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

Aleksei Zhuravlev; Lennart Bastian; Dongliang Cao; Nafie El Amrani; Paul Roetzer; Viktoria Ehm; Riccardo Marin; Hiroki Nishizawa

Figure 1

Figure 2

Figure 3

Figure 4

Symbol	Domain	Description
$\mathcal{M},\mathcal{N}$		3D shapes with $n_{\mathcal{M}}$ and $n_{\mathcal{N}}$ vertices
$X_{\mathcal{M}}$	$\mathbb{R}^{n_{\mathcal{M}}\times 3}$	Vertex positions of $\mathcal{M}$
$E_{\mathcal{M}}$	$\subseteq\{1,\dots,n_{\mathcal{M}}\}^{2}$	Edges of $\mathcal{M}$
$T_{\mathcal{M}}$	$\subseteq\{1,\dots,n_{\mathcal{M}}\}^{3}$	Faces (triangles) of $\mathcal{M}$
$f_{\mathcal{M}}$	$\mathbb{R}^{n_{\mathcal{M}}\times d}$	Vertex-wise features on $\mathcal{M}$
$L_{\mathcal{M}}$	$\mathbb{R}^{n_{\mathcal{M}}\times n_{\mathcal{M}}}$	Discrete Laplace–Beltrami operator on $\mathcal{M}$

Table 1: Summary of notation used throughout the report.

Figure 2: 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: [ GRE∗23 ] .

Figure 3: 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 [ ZLG25 , VOM25 ] .

Figure 4: 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 5: 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.

Method	Cat.	MS	NI	P	TN
ZoomOut [ MRR∗19 ]	A	✗	✓	✓	✗
ConsistentZoomOut [ HRWO20 ]	A	✓	✓	✓	✗
RHM [ ESBC19 ]	A	✗	✓	✗	✗
ComplexFMaps [ DCMO22 ]	A	✗	✓	✗	✗
DiscreteOpt [ RMWO21 ]	A	✗	✓	✗	✗
SmoothFM [ MRSHO22 ]	A	✗	✓	✗	✗

Table 2: Overview of spectral shape correspondence methods. Each method is categorized by approach ( Cat. ): axiomatic ( A ), supervised ( S ), unsupervised ( U ). The remaining columns indicate supported properties: multi-shape consistency ( MS ), non-isometric deformations ( NI ), partiality ( P ), topological noise ( TN ).

Figure 6: 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 [ PLDO25 ] .

Method	LB	S2S	MS	Q	LO	RF
Maron et al. [ MDK∗16 ]	✓	✓	✗	✗	✗	✓
MINA [ BST20 ]	✓	✗	✗	✗	✗	✗
SIGMA [ GRE∗23 ]	✓	✗	✗	✗	✗	✗
Ren et al. [ RMWO21 ]	✗	✓	✗	✗	✗	✗
Pai et al. [ PRM∗21 ]	✗	✓	✗	✗	✗	✗
SM-Comb [ RSCB22 ]	✗	✓	✗	✗	✓	✗

Table 3: Overview over combinatorial matching methods and classification whether methods are (LB) landmark-based, are (S2S) Surface-to-Surface methods, are applicable to MS multiple shapes, are (Q) Quantum, have a (LO) linear objective

f(x)

, or have a (RF) relax the feasible set

\mathcal{F}

.

Figure 7: Visualisation of correspondence between landmarks (in colour) lying on the surface (grey) of two 3D shapes. Image source: [ GRE∗23 ]

Figure 8: Visualisation of the (in-layers-structured) product graph

\mathcal{P}

between a cyclic chain graph

\mathcal{C}

and the graph of a triangle mesh of a 3D shape

\mathcal{N}

. Correspondences between

\mathcal{C}

and

\mathcal{N}

can be computed by finding the shortest cyclic path that visits all layers. Image source: [ RB24 ] .

Figure 9: Deformation methods recover the correspondence between shapes by finding a deformation that aligns them spatially.

Method	Category	Correspondence	Embedding Space	Deformation Model	Training
Revisited-CPD [ FMT∗22 ]	O	Soft	Euclidean	Coherent	N
Geo-CPD [ Hir23 ]	O	Soft	Manifold	Coherent	N
Tajdari et al. [ THYS22 ]	O	Hard	Euclidean	Affine	N
Yao et al. [ YDXZ23 ]	O	Hard	Euclidean	Locally-affine	N
Zhao et al. [ ZJM∗24 ]	O	Soft	Euclidean	Kernel-based	N
Merrouche et al. [ MWB25b ]	O	Hard	Neural	ARAP	N

Table 4: Overview of deformation-based matching methods. Methods are categorized by their approach ( O ptimisation, T emplate, L earning, or H ybrid), correspondence type (Hard, Soft, Neural, Global), feature space, deformation model, and training requirements ( N one, S upervised, or U nsupervised).

Figure 10: An example of deformation from GeoCPD [ Hir23 ] . The source shape (black) is deformed toward the target one (blue), toward multiple steps till final convergence.

Figure 11: Zero-shot matching involves establishing correspondences between structurally distinct shape categories (such as different animal species), which often contain unique regions present on only one of the shapes (shown in red). In the absence of training data, feature descriptors from 2D vision models are particularly effective for this task. Shapes are taken from SHREC’20 [ DLR∗20 ] and BeCoS [ EEAX∗25 ] datasets.

