Analyse Harmonique et Problèmes Inverses – AHPI

The endeavour of this project is to develop some methodology for modelling and solving inverse problems of a certain type using tools from harmonic and complex analysis. These problems pertain to deconvolution issues, identification of fractal dimension for Gaussian fields, and free boundary problems for propagation and diffusion phenomena. The target applications concern radar detection, clinical investigation of the human body (e.g. to diagnose osteoporosis from X-rays or epileptic foci from electro/magneto encephalography), sismology, and the computation of free boundaries of plasmas subject to magnetic confinment in a tokamak. Such applications share as a common feature that they can be modeled through measurements of some transform of an initial signal. This transform is of the Fourier type in Gaussian modelling of X-rays pictures or in sismology, of the Fourier-Wigner type in radar detection, and of the Riesz type in electro/magneto encephalography or in plasma-boundary control. Its non-local character generates various forms of the uncertainty principle. Either signals are partially observed (as in radar detection where only moduli can be recovered and in free boundary problems where the free boundary remains of course unobserved), or else certain structural parameters of the kernel are unknown (the diposal of geological layers in sismology, the location of the sources in electro/magneto-encephalography, the parameters of the underlying Gaussian field in the modelling of X-ray pictures). Therefore, the uncertainty makes the problem ill-posed. To approximate the solution, some kind of regularization is then required whose effect has to be analysed under suitable assumptions. Various regularization methods will be used depending on the situation. In free boundary issues, we shall rely on extremal problems where data get approximated on the observed part of the boundary by solutions of the underlying equation, meeting some constraints on the unobserved part. This is germane but not identical to Tikhonov-like regularization for most choices of constraints. In other cases, one uses particular bases expansions for truncation and one restricts the class of input signals. This is the case in radar detection where the use of discrete or Hermite signals reduces the problem to bilinear Fourier analysis with geometric flavour. Likewise, wavelets are particularly adapted to discrete sismological models, while wave propagation is an invitation to microlocal analysis and asymptotic expansions of kernels of pseudo differential operators. Modelling aspects are also part of the proposal, in particular when describing grey levels of X-ray pictures through gaussian fields where geometric averaging techniques causes anisotropy that impinges on the fractal dimension. In all cases the main tools from harmonic analysis are used, and they team up with complex analysis, analytic operator theory, and approximation theory in radar detection and in plasma boundary control (where the circular symmetry makes the model into a 2-dimensional real Beltrami equation). Complex analytic tools actually also arise in 3-dimensional source detection for some particular geometries (e.g. ellipsoids) that reduce the problem to a sequence of 2-dimensional branchpoint recovery issues for analytric functions in generic planar cross sections. Our goal is to obtain, in the above-mentioned cases, constructive and efficient algorithms on structured models, that should be used to initialize more versatile but heavier computationally dedicated heuristics. Three of the teams (Orléans, Pau, Sophia) maintain close contacts with academic and industrial laboratories involved with the target applications. The Bordeaux team, with his well-established expertise in harmonic and complex analysis and long tradition of industrial cooperation in signal processing and medical imaging, naturally complements the other three.