\section{Background \& fundamentals}
\label{sec:fundamentals}









This section provides an overview of the shape correspondence problem setting. 
We begin by discussing how surfaces can be characterised locally using \textit{feature descriptors}, which is essential for estimating correspondence between shapes.
These descriptors serve as an initial, local similarity measure but are often insufficient on their own due to ambiguity and limited discriminative power.
Additional \textit{structural priors} can be introduced to constrain the solution space and improve the faithfulness of the recovered correspondences.
These priors provide the conceptual foundation for the three methodological categories discussed in later sections.
However, no type of structural prior has emerged as a silver bullet; each has its \textit{limitations} and is best suited to specific settings.
The notation we adopt throughout this report is summarised in Table~\ref{tab:notation}.

\begin{table}[t]
\centering
\footnotesize
\begin{tabularx}{\linewidth}{lll}
\toprule
\textbf{Symbol} & \textbf{Domain} & \textbf{Description} \\
\midrule
$\mathcal{M}, \mathcal{N}$ & & 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 $\mathcal{M}$ \\
$\phi_\mathcal{M}$ & $\mathbb{R}^{n_\mathcal{M} \times k}$ & Eigenfunctions of $L_\mathcal{M}$ (spectral basis) \\
$C_{\mathcal{M} \mathcal{N}}$ & $\mathbb{R}^{k \times k}$ & Functional map from $\mathcal{M}$ to $\mathcal{N}$ \\
$\Pi_{\mathcal{M} \mathcal{N}}$ & $\{0,1\}^{n_\mathcal{M} \times
n_\mathcal{N}}$ & Point-wise map from $\mathcal{M}$ to $\mathcal{N}$ \\
$\Energy(\cdot)$ & & Energy or cost function to be minimized \\
$\Loss(\cdot)$ & & Loss for neural network training \\
\bottomrule
\end{tabularx}
\caption{Summary of \textbf{notation} used throughout the report.}
\label{tab:notation}
\end{table}

\subsection{The shape correspondence problem}

In the context of 3D geometry processing, a shape is typically defined as a two-dimensional surface embedded in three-dimensional space. 
Such a surface is often discretised as a triangular mesh, consisting of vertices \( X_\mathcal{M} \), edges \( E_\mathcal{M} \), and faces \( T_\mathcal{M} \). 
When a surface triangulation is not given, the vertices \( X_\mathcal{M} \) can be considered as an unstructured point cloud.
Given two shapes, denoted \( \mathcal{M} \) and \( \mathcal{N} \), the goal of shape correspondence is to determine which points on \( \mathcal{M} \) match semantically or geometrically to points on \( \mathcal{N} \). 
This relationship can be expressed as a point-wise map \( \Pi_{\mathcal{M} \mathcal{N}}: \mathcal{M} \rightarrow \mathcal{N} \), assigning to each point on the source shape a corresponding point on the target shape.

Throughout the report, we use the terms \textit{correspondence} and \textit{matching} interchangeably.
We focus primarily on the dense correspondences setting, which establishes a matching for every point on the source shape to the target shape (formally, we consider $n_{\mathcal{M}} \leqslant n_{\mathcal{N}}$ with $\Pi_{\mathcal{M}\mathcal{N}}$ encoding an injective map, i.e., $\Pi_{\mathcal{M}\mathcal{N}} \mathds{1} = \mathds{1}$ and $\Pi_{\mathcal{M}\mathcal{N}}^\top \mathds{1} \leqslant \mathds{1}$, where we denote the all-ones vector as $\mathds{1}$).
Sparse, landmark-based approaches can also be useful in certain applications, specifically as they can be annotated more easily than dense maps (see \cref{sec:combinatorial}).
The deformation-based family of works also considers the registration settings, where a deformation field is recovered in the ambient domain without first matching points locally (see \cref{sec:deformation}).
When two shapes are aligned in this manner, dense correspondence maps can typically be recovered via a heuristic.






