Isospectralization, or how to hear shape, style, and correspondence
This addresses classical challenges in geometry processing, computer vision, and graphics, offering a practical tool for shape analysis and manipulation.
The paper tackles the problem of reconstructing and optimizing shapes using their Laplacian spectra, introducing a numerical procedure called isospectralization that deforms shapes to match spectra, with applications in shape reconstruction, pose and style transfer, and dense correspondence.
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 paper, we introduce a numerical procedure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of another. We implement the isospectralization procedure using modern differentiable programming techniques and exemplify its applications in some of the classical and notoriously hard problems in geometry processing, computer vision, and graphics such as shape reconstruction, pose and style transfer, and dense deformable correspondence.