Fedor Goncharov

Centre de Mathématique Appliquées (CMAP), Ecole Polytechnique

PhD thesis is in preparation under the supervision of Roman Novikov

Ecole Polytechnique,
Route de Saclay
91120 PALAISEAU Cedex

fedor.goncharov [at] cmap [pt] polytechnique [pt] fr

Publications / Preprints

Main research projects

Weighted Radon transforms and their applications

In this project we aim to develop new inversion methods of weighted (generaliezed) Radon transforms which are
of particular importance in various applications in the domain of inverse problems (e.g., in tomographies, geophysics).
In particular, we work on methods which could be numerically more stable against the noise in tomographical data
compared to existing methods. For our recent progress in this direction one can look at [2, 3].

At the same time, I'm interested in pure mathematical studies of weighted (generalized) ray/Radon transforms.
Not a long time ago, with professor R. Novikov we had obtained a series of unexpected counterexamples to injectivity of the latter
under minor assumptions. More precisely about these counterexamples one can look at [4-6].

Numerical inversion of (weighted) ray / Radon transforms in 3D

Here I will put some pictures related to numerical realizations of our algorithms.
The version of my code and the related project can be found on GitHub (the project is still under development).
Smooth "hats" - inversion of Radon transforms in 3D
(projection theorem/NFFT TU Chemnitz)
Shepp-Logan phantom - reduction of ray transforms to Radon transforms in 3D and their subsequent inversion
(projection theorem/NFFT TU Chemnitz)

Distribution of the nucleotide inside the monkey's brain
(3D vs. 2D reconstructions)