\begin{figure}[t]
  \centering

  \begin{subfigure}{0.98\linewidth}
    \centering
    \includegraphics[width=0.36\linewidth]{figures/02/dt4d-Standing2HMagicAttack01015-BrooklynUprock077_M}
    \includegraphics[width=0.36\linewidth]{figures/02/dt4d-Standing2HMagicAttack01015-BrooklynUprock077_N}
    \caption{Feature functions on source and target shape.}
    \label{fig:02_features}
  \end{subfigure}
  \hfill
  \vspace{2mm} %

  \begin{subfigure}{0.6\linewidth}
    \centering
    \includegraphics[width=\linewidth]{figures/02/out_single_plot_front_lines_no_shadow}
    \caption{Visualised correspondences.}
    \label{fig:02_matches}
  \end{subfigure}

  \caption{
    \textbf{Visualising} and \textbf{matching features} between shapes. 
    \textbf{(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.
    \textbf{(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:~\cite{gao_sigma_2023}.
    }
  \label{fig:fundamentals}
\end{figure}


\subsection{Characterising a shape's surface}

A central goal in correspondence estimation is to quantify the similarity between shapes in a tractable form. 
Given shapes \( \mathcal{M} \) and \( \mathcal{N} \), per-vertex feature functions \( f_\mathcal{M} \in \mathbb{R}^{n_\mathcal{M} \times d} \) and \( f_\mathcal{N} \in \mathbb{R}^{n_\mathcal{N} \times d} \), also referred to as descriptors, can be specified based on a heuristic or estimated in a data-driven manner. 
These descriptors are intended to encode geometric or semantic properties of the shapes, see Fig.~\ref{fig:02_features} for an illustration. 
Then, for any point on shape \( \mathcal{M} \) and a ground-truth point-wise map \( \Pi_{\mathcal{M}\mathcal{N}} \in \{0,1\}^{n_\mathcal{M} \times n_\mathcal{N}} \), the descriptor preservation principle assumes that the corresponding point on \( \mathcal{N} \) should have similar descriptors:
\begin{equation}
\label{eq:descr_preservation}
    f_\mathcal{M} \approx \Pi_{\mathcal{M}\mathcal{N}} f_\mathcal{N}.
\end{equation}


Several types of descriptors have been proposed for both structured and unstructured shape representations. 
They can be broadly categorised by their complexity and level of abstraction.

The most basic descriptors are derived directly from \emph{geometric properties} of the shapes themselves. 
These include both extrinsic properties, such as 3D vertex coordinates~\cite{tombari2010unique} or surface normals~\cite{salti2014shot}, and intrinsic properties, such as local curvature~\cite{meyer_discrete_2003,windheuser2011geometrically}.
While the resulting \textit{hand-crafted} descriptors capture local geometric information, they each have inherent limitations. 
Extrinsic descriptors such as SHOT \cite{tombari2010unique} characterise surfaces through local coordinate systems but are not isometrically invariant. 
Conversely, intrinsic descriptors such as the Heat and Wave Kernel Signatures \cite{sun2009concise,aubry2011wave} naturally inherit the Laplace-Beltrami operators' isometric invariance but are not symmetry aware, can change drastically under non-isometric deformations, and remain sensitive to noise.
Despite these limitations, these descriptors can still yield effective matches when combined with strong geometric constraints on the point-wise map~\cite{windheuser2011geometrically}.
Moreover, they are easily applicable in various domains, in contrast to the more discriminative learned descriptors.


Recent approaches increasingly rely on \emph{learning} geometric descriptors by optimising for correspondence on collections of shapes~\cite{litany2017deep}. 
These models can transform raw geometric inputs (e.g. vertex positions) or axiomatic descriptors (e.g. HKS, WKS, SHOT) to more expressive, domain-specific features \cite{sharp2022diffusionnet, saleh2020graphite}.
Learned descriptors can overcome many of the limitations of hand-crafted features and adapt to challenging non-isometric or partial matching scenarios.
In practice, however, learning sufficiently discriminative feature descriptors remains challenging.
Surface ambiguities, noise, symmetries, shape partiality, and discretisation artefacts can yield descriptors with numerous plausible matching candidates on the target shape.


