Edge-preserving and peak-preserving smoothing
TL;DR: An alternative procedure is developed to the smoothed linear fitting method of McDonald and Owen based on the detection of discontinuities by comparing, at any given position, three smooth fits.
Abstract: An alternative procedure is developed to the smoothed linear fitting method of McDonald and Owen. The procedure is based on the detection of discontinuities by comparing, at any given position, three smooth fits. Diagnostics are used to detect discontinuities in the regression function itself (edge detection) or in its first derivative (peak detection). An application in electron microscopy is discussed.
Citations
More filters
TL;DR: In this paper, a method is proposed to detect jumps and sharp cusps in a function which is observed with noise, by checking if the wavelet transformation of the data has significantly large absolute values across fine scale levels.
Abstract: SUMMARY A method is proposed to detect jumps and sharp cusps in a function which is observed with noise, by checking if the wavelet transformation of the data has significantly large absolute values across fine scale levels. Asymptotic theory is established and practical implementation is discussed. The method is tested on simulated examples, and applied to stock market return data. The analysis of change-points, which describe sudden localised changes, has recently found increasing interest. Change-points can be used to model practical problems arising in fields such as quality control, economics, medicine, signal and image processing, and physical sciences. For example, in electroencephalogram signals, sharp cusps exhibit the accelerations and decelerations in the beating of the hearts. Many practical problems like this involve functions which have jumps and sharp cusps. The recently developed theory of wavelets has drawn much attention from both math- ematicians, statisticians and engineers. In the seminal work of Donoho (1993), Donoho & Johnstone (1994, 1995a,b) and Donoho, Johnstone et al. (1995), orthonormal bases of compactly supported wavelets have been used to estimate functions. The theory of wavelets permits decomposition of functions into localised oscillating components. This is an ideal tool to study localised changes such as jumps and sharp cusps in one dimension as well as several dimensions. Unlike traditional smoothing methods based on a fixed spatial scale, the wavelet method is a multiresolution approach and has local adaptivity. In this
332 citations
TL;DR: In this paper, the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions are studied and it is shown that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces."
Abstract: We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in L 2 ([0,1)) our results cover other metrics like Skorokhod metric on the space of cadlag functions and uniform metrics on C([0, 1]). We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) Holder continuous functions of order 0 < α ≤ 1 and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.
181 citations
TL;DR: In this article, the authors consider a regression model in which the mean function may have a discontinuity at an unknown point and propose an estimate of the location of the discontinuity based on one-side nonparametric regression estimates of the mean functions.
Abstract: We consider a regression model in which the mean function may have a discontinuity at an unknown point. We propose an estimate of the location of the discontinuity based on one-side nonparametric regression estimates of the mean function. The change point estimate is shown to converge in probability at rate 0(n-1) and to have the same asymptotic distribution as maximum likelihood estimates considered by other authors under parametric regression models. Confidence regions for the location and size of the change are also discussed.
176 citations
TL;DR: An approach to this problem called “sigma filtering” is discussed and an improvement based on running M estimation is proposed, which has a niche between standard filtering approaches and Bayes–Markov random-field analysis.
Abstract: Classical smoothers have limited usefulness in image processing, because sharp “edges” tend to be blurred. There is a literature on edge-preserving smoothers, but these all require moderately large “smooth stretches.” Here we discuss an approach to this problem called “sigma filtering” and propose an improvement based on running M estimation. Both computational and theoretical aspects are developed. For image processing, the methods have a niche between standard filtering approaches and Bayes–Markov random-field analysis.
163 citations
TL;DR: In this paper, kernel-type estimators of the locations of jump points and the corresponding sizes of jump values of the regression function are proposed and analyzed with almost sure results and limiting distributions.
Abstract: In the fixed-design nonparametric regression model, kernel-type estimators of the locations of jump points and the corresponding sizes of jump values of the regression function are proposed. These kernel-type estimators are analyzed with almost sure results and limiting distributions. Using the limiting distributions, we are able to test the number of jump points and give asymptotic confidence intervals for the sizes of jump values of the regression function. Simulation studies demonstrate that the asymptotic results hold for reasonable sample sizes.
