
\section{Challenges and emerging trends}
\label{sec:emerging}

In this section, we explore several opportunities that have recently emerged in shape correspondence research.
These include leveraging foundation models, preserving local neighbourhoods during matching, and addressing partial shape scenarios.
Such directions extend the capabilities of conventional methods, enabling them to handle more challenging and previously inaccessible settings.

\subsection{Feature aggregation from foundation models}


























\newcommand{\nonisometricWidth}{0.265\columnwidth}
\newcommand{\nonisometricHeight}{2.65cm}
\begin{figure}[t]
    \centering

    \begin{tabular}{cc}
        \small

         \includegraphics[height=\nonisometricHeight,width=\nonisometricWidth]{figures/06/images_shrec20/2-0}&
         \includegraphics[height=\nonisometricHeight,width=0.26\columnwidth]{figures/06/images_shrec20/2-1}\\

         \includegraphics[height=\nonisometricHeight,width=\nonisometricWidth]{figures/06/images_shrec20/3-0}&
         \includegraphics[height=\nonisometricHeight,width=\nonisometricWidth]{figures/06/images_shrec20/3-1}\\
         \textbf{Source} & \textbf{Target} \\
    \end{tabular}
    
    \caption{\textbf{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~\cite{dyke2020shrec} and BeCoS~\cite{ehm2025beyond} datasets.}
    \label{fig:6_2}
\end{figure}

Axiomatic and learned feature descriptors are fundamental to many correspondence estimation pipelines.
However, their key limitations lie in the former's limited expressiveness and the latter's poor generalisation to out-of-distribution shapes.
As an alternative, an increasingly popular direction is to utilise features from 2D foundation vision models trained on large-scale image datasets, such as self-supervised transformers \cite{caron2021emerging,oquab2023dinov2}, image diffusion models \cite{rombach2022high}, or segmentation models \cite{kirillov2023segment}. 
These distilled descriptors have the potential to generalise zero-shot across shape classes, without category-specific training data. 


In a typical pipeline for adapting 2D vision features to 3D shape correspondence, a mesh $\mathcal{M}$ is projected to image space from $n$ viewpoints using a projection operator $\text{Proj}$:
\begin{equation}
    \text{Proj}_{\mathcal{C}_j}\left( \mathcal{M}\right)= I_j \in \mathbb{R} ^{H \times W},    
\end{equation}
\noindent where $H, W$ denote the height and width of the rendered image, with $\mathcal{C}_j$ representing the $j^{\text {th }}$ camera producing the image $I_j$.
Together with a set of additional information $\mathcal{G}$ such as depth, normal maps, or a text prompt, the rendered image $I_j$ is processed by a 2D vision model $\Psi$ to obtain $d$-dimensional features $f _j^{2D}$,
\begin{equation}
    \Psi(I_j, \mathcal{G})=  f _j^{2D} \in \mathbb{R} ^{H \times W \times d}.
\end{equation}
\noindent Known camera parameters are then leveraged to unproject features from image space to mesh vertices $f_j^{3D}$. 
These are aggregated across viewpoints to obtain shape descriptors $f_\mathcal{M}$, 
\begin{equation}
\begin{aligned}
    f _j^{2D} &\xrightarrow{\text{Proj}^{-1}} f _j^{3 D}, \quad & f_\mathcal{M} &=\frac{1}{n} \sum_{j=1}^n f _j^{3 D}.
\end{aligned}
\end{equation}

\noindent The latter can be passed to a downstream matching pipeline for computing correspondences, or combined with other axiomatic or learned descriptors.

Among the earliest works to leverage foundation models for 3D shape correspondence, Abdelreheem et al.~\cite{abdelreheem2023zero} use vision-language models~\cite{li2023blip,brown2020language} to infer shape categories and semantic regions, which are localised by a multiview object detector~\cite{liu2024grounding} and segmented using Segment Anything~\cite{kirillov2023segment}. 
The resulting masks are aggregated onto the mesh to guide the initialisation of the functional map with WKS descriptors~\cite{aubry2011wave}, and are then refined via BCICP~\cite{ren2018continuous}. 
In contrast, Diff3F renders textureless shape views and uses Stable Diffusion~\cite{rombach2022high} to synthesise textured images, extracting intermediate features and fusing them with DINO~\cite{oquab2023dinov2}. 
These are unprojected onto the mesh and used in a standard functional map pipeline~\cite{ovsjanikov_functional_2012} to compute dense correspondences. 





