Non-Rigid 3D Shape Correspondences: From Foundations to Open Challenges and Opportunities

Abstract

Estimating correspondences between deformed shape instances is a long-standing problem in computer graphics; numerous applications, from texture transfer to statistical modelling, rely on recovering an accurate correspondence map. Many methods have thus been proposed to tackle this challenging problem from varying perspectives, depending on the downstream application. This state-of-the-art report is geared towards researchers, practitioners, and students seeking to understand recent trends and advances in the field. We categorise developments into three paradigms: spectral methods based on functional maps, combinatorial formulations that impose discrete constraints, and deformation-based methods that directly recover a global alignment. Each school of thought offers different advantages and disadvantages, which we discuss throughout the report. Meanwhile, we highlight the latest developments in each area and suggest new potential research directions. Finally, we provide an overview of emerging challenges and opportunities in this growing field, including the recent use of vision foundation models for zero-shot correspondence and the particularly challenging task of matching partial shapes.

Figures (14)

• Figure 1: Visualising and matching features between shapes. (a) Per-vertex feature functions are computed on two non-rigidly deformed shapes visualised by colour. When these features are sufficiently descriptive (as evidenced by similar regions coloured identically), a correspondence mapping can be derived. (b) Dense correspondences (visualised by transferring colours from one to the other shape and by connecting lines) computed between two non-rigidly deformed shapes, demonstrating geometric and semantic alignment. Image source: gao_sigma_2023.
• Figure 2: Qualitative visualisation of correspondence quality using different transfer strategies. (a) Color transfer provides an overview of the global correspondence structure. (b) High-frequency texture transfer can reveal small local misalignments. (c) Deformation transfer can similarly visualise small topological inconsistencies caused by inaccurate correspondences. Figures adapted from zhuravlev_denoising_2025vigano2025nam.
• Figure 3: Illustration of the notation used for computing the cotangent weight matrix $W_{ij}$ and the diagonal area matrix $A_{ij}$. The cotangent weight matrix $W_{ij}$ uses the angles opposite the edge between $v_i$ and $v_j$; $T_1$ and $T_2$ denote the areas of the corresponding triangles. The diagonal area matrix $A_{ij}$ assigns each vertex one-third of the total area of its adjacent triangles.
• Figure 4: Visualisation of the functional map framework as an analogy of Fourier analysis in signal processing. Given two functions $f_{\mathcal{M}}$ and $f_{\mathcal{N}}$ defined on shapes $\mathcal{M}$ and $\mathcal{N}$, respectively, the functional map $C_{\mathcal{MN}}$ transfers $f_{\mathcal{M}}$ from $\mathcal{M}$ to $f_{\mathcal{N}}$ on $\mathcal{N}$, each function is approximated using the first $k$ LBO eigenfunctions $\phi$ of the corresponding shape, which is similar to Fourier transform that uses sinusoidal functions with different frequencies as basis functions. In this representation, the functional map transfers the coefficients $a$ of the basis functions on $\mathcal{M}$ to the coefficients $b$ on $\mathcal{N}$, thereby encoding the point-wise correspondences $\Pi_{\mathcal{NM}}$ in a compact $k \times k$ matrix.
• Figure 5: Functional maps provide a suitable domain for 2D diffusion models due to their compact matrix representation. Such spectral diffusion models, trained on large collections of registered shapes, introduce a learned prior for functional map solvers. Adapted from pierson2025diffumatch.