Image Denoising: Pointwise Adaptive Approach

. The paper is concerned with the problem of image denoising. We consider the case of black-white type images consisting of a finite number of regions with smooth boundaries and the image value is assumed to be piecewise constant within each region. New method of image denoising is proposed which is adaptive (assumption free) to the number of regions and smoothness properties of edges. The method is based on a pointwise image recovering and it relies on an adaptive choice of a smoothing window. It is shown that the attainable quality of estimation depends on the distance from the point of estimation to the closest boundary and on the smoothness properties of this boundary. As a consequence, it turns out that the proposed method provides the optimal rate of the edge estimation. 1. Introduction One of the main problem of image analysis is reconstruction of an image (a picture) from noisy data. It has been intensively studied last years, see e.g. the books of Pratt (1978), Grenander (19..

Estimation Of A Function With Discontinuities Via Local Polynomial Fit With An Adaptative Window Choice

. New method of adaptive estimation of a regression function is proposed. The resulting estimator achieves near optimal rate of estimation in the classical sense of mean integrated squared 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 the location of this jump 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..

Exact Asymptotics of Minimax Bahadur Risk in Lipschitz Regression

this paper was made possible in part by Grant MCF000 from the International Scientic Foundation, in part within the Sonderforschungbereich 373 at Humbolt Unversity Berlin and was printed using funds made available by the Deutsche Forschungsgemeinschaft. 1<F3.065e+05> # # # # # # ## #<F3.411e+05> ## ## ## # # # # # # # # # #<F2.421e+05> #<F4.353e+05> #<F3.813e+05> ########### ####### ############# ######## ### ##### ### ########<F5.275e+05> June, 15, 1994 1 Consider a nonparametric regression model with observations = + = 1 0 1 ; = 1 2 (1.1) The regression function ( ) belongs a priori to the class of Lipschitz functions, i.e. ( ), ( ) = : ( ) ( ) where is a given positive. For each the random variables are i.i.d. with a probability density ( ). Our goal is to estimate the value (0) of the regression function at the origin from the observations (1.1). Let ^ be an estimator, i.e. an arbitrary function of the observation (1.1). We would like to nd an estimator which minimizes the probability ^ (0) for a xed positive . Here denotes the probability of the observations (1.1) corresponding to the true regression . We follow Bahadur(1960,1967) whose approach we modify in the spirit of the minimax theory (see Ibragimov and Khasminskii, 1981, Ch.1). Introduce the minimax Bahadur risk by ( ) = inf sup 1 log ^ (0) (1.2)<F5.034e+05> Y f i n ; i : : : ; ; ; ; : : : n ; ; : : : f t ; t R ; f L L f f t f t L t t L n p x f f P f f > c c P f c n P f f > c :<F6.144e+05> # # # # # # # # # # ## # # # # # #<F4.25e+05> ########<F5.851e+05> ##### ########### ## ####### ####### #### ## ######### ##########<F5.045e+05> . Introduction <F5.661e+05># # # # # # # # # #<F4.111e+05> 2 KOROSTELEV, A.P. AND SPOKOINY, V.G.<F3.065e+05> # # # # # # # # # # # # # # # # # ## #<F2.421e+05> # # #<..

Optimal Pointwise Adaptive Methods In Nonparametric Estimation

. 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..

Image denoising: pointwise adaptive approach

The paper is concerned with the problem of image denoising. We consider the case of black-white type images consisting of a finite number of regions with smooth boundaries and the image value is assumed to be piecewise constant within each region. New method of image denoising is proposed which is adaptive (assumption free) to the number of regions and smoothness properties of edges. The method is based on a pointwise image recovering and it relies on an adaptive choice of a smoothing window. It is shown that the attainable quality of estimation depends on the distance from the point of estimation to the closest boundary and on the smoothness properties of this boundary. As a consequence, it turns out that the proposed method provides the optimal rate of the edge estimation. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(332)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice

New method of adaptive estimation of a regression function is proposed. The resulting estimator achieves near optimal rate of estimation in the classical sense of mean integrated squared 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"-"1 log n around change-points the quality of estimation is essentially the same as if the location of this jump 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. (orig.)Available from TIB Hannover: RR 5549(291)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Optimal pointwise adaptive methods in nonparametric estimation

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 estimators 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 accuray 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 first one under the qualitative assumption that the function to be estimated belongs to some Hoelder class #SIGMA#(#beta#, L) with unknown parameters #beta#, L. This assumption allows to choose a family of kernel in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(229)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Adaptive hypothesis testing using wavelets

The present paper continues studying the problem of minimax nonparametric hypothesis testing started in Lepski and Spokoiny (1995). The null hypothesis assumes that the function observed with a noise is identically zero i.e. no signal is present. The alternative is composite and minimax: the function is assumed to be separated away from zero in an integral (L_2-) norm and also to possess some smoothness properties. The minimax rate of testing for this problem was evaluated by Ingster for the case of Sobolev smoothness classes. Then this problem was studied by Lepski and Spokoiny in the sutiation of an alternative with inhomogeneous smoothness properties that leads to considering Besov smoothness classes. But for both cases the optimal rate and the structure of optimal (in rate) tests depends on smoothness parameters which are usually unknown in practical applications. In this paper the problem of adaptive (assumption free) testing is considered. It is schown that the adaptation without loss of efficiency is impossible. An extra (log log)-factor is nonsignificant 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. (orig.)Available from TIB Hannover: RR 5549(176)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman