Isospectralization, or How to Hear Shape, Style, and Correspondence

Abstract

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.

Correspondence-Free Region Localization for Partial Shape Similarity via Hamiltonian Spectrum Alignment

“…A similar property was also observed in [13], and might be a feature common to eigenvalue alignment approaches. A deeper comparison to [13] will be provided in Section 5.2. Finally, invariance to deformations is put in evidence below: Property 3 Since the eigenvalues of ∆ X and ∆ Y are isometry-invariant, so are all solutions to problem (15).…”

“…As a second application we address the shape-fromspectrum problem. This task was recently introduced in [13] and is phrased as follows: Given as input a short sequence of Laplacian eigenvalues {µ i } k i=1 of an unknown shape Y, recover a geometric embedding of Y. We stress that the Laplacian ∆ Y is not given; if given, it would lead to a shape-from-operator problem [6].…”

“…The initial shape used in [13] is also required to have the correct topology (e.g., disclike or annulus-like), while by Property 4 our optimization Figure 9. Shape recovery comparisons with [13]. We show the ground truth shapes (blue outline) and our solutions (potential on the square, growing from white to red), all obtained in 2-3 minutes from k = 50 input eigenvalues.…”

“…We always use a constant initialization. On the bottom we show results for [13] with their initial templates, which are required to always have the correct topology. is completely oblivious to the topology of the shape to recover.…”

“…We illustrate this with an example in the rightmost column of Figure 9, where we directly compare with [13]. Finally, optimizing over embedding coordinates as in [13] requires the adoption of additional regularizers to avoid flipped triangles, and careful parameter tuning to avoid rough shapes and collapsing edges; this makes their energy more difficult to minimize, which is done using stochastic optimization tools [21] and automatic differentiation as implemented in TensorFlow [1]. By optimizing over scalar functions instead of deforming a given mesh, we do not encounter any of these issues; for this reason, our optimization can be made much simpler as detailed in Section 4.…”