Orthogonalized Fourier Polynomials for Signal Approximation and Transfer

F Maggioli (1) , S Melzi (1) , M Ovsjanikov (2, 3) , M M Bronstein (4) , E Rodolà (1)

1 UNIROMA - Università degli Studi di Roma "La Sapienza" = Sapienza University [Rome]
2 X - École polytechnique
3 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
4 Imperial College London

We propose a novel approach for the approximation and transfer of signals across 3D shapes. The proposed solution is based on taking pointwise polynomials of the Fourier-like Laplacian eigenbasis, which provides a compact and expressive representation for general signals defined on the surface. Key to our approach is the construction of a new orthonormal basis upon the set of these linearly dependent polynomials. We analyze the properties of this representation, and further provide a complete analysis of the involved parameters. Our technique results in accurate approximation and transfer of various families of signals between near-isometric and non-isometric shapes, even under poor initialization. Our experiments, showcased on a selection of downstream tasks such as filtering and detail transfer, show that our method is more robust to discretization artifacts, deformation and noise as compared to alternative approaches.

Mots clés: CCS Concepts • Computing methodologies → Shape analysis; • Theory of computation → Computational geometry; • Mathematics of computing → Functional analysis; CCS Concepts • Computing methodologies → Shape analysis; CCS Concepts • Computing methodologies → Shape analysis

Domaines: Vision par ordinateur et reconnaissance de formes [cs.CV]

F Maggioli, S Melzi, M Ovsjanikov, M M Bronstein, E Rodolà. Orthogonalized Fourier Polynomials for Signal Approximation and Transfer. Computer Graphics Forum, 2021. ⟨hal-03781613⟩