I started my secondment at the University of Trento in September 2025 under the supervision of Alessandra Bernardi.
The project focused on mitigating beam hardening and scattering artefacts in industrial X-ray computed tomography (CT), while preserving structural information relevant for non-destructive testing. Using Python, I developed an approach combining tensor decomposition techniques, in particular TRPCA, Independent Component Analysis (ICA), and polynomial modelling to identify and approximate the global bias associated with these artefacts. The analysis was carried out on Beer–Lambert and Monte Carlo simulated images, which share the same geometry and therefore allow a controlled comparison.
After the industrial secondment project, I continued my second-year research through three projects, two of which are still ongoing. One of them resulted in the recent publication “Eigensolvers for Polynomial Roots and Tensor Decomposition”, joint work with B. Mourrain. In this work, we develop a symbolic-numeric framework based on Truncated Normal Forms and joint eigencomputation for solving polynomial systems and symmetric tensor decomposition problems. A central contribution of the work is the Julia package AlgebraicSolvers.jl, which implements these algebraic and numerical methods.
The project I started with Alessandra Bernardi and Daniele Taufer concerns the smoothability of zero-dimensional schemes from the viewpoint of multiplication matrices. We study how a smoothing can be described through formal deformations of commuting multiplication operators together with common eigenvectors corresponding to evaluations at the points of the generic fibre. This leads to an order-by-order formulation of the deformation problem, where the lifting equations are linear in the new coefficients once the lower-order terms are fixed. We are also investigating how the intrinsic filtration of generalized eigenvectors and the associated multiplication maps can be used to organize these deformations and detect possible obstructions. In particular, adapted grids of colliding points are studied as a way to recover the actual multiplication structure of non-monomial algebras.
The other ongoing project, together with Bernard Mourrain, concerns a dimension-extension approach for tensor decomposition. The idea is to introduce additional variables and search for a flat extension of the associated Hankel matrix, from which the tensor decomposition can then be recovered. This project will be one of the starting points of my third year, when I return to Inria in September, and I am looking forward to continuing this work there!