Spherical implicit neural representations

Details

Supervisors

Jacques, Laurent

Faculty

Ecole polytechnique de Louvain

Degree label

Master [120] : ingénieur civil en mathématiques appliquées, à finalité spécialisée

Abstract

enRepresenting and processing data in spherical domains presents unique challenges, primarily due to the curvature of the domain, which complicates the application of classical Euclidean techniques. Implicit neural representations (INRs) have emerged as a promising alternative to high-fidelity data representation; however, to effectively handle spherical domains, these methods must be adapted to the inherent geometry of the sphere to maintain both accuracy and stability. In this thesis, we introduce Herglotz-NET (HNET), a novel INR architecture that employs harmonic positional encoding based on complex Herglotz mappings. This approach yields a robust representation of spherical domains, offering interpretable and scalable spectral properties. Furthermore, we provide a unified expressivity analysis showing that any spherical‑based INRs satisfying a mild condition exhibits a predictable spectral expansion that scales with network depth. Additionally, we extend HNET from the sphere to full three‑dimensional spaces and evaluate it with the challenging task of reconstructing the gravitational potential of celestial bodies. Our results position HNET as a versatile, computationally efficient framework for spherical data, and demonstrate promising performance on gravitational-potential reconstruction.