Discrete inverse spectral geometry for shape analysis

Spectral quantities as the eigenvalues of the Laplacian operator are widely used in geometry processing since they provide a very informative summary of the intrinsic geometry of deformable shapes. Typically, the intrinsic properties of shapes are computed from their representation in 3D space and are used to encode compact geometric features, thus adopting a data-reduction principle. On the contrary, this talk focuses on the inverse problem: namely, recovering an extrinsic embedding from a purely intrinsic encoding, like in the classical “hearing the shape of the drum” problem. I will start by addressing the question of whether one can recover the shape of a geometric object solely from its vibration frequencies. Theoretically, the answer to this question is negative; however, little is known about the practical possibility of using the spectrum for shape reconstruction and optimization. I will introduce a numerical procedure called isospectralization, showing how to solve the problem in practice. Finally, I will discuss some recent advances tackling this recovery problem from a learning-based perspective, leading to unprecedented results and enabling novel applications in 3D vision and graphics.