# A scale-space approach with wavelets to singularity estimation

ESAIM: Probability and Statistics (2010)

- Volume: 9, page 143-164
- ISSN: 1292-8100

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topBigot, Jérémie. "A scale-space approach with wavelets to singularity estimation." ESAIM: Probability and Statistics 9 (2010): 143-164. <http://eudml.org/doc/104327>.

@article{Bigot2010,

abstract = {
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.
},

author = {Bigot, Jérémie},

journal = {ESAIM: Probability and Statistics},

keywords = {Lipschitz singularity; continuous wavelet transform; scale-space representation; zero-crossings; wavelet maxima; feature extraction; non parametric estimation; bagging; landmark-based matching.; landmark-based matching},

language = {eng},

month = {3},

pages = {143-164},

publisher = {EDP Sciences},

title = {A scale-space approach with wavelets to singularity estimation},

url = {http://eudml.org/doc/104327},

volume = {9},

year = {2010},

}

TY - JOUR

AU - Bigot, Jérémie

TI - A scale-space approach with wavelets to singularity estimation

JO - ESAIM: Probability and Statistics

DA - 2010/3//

PB - EDP Sciences

VL - 9

SP - 143

EP - 164

AB -
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.

LA - eng

KW - Lipschitz singularity; continuous wavelet transform; scale-space representation; zero-crossings; wavelet maxima; feature extraction; non parametric estimation; bagging; landmark-based matching.; landmark-based matching