Recent methods extend the early use of 2D vision features by integrating them into more advanced correspondence pipelines.
Morreale et al.~\cite{morreale2024neural} employ DINO to co-align shapes by estimating their upright orientation from multiview renderings and to provide semantic descriptors. 
These features are then integrated into Neural Surface Maps \cite{morreale2021neural}, a framework that computes continuous correspondences between surfaces by encoding them with dedicated neural functions. 
DenseMatcher \cite{zhu_densematcher_2024} proposes to fuse features from Stable Diffusion and DINO with HKS~\cite{tombari2010unique} descriptors and refine them using a trainable DiffusionNet \cite{sharp2022diffusionnet}, supervised by a semantic distance loss that encourages feature distances to align with semantic part similarity. 
In \cite{uzolas2025surface}, the problem of disambiguating symmetric parts on a shape is addressed by training an autoencoder network to refine Diff3F \cite{dutt_diffusion_2024} features through an objective that encourages the preservation of geodesic distances in the learned embedding space.

Several works integrate vision features into deformation-based correspondence pipelines.
In SRIF \cite{sun_srif_2024}, multiview renderings of the source and target meshes are processed by a diffusion-based image morphing model \cite{zhang2024diffmorpher}, which generates smooth transitional frames between the two shapes.
A continuous deformation field is then estimated via a normalising flow \cite{yang2019pointflow}, gradually warping the source mesh to match the target while following the reconstructed trajectory. 
In DV-Matcher \cite{chen_dv-matcher_2025}, DINO features are combined with positional encodings and processed by a Siamese network with a dual-path attention mechanism, trained to predict the deformation between source and target shapes. 
Stable-SCore \cite{liu_stable-score_2025} utilises vision features to obtain 2D correspondences between rendered views of meshes, which are then used to guide a mesh deformation model using Neural Jacobian Fields (NJF)~\cite{aigerman_neural_2022}.


We summarise the discussed methods in Table~\ref{tab:foundational}.
These approaches demonstrate the strong potential of 2D vision models for matching structurally distinct shape categories, such as animal species.
However, their limitation lies in the lack of fine-grained geometric awareness, where axiomatic or learned descriptors continue to perform strongly. 
Moreover, certain shapes may lack sufficiently discriminative textures (e.g., anatomical organs) or may not be sufficiently represented in the training distributions of these models, making them inapplicable.
Future research could investigate hybrid strategies that integrate foundation model features with other types of descriptors to leverage their complementary strengths, or explore features coming from 3D foundation models \cite{du2025hierarchical}.









\begin{table}[t]
\centering
\small

\begin{tabular}{lcc}

    \toprule
    \small
    \textbf{Method} & \textbf{FM} & \textbf{CP} \\
    \midrule
    Abdelreheem et al.~\cite{abdelreheem2023zero} & SAM + DINO & \textbf{\textcolor{red}{SP}} \\
    Diff3F~\cite{dutt_diffusion_2024} & SD + DINO & \textbf{\textcolor{red}{SP}} \\
    DenseMatcher~\cite{zhu_densematcher_2024} & SD + DINO & \textbf{\textcolor{red}{SP}} \\
    Echomatch~\cite{xie_echomatch_2025} & DINO & \textbf{\textcolor{red}{SP}} \\
    SRIF~\cite{sun_srif_2024} & DiffMorpher & \textbf{\textcolor{green!50!black}{D}} \\
    DV-Matcher~\cite{chen_dv-matcher_2025} & DINO & \textbf{\textcolor{green!50!black}{D}} \\
    Stable-SCore~\cite{liu_stable-score_2025} & SD + DINO & \textbf{\textcolor{green!50!black}{D}} \\
    Morreale et al.~\cite{morreale2024neural} & DINO & \textbf{\textcolor{blue!70!black}{O}} \\
    Uzolas et al.~\cite{uzolas2025surface} & SD + DINO & \textbf{\textcolor{blue!70!black}{O}} \\
    \bottomrule
\end{tabular}
\caption{Overview of methods using \textbf{foundation models} for shape correspondence. They are categorized by the foundation model used for descriptor extraction \textbf{(FM)}: DINO~\cite{caron2021emerging,oquab2023dinov2}, Stable Diffusion~\cite{rombach2022high}, SAM~\cite{kirillov2023segment}, DiffMorpher~\cite{zhang2024diffmorpher}, as well as the downstream correspondence pipeline \textbf{(CP)}: Spectral (\textbf{\textcolor{red}{SP}}), Deformation-based (\textbf{\textcolor{green!50!black}{D}}), or Other (\textbf{\textcolor{blue!70!black}{O}}).
}
\label{tab:foundational}
\end{table}