While obtaining informative descriptors is essential for solving the correspondence problem, recovering a point-wise map (visualised in Fig.~\ref{fig:02_matches}) is non-trivial even from descriptors learned by state-of-the-art models.
A naive approach to obtain a correspondence map would be to match each point on shape \( \mathcal{M} \) to the most similar descriptor on shape \( \mathcal{N} \) via nearest-neighbour search:
\begin{equation}\label{eq:matching-intro}
\underset{\Pi_{\mathcal{M}\mathcal{N}} \in \{0, 1\}^{n_{\mathcal{M}} \times n_{\mathcal{N}}}}{\arg \min} \| f_\mathcal{M} - \Pi_{\mathcal{M}\mathcal{N}} f_\mathcal{N} \|_F^2. 
\end{equation}

\noindent However, this formulation imposes no constraints on the resulting map. 
Inaccuracies or ambiguities in the feature descriptors representation can lead to errancies: for example, neighbouring points on \( \mathcal{M} \) may be mapped to distant locations on \( \mathcal{N} \), yielding an undesirable \textit{geometric inconsistency}. 
To overcome this, many recent methods impose additional assumptions or structural priors to obtain more accurate correspondences.





\subsection{Methodological Overview}



Solving the matching problem in \cref{eq:matching-intro}, while straightforward, typically yields point-wise maps with errant feature pairings.
To address this, correspondence optimisation is typically augmented with prior knowledge about the structure and geometry of the shapes, constraining solutions to more plausible point-wise maps. 
We group recent developments into three main categories:



\paragraph{Spectral methods.} 
A common assumption is that the desired map is smooth and can be compactly represented in a low-frequency basis.
The resulting insight that such a smooth map can be represented in a low-frequency function space has led to the rise of the seminal \textit{functional maps} framework~\cite{ovsjanikov_functional_2012}.
Most commonly used to construct this function space are the first $k$ eigenfunctions of the Laplace-Beltrami operator (LBO), which form an orthonormal basis that captures the intrinsic geometry of the surface and remains invariant under isometric deformations~\cite{belkin2003laplacian}.
The key insight of the functional map framework is that instead of directly matching points between shapes, one can align the eigenfunctions $\phi = [\phi_1, \phi_2, \ldots, \phi_k] \in \mathbb{R}^{n \times k}$ for a pair of shapes $\mathcal{M}$ and $\mathcal{N}$ in spectral space through a low-rank linear map $C_{\mathcal{MN}} \in \mathbb{R}^{k \times k}$. 
Given eigenfunctions $\phi_{\mathcal{M}}$ and $\phi_{\mathcal{N}}$ on shapes $\mathcal{M}$ and $\mathcal{N}$, and a pointwise correspondence $\Pi_{\mathcal{NM}}$, the relationship between them is captured as:
\begin{equation}
\label{eq:intro-approx-fm}
\Pi_{\mathcal{NM}}\phi_{\mathcal{M}} \approx \phi_{\mathcal{N}}C_{\mathcal{MN}}.
\end{equation}
\noindent Intuitively, this aligns the spectral bases between shapes by transferring an indicator function from shape $\mathcal{M}$ through the point-wise map, yielding a function on $\mathcal{N}$ whose spectral coefficients are determined by the functional map $C_{\mathcal{MN}}$.

The LBO eigenfunctions are famously invariant under non-isometric deformation, and robust even under mild deviations from isometry \cite{reuter2006laplace}. 
Numerous recent works leverage the insight that shape descriptors, whether hand-crafted or learned, can be projected onto the basis functions to establish correspondences in spectral space.
The functional map $C$ naturally regularises the point-wise map due to its low dimensionality and the smoothness of the LBO eigenfunctions.
Structural priors have also been derived for $C_{\mathcal{MN}}$, encouraging desirable properties such as promoting approximately isometric mappings or bijectivity~\cite{roufosse2019unsupervised}.
The use of such geometric priors has led to unsupervised paradigms that enable state-of-the-art pipelines to learn robust and discriminative feature descriptors from collections of shapes~\cite{cao_unsupervised_2023}.

However, the low-rank spectral representation inherently compresses the recovered map, limiting the ability to represent high-frequency geometric details.
Moreover, the quality of the LBO eigenfunctions degrades in the presence of non-isometries \cite{bastian2024hybrid} and holes or partiality~\cite{ehm2025beyond}, and extracting a faithful point-wise map from a functional map remains a non-trivial step that can introduce additional error \cite{vigano_adjoint_2023}.
These trade-offs are discussed in detail in \cref{sec:spectral}.

