Renormalizing Experiments for Nonlinear Functionals

Donoho, David L.

doi:10.1007/978-1-4612-1880-7_11

Cited by 5 publications

(2 citation statements)

References 13 publications

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“…Nonparametric statistical methods for the analysis of discontinuities in images have been studied in a number of papers (Tsybakov, 1989(Tsybakov, , 1991(Tsybakov, , 1994Korostelev, 1991;Korostelev andTsybakov, 1992, 1993a,b;Carlstein and Krishnamoorthy, 1992;Müller and Song, 1994a,b;Rudemo and Stryhn, 1994a,b;Freidlin and Korostelev, 1995;Mammen and Tsybakov, 1995;Jacob and Suquet, 1996;Donoho, 1997;Hall and Raimondo, 1997a,b;Qiu, 1997; Qiu and Yandell, 1997; Chu et al, 1998;Polzehl and Spokoiny (1998a,b)). These papers deal with statistical estimation of discontinuity contours (edges) in images.…”

Section: Introductionmentioning

confidence: 98%

Testing Hypotheses About Contours in Images

Gayraud

Tsybakov

2002

Journal of Nonparametric Statistics

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We consider the problem of testing hypotheses about the contours in binary images observed on the regular grid. We propose a simple goodness-of-fit test of the hypothesis that a contour belongs to a given parametric family against a nonparametric alternative. We analyze the behavior of the test under the null hypothesis, and under the alternative separated from the null parametric family by a distance of order n À1=2 (n is the total number of observations and the distance is defined as the measure of symmetric difference between the sets whose boundaries are the contours of interest). Finally, we prove the lower bound showing that no test can be consistent if the distance between the hypothesis and the alternative is of the order smaller than n À1=2 .

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Section: Introductionmentioning

confidence: 98%

Testing Hypotheses About Contours in Images

Gayraud

Tsybakov

2002

Journal of Nonparametric Statistics

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“…The use of geometric tools from Hilbert space seems promising for renormalization questions (see, e.g., [64], [15], [23], and [74]) since notions of repetition up to similarity of form at infinitely many scales are common in the Hilbert-space approach to multiresolutions; see, e.g., [35]. And this self-similarity up to scale parallels a basic feature of renormalization questions in physics: for a variety of instances of dynamics in physics, and in telecommunication [64,15,20], we encounter scaling laws of self-similarity; i.e., we observe that a phenomenon reproduces itself on different time and/or space scales.…”

Section: The Renormalization Questionmentioning

confidence: 99%

Methods from Multiscale Theory and Wavelets Applied to Nonlinear Dynamics

Dutkay

Jørgensen

Wavelets, Multiscale Systems and Hypercomplex Analysis

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We show how fundamental ideas from signal processing, multiscale theory and wavelets may be applied to nonlinear dynamics.The problems from dynamics include iterated function systems (IFS), dynamical systems based on substitution such as the discrete systems built on rational functions of one complex variable and the corresponding Julia sets, and state spaces of subshifts in symbolic dynamics. Our paper serves to motivate and survey our recent results in this general area. Hence we leave out some proofs, but instead add a number of intuitive ideas which we hope will make the subject more accessible to researchers in operator theory and systems theory. (2000). 42C40, 42A16, 43A65, 42A65. Mathematics Subject Classification

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