Abstract—A novel adaptive and patch-based 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 parameter-free 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.

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

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 minimax-rate adaptive estimator can be constructed such that it automatically converges at the right rate for each of the classes being considered.

. 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 kernel. The resulting estimator is optimal among all feasible estimators. The important feature of this procedure is that no prior information is used about smoothness properties of the estimated function i.e. the procedure is completely adaptive and "works" for the class of all functions. With it the attainable accuracy of estimation depends on the function itself and it is expressed in terms of "ideal" bandwidth corresponding to this function. The second procedure can be considered as a specification of the firs...

. 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 change-points 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 change-points 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 non-asymptotic way and can be therefore applied for an arbitrary sample size. 1. Introduction The change-point 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...

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We consider estimating an unknown function f from indirect white noise observations with particular emphasis on the problem of nonparametric deconvolution. Nonparametric estimators that can adapt to unknown smoothness of f are developed. The adaptive estimators are specified under two sets of assumptions on the kernel of the convolution transform. In particular, kernels having the Fourier transform with polynomially and exponentially decaying tails are considered. It is shown that the proposed estimates possess, in a sense, the best possible abilities for pointwise adaptation. Keywords: adaptive estimation; deconvolution; rates of convergence Running title: Adaptive nonparametric deconvolution Department of Statistics, University of Haifa, Mount Carmel, 31905 Haifa, Israel y e-mail: goldensh@rstat.haifa.ac.il 1 Introduction This paper investigates the problem of pointwise adaptive nonparametric estimation from indirect white noise observations. Let f 2 L 2 (R) be an unknown func...

A new method of pointwise adaptation has been proposed and studied in Spokoiny (1998) in context of estimation of piecewise smooth univariate functions. The present paper extends that method to estimation of bivariate grey-scale images composed of large homogeneous regions with smooth edges and observed with noise on a gridded design. The proposed estimator # f(x) at a point x is simply the average of observations over a window # U(x) selected in a data-driven way. The theoretical properties of the procedure are studied for the case of piecewise constant images. We present a nonasymptotic bound for the accuracy of estimation at a specific grid point x as a function of the number of pixel n, of the distance from the point of estimation to the closest boundary and of smoothness properties and orientation of this boundary. It is also shown that the proposed method provides a near optimal rate of estimation near edges and inside homogeneous regions. We briefly discuss algorithmic aspects and the complexity of the procedure. The numerical examples demonstrate a reasonable performance of the method and they are in agreement with the theoretical issues. An example from satellite (SAR) imaging illustrates the applicability of the method. # The authors thank A.Juditski, O. Lepski, A.Tsybakov and Yu.Golubev for important remarks and discussion. polzehl, j. and spokoiny, v. 1 1

image deblurring

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 wavelet-based approaches, which impose an upper bound on the degree of smoothness to which they can adapt, the estimators studied here guarantee non-negativity and do not require any a priori knowledge of the underlying signal’s smoothness to guarantee near-optimal performance. At the heart of these methods lie multiscale decompositions based on free-knot, free-degree piecewise-polynomial 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 platelet-based estimators in two dimensions exhibit similar near-optimal error convergence rates for images consisting of smooth surfaces separated by smooth boundaries.