Method	FM	CP
Abdelreheem et al. [ AEOW23 ]	SAM + DINO	SP
Diff3F [ DMM24 ]	SD + DINO	SP
DenseMatcher [ ZJZ∗25 ]	SD + DINO	SP
Echomatch [ XER∗25 ]	DINO	SP
SRIF [ SGJ∗24 ]	DiffMorpher	D
DV-Matcher [ CJH25 ]	DINO	D

Table 5: Overview of methods using foundation models for shape correspondence. They are categorized by the foundation model used for descriptor extraction (FM) : DINO [ CTM∗21 , ODM∗23 ] , Stable Diffusion [ RBL∗22 ] , SAM [ KMR∗23 ] , DiffMorpher [ ZZX∗24 ] , as well as the downstream correspondence pipeline (CP) : Spectral ( SP ), Deformation-based ( D ), or Other ( O ).

Figure 12: Visualisation of the importance of neighbourhood preservation for correspondence computation by transferring triangulation from source to target shape via computing matchings. Matchings that violate neighbourhood preservation can lead to arbitrary distortions of triangles. Image source: [ RB25 ] .

Method	IF	S	TF	GNP	P
Takayama [ Tak22 ]	✗	✗	✗	✓	✗
Schmidt et al. [ SPK23 ]	✗	✗	✗	✓	✗
Rorberg et al. [ RVBC23 ]	✗	✓	✗	✗	✗
Born et al. [ BSCK21 ]	✗	✓	✗	✗	✗
Magnet et al. [ MRSHO22 ]	✗	✓	( ✓ )	✗	✓
Xia et al. [ XLGM24 ]	✗	✓	( ✓ )	✗	✗

Table 6: Overview of neighbourhood preserving approaches and classification if methods are (IF) initialisation-free, are (S) scalable, are (TF) topologically flexible, are (GNP) globally neighbourhood preserving, or can solve (P) shape matching problems where at least one shape is partial. The scalability of SpiderMatch [ RB24 ] depends on the input representation; see [ RB25 ] for more details. Approaches [ MRSHO22 , XLGM24 ] do not show experiments on topologically noisy shapes but would, in theory, support such inputs. The formalism solved by SM-Comb [ RSCB22 ] is not designed for partial-to-full or partial-to-partial shape matching, yet the authors show that it is applicable when closing shapes using triangle fans. Further, the approaches [ RB24 , REC∗25 ] do not show partial-to-full experiments, but their formalisms naturally support this case.

Figure 13: Partial shape matching settings : In the partial-to-full setting (left), a partial shape is matched to a template shape. In comparison, in the partial-to-partial setting, both shapes are only partially observed (right). The grey grid indicates that this part is missing, where red indicates that this part is observed but not in the overlapping region. Shapes are taken from FAUST [ BRLB14 ] , KIDS [ RRBW∗14 ] and BeCoS [ EEAX∗25 ] datasets.

Method	Setting	Framework	Type
DPFM [ APO21 ]	P2F,P2P	PFM	S , U
Bensaid et al. [ BRGK23 ]	P2F, P2P	PFM	S
Bensaid et al. [ BK24 ]	P2F	Spectral	A
Bracha et al. [ BDK24a ]	P2F	Dist	U
Wormhole [ BDK24b ]	P2F	Dist	U
ULRSSM [ CRB23 ]	P2F	PFM	U

Table 7: Categorization of Partial Shape Matching Methods: We categorize partial shape matching methods in their setting (partial-to-full ( P2F ) or partial-to-partial ( P2P )), their underlying framework ( P artial F unctional M aps, Spectral alignment, Dist ance-based or Neigh bourhood-preserving) and their optimisation type (axiomatic ( A ), supervised ( S ), unsupervised ( U )).

Figure 14: The BeCoS dataset [ EEAX∗25 ] connects seven existing 3D shape matching datasets [ BRLB14 , ZKJB17 , ASK∗05 , BBK08 , LTT∗21 , RRBW∗14 , DLR∗20 ] , enabling the propagation of dense correspondences across a wide range of categories, including humanoids, cats, elephants, and others. Furthermore, it provides annotations for partial correspondences, supporting both partial-to-full ( P2F ) and partial-to-partial ( P2P ) matching settings. Image source: [ EEAX∗25 ] .

Table 8: An overview of 3D deformable shape datasets . The datasets are categorized according to their shape matching setting: full-to-full ( F2F ), partial-to-full ( P2F ), and partial-to-partial ( P2P ). For each dataset, we report several characteristics, including non-isometry ( NS ), number of unique shapes ( #US ), topological noise ( TN ), and type of shape categories ( SC ), indicating whether the dataset contains humanoid shapes () or animal shapes ().

Figure 15: Practical applications of shape correspondence span numerous tasks in graphics and geometry processing. (a) Segmentation transfer propagates semantic part annotations across a collection of registered shapes. (b) Shape interpolation uses correspondences to generate smooth transitions between source and target shapes. (c) Statistical shape modelling uses correspondences to build low-dimensional shape spaces, including anatomical models of organs and tissues. (d) Animation transfer allows to propagate motion sequences across registered shapes. Figures adapted from [ CEEA∗24 , AEOW23 , EACB24 , MMMC21b ] .

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

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