Reading and Understanding Continuous Wavelet Transforms

Authors: Grossmann A., Kronland-Martinet R., Morlet J.
Publication Date: May 1989
Journal: Wavelets - Time-Frequency Methods and Phase Space (IPTI, Springer-Verlag Berlin Heidelberg, pp 2-20, 1989)

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One of the aims of wavelet transforms is to provide an easily interpretable visual representation of signals. This is a prerequisite for applications such as selective modifications of signals or pattern recognition.

This paper contains some background material on continuous wavelet transforms and a description of the representation methods that have gradually evolved in our work. A related topic, also discussed here,is the influence of the choice of the wavelet in the interpretation of wavelet transforms. roughly speaking, there are many qualitative features (in particularly concerning the phase) which are independant of the choice of analyzing wavelet; however, in some situations (such as detection of “musical chords”) an appropriate choice of wavelet is essential.

We also briefly discuss the finite interpolation problem for wavelet transforms with respect to a given analyzing wavelet, and give some details about analyzing wavelets of gaussian type.