Jean-Luc Falcone

Eugen Bronasco, Jean-Luc Falcone, Gilles Vilmart

We design the Arboretum.hs package for symbolic computations with algebras of trees and more general graphs in Haskell. Thanks to the declarative nature of functional programming, the package's implementation closely follows mathematical definitions, making the code intuitive and transparent for users working with algebraic and combinatorial structures. To assist with current mathematical research, Arboretum.hs supports experimentation by facilitating the introduction of new algebraic operations, as well as providing functionality for rendering trees and forests through LaTeX integration. Compared to recent imperative implementations in languages such as Julia or Python, Arboretum.hs offers greater flexibility for manipulating and extending tree-based structures. Its use of Haskell enables safe programming and strong compile-time guarantees, serving both as a practical computational tool and a foundation for further research in algebraic combinatorics, beyond the setting of trees usually considered in the implementation of Butcher series, which are a fundamental tool for the analysis of numerical integrators.

Oscar Dabrowski, Jean-Luc Falcone, Antoine Klauser et al.

MRI, a widespread non-invasive medical imaging modality, is highly sensitive to patient motion. Despite many attempts over the years, motion correction remains a difficult problem and there is no general method applicable to all situations. We propose a retrospective method for motion estimation and correction to tackle the problem of in-plane rigid-body motion, apt for classical 2D Spin-Echo scans of the brain, which are regularly used in clinical practice. Due to the sequential acquisition of k-space, motion artifacts are well localized. The method leverages the power of deep neural networks to estimate motion parameters in k-space and uses a model-based approach to restore degraded images to avoid ''hallucinations''. Notable advantages are its ability to estimate motion occurring in high spatial frequencies without the need of a motion-free reference. The proposed method operates on the whole k-space dynamic range and is moderately affected by the lower SNR of higher harmonics. As a proof of concept, we provide models trained using supervised learning on 600k motion simulations based on motion-free scans of 43 different subjects. Generalization performance was tested with simulations as well as in-vivo. Qualitative and quantitative evaluations are presented for motion parameter estimations and image reconstruction. Experimental results show that our approach is able to obtain good generalization performance on simulated data and in-vivo acquisitions. We provide a Python implementation at https://gitlab.unige.ch/Oscar.Dabrowski/sismik_mri/.