## Wavelets: The Key to Intermittent Information?In recent years there has been an explosion of interest in wavelets, in a wide range of fields in science and engineering and beyond. This book brings together contributions from researchers from disparate fields, both in order to demonstrate to a wide readership the current breadth of work in wavelets, and to encourage cross-fertilization of ideas. It demonstrates the genuinely interdisplinary nature of wavelet research and applications. Particular areas covered include turbulence, statistics, time series analysis, signal and image processing, the physiology of vision, astronomy, economics and acoustics. Some of the work uses standard wavelet approaches and in other cases new methodology is developed. The papers were originally presented at a Royal Society Discussion Meeting, to a large and enthusiastic audience of specialists and non-specialists. |

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### Contents

Wavelets in timeseries analysis | 1 |

The onedimensional case | 7 |

Applications | 16 |

Experimental evidence for lognormal statistics | 25 |

Experimental evidence for a nonscaleinvariant lognormal | 32 |

The multifractal description of intermittency revisited with wavelets | 38 |

Conclusions and perspectives | 44 |

Intermittency and eddy capacity | 50 |

Dealing with irregular data | 77 |

Acknowledgements | 85 |

Optimality in the whitenoise model | 99 |

The continuous ridgelet transform | 115 |

Wavelets vision and the statistics of natural scenes | 149 |

Image processing with complex wavelets | 165 |

Approximation and compression of piecewise smooth functions | 199 |

2 The contribution of wavelets to the analysis of economic | 221 |

Application to high resolution turbulent velocity signals | 63 |

References | 70 |

3 Harmonic wavelets in vibrations and acoustics | 237 |

### Common terms and phrases

algorithm Allan variance amplitude approximation Arneodo asymptotic autocovariance behaviour Besov spaces cascade process Castaing coding computed correlation functions corresponding curve datasets Daubechies defined dimensions discrete wavelet transform distribution divided differences Donoho eddy capacity equation estimation example experimental filters Fourier transform fractal frequency Gaussian Haar wavelet inertial range intermittency interpolating irregular grid Johnstone kernel l.et l.ett linear log-Poisson lognormal mesh method minimax multiresolution multiresolution analysis natural scenes NDWT noise non-isolated non-parametric obtained optimal orthogonal orthonormal orthonormal basis parameter PDFs Phys plotted points polynomial properties Radon transform receptive fields reconstruction regression representation Reynolds number ridge functions ridgelet sample scale self-similar sequence shows Silverman singularities smoothness space space-scale sparse sparse code sparsity spatial spectral spectrum stationary statistical structure subdivision scheme theory thresholding time-series tsee figure two-dimensional values Vassilicos vectors visual wavelet analysis wavelet basis wavelet coefficients wavelet transform WTMM zero zero-crossings