\subsection{Neighbourhood-preservation}
\label{sec:neigh}
Neighbourhood preservation is an important aspect in shape matching as it leads to smoother correspondences (see~\cref{fig:geometric-consistency}).
Furthermore, many downstream applications rely on the fact that matchings preserve neighbourhoods, including texture transfer, shape reconstruction, and many others.
In the literature, neighbourhood preservation is also referred to as \emph{geometric consistency}, a term which was introduced by Windheuser et al.~\cite{windheuser2011geometrically}.


\newcommand{\geoConsFigWidth}{0.3\columnwidth}
\newcommand{\geoConsFigHeight}{3cm}
\begin{figure}[t]
    \centering
    \begin{tabular}{ccc}
        \small
         \includegraphics[height=\geoConsFigHeight,width=\geoConsFigWidth]{figures/paul/src}&
         \includegraphics[height=\geoConsFigHeight,width=\geoConsFigWidth]{figures/paul/cao}&
         \includegraphics[height=\geoConsFigHeight,width=\geoConsFigWidth]{figures/paul/geco}\\
         Source& \xmark\hspace{0.2cm} Neighbourhood- & \cmark\hspace{0.2cm} Neighbourhood-\\
         & preservation & preservation
    \end{tabular}
    
    \caption{Visualisation of the importance of \textbf{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:~\cite{roetzer2025geco}.}
    \label{fig:geometric-consistency}
\end{figure}


Despite its importance, neighbourhood preservation is often overlooked. 
The resulting optimisation problems are usually harder to solve, and, as a consequence, scalability is limited, as neighbourhood-preserving approaches typically do not scale to shapes with thousands of triangles.
For example, the LAP (not neighbourhood preserving) can be solved in polynomial time while the QAP (incorporates neighbourhood relations via costs) is generally NP-hard~\cite{rendl1994quadratic}.

Approaches that do incorporate neighbourhood preservation often use combinatorial optimisation techniques.
Thus, we pick up many of the aforementioned combinatorial methods in this section.
Nevertheless, there are other lines of work which build on other ideas.

In the following, we categorise neighbourhood-preserving methods into local and global approaches; see also \cref{tab:neighbourhood}.
Local approaches rely on initialisation in the form of sparse landmark correspondences or noisy input maps, whereas global approaches consider neighbourhood relations directly during optimisation of shape correspondences.

\begin{table}[t]
    \centering
    \small
    \setlength{\tabcolsep}{8pt}
    \begin{tabular}{l@{}ccccc}
        \toprule
        \textbf{Method} & \textbf{IF} & \textbf{S} & \textbf{TF} & \textbf{GNP} & \textbf{P}\\
        \midrule
        Takayama \cite{takayama2022compatible} & \xmark &\xmark & \xmark & \cmark & \xmark\\
        Schmidt et al.~\cite{schmidt2023surface} & \xmark &\xmark & \xmark & \cmark & \xmark\\
        Rorberg et al.~\cite{rorberg2023bpm} & \xmark &\cmark & \xmark & \xmark & \xmark\\
        Born et al.~\cite{born2021surface} & \xmark &\cmark & \xmark & \xmark & \xmark\\
        Magnet et al.~\cite{magnet2022smooth} & \xmark &\cmark  &(\cmark) & \xmark & \cmark\\
        Xia et al.~\cite{xia2024locality} & \xmark &\cmark &(\cmark) & \xmark & \xmark\\
        SM-Comb~\cite{roetzer_scalable_2022} & \cmark & \xmark & \xmark & \cmark & (\cmark)\\
        DiscoMatch~\cite{roetzer2024discomatch} & \cmark & \xmark & \xmark & \cmark & \xmark\\
        SpiderMatch~\cite{roetzer_spidermatch_2024} & \cmark & (\cmark) & \xmark & \xmark & (\cmark)\\
        Roetzer et al.~\cite{roetzer2025higherorder} & \cmark & \cmark & \xmark & \xmark & (\cmark) \\
        GeCo3D~\cite{roetzer2025geco} & \cmark & \cmark & \xmark & \cmark & \cmark\\
        SuPaMatch~\cite{amrani2025highres} & \cmark & \cmark & \xmark & \cmark & \cmark\\
        GC-PSM~\cite{ehm_geometrically_2024}& \cmark & \xmark & \xmark & \cmark & \cmark\\
        GC-PPSM~\cite{ehm_partial--partial_2024}& \cmark & \xmark & \xmark & \cmark & \cmark\\
        Partial-GeCo~\cite{ehm2026partialgeco} & \cmark & \cmark & \xmark & \cmark & \cmark\\
        \bottomrule
    \end{tabular}
    \caption{Overview of \textbf{neighbourhood preserving} approaches and classification if methods are \textbf{(IF)} initialisation-free, are \textbf{(S)} scalable, are \textbf{(TF)} topologically flexible, are \textbf{(GNP)} globally neighbourhood preserving, or can solve \textbf{(P)} shape matching problems where at least one shape is partial.
    The scalability of SpiderMatch~\cite{roetzer_spidermatch_2024} depends on the input representation; see \cite{roetzer2025geco} for more details.
    Approaches~\cite{magnet2022smooth,xia2024locality} do not show experiments on topologically noisy shapes but would, in theory, support such inputs.
    The formalism solved by SM-Comb~\cite{roetzer_scalable_2022} 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~\cite{roetzer_spidermatch_2024, roetzer2025higherorder} do not show partial-to-full experiments, but their formalisms naturally support this case.
    }
    \label{tab:neighbourhood}
\end{table}






