Denoising Functional Maps: Diffusion Models for Shape Correspondence

TL;DR

DenoisFM introduces a fundamentally new approach to shape correspondence by directly predicting functional maps with denoising diffusion models. It leverages template-based training, a sign-corrected Laplacian eigenbasis, and a diffusion-driven conditioning mechanism to produce accurate and generalizable maps across near-isometric, anisotropic, and cross-domain shapes, including animals. A key contribution is the unsupervised sign correction that canonicalizes eigenvectors, enabling stable learning of $C_{1T}$ and subsequent pairwise maps via composition and ZoomOut refinement. The results demonstrate competitive performance on standard datasets (FAUST, SCAPE, SHREC'19), strong cross-category generalization (zero-shot DT4D), and applicability to animal shapes, highlighting the potential of diffusion models in geometric matching and broader generalization capabilities.

Abstract

Estimating correspondences between pairs of deformable shapes remains a challenging problem. Despite substantial progress, existing methods lack broad generalization capabilities and require category-specific training data. To address these limitations, we propose a fundamentally new approach to shape correspondence based on denoising diffusion models. In our method, a diffusion model learns to directly predict the functional map, a low-dimensional representation of a point-wise map between shapes. We use a large dataset of synthetic human meshes for training and employ two steps to reduce the number of functional maps that need to be learned. First, the maps refer to a template rather than shape pairs. Second, the functional map is defined in a basis of eigenvectors of the Laplacian, which is not unique due to sign ambiguity. Therefore, we introduce an unsupervised approach to select a specific basis by correcting the signs of eigenvectors based on surface features. Our model achieves competitive performance on standard human datasets, meshes with anisotropic connectivity, non-isometric humanoid shapes, as well as animals compared to existing descriptor-based and large-scale shape deformation methods. See our project page for the source code and the datasets.

Figures (8)

• Figure 1: We propose DenoisFM, a novel method for predicting shape correspondences in the form of functional maps using denoising diffusion models. (Left:) Challenging examples our method can handle, with color-coded correspondences. (Right:) By sequentially denoising samples of random noise, the diffusion model can predict the correct functional map between a pair of shapes.
• Figure 2: Overview of DenoisFM. (a) A pair of shapes is matched using a DDPM model. (b) The signs of eigenvectors are corrected through learned features.
• Figure 3: Qualitative results by our approach on SHREC'19 (top) and DT4D (bottom) datasets; see also Supp. \ref{['sec:qualitative_supp']}.
• Figure 5: Learned feature vectors and their corresponding eigenvectors. Positive and negative values are shown in red and blue, respectively. The low-order feature vectors resemble the eigenvectors themselves, while the high-order ones are mainly concentrated in the arm and hand regions.
• Figure 6: Illustration of the Dirichlet Medoid map selection. Given $k=6$ candidate matches on the target shape, we report the one that has the lowest total distance to the other $k-1$ candidates (shown in green).