INTERNSHIP FOR A MASTER STUDENT

RADON TYPE TRANSFORMS AND TOMOGRAPHY Research area : Tomography, integral geometry, numerical analysis. Place : CMAP (Centre de Math´ematiques Appliqu´ees), Ecole Polytechnique, 91128,

Palaiseau Cedex.

Advisers :

Habib Ammari (ammari@cmapx.polytechnique.fr),

Roman Novikov (novikov@cmapx.polytechnique.fr)

Duration of the internship : 4to6months. This internship canbefollowedbya PhD thesis.

Topic : The tomography is known, first of all, as a research area related with the problem of finding a structure of some object from its radiographs. Besides this X-ray tomography, several other tomographies are known at present, where instead of X-rays some other types of radiation are used. These problems arise in medicine, biology, industry and different domains of physics. Related mathematics involves, in particular, integral geometry (or, by other words, the theory of Radon type transforms), numerical analysis, partial differential equations.

At the beginning of the internship the student will perform the the following tasks:

1. Learn some first sections of the book [1] for introduction in computerized tomography.

2. Make some numerical computations in the framework of the filtered back projection algorithm.

3. Make some theoretical comparison between iterative [2] and exact [3] methods for inversion of the attenuated Radon transform. A good acquaintance with the Fourier analysis, in particular, and functional analysis,

in general, is necessary. A basic knowledge of numerical analysis is sufficient.

References

1. F. Natterer, The Mathematics of Computerized Tomography (Stuttgart: Teubner), 1986

2. L.A. Kunyansky, Generalized and attenuated Radon transforms: restorative approach to the numerical inversion,Inverse Problems 8 809-19, 1992

3. R.G. Novikov, An inversion formula for the attenuated X-ray transformation,Arkiv f¨or matematik 40(1), 145-167, 2002

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