\paragraph{Combinatorial methods.}
Solving the naive nearest-neighbour matching in \cref{eq:matching-intro} yields an initial point-wise map, but imposes no structure on the solution.
A natural extension is the Linear Assignment Problem (LAP) which considers recovering a bijective mapping (i.e., $n_{\mathcal{M}} = n_{\mathcal{N}}$, obtainable through re-meshing or subsampling) that minimises a linear dissimilarity cost $D \in \mathbb{R}^{n_{\mathcal{M}} \times n_{\mathcal{N}}}$ over the set of permutation matrices $\mathbb{P}$:
\begin{equation}\label{eq:intro-lap}
\underset{\Pi \in \mathbb{P}}{\arg \min} \; \langle D, \Pi \rangle \quad \text{where, e.g.} \quad D_{ij} = \| f_\mathcal{M}(i) - f_\mathcal{N}(j) \|^2.
\end{equation}
\noindent This formulation has been extensively studied and can be solved efficiently through the Auction algorithm~\cite{bertsekas1992auction}.
Nevertheless, it still relies entirely on feature similarity and neglects neighbourhood relationships between features.
This can lead to \textit{geometrically inconsistent} matchings where neighbouring points on $\mathcal{M}$ map to distant locations on $\mathcal{N}$.

On the other hand, the Quadratic Assignment Problem (QAP) incorporates a metric that is preserved between matching pairs of points (geodesic distances are commonly chosen) but is famously NP-hard~\cite{rendl1994quadratic}, making solutions computationally intractable for most meshes of practical size.
Many recent combinatorial shape matching methods, therefore, seek a middle ground, incorporating geodesic constraints by exploiting the surface's discrete structure while avoiding the full quadratic cost.
Recent approaches have improved scalability through convex relaxations~\cite{kushinsky2019sinkhorn,droge_kissing_2023}, proposed heuristics~\cite{vestner2017efficient, haller2022comparative, roetzer_scalable_2022}, or additional constraints like neighbourhood preservation~\cite{windheuser2011geometrically,ehm_geometrically_2024,roetzer_spidermatch_2024}.

Despite these scalability challenges, the combinatorial family of methods discussed in \cref{sec:combinatorial} has recently gained traction, as they allow enforcing that solutions have certain geometric properties by constraining the feasible set.
Beyond neighbourhood preservation, this also extends to cycle consistency across shape collections~\cite{bhatia_ccuantumm_2023,kahl2025towards}.
However, combinatorial methods require reasonably discriminative feature representations as input and are typically non-differentiable, thus serving as a complement rather than a replacement to learned methods.


\paragraph{Deformation-based methods.} This class of methods directly recovers a global deformation in the ambient embedding space without first decomposing the problem into one of matching local elements.
In general, one may seek to recover a global deformation field $\mathcal{F}: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ that aligns the source shape $\mathcal{M}$ to the target shape $\mathcal{N}$:
\begin{equation}
\mathcal{F}(X_{\mathcal{M}}) \approx {\Pi_{\mathcal{MN}}}X_{\mathcal{N}}. 
\end{equation}
Pioneered by iterative closest point (ICP) \cite{besl1992method} and coherent point drift (CPD) \cite{myronenko2010point}, this class of methods remains widely applicable in settings where obtaining discriminative feature descriptors is challenging.
As the general formulation is highly ill-posed, the deformation field $\mathcal{F}$ is typically regularised with priors that induce spatial smoothness, or require that it is locally rigid.
Recent methods have emphasised more robust regularisation techniques~\cite{yao2023fast,yao2025spare}, parametrising the deformation with neural networks~\cite{li2022non,sundararaman_deformation_2024}, and the incorporation of parametric templates into the optimisation or learning problem~\cite {Feng_2023_ICCV,li2025etchgeneralizingbodyfitting}.

The deformation-based methods discussed in \cref{sec:deformation} operate in the ambient Euclidean space and are thus broadly applicable to various 3D data representations, including unstructured point clouds, without requiring manifold connectivity or consistent topology.
This extrinsic paradigm is particularly suited to settings where noise, topological changes, or incomplete data pose challenges for methods requiring clean surface geometry.
Recent works have further extended the framework with robust norms~\cite{yao2023fast}, parametric templates~\cite{Feng_2023_ICCV,li2025etchgeneralizingbodyfitting}, and learned deformation priors~\cite{li2022non,marin2024nicp}.
However, the joint recovery of correspondence and deformation is inherently ill-posed, and alternating optimisation schemes inherited from ICP~\cite{besl1992method} remain sensitive to initialisation and prone to local minima.

