## Section: Partnerships and Cooperations

### Regional Initiatives

#### Inria SAM Action Transverse

Participants : Paul Görlach, Evelyne Hubert.

Finding biomarkers of abnormalities of the white matter is one important problem in dMRI processing. As these biomarkers need to be independent of the orientation of the head, they are functions of the rotational invariants of the shapes that characterize the diffusion probabilities in the white matter. While the situation is well understood for second order tensors, these are not powerful enough to represent crossings in the white matter. Acquisitions made with the HARDI scheme allow for a richer description of probabilities. In particular, the project-team Athena has modelled them as (positive) ternary quartics (symmetric tensors of order 4). But invariants of these quartics are not well known. For a long period, only six were known, when there should be at least 12. Strategies were developed in the project-team Athena to compute more invariants, either algebraic [25] or polynomial [21]. The former suffered some instability issues in their evaluations, the latter did not form a minimal set. The goal of this "Transverse action" was to team up with expertise in algebraic computation and leverage the methods [23], [24], [22] [19], [7] developed in the project team Aromath to gain more insight in this problem of rotational invariants of ternary quartics.

This action is done in collaboration with Théodore Papadopoulo (Athena team).

#### CIMI thematic project

Participant : Evelyne Hubert.

Labex CIMI Toulouse supports the project *Joint Implicit and
Parametric Representation based on Skeleton* where the PI are
Géraldine Morin (IRIT, Vortex team) and Evelyne Hubert. This project
aims at developing a mathematical model and software for surfaces,
based on a joint parametric and implicit representation, with a
skeleton.