155 citations
References
More filters
TL;DR: In this article, a class of strongly consistent estimators for the change-point (0, 1) was proposed, which require no knowledge of the functional forms or parametric families of the variables.
Abstract: Consider a sequence of independent random variables $\{X_i: 1 \leq i \leq n\}$ having cdf $F$ for $i \leq \theta n$ and cdf $G$ otherwise. A class of strongly consistent estimators for the change-point $\theta \in (0, 1)$ is proposed. The estimators require no knowledge of the functional forms or parametric families of $F$ and $G$. Furthermore, $F$ and $G$ need not differ in their means (or other measure of location). The only requirement is that $F$ and $G$ differ on a set of positive probability. The proof of consistency provides rates of convergence and bounds on the error probability for the estimators. The estimators are applied to two well-known data sets, in both cases yielding results in close agreement with previous parametric analyses. A simulation study is conducted, showing that the estimators perform well even when $F$ and $G$ share the same mean, variance and skewness.
235 citations
"Edge-preserving and peak-preserving..." refers background in this paper
...For recent discussion of these two branches of research, see Carlstein (1988) and Lee (1990)....
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TL;DR: In this paper, kernel-type estimators of the locations of jump points and the corresponding sizes of jump values of the regression function are proposed and analyzed with almost sure results and limiting distributions.
Abstract: In the fixed-design nonparametric regression model, kernel-type estimators of the locations of jump points and the corresponding sizes of jump values of the regression function are proposed. These kernel-type estimators are analyzed with almost sure results and limiting distributions. Using the limiting distributions, we are able to test the number of jump points and give asymptotic confidence intervals for the sizes of jump values of the regression function. Simulation studies demonstrate that the asymptotic results hold for reasonable sample sizes.
155 citations
TL;DR: A family of smoothing algorithms that can produce discontinuous output are introduced that can be used for smoothing with edge detection in image processing and applied to sea surface temperature data where the discontinuities arise from changes in ocean currents.
Abstract: We introduce a family of smoothing algorithms that can produce discontinuous output. Unlike most commonly used smoothers, that tend to blur discontinuities in the data, this smoother can be used for smoothing with edge detection. We cite examples of other approaches to (two-dimensional) smoothing with edge detection in image processing, and apply our one-dimensional smoother to sea surface temperature data where the discontinuities arise from changes in ocean currents.
102 citations
TL;DR: The general principles of detection, classification, and measurement of discontinuities are studied and a statistical method is proposed for the approximate pattern matching.
Abstract: The general principles of detection, classification, and measurement of discontinuities are studied. The following issues are discussed: detecting the location of discontinuities; classifying discontinuities by their degrees; measuring the size of discontinuities; and coping with the random noise and designing optimal discontinuity detectors. An algorithm is proposed for discontinuity detection from an input signal S. For degree k discontinuity detection and measurement, a detector (P, Phi ) is used, where P is the pattern and Phi is the corresponding filter. If there is a degree k discontinuity at location t/sub 0/, then in the filter response there is a scaled pattern alpha P at t/sub 0/, where alpha is the size of the discontinuity. This reduces the problem to searching for the scaled pattern in the filter response. A statistical method is proposed for the approximate pattern matching. To cope with the random noise, a study is made of optimal detectors, which minimize the effects of noise. >
64 citations
TL;DR: In this paper, a method based on partial spline models is developed for including specified discontinuities in otherwise smooth two-and three-dimensional objective analyses, and a prototype two-dimensional analysis based on simulated radiosonde and tropopause height data is shown.
Abstract: A new method, based on partial spline models, is developed for including specified discontinuities in otherwise smooth two- and three-dimensional objective analyses. The method is appropriate for including tropopause height information in two- and three-dimensinal temperature analyses, using the O'Sullivan-Wahba physical variational method for analysis of satellite radiance data, and may in principle be used in a combined variational analysis of observed, forecast, and climate information. A numerical method for its implementation is described and a prototype two-dimensional analysis based on simulated radiosonde and tropopause height data is shown. The method may also be appropriate for other geophysical problems, such as modeling the ocean thermocline, fronts, discontinuities, etc.
49 citations