\subsection{Challenges across matching scenarios}

Choosing a suitable correspondence method for a given application largely depends on the characteristics of the shapes being matched. 
As shape correspondence methods are applied to an increasingly wide range of geometric inputs, no single approach has remained universally applicable. 
In practice, the assumptions underlying a given method may hold only approximately, or be entirely violated, depending on the specific matching scenario. 
Therefore, it is essential to consider the nature of the shapes, particularly their degree of similarity, partiality, and the availability of training data (with or without annotations), when deciding on the appropriate methodology. 

A key factor is the nature of the deformation. 
When the shapes are represented as water-tight meshes and near-isometric, as in cases of simulated pose variation or mild articulation, most structural and spectral assumptions of matching methods remain valid. 
In contrast, non-isometric scenarios (e.g., matching across different animal species) are more challenging.
As geometric similarity decreases, the availability of sufficiently discriminative features decreases as well, requiring stronger constraints in correspondence optimisation, such as strictly preserving local neighbourhoods.
Especially difficult non-isometric cases occur when shapes deviate from idealised, clean, and manifold conditions. 
Real-world 3D data often contains topological noise, including non-uniform sampling, small holes, or self-intersections. 
Such violations of smooth manifold structure can compromise correspondence methods that depend on local geometric information, resulting in unstable or inconsistent matches between shapes.


A further challenge arises in the case of \textit{partiality}, where one or both shapes are incomplete.
In full-to-full matching, it is generally assumed that every point of one shape has a corresponding partner on the other.
In the case of partial shape matching, this assumption is often grossly violated.
In the partial-to-full matching scenario, a mapping can be defined from the partial shape to the full one, but not the other way round, since not every region of the full shape is observed in the partial shape.
This becomes even more challenging in the partial-to-partial case, where both shapes are incomplete. 
Here, it is necessary not only to establish correspondences but also to identify the overlapping region between the two shapes.
These difficulties demand the development of more sophisticated matching strategies, which we emphasise in Section~\ref{sec:partiality} as the problem is largely unresolved.




\begin{figure*}[t]
  \centering
  \begin{subfigure}[t]{0.32\textwidth}
    \centering
    \includegraphics[width=0.80\linewidth]{figures/02/color_transfer}
    \vfill
    \caption{Color transfer}
    \label{fig:vis_color}
  \end{subfigure}
  \hfill
  \begin{subfigure}[t]{0.32\textwidth}
    \centering
    \includegraphics[width=0.75\linewidth]{figures/02/texture_transfer}
    \vfill
    \caption{Texture transfer}
    \label{fig:vis_texture}
  \end{subfigure}
  \hfill
  \begin{subfigure}[t]{0.32\textwidth}
    \centering
    \includegraphics[width=0.60\linewidth]{figures/02/deformation_transfer_nam}
    \caption{Deformation transfer}
    \label{fig:vis_deformation}
  \end{subfigure}

  \caption{
    Qualitative visualisation of \textbf{correspondence quality} using different transfer strategies.
    \textbf{(a)} Color transfer provides an overview of the global correspondence structure.
    \textbf{(b)} High-frequency texture transfer can reveal small local misalignments.
    \textbf{(c)} Deformation transfer can similarly visualise small topological inconsistencies caused by inaccurate correspondences. 
    Figures adapted from~\cite{zhuravlev_denoising_2025,vigano2025nam}.
    }
  \label{fig:02:correspondence_visualisation}
\end{figure*}

\subsection{Correspondence evaluation}
\label{sub_sec:eval_metrics}

Once a correspondence map $\Pi_{\mathcal{M}\mathcal{N}} \in \{0,1\}^{n_\mathcal{M} \times n_\mathcal{N}}$ between a pair of shapes is recovered, it can be evaluated through various metrics.