UR - http://eudml.org/doc/104327

ER -

## References

top- A. Antoniadis, J. Bigot and T. Sapatinas, Wavelet estimators in nonparametric regression: a comparative simulation study. J. Statist. Software6 (2001) 1–83.
- A. Antoniadis and I. Gijbels, Detecting abrupt changes by wavelet methods. J. Nonparam. Statist14 (2001) 7–29. Zbl1017.62033
- A. Arneodo, E. Bacry, S. Jaffard and J.F. Muzy, Oscillating singularities and fractal functions, in Spline functions and the theory of wavelets (Montreal, PQ, 1996), Amer. Math. Soc., Providence, RI. CRM Proc. Lect. Notes18 (1999) 315–329.. Zbl1003.28007
- A. Arneodo, E. Bacry, S. Jaffard and J.F. Muzy, Singularity spectrum of multifractal functions involving oscillating singularities. J. Fourier Anal. Appl.4 (1998) 159–174. Zbl0914.28005
- A. Arneodo, E. Bacry, S. Jaffard and J.F. Muzy, Oscillating singularities on Cantor sets: a grand-canonical multifractal formalism. J. Statist. Phys.87 (1997) 179–209. Zbl0917.28007
- A. Arneodo, E. Bacry and J.F. Muzy, The thermodynamics of fractals revisited with wavelets. Physica A213 (1995) 232–275. Zbl1064.80002
- E. Bacry, J.F. Muzy and A. Arneodo, Singularity spectrum of fractal signals: exact results. J. Statist. Phys.70 (1993) 635–674. Zbl0943.37500
- J. Bigot, Automatic landmark registration of 1D curves, in Recent advances and trends in nonparametric statistics, M. Akritas and D.N. Politis Eds., Elsevier (2003) 479–496.
- J. Bigot, Landmark-based registration of 1D curves and functional analysis of variance with wavelets. Technical Report TR0333, IAP (Interuniversity Attraction Pole network) (2003).
- L. Breiman, Bagging Predictors. Machine Learning24 (1996) 123–140.
- L.D. Brown and M.G. Lo, Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist.3 (1996) 2384–2398. Zbl0867.62022
- P. Chaudhuri and J.S.Marron, SiZer for exploration of structures in curves. J. Am. Statist. Ass.94 (1999) 807–823. Zbl1072.62556
- P. Chaudhuri and J.S. Marron Scale space view of curve estimation. Ann. Statist.28 (2000) 408–428. Zbl1106.62318
- R.R. Coifman and D.L. Donoho, Translation-invariant de-noising, in Wavelets and Statistics, A. Antoniadis and G. Oppenheim, Eds., New York: Springer-Verlag. Lect. Notes Statist.103 (1995) 125–150. Zbl0866.94008
- I. Daubechies, Ten Lectures on Wavelets. Philadelphia, SIAM (1992). Zbl0776.42018
- D.L. Donoho and I.M. Johnstone, Ideal spatial adaptation by wavelet shrinkage. Biometrika81 (1994) 425–455. Zbl0815.62019
- D.L. Donoho and I.M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage. J. Am. Statist. Ass.90 (1995) 1200–1224. Zbl0869.62024
- D.L. Donoho and I.M. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Statist.26 (1998) 879–921. Zbl0935.62041
- D.L. Donoho and I.M. Johnstone, Asymptotic minimality of wavelet estimators with sampled data. Stat. Sinica9 (1999) 1–32. Zbl1065.62518
- D.L. Donoho, I.M. Johnstone, G. Kerkyacharian and D. Picard, Wavelet shrinkage: Asymptotia? (with discussion). J. R. Statist. Soc. B57 (1995) 301–337. Zbl0827.62035
- N.I. Fisher and J.S. Marron, Mode testing via the excess mass estimate. Biometrika88 (2001) 499–517. Zbl0985.62034
- T. Gasser and A. Kneip, Searching for Structure in Curve Samples. J. Am. Statist. Ass.90 (1995) 1179–1188. Zbl0864.62019
- B. Hummel and R. Moniot, Reconstruction from zero-crossings in scale-space. IEEE Trans. Acoust., Speech, and Signal Proc.37 (1989) 2111–2130.
- S. Jaffard, Mathematical Tools for Multifractal Signal Processing. Signal Processing for Multimedia, J.S Byrnes Ed., IOS Press (1999) 111–128. Zbl0991.94012
- A. Kneip and T. Gasser, Statistical tools to analyze data representing a sample of curves. Ann. Statist.20 (1992) 1266–1305. Zbl0785.62042
- M.R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag (1983). Zbl0518.60021
- T. Lindeberg, Scale Space Theory in Computer Vision. Kluwer, Boston (1994). Zbl0812.68040
- S. Mallat, Zero-Crossings of a Wavelet Transform. IEEE Trans. Inform. Theory37 (1991) 1019–1033.
- S. Mallat, A Wavelet Tour of Signal Processing. Academic Press (1998). Zbl0937.94001
- S. Mallat and W.L. Hwang, Singularity Detection and Processing with Wavelets. IEEE Trans. Inform. Theory38 (1992) 617–643. Zbl0745.93073
- S. Mallat and S. Zhong, Characterization of Signals From Multiscale Egde. IEEE Trans. Pattern Anal. Machine Intelligence14 (1992) 710–732.
- S. Mallat and S. Zhong, Wavelet Transformation Maxima and Multiscale Edges, in Wavelets: A Tutorial in Theory and Applications, C.K. Chui Ed. Boston, Academic Press (1992) 66–104. Zbl0804.68158
- S. Mallat and S. Zhong, Wavelet Maxima Representation, in Wavelets and Applications, Y. Meyer Ed. New York, Springer-Verlag (1992) 207–284. Zbl0804.68157
- M.C. Minnotte and D.W. Scott, The mode tree: a tool for visualization of nonparametric density features. J. Computat. Graph. Statist.2 (1993) 51–68.
- M.C. Minnotte, D.J. Marchette and E.J. Wegman, The bumpy road to the mode forest. J. Comput. Graph. Statist.7 (1998) 239–251.
- M. Misiti, Y. Misiti, G. Oppenheim and J.-M. Poggi, Décomposition en ondelettes et méthodes comparatives : étude d'une courbe de charge éléctrique. Revue de Statistique Appliquée17 (1994) 57–77.
- J.F. Muzy, E. Bacry and A. Arneodo, The multifractal formalism revisited with wavelets. Int. J. Bif. Chaos4 (1994) 245–302. Zbl0807.58032
- D. Picard and K. Tribouley, Adaptive confidence interval for pointwise curve estimation. Ann. Statist.28 (2000) 298–335. Zbl1106.62331
- M. Raimondo, Minimax estimation of sharp change points. Ann. Statist.26 (1998) 1379–1397. Zbl0929.62039
- J.O. Ramsay and X. Li, Curve registration. J. R. Statist. Soc. B60 (1998) 351–363. Zbl0909.62033
- J.O. Ramsay and B.W. Silverman, Functional data analysis. New York, Springer Verlag (1997). Zbl0882.62002
- Y. Raviv and N. Intrator, Bootstrapping with Noise: An Effective Regularization Technique. Connection Science, Special issue on Combining Estimator8 (1996) 356–372.
- M. Unser, A. Aldroubi and M. Eden, On the Asymptotic Convergence of B-Spline Wavelets to Gabor Functions. IEEE Trans. Inform. Theory38 (1992) 864–872. Zbl0757.41022
- Y. Wang, Jump and Sharp Cusp Detection by Wavelets. Biometrica82 (1995) 385–397. Zbl0824.62031
- K. Wang and T. Gasser, Alignment of curves by dynamic time warping. Ann. Statist.25 (1997) 1251–1276. Zbl0898.62051
- K. Wang and T. Gasser, Synchronizing sample curves nonparametrically. Ann. Statist.27 (1999) 439–460. Zbl0942.62043
- Y.P. Wang and S.L. Lee, Scale-Space Derived From B-Splines. IEEE Trans. on Pattern Analysis and Machine Intelligence20 (1998) 1040–1055.
- L. Younes, Deformations, Warping and Object Comparison. Tutorial (2000) . URIhttp://www.cmla.ens-cachan.fr/~younes
- A.L. Yuille and T.A. Poggio, Scaling Theorems for Zero Crossings. IEEE Trans. Pattern Anal. Machine Intelligence8 (1986) 15–25. Zbl0575.94001

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