A scale-space approach with wavelets to singularity estimation

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In order to identify the singularities of the unknown signal, we introduce a new tool, "the structural intensity", that computes the "density" of the location of the modulus maxima of a wavelet representation along various scales. This approach is shown to be an effective technique for detecting the significant singularities of a signal corrupted by noise and for removing spurious estimates. The asymptotic properties of the resulting estimators are studied and illustrated by simulations. An application to a real data set is also proposed.

[1]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[2]  Tapabrata Maiti.  Recent Advances and Trends in Nonparametric Statistics , 2005 .

[3]  Leo Breiman,et al.  Bagging Predictors , 1996, Machine Learning.

[4]  D. Donoho,et al.  Translation-Invariant De-Noising , 1995 .

[5]  T. Gasser,et al.  Synchronizing sample curves nonparametrically , 1999 .

[6]  Jean-Michel Poggi,et al.  Décomposition par ondelettes et méthodes comparatives : étude d'une courbe de charge électrique , 1994 .

[7]  I. Johnstone,et al.  ASYMPTOTIC MINIMAXITY OF WAVELET ESTIMATORS WITH SAMPLED DATA , 1999 .

[8]  Mathematical Tools for Multifractal Signal Processing , 1999 .

[9]  Alan L. Yuille,et al.  Scaling Theorems for Zero Crossings , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  Serge Dubuc,et al.  Spline functions and the theory of wavelets , 1999 .

[11]  E. Bacry,et al.  The Multifractal Formalism Revisited with Wavelets , 1994 .

[12]  I. Johnstone,et al.  Wavelet Shrinkage: Asymptopia? , 1995 .

[13]  E. Bacry,et al.  Singularity spectrum of fractal signals from wavelet analysis: Exact results , 1993 .

[14]  David J. Marchette,et al.  The Bumpy Road to the Mode Forest , 1998 .

[15]  Nathan Intrator,et al.  Bootstrapping with Noise: An Effective Regularization Technique , 1996, Connect. Sci..

[16]  James Stephen Marron,et al.  Mode testing via the excess mass estimate , 2001 .

[17]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[18]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[19]  Emmanuel Bacry,et al.  Singularity spectrum of multifractal functions involving oscillating singularities , 1998 .

[20]  Anestis Antoniadis,et al.  Detecting Abrupt Changes by Wavelet Methods , 2002 .

[21]  T. Gasser,et al.  Alignment of curves by dynamic time warping , 1997 .

[22]  Emmanuel Bacry,et al.  Oscillating singularities on cantor sets: A grand-canonical multifractal formalism , 1997 .

[23]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[24]  T. Gasser,et al.  Searching for Structure in Curve Samples , 1995 .

[25]  Robert Hummel,et al.  Reconstructions from zero crossings in scale space , 1989, IEEE Trans. Acoust. Speech Signal Process..

[26]  L. Brown,et al.  Asymptotic equivalence of nonparametric regression and white noise , 1996 .

[27]  Thomas G. Dietterich.  What is machine learning? , 2020, Archives of Disease in Childhood.

[28]  Michael Unser,et al.  On the asymptotic convergence of B-spline wavelets to Gabor functions , 1992, IEEE Trans. Inf. Theory.

[29]  Emmanuel Bacry,et al.  THE THERMODYNAMICS OF FRACTALS REVISITED WITH WAVELETS , 1995 .

[30]  D. W. Scott,et al.  The Mode Tree: A Tool for Visualization of Nonparametric Density Features , 1993 .

[31]  T. Gasser,et al.  Statistical Tools to Analyze Data Representing a Sample of Curves , 1992 .

[32]  I. Johnstone,et al.  Minimax estimation via wavelet shrinkage , 1998 .

[33]  Yazhen Wang.  Jump and sharp cusp detection by wavelets , 1995 .

[34]  Stéphane Mallat,et al.  Characterization of Signals from Multiscale Edges , 2011, IEEE Trans. Pattern Anal. Mach. Intell..

[35]  Marc Raimondo,et al.  Minimax estimation of sharp change points , 1998 .

[36]  Yu-Ping Wang,et al.  Scale-Space Derived From B-Splines , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[37]  Stéphane Mallat,et al.  Zero-crossings of a wavelet transform , 1991, IEEE Trans. Inf. Theory.

[38]  Anestis Antoniadis,et al.  Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study , 2001 .

[39]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[40]  D. Picard,et al.  Adaptive confidence interval for pointwise curve estimation , 2000 .

[41]  Hsiuying Wang.  Brown's paradox in the estimated confidence approach , 1999 .

[42]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[43]  Stéphane Mallat,et al.  Singularity detection and processing with wavelets , 1992, IEEE Trans. Inf. Theory.

[44]  Tony Lindeberg,et al.  Scale-space theory , 2001 .

[45]  J. Marron,et al.  SCALE SPACE VIEW OF CURVE ESTIMATION , 2000 .

[46]  J. Marron,et al.  SiZer for Exploration of Structures in Curves , 1999 .

[47]  S. Mallat.  VI – Wavelet zoom , 1999 .