\subsubsection{Local optimisation}
Several lines perform local optimisation, or \emph{refinement}, to obtain neighbourhood-preserving maps, or rather to convert existing maps into neighbourhood-preserving ones.

In recent years, Takayama~\cite{takayama2022compatible} and Schmidt et al.~\cite{schmidt2023surface} propose using piece-wise linear transformations (which can be seen as piece-wise plane embeddings) to refine sparse landmark correspondences into dense, neighbourhood-preserving matchings by building on so-called intrinsic triangulations, see~\cite{sharp2019navigating}.
Other works use piece-wise Möbius transformations to obtain smoother maps~\cite{rorberg2023bpm}, inspired by ~\cite{lipman2009mobius}.

For shape pairs with genus greater than zero, \cite{born2021surface} propose to use homology transfer to obtain high-quality maps. 
Here, the homology bases and their respective transfer functions act as low-pass filters, making the resulting maps smooth by filtering out local noisy/inconsistent matches.
In~\cite{magnet2022smooth}, authors propose to use Dirichlet energy minimisation in a functional map framework to obtain smooth maps.
Further, \cite{xia2024locality} finds inliers of given noisy input correspondences by exploiting neighbourhood information, and subsequently, by utilising these inliers, they refine the outliers.

These refinement methods, by definition, rely on the quality of the input map and, therefore, matchings could in theory be arbitrarily bad.
In contrast, global approaches are not dependent on initialisation.

\subsubsection{Global approaches}\label{sec:glob-nei-nei}

Global neighbourhood preserving approaches refer to 
approaches that are initialisation-free, explicitly consider neighbourhood relations during optimisation, and provide (some form of) optimality certificates.
Among the most important approaches in this category is the integer linear programming formalism proposed in~\cite{windheuser2011geometrically} in which neighbourhoods of surface elements are preserved as a hard constraint and, as already mentioned, which shaped the term ``geometric consistency''.
Yet, this formalism is hard to solve and does not scale to higher shape resolutions.
Motivated by this, \cite{roetzer_scalable_2022, roetzer2024discomatch} proposed approximate solvers which yield improved scalability.

Other lines of work aim for novel formalisms.
Among these, SpiderMatch~\cite{roetzer_spidermatch_2024} (see \cref{sec:combinatorial}) achieves geometric consistency by representing a 3D shape as a cyclic, self-intersecting curve and by preserving intersections in their matching formalism.
Building on this idea, \cite{roetzer2025geco} encodes each triangle of a 3D shape with a single cyclic curve, leading to independent matching problems for each curve. 
Geometric consistency is then enforced by coupling the problems at opposite edges such that the constraints always lead to a faithful surface-to-surface matching and, with that, to better geometric consistency compared to \cite{roetzer_spidermatch_2024}.
In addition, this formalism is faster to solve and, in practice, can be solved to global optimality.
Motivated by the improved scalability, follow-up work~\cite{amrani2025highres} builds on the idea of an alternative shape representation and proposes a two-stage formalism: first, transfer patches from source to target shape and second, find a dense surface-to-surface matching for each pair of corresponding patches.
Yet, for both approaches, there are no theoretical guarantees of polynomial runtime, and thus it remains an open research question whether there exists a neighbourhood-preserving shape-matching approach that is globally optimal in polynomial time.

