Everything you always wanted to know about wavelets and did not
dare to ask: application of wavelets to multiscale analysis and
non-linear estimation
dr. Felix J. Herrmann
February 6, 2002
Abstract
In this 4-week 16-hour special course we will review the application of wavelets and related
transforms to geophysical data analysis and estimation. Extensive use will be made of Mal-
lat’s “A Wavelet tour of signal processing”. Topics include: fractal analysis by the continuous
wavelet transform; construction of discrete wavelets from (fractional) splines (including a lec-
ture by Michael Unser on Thursday March 7 at 11:00); design of non-linear mini-max estimators
by thresholding. Applications include: scaling analysis sedimentary records and seismic data;
edge-preserving denoising and deconvolution; hands-on examples using Donoho’s WaveLab (in
matlab). For further information check www-erl.mit.edu/ felix/Teaching/Wavelets/MIT
Preliminary overview
Topic Overview: why wavelets?
• from complex exponentials to splines Vetterli [2001], Unser [1999b,a]
• filter banks Strang and Nguyen [1997]
• multiresolution analysis Vetterli [2001], Mallat [1997]
• from linear to non-linear estimation Vetterli [2001], Mallat [1997]
• estimation by thresholding Vetterli [2001], Mallat [1997]
Topic Multiscale analysis by the Continuous Wavelet Transform
• Continuous Wavelet Transform Mallat [1997], Herrmann [1997]
• Modulus Maxima Method Mallat [1997], Herrmann [1997]
• Application to well-log (sedimentary record) and turbulence analysis
Topic Multifractal Analysis
• Hausdorff dimensions Mallat [1997], Herrmann [1997]
• Partition functions Mallat [1997], Herrmann [1997]
• Singularity Spectra Mallat [1997], Herrmann [1997]
Topic Multifractal analysis by the Continuous Wavelet Tranform
1
• Modulus maxima partitioning Mallat [1997], Herrmann [1997]
• Partition function by Modulus Maxima Mallat [1997], Herrmann [1997]
• Application to well-log analysis Mallat [1997], Herrmann [1997]
• Some remarks on link between scaling and regularity (Besov