0.4 Time-frequency dictionaries  (Page 2/3)

It can be interpreted as a Fourier transform of $f$ at the frequency ξ , localized by the window $g(t-u)$ in the neighborhood of u . This windowed Fourier transform is highly redundant and represents one-dimensional signalsby a time-frequency image in $(u,\xi )$ . It is thus necessary to understand how to select many fewertime-frequency coefficients that represent the signal efficiently.

When listening to music, we perceive sounds that have a frequency that varies in time. Chapter 4 showsthat a spectral line of $f$ creates high-amplitude windowed Fourier coefficients $Sf(u,\xi )$ at frequencies $\xi \left(u\right)$ that depend on time u . These spectral components are detected and characterized byridge points, which are local maxima in this time-frequency plane. Ridge points define a time-frequency approximation support λ of $f$ with a geometry that depends on the time-frequency evolution of the signal spectral components. Modifying thesound duration or audio transpositions are implemented by modifying the geometry of the ridge support in time frequency.

A windowed Fourier transform decomposes signals over waveforms that have the same time and frequency resolution. It is thus effectiveas long as the signal does not include structures having different time-frequency resolutions, some being very localizedin time and others very localized in frequency.  Wavelets address this issue by changing the time and frequency resolution.

Continuous wavelet transform

In reflection seismology, Morlet knew that the waveforms sent underground have a duration that is too longat high frequencies to separate the returns of fine, closely spaced geophysical layers. Such waveforms are called wavelets in geophysics. Instead of emitting pulses of equal duration,he thought of sending shorter waveforms at high frequencies. These waveforms were obtained by scaling the motherwavelet, hence the name of this transform. Although Grossmann was working in theoretical physics, he recognized in Morlet's approach some ideasthat were close to his own work on coherent quantum states.

Nearly forty years after Gabor, Morlet and Grossmann reactivated a fundamentalcollaboration between theoretical physics and signal processing, whichled to the formalization of the continuous wavelet transform(GrossmannM:84). These ideas were not totally new to mathematicians working in harmonic analysis, or to computer visionresearchers studying multiscale image processing. It was thus only the beginning of a rapid catalysis that brought togetherscientists with very different backgrounds.

A wavelet dictionary is constructed from a mother wavelet ψ of zero average

which is dilated with a scale parameter s , and translated by u :

The continuous wavelet transform of $f$ at any scale s and position u is the projection of $f$ on the corresponding wavelet atom:

It represents one-dimensional signals by highly redundant time-scale images in $(u,s)$ .

Varying time-frequency resolution

As opposed to windowed Fourier atoms, wavelets have a time-frequency resolution that changes.The wavelet ${\psi }_{u,s}$ has a time support centered at u and proportional to s . Let us choose a wavelet ψ whose Fourier transform $\widehat{\psi }\left(\omega \right)$ is nonzero in a positive frequency interval centered at η . The Fourier transform ${\widehat{\psi }}_{u,s}\left(\omega \right)$ is dilated by $1/s$ and thus is localized in a positive frequencyinterval centered at $\xi =\eta /s$ ; its size is scaled by $1/s$ . In the time-frequency plane, the Heisenberg boxof a wavelet atom ${\psi }_{u,s}$ is therefore a rectangle centered at $(u,\eta /s)$ , with time and frequency widths, respectively,proportional to s and $1/s$ . When s varies, the time and frequency width of this time-frequency resolution cell changes, butits area remains constant, as illustrated by [link] .