In contrast, \cite{ehm_geometrically_2024,ehm_partial--partial_2024} propose extensions of the formalism by Windheuser et al.~\cite{windheuser2011geometrically} to partial-to-full~\cite{ehm_geometrically_2024} and partial-to-partial shape matching~\cite{ehm_partial--partial_2024}.
For partial-to-full shape matching \cite{ehm_geometrically_2024} relaxes the hard neighbourhood constraints proposed in~\cite{windheuser2011geometrically}.
For the more challenging partial-to-partial shape-matching task, \cite{ehm_partial--partial_2024} proposes solving for the mean matching cost rather than the sum of matching costs.
This effectively allows for solving for correspondences and unknown overlap at the same time.
Yet, it is harder to solve due to the resulting non-linear objective function and requires solving multiple integer linear programs.
To address the challenges posed by the non-linear objective, \cite{ehm2026partialgeco} adapts \cite{roetzer2025geco} to the partial-to-partial setting and introduces an integer linear programming formalism that explicitly enforces geometric consistency.
By integrating a predicted overlap region directly into the objective function, the resulting approach scales significantly better than the previous geometrically consistent method for partial-to-partial shape matching.
In Section~\ref{sec:partiality}, we discuss additional approaches to partial shape matching.

\subsection{Neural correspondence optimization}

The machine learning and computer vision communities alike have become increasingly interested in approximating complex optimisation problems with neural networks~\cite{wang2025vggt}.
Deep learning provides a flexible mechanism for integrating intricate, data-driven priors to approximate what would otherwise be solved iteratively, such as the correspondence between two shapes.
As we saw in~\cref{subsec:fm_diffusion}, the optimisation to recover a functional map for a given set of spectral correspondences can be approximated with a diffusion model trained on a large collection of such maps \cite{zhuravlev_denoising_2025,pierson2025diffumatch}.
\cite{wu2024diff} recently observed that the analogous can be achieved for doubly stochastic ``soft'' correspondence matrices.
This raises the question of whether more complex costs and objectives, such as neighbourhood constraints, can be incorporated into such a formulation, paying the cost of computationally intensive numerical optimisation upfront during training time. 

\subsection{Partial shape matching}
\label{sec:partiality}

As mentioned in previous chapters, the problem of matching two complete shapes has been explored in many works.
However, partial shape matching, in which at least one shape is incomplete, has so far received less attention.
In the context of partial shape matching, we distinguish between two cases: partial-to-full shape matching, in which only one of the shapes is partially observed, and partial-to-partial shape matching, in which both shapes are incomplete.

\begin{figure}[t]
    \centering
    \includegraphics[width=1\linewidth]{figures/emerging_trends_partiality}
    \caption{\textbf{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~\cite{bogo_faust_2014}, KIDS~\cite{rodola2014dense} and BeCoS~\cite{ehm2025beyond} datasets.}
    \label{fig:partial_settings}
\end{figure}

\subsubsection{Partial-to-full shape matching} \label{sub:partial-to-full}
In partial-to-full shape matching, we explore the task of a partially observed shape $\mathcal{N}$ being matched on a full template shape $\mathcal{M}$ (see Fig.~\ref{fig:partial_settings} left).

