Watch the lectures here. Detection of gravitational waves and time-frequency wavelets Abstract: After describing the now famous discovery of gravitational waves the focus will be on time-frequency analysis.
Complex physical phenomena, signals and images involve structures of very different scales. A wavelet transform operates as a zoom, which simplifies the analysis by separating local variations at different scales. Yves Meyer found wavelet orthonormal bases having better properties than Fourier bases to characterize local properties of functions, physical measurements and signals.
This discovery created a major scientific catalysis, which regrouped physicists, engineers and mathematicians, leading to a coherent theory of multiscale wavelet decompositions with a multitude of applications. This lecture will explain the construction of Meyer wavelet bases and their generalization with fast computations. We shall follow the path of this human adventure, with ideas independently developed by scientists working in quantum physics, geophysics, image and signal processing but also neurophysiology of perception.
The synthesis in the 's provoked by Yves Meyer's work was an encounter between applications and a pure harmonic analysis research program, initiated by Littlewood-Paley in the 's. Professor Ingrid Daubechies, Duke University: Yves Meyer's surprising construction of orthonormal bases consisting of dilates and translates of a single smooth function was followed soon after by the development of the Multiresolution Analysis framework in collaboration with Stephane Mallat.
As already shown in the presentation by Stephane Mallat, this development was rooted in and used insights from a variety of fields -- ranging from pure harmonic analysis to statistics, quantum physics, geophysics and computer vision.
The lecture will discuss some of those diverse roots in more detail, and also show how the new wavelet synthesis, sparked by Yves Meyer's seminal work, led to further progress in all those fields as well as others. Wavelets, sparsity and its consequences Abstract: Soon after they were introduced, it was realized that wavelets offered representations of signals and images of interest that are far more sparse than those offered by more classical representations; for instance, Fourier series.
Owing to their increased spatial localization at finer scales, wavelets prove to be better adapted to represent signals with discontinuities or transient phenomena because only a few wavelets actually interact with those discontinuities. It turns out that sparsity has extremely important consequences and this lecture will briefly discuss three vignettes.
First, enhanced sparsity yields the same quality of approximation with fewer terms, a feat which has implications for lossy image compression since it roughly says that fewer bits are needed to achieve the same distortion. Second, enhanced sparsity yields superior statistical accuracy since there are fewer degrees of freedom or parameters to estimate.
This gives scientists better methods to tease apart the signal from the noise. Third, enhanced sparsity has important consequences for data acquisition itself: