IDL Wavelet Toolkit User's Guide

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54 Chapter 3: Theory and ExamplesMultiresolution AnalysisBackgroundThe wavelet transform can be thought of as a band-pass filter, where the location andwidth in Fourier space depends on the wavelet scale. Larger scales imply a lowerfrequency and small bandwidth.In computing the wavelet transform, you change from small scales to larger scales. Ateach stage you can stop and compute the inverse wavelet transform using theremaining coefficients, while setting the small-scale coefficients to zero. You canthen build up a series of smooth (or low-passed), detailed (or band-passed), or rough(high-passed) versions of your original data.MethodDetails on computing the multiresolution analysis can be found in Lindsay et al.(1996).ExampleUse the “Mantle convection” dataset that is included in the Wavelet sample file. Thisdataset contains an image of convection within the Earth’s mantle.Try the following steps:1. Select the Convection dataset and start up the Multiresolution Analysis viewerusing either the Visualize Menu or the Toolbar button.2. As you progress from top to bottom the wavelet scale increases in powers oftwo. At the smallest scale most of the image is still in the Smooth image.Notice that the Rough image contains only the edges or discontinuities whichthe small scales can pick out.3. Change to the Haar wavelet and observe the different structure of the images.Multiresolution AnalysisIDL Wavelet Toolkit
Chapter 3: Theory and Examples 55BibliographyDaubechies, I., 1992: Ten Lectures on Wavelets. Society for Industrial and AppliedMathematics, 357 pp.Donoho, D. L. and I. M. Johnstone, 1994: Ideal spatial adaptation by waveletshrinkage. Biometrika, 81, 425–455.Hubbard, B. B., 1998: The World According to Wavelets, 2nd ed. A. K. Peters,Wellesley, Mass., 331 pp.Lindsay, R. W., D. B. Percival, and D. A. Rothrock, 1996: The discrete wavelettransform and the scale analysis of the surface properties of sea ice. IEEE Trans.Geosci. Remote Sens., 34, 771–787.Mallat, S., 1989: Multiresolution approximation and wavelets. Trans. Amer. Math.Soc., 315, 69–88.Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992: NumericalRecipes in C: The Art of Scientific Computing, 2nd ed. Cambridge University Press,994 pp.Torrence, C., and G. P. Compo, 1998: A practical guide to wavelet analysis. Bull.Amer. Meteor. Soc., 79, 61–78.IDL Wavelet ToolkitBibliography
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