\begin{table}[t]
    \centering
    \small
    \begin{tabular}{l@{}ccc}
    \toprule
        \textbf{Method} & \textbf{Setting} & \textbf{Framework} & \textbf{Type} \\
        \midrule
        DPFM~\cite{attaiki2021dpfm} & P2F,P2P & PFM & \textcolor{orange!90!black}{S},\textbf{\textcolor{green!50!black}{U}}\\
        Bensaid et al.~\cite{bensaid_partial_2023} & P2F, P2P & PFM & \textcolor{orange!90!black}{S} \\
        Bensaid et al.~\cite{bensaid_multi-spectral_2024} & P2F & Spectral & \textbf{\textcolor{red!70!black}{A}} \\
        Bracha et al.~\cite{bracha_unsupervised_2024} & P2F & Dist & \textbf{\textcolor{green!50!black}{U}} \\
        Wormhole\cite{bracha_wormhole_2024} & P2F & Dist & \textbf{\textcolor{green!50!black}{U}} \\
        ULRSSM~\cite{cao_unsupervised_2023} & P2F & PFM & \textbf{\textcolor{green!50!black}{U}} \\
        Magnet et al.~\cite{magnet2022smooth}& P2F & Neigh & \textbf{\textcolor{red!70!black}{A}} \\
        GC-PSM~\cite{ehm_geometrically_2024} & P2F & Neigh & \textbf{\textcolor{red!70!black}{A}} \\
        GeCo3D~\cite{roetzer2025geco} & P2F & Neigh & \textbf{\textcolor{red!70!black}{A}}\\
        El Amrani et al.~\cite{amrani2025highres} & P2F & Neigh & \textbf{\textcolor{red!70!black}{A}}\\ EchoMatch~\cite{xie_echomatch_2025} & P2P & PFM & \textcolor{orange!90!black}{S} \\
        GC-PPSM~\cite{ehm_geometrically_2024} & P2P & Neigh & \textbf{\textcolor{red!70!black}{A}} \\
        Partial-GeCo~\cite{ehm2026partialgeco} & P2P & Neigh & \textbf{\textcolor{red!70!black}{A}}\\
    \bottomrule
    \end{tabular}
    \caption{Categorization of Partial Shape Matching Methods: We categorize partial shape matching methods in their setting (partial-to-full (\textbf{P2F}) or partial-to-partial (\textbf{P2P})), their underlying framework (\textbf{P}artial \textbf{F}unctional \textbf{M}aps, \textbf{Spectral} alignment, \textbf{Dist}ance-based or \textbf{Neigh}bourhood-preserving) and their optimisation type (axiomatic (\textbf{\textcolor{red!70!black}{A}}), 
supervised (\textbf{\textcolor{orange!90!black}{S}}), unsupervised (\textbf{\textcolor{green!50!black}{U}})).
}
\label{tab:partial}
\end{table}

\paragraph{Partial functional maps.}
In one of the first works to address partial-to-full shape matching, the authors observe a partial setting in the functional map context~\cite{rodola2017partial} for isometric shape matching.
For reference, for isometric shapes we observe a close to diagonal structure of the functional map $C_{\mathcal{MN}} = \phi_\mathcal{N}^{\dagger}\Pi_{\mathcal{NM}}\phi_\mathcal{M}$ with an optimal point-wise map $\Pi_{\mathcal{NM}}$.
In the partial-to-full scenario, this is not the case, as some eigenfunctions of the full shape do not have a corresponding eigenfunction in the partial shape.
Therefore, instead of a full diagonal, we observe a slanted diagonal structure with rank $r$ of the partial functional map $C_{\mathcal{MN}}$.
The rank $r$ can be estimated by comparing the surface areas of the two shapes~\cite{weyl1911asymptotische}.

Similar to the full-to-full case, the partial functional map has several key properties.
One is semi-orthogonality, which ensures that the map preserves volume when transferring functions from the partial shape to the full one, but not necessarily in the opposite direction. The other is bijectivity, which requires that mapping a function from the partial shape to the full shape and then back recovers the original function.
While supervised methods rely on the ground-truth functional map to guide training~\cite{attaiki2021dpfm}, these structural properties provide a foundation for unsupervised learning~\cite{attaiki2021dpfm, cao_unsupervised_2023}, including the orthogonality losses

\begin{equation}
\left\lVert C_{\mathcal{M}\mathcal{N}} C_{\mathcal{M}\mathcal{N}}^{\top} - I_r \right\rVert_F^2 
\text{ and } 
\left\lVert C_{\mathcal{N}\mathcal{M}}^{\top} C_{\mathcal{N}\mathcal{M}} - I_r \right\rVert_F^2,
\end{equation}
\noindent and the bijectivity loss,
\begin{equation}
\left\|C_\mathcal{MN}C_\mathcal{NM} - I_r\right\|^2_F.
\end{equation}
Here, $I_r$ denotes the identity matrix with only the first $r$ elements being 1.
DPFM~\cite{attaiki2021dpfm} introduces a cross-attention refinement module that enables feature communication across shapes.
Bensaid et al.~\cite{bensaid_partial_2023} propose learning a piecewise-smooth function over the surface: sparse landmark correspondences between a full shape and its partial counterpart are first established via feature similarity, and a neural representation is then optimised to fit these landmarks, enabling smooth interpolation between matched features that serve as anchors.

