 1996
"... Let a function f be observed with a noise. We wish to test the null hypothesis that the function is identically zero, against a composite nonparametric alternative: functions from the alternative set are separated away from zero in an integral Ž e.g., L. 2 norm and also possess some smoothness prope ..."
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Cited by 46 (6 self)
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Let a function f be observed with a noise. We wish to test the null hypothesis that the function is identically zero, against a composite nonparametric alternative: functions from the alternative set are separated away from zero in an integral Ž e.g., L. 2 norm and also possess some smoothness properties. The minimax rate of testing for this problem was evaluated in earlier papers by Ingster and by Lepski and Spokoiny under different kinds of smoothness assumptions. It was shown that both the optimal rate of testing and the structure of optimal Ž in rate. tests depend on smoothness parameters which are usually unknown in practical applications. In this paper the problem of adaptive Ž assumption free. testing is considered. It is shown that adaptive testing without loss of efficiency is impossible. An extra log logfactor is inessential but unavoidable payment for the adaptation. A simple adaptive test based on wavelet technique is constructed which is nearly minimax for a wide range of Besov classes. 1. Introduction. Suppose
Local adaptivity to variable smoothness for exemplarbased image denoising and representation
, 2005
"... ..."
Optimal pointwise adaptive methods in nonparametric estimation
 ANN. STATIST
, 1997
"... The problem of optimal adaptive estimation of a function at a given point from noisy data is considered. Two procedures are proved to be asymptotically optimal for different settings. First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel. ..."
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Cited by 38 (9 self)
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The problem of optimal adaptive estimation of a function at a given point from noisy data is considered. Two procedures are proved to be asymptotically optimal for different settings. First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel. We propose a bandwidth selection procedure and prove its optimality in the asymptotic sense. Moreover, this optimality is stated not only among kernel estimators with a variable bandwidth. The resulting estimator is asymptotically optimal among all feasible estimators. The important feature of this procedure is that it is fully adaptive and it “works” for a very wide class of functions obeying a mild regularity restriction. With it the attainable accuracy of estimation depends on the function itself and is expressed in terms of the “ideal adaptive bandwidth” corresponding to this function and a given kernel. The second procedure can be considered as a specialization of the first one under the qualitative assumption that the function to be estimated belongs to some Hölder class ��β � L � with unknown parameters β � L. This assumption allows us to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense in any range of adaptation with β ≤ 2.
Combining Different Procedures for Adaptive Regression
 Journal of Multivariate Analysis
, 1998
"... Given any countable collection of regression procedures (e.g., kernel, spline, wavelet, local polynomial, neural nets, etc), we show that a single adaptive procedure can be constructed to share the advantages of them to a great extent in terms of global squared L 2 risk. The combined procedure basic ..."
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Cited by 36 (7 self)
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Given any countable collection of regression procedures (e.g., kernel, spline, wavelet, local polynomial, neural nets, etc), we show that a single adaptive procedure can be constructed to share the advantages of them to a great extent in terms of global squared L 2 risk. The combined procedure basically pays a price only of order 1=n for adaptation over the collection. An interesting consequence is that for a countable collection of classes of regression functions (possibly of completely different characteristics), a minimaxrate adaptive estimator can be constructed such that it automatically converges at the right rate for each of the classes being considered.
Uniform in bandwidth consistency of kerneltype function estimators
 Ann. Stat
, 2005
"... We introduce a general method to prove uniform in bandwidth consistency of kerneltype function estimators. Examples include the kernel density estimator, the Nadaraya–Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of dat ..."
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Cited by 29 (4 self)
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We introduce a general method to prove uniform in bandwidth consistency of kerneltype function estimators. Examples include the kernel density estimator, the Nadaraya–Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of datadriven bandwidth kerneltype function estimators. 1. Introduction and statements of main results. Let X,X1,X2,... be i.i.d. Rd, d ≥ 1, valued random variables and assume that the common distribution function of these variables has a Lebesgue density function, which we shall denote by f. A kernel K will be any measurable function which
Estimation Of A Function With Discontinuities Via Local Polynomial Fit With An Adaptive Window Choice
, 1996
"... . We propose a method of adaptive estimation of a regression function and which is near optimal in the classical sense of the mean integrated error. At the same time, the estimator is shown to be very sensitive to discontinuities or changepoints of the underlying function f or its derivatives. For ..."
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Cited by 27 (2 self)
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. We propose a method of adaptive estimation of a regression function and which is near optimal in the classical sense of the mean integrated error. At the same time, the estimator is shown to be very sensitive to discontinuities or changepoints of the underlying function f or its derivatives. For instance, in the case of a jump of a regression function, beyond the interval of length (in order) n \Gamma1 log n around changepoints the quality of estimation is essentially the same as if locations of jumps were known. The method is fully adaptive and no assumptions are imposed on the design, number and size of jumps. The results are formulated in a nonasymptotic way and can be therefore applied for an arbitrary sample size. 1. Introduction The changepoint analysis which includes sudden, localized changes typically occurring in economics, medicine and the physical sciences has recently found increasing interest, see Muller (1992) for some examples and discussion of the problem. Let...
Multiscale Poisson intensity and density estimation
 IEEE TRANS. INFO. TH
, 2005
"... The nonparametric Poisson intensity and density estimation methods studied in this paper offer near minimax convergence rates for broad classes of densities and intensities with arbitrary levels of smoothness. The methods and theory presented here share many of the desirable features associated with ..."
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Cited by 25 (12 self)
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The nonparametric Poisson intensity and density estimation methods studied in this paper offer near minimax convergence rates for broad classes of densities and intensities with arbitrary levels of smoothness. The methods and theory presented here share many of the desirable features associated with waveletbased estimators: computational speed, spatial adaptivity, and the capability of detecting discontinuities and singularities with high resolution. Unlike traditional waveletbased approaches, which impose an upper bound on the degree of smoothness to which they can adapt, the estimators studied here guarantee nonnegativity and do not require any a priori knowledge of the underlying signal’s smoothness to guarantee nearoptimal performance. At the heart of these methods lie multiscale decompositions based on freeknot, freedegree piecewisepolynomial functions and penalized likelihood estimation. The degrees as well as the locations of the polynomial pieces can be adapted to the observed data, resulting in near minimax optimal convergence rates. For piecewise analytic signals, in particular, the error of this estimator converges at nearly the parametric rate. These methods can be further refined in two dimensions, and it is demonstrated that plateletbased estimators in two dimensions exhibit similar nearoptimal error convergence rates for images consisting of smooth surfaces separated by smooth boundaries.
A spatially adaptive nonparametric regression image deblurring
 IEEE TRANS. IMAGE PROCESS
, 2005
"... (slides) ..."