Positive Wavelet Representation of Fractal Signals and Images
With appropriate choice of an analysing wavelet in the form of a positive pulse, information concerning the structure of a signal is concentrated economically in the local maxima and minima of the function of two variables, position and scale, given by the wavelet transformation. This information is extracted by a process of correlation detection, in which the analysing wavelet is regarded as a multiple-scale matched filter. Identification of local extrema corresponds to the detection of signal wavelets. The ensemble of such signal wavelets provides a discrete feature-based representation of the given function. In contrast to the standard method of reconstruction by means of the linear inverse wavelet transform, an alternative method of reconstruction from the feature-based representation is demonstrated. Applications of the positive wavelet representation to an elucidation of the fractal structure of measured and simulated turbulence are illustrated. The method extends in a straightforward manner to two dimensions. An application which demonstrates the multiple-scale feature-based representation of a two-dimensional image is presented.
Unable to display preview. Download preview PDF.
- [Earw88]Earwicker, P.G., Correlation detection in self-similar noise, Royal Aerospace Establishment, Tech. Report 88032, 1988.Google Scholar
- [Earw90]Earwicker, P.G., and Jones, J.G., Correlation detection using multiple-scale filters and self-similar noise models, in Mathematics in Signal Processing II, McWhirter, J.G., Ed., Oxford, UK: Clarendon Press, 1990.Google Scholar
- [Fost89]Foster, G.W., and Jones, J.G., Analysis of atmospheric turbulence measurements by spectral and discrete gust methods, Aero. Jour. Royal Aeronautical Society, Vol. 93, pp. 162–176, 1989.Google Scholar
- [Gros85]Grossman, A., and Morlet, J., Decomposition of functions into wavelets of constant shape, and related transforms, in Mathematics and Physics, Lectures on Recent Results, Vol. 1, Streit, L., Ed., Singapore: World Scientific, 1985.Google Scholar
- [Jone90]Jones, J.G., Earwicker, P.G., and Foster, G.W., Multiple-scale correlation detection, wavelet transforms and multifractal turbulence, IMA Conf. on Wavelets, Fractals and Fourier Transforms, Cambridge, UK, December 1990.Google Scholar
- [Jone92]Jones, J.G., Positive wavelet representation, Defence Research Agency, Farnbor-ough, England, Tech. Memo. TM MTC28, 1992.Google Scholar