\paragraph{Spectral alignment methods.}
Rampini et al.~\cite{rampini2019correspondence} align spectral representations derived from the Laplace–Beltrami operator (LBO).
Building on this idea, Bensaid et al.~\cite{bensaid_multi-spectral_2024} combine the standard LBO with its scale-invariant variant (SI-LBO), leveraging the complementary information from both spectra to better capture semantically meaningful curved regions.

\paragraph{Distance-based methods.}
Another line of work shows that using functional maps in the partial shape matching setting introduces unavoidable errors~\cite{bracha_unsupervised_2024}.
Therefore, the method primarily operates in the spatial domain rather than the spectral domain.
The main loss preserves the geodesic distance (surface distance) between points on both shapes.
Bracha et al.~\cite{bracha_wormhole_2024} tackle the problem that the geodesic distance of a partial shape of two points does not necessarily have the same length as the same two points on a full shape, due to possible holes.
Therefore, reliable point pairs are identified and are later used in the loss formulation.

\paragraph{Neighbourhood-preserving methods.}
\cite{magnet2022smooth,ehm_geometrically_2024,amrani2025highres,roetzer2025geco} use integer neighbourhood-preserving techniques to find solutions for the partial-to-full shape matching problem, see also \cref{sec:glob-nei-nei}.





\subsubsection{Partial-to-partial shape matching} \label{sub:partial-to-partial}
The more challenging problem in partial shape matching is the partial-to-partial setting, where we have to solve two problems in parallel: finding the overlapping region and determining correspondences in it (see Figure~\ref{fig:partial_settings} right). Due to its complexity, it remains an underexplored problem and warrants further consideration.


So far, only a few supervised learning-based methods~\cite{attaiki2021dpfm, bensaid_partial_2023,xie_echomatch_2025} exist that tackle the partial-to-partial shape matching problem.
Two methods~\cite{xie_echomatch_2025, attaiki2021dpfm} mainly operate similarly to the partial-to-full or full-to-full setting, but add an overlap prediction module to determine the overlapping region.
For two partial shapes $\mathcal{M}$ and $\mathcal{N}$ define a overlap prediction vector $p_\mathcal{M} \in [0,1]^{n_\mathcal{M}}$ and $p_\mathcal{N} \in [0,1]^{n_\mathcal{N}}$, indicating the probability that a vertex is in the overlapping region or not.
These predictions are compared with a binary cross entropy loss (BCE) with the ground truth overlap $\text{gt}_\mathcal{M}$ and $\text{gt}_\mathcal{N}$: 


\begin{equation}
    \Loss_{\text{overlap}} = 
    \frac{1}{2} 
    \Bigl[ 
      \text{BCE}\!\left(p_\mathcal{M},\, \text{gt}_\mathcal{M} \right)
      + 
      \text{BCE}\!\left(p_\mathcal{N},\, \text{gt}_\mathcal{N} \right)
    \Bigr].
\end{equation}

DPFM~\cite{attaiki2021dpfm} uses the predicted features directly in an MLP to predict the overlap predictions $p_\mathcal{M}$ and $p_\mathcal{N}$.
In contrast, EchoMatch~\cite{xie_echomatch_2025} introduces a correspondence reflection module that leverages cycle consistency of the point-wise maps, \( P_\mathcal{M} = \Pi_{\mathcal{M}\mathcal{N}} \Pi_{\mathcal{N}\mathcal{M}} \), to verify whether a point returns to its original neighbourhood.  
It is further demonstrated that image-based DINOv2 features are effective for handling partial shapes.  
A different strategy is proposed in~\cite{bensaid_partial_2023}, where anchor correspondences are first established to enable smooth interpolation.  
For the partial-to-partial setting, two additional area-based losses are introduced: one to enforce similarity of overlapping regions across shapes, and another to encourage large overlap.
\cite{ehm_partial--partial_2024, ehm2026partialgeco} propose an axiomatic neighbourhood-preserving solution for partial-to-partial shape matching, as mentioned in described ~\ref{sec:neigh}.

