Personal Webpage of Francesco Ballerin

PhD in Mathematics

University of Bergen · August 2022 – October 2026

Thesis

Geometric Methods for Image Restoration, Spherical Data, and Data-Driven Discovery of PDEs

Supervisors

Erlend Grong (main), Nello Blaser (co-supervisor)

Curvature and symmetry are among the most fundamental properties a space can possess, and classical methods that assume a flat domain are forced to discard them. This thesis develops geometric tools that encode these properties, using differential geometry and Lie theory, and applies them as inductive biases in three settings. First, it addresses image restoration and enhancement through diffusion on a sub-Riemannian model of the primary visual cortex V1. Second, it introduces SO(3)-equivariant neural networks for scalar and vector fields on the sphere. Last, it provides a complete classification of equivariant nonlinear differential operators on constant curvature spaces, which serves as a prior for the data-driven discovery of PDEs.

Master's Degree in Mathematical Analysis

University of Bergen · August 2020 – June 2022

Thesis

Sub-Riemannian geometry and its applications to image processing

Supervisor

Erlend Grong

Average grade

A

After an introduction on sub-Riemannian geometry, focusing on the examples of the Lie group \(SE(2)\) and the projective tangent bundle \(PT\mathbb{R}^2\), applications to the field of image processing are discussed. In particular we study a model of geometry of vision for image restoration due to Petitot, Citti and Sarti and further developments by Boscain, Duplaix, Gauthier and Rossi on hypoelliptic operators. New tools and techniques based on such work are developed and discussed.

Bachelor's Degree in Mathematics

University of Trento · September 2017 – September 2020

Thesis

Active Contour Models for Image Segmentation and Motion Tracking

Final grade

107/110

Traditional image segmentation methods based on global and local thresholds and the detection of intensity discontinuities have been widely used for numerous applications, but they intrinsically have multiple limitations. Two PDE-based active contours approaches, inspired by the fundamental snake equation presented by Kass, Witkin and Terzopoulos, are discussed. An implementation is then produced, as well as an extension of the algorithm to realize active tracking on video files.

NORA Summer School

NORA AI · June 2024

School covering advanced topics in machine learning with a focus on geometric deep learning, symmetry-aware models, and their applications to structured data. Included both theoretical foundations and hands-on work with modern ML methods.

Geilo Winter School

SINTEF · January 2024

School focusing on graph-based data analysis and machine learning. Covered spectral graph theory, graph signal processing, and graph neural networks.