Arianna Rampini (Sapienza, University of Rome)
The question whether one can recover the shape of a geometric object from its
Laplacian spectrum (‘hear the shape of the drum’) is a classical problem in spectral geometry
with a broad range of implications and applications. While theoretically the answer to this
question is negative (there exist examples of iso-spectral but non-isometric manifolds) little
is known about the practical possibility of using the spectrum for shape reconstruction and
optimization. In this talk, I will introduce a numerical procedure called isospectralization,
consisting of deforming one shape to make its Laplacian spectrum match that of another.
By implementing isospectralization using modern differentiable programming techniques,
we showed that the *practical* problem of recovering shapes from the Laplacian spectrum
is solvable. I will finally exemplify the applications of this procedure in some of the classical
and notoriously hard problems in geometry processing, computer vision, and graphics such
as shape reconstruction, style transfer, and non-isometric shape matching.