\paragraph{Metric adaptations.}

Metrics must be properly adapted for the partial correspondence settings.
In the partial-to-full (P2F) case, where the source shape $\mathcal{M}$ is a partial observation and the target shape $\mathcal{N}$ is complete, the metrics are computed identically to the full-to-full case.
In contrast, the partial-to-partial (P2P) setting assumes that both the predicted and ground-truth correspondences are defined only on the overlapping regions of the source and target shapes. 
Let $\Omega^* \subseteq \mathcal{M}$ denote the ground-truth overlap region, and $\Omega \subseteq \mathcal{M}$ the predicted one. 
To ensure consistency across partial and full scenarios, all geodesic distances are normalised by the square root of the area of the full target shape, $\sqrt{\text{Area}(\mathcal{N}_{\text{full}})}$.
Vertices in the ground-truth overlap that were not matched ($i \in \Omega^* \setminus \Omega$), as well as vertices falsely matched outside the overlap ($i \in \Omega \setminus \Omega^*$), are penalised with infinite error: $e(i) = \infty$. 
When evaluating the mean geodesic error, PCK, or AUC, only correspondences with finite error ($e(i) < \infty$) are considered. 
This ensures that the metrics measure accuracy within the overlap area, while ignoring unmatched or undefined regions. 
Additional evaluation metrics for the P2P setting are discussed in~\cite{ehm2025beyond,ehm_partial--partial_2024}.

While previously discussed mesh-based approaches rely on explicit surface connectivity, the next section focuses on matching pairs of point clouds, where such connectivity information is unavailable. 
Structural priors can therefore only approximate connectivity.

\subsection{Point cloud matching}


Several works have focused specifically on the point cloud setting.
Such methods are generally also applicable to structured mesh representations but cannot leverage as strong a structural prior; thus, methods like those in \cref{sec:spectral} and \ref{sec:combinatorial} are preferred.
Nevertheless, unstructured point cloud acquisitions are ubiquitous in real-world settings, and thus these methods remain of fundamental importance.

Point cloud correspondence methods generally rely on strong feature representations for matching because there are few structural priors to exploit.
Many require ground-truth supervision; those that don't typically regularise the resulting deformation using the chamfer loss, which requires that the shapes already have a reasonable extrinsic alignment, or using piecewise ARAP deformations.
Corrnet3d~\cite{zeng2021corrnet3d} predicts high-dimensional point-wise features for both point clouds, which are used to regress the permutation matrix between them. 
This permutation matrix is used to deform the point clouds.
DPC~\cite{lang2021dpc} uses point-wise features for self-construction and cross-construction, where self-construction encourages smooth correspondence maps and cross-construction unique matches.
In~\cite{netto2022robust}, the authors use self- and cross-attention modules to predict a soft correspondence matrix, where, in comparison to Corrnet3d and DCP, they also adopt a \textit{dustbin} paradigm to discard correspondences.
In~\cite{he2023hierarchical}, the authors use \textit{multi-receptive-field point encoders} with different receptive radii to compute hierarchical features.
SE-ORNet~\cite{deng2023se} aligns the orientation of shapes to tackle the symmetry problem in human and animal shapes.
In~\cite{deng2024unsupervised}, the authors propose learning a set of templates with explicit structures that can later be used for better shape correspondences.
Spectral methods can indeed also be applied in point cloud settings~\cite{cao_self-supervised_2023} due to the development of point cloud Laplacians~\cite{sharp2020laplacian}, yielding feature encoders that can operate on both mesh and point clouds.
The same holds for the Gaussian splatting representation~\cite{zhou2025gaussLBO}.
Alternatively, graphs are also well-suited for such multi-modal  encoders~\cite{saleh_bending_2022}.

While these methods mostly handle the task of matching complete, noisy point clouds, some also target partiality.
Lepard~\cite{li2022lepard} uses a Transformer architecture with self and cross-attention layers. 
While primarily designed for the rigid case, design a deformation module which predicts a warp function for every point in a dense point cloud for non-rigid settings. 
Training also proceeds in a supervised manner. 
Later, a rotation invariant method~\cite{yu2023rotation} was introduced to prevent instabilities for unseen rotations. 
They introduce a rotation-invariant attention-based encoder-decoder architecture for local geometry and a global transformer with rotation-invariant cross-frame position awareness. 
Similar to Lepard, they predict a per-point flow to also tackle the deformable case.
DiffReg~\cite{wu2024diff} builds upon these techniques, but proposes to recover a correspondence map through a diffusion process in the space of double-stochastic matrices.





