Synchronous Diffusion for Unsupervised Smooth Non-Rigid 3D Shape Matching

TL;DR

This work tackles unsupervised non-rigid 3D shape matching by addressing the lack of spatially smooth pointwise correspondences in functional-map based methods. It introduces a synchronous diffusion regularisation that enforces cross-shape consistency by diffusing the same input function on both shapes and penalising the mismatch after mapping back, integrated into a deep functional map framework with multiscale diffusion times. Theoretical grounding connects the approach to quadratic assignment and a continuous Weisfeiler-Leman perspective, and comprehensive experiments across near-isometric, topological-noise, non-isometric, and partial settings demonstrate substantial performance gains over state-of-the-art alternatives, with ablations highlighting the benefits of random diffusion times and random initial functions. The method is efficient, scalable, and broadly applicable to unsupervised smooth matching, offering a practical path toward robust shape correspondence in challenging real-world data and enabling potential extensions to related matching tasks in other domains.

Abstract

Most recent unsupervised non-rigid 3D shape matching methods are based on the functional map framework due to its efficiency and superior performance. Nevertheless, respective methods struggle to obtain spatially smooth pointwise correspondences due to the lack of proper regularisation. In this work, inspired by the success of message passing on graphs, we propose a synchronous diffusion process which we use as regularisation to achieve smoothness in non-rigid 3D shape matching problems. The intuition of synchronous diffusion is that diffusing the same input function on two different shapes results in consistent outputs. Using different challenging datasets, we demonstrate that our novel regularisation can substantially improve the state-of-the-art in shape matching, especially in the presence of topological noise.

Figures (13)

• Figure 1: Left: We present an unsupervised regularisation based on synchronous diffusion for spatially smooth non-rigid 3D shape matching. This is based on the motivation that diffusing a function $f$ on shape $\mathcal{M}$ should lead to comparable results when diffusing $\pi(f)$ (i.e. $f$ transferred to shape $\mathcal{N}$ using the map $\pi$) on shape $\mathcal{N}$. Coloured points are used to illustrate a (subset of) random function values. Right: Our approach can be applied in a broad range of challenging scenarios, including topological noise, non-isometry and partiality.
• Figure 1: Qualitative results of our method on TOPKIDS. Our method obtains accurate correspondences for shapes with topological noise.
• Figure 2: Illustration of our synchronous diffusion process for smooth matching. First, we transfer the function $\mathbf{F}_{\mathcal{M}}$ from shape $\mathcal{M}$ to shape $\mathcal{N}$ and perform synchronous diffusion on both shapes. Afterwards, the diffused function $\mathbf{F}_{\mathcal{N}}(t)$ from shape $\mathcal{N}$ is transferred back to shape $\mathcal{M}$. The difference between the two diffused functions is used to penalise spatially unsmooth pointwise correspondences.
• Figure 2: Qualitative results of our method on SHREC'19. Our method obtains accurate correspondences for human shapes with diverse poses and shapes.
• Figure 3: Left: Near-isometric shape matching and cross-dataset generalisation on FAUST, SCAPE and SHREC'19 datasets. The mean geodesic error kim2011blended is used as quantitative evaluation metric. The best results in each column are highlighted. Right: Qualitative results of our method on the challenging SHREC'19 dataset trained on FAUST and SCAPE datasets. Our method outperforms existing state-of-the-art methods and demonstrates superior cross-dataset generalisation ability.

Theorems & Definitions (1)

• definition thmcounterdefinition: Synchronous diffusion