SFSASDA

Spline-like function spaces with applications to scattered data approximations

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 Obiettivo del progetto (Objective)

'We will develop mathematical methods for scattered data approximation that can be used to implement efficient algorithms in signal processing and related areas. So far, the problem of reconstructing an unknown function from a given discrete set of sampling points has been studied in shift invariant spaces, i.e. spaces generated by translates of a function (or finitely many functions). We propose to consider the problem in spaces generated by translates of compressed copies of a function (or finitely many functions) along irregularly spaced points. We are convinced that with our model we will obtain better approximation properties, since it takes into account the structure of the sampling data. The problem will be first studied on the real line, then in the plane and finally on the sphere, where the lack of regular grids requires certainly some modifications. Both the functional analytic side and the computational aspects of the problem will be covered. The candidate has experience in the field of the proposed topic, since her research area is time-frequency analysis, frame theory, sampling theory, shift invariant spaces, etc., which constitute the background of this project. She was member of several research projects related to the topic, her master and Ph.D thesis are connected to the proposed research. The project will be carried out at the Numerical Harmonic Analysis Group (NuHAG), University of Vienna, whose key scientists are Prof. H.G. Feichtinger and Prof. K. Gröchenig. The proposed research center has a leading position in the area of sampling theory, time-frequency analysis, frame theory and wavelet analysis, which makes it the proper environment for the proposed project. The experience that NuHAG has in the mentioned areas and also in training researchers will be helpful in the developing of the professional skills of the proposer. The MC Exc. Grant EUCETIFA (2005-2009) and the cooperation with other PostDocs at NuHAG will support the project.'

Introduzione (Teaser)

We live in a world of mixed signals - from sound to light - that often don't make any sense. Through advanced mathematics, we can now analyse and exploit them to the benefit of advancing technology.

Descrizione progetto (Article)

The world is bombarding us with different intermittent signals from everywhere. These can be audio signals, radio signals, light signals, image signals, seismic signals or even signals defined by space or time. Quantifying these signals and studying them helps us understand the world we live in and has many applications. But doing so requires sophisticated mathematical calculations and algorithms.

Although the average person may not know much about this discipline, behind the scenes there are scientists and mathematicians trying to make sense of the 'signal-filled' world we live in. The EU has fully funded the SFSASDA project to help understand these signals, creating more accurate models and giving more meaning to signals.

SFSASDA used a complex way to study these signals - a method called scattered data approximation. The method has been used in the past for reconstructing uneven surfaces, modelling terrain, defining fluid interaction and estimating parameters, among other applications. The project's challenge is to make it work for signal processing, a discipline in itself that falls somewhere between mechanical engineering and mathematics.

Scattered data approximation has been ideal for computing undefined phenomena in biology, engineering geology and mathematics to name a few. In theory it should work well in defining and approximating the intermittent data that signals emit. After intense testing of different mathematical models, the project successfully developed ways to apply signal analysis in the fields of geophysics, wireless communication and medical imaging. The new results also help in filtering out noise and improving acoustics.

In the near future, when better maps, medical equipment, hearing aids, or sound systems are created, some of this 'behind-the-scenes' computation and technology may well have much to do with it.