\paragraph{Accuracy.}
When working with standard synthetic datasets, ground-truth correspondence matrices
$\Pi_{\mathcal{M}\mathcal{N}}^*$ are generally available.
This allows one to directly evaluate the accuracy of the predicted map  $\Pi_{\mathcal{M}\mathcal{N}}$.
Let $D_\mathcal{N} \in \mathbb{R}^{n_\mathcal{N} \times n_\mathcal{N}}$ be the pairwise
geodesic distance matrix on the target shape $\mathcal{N}$, and
$\operatorname{Area}(\mathcal{N})$ the total surface area of the target shape.
Then, the vector of normalised geodesic errors over all vertices in $\mathcal{M}$
is defined as~\cite{kim2011blended}:

\begin{equation}
    e = 
    \frac{1}{\sqrt{\operatorname{Area}(\mathcal{N})}}
    \cdot 
    \operatorname{diag}\!\left(
        \Pi_{\mathcal{M}\mathcal{N}} \,
        D_\mathcal{N} \,
        \Pi_{\mathcal{M}\mathcal{N}}^{*\top}
    \right).
\end{equation}

\noindent
Here, 
$\Pi_{\mathcal{M}\mathcal{N}} \, D_\mathcal{N} \, \Pi_{\mathcal{M}\mathcal{N}}^{*\top} \in \mathbb{R}^{n_\mathcal{M} \times n_\mathcal{M}}$
contains the geodesic distances between the predicted and ground-truth correspondences for each source vertex, 
and $\operatorname{diag}(\cdot)$ extracts diagonal entries.

From the vector of normalised geodesic errors $e \in \mathbb{R}^{n_\mathcal{M}}$, 
three standard evaluation metrics are typically computed.
\textit{Mean geodesic error (MGE)} over all source vertices is defined as:
\begin{equation}
    \mathrm{MGE} = \frac{1}{n_\mathcal{M}} \sum_{i=1}^{n_\mathcal{M}} e(i).
\end{equation}

\noindent
\textit{Percentage of correct keypoints (PCK)} measures the fraction of correspondences whose normalised geodesic error is below a threshold $\tau > 0$. 
Using $\mathbf{1}[\cdot]$ as the indicator function,
\begin{equation}
    \mathrm{PCK}(\tau) = \frac{1}{n_\mathcal{M}} \sum_{i=1}^{n_\mathcal{M}} \mathbf{1}\left[ e(i) < \tau \right].
\end{equation}

\noindent
\textit{Area under the PCK curve (AUC)} captures the overall accuracy of the predicted correspondences for varying thresholds, typically up to the maximum threshold $\tau_{\max}$. This is defined as:
\begin{equation}
    \mathrm{AUC}(\tau_{\max}) = \frac{1}{\tau_{\max}} \int_0^{\tau_{\max}} \mathrm{PCK}(\tau) \, d\tau.
\end{equation}

\paragraph{Smoothness.}
Smoothness metrics assess whether neighbouring surface elements on the source and target shapes are consistently mapped under the predicted correspondence.
\textit{Geodesic distortion}~\cite{Ren2021ShapeMatching} measures how well intrinsic distances are preserved
by comparing pairwise geodesic distances under the correspondence:
\[
\sum_{i,j} \bigl( D_\mathcal{M}(i,j) - D_\mathcal{N}(\Pi_{\mathcal{M}\mathcal{N}}(i), \Pi_{\mathcal{M}\mathcal{N}}(j)) \bigr)^2 .
\]

The \textit{Dirichlet energy}~\cite{magnet2022smooth,ezuz2019reversible} can be used to evaluate the smoothness of the induced map, essentially penalising large differences between the mapped positions of neighbouring vertices.
\textit{Conformal distortion}~\cite{ezuz2019reversible,ren2020maptree} quantifies deviations from local angle preservation, indicating the extent to which small surface elements are stretched or sheared under the mapping.
Lower values of these metrics indicate smoother and more geometrically consistent correspondences, which has important ramifications for downstream applications (see \cref{sec:challenges}).

\paragraph{Coverage.}

Coverage metrics evaluate how completely the target shape is reached by the correspondence and are often used in conjunction with smoothness.
A commonly used criterion is the \emph{un-coverage rate}, defined as the fraction of target vertices (or surface area) that are not mapped to by any source vertex under the correspondence.
Low un-coverage means that the correspondence utilises the target shape more uniformly, while high
un-coverage indicates a collapsing or highly non-injective mapping.

\paragraph{Bijectivity.}
Bijectivity metrics assess whether a correspondence behaves approximately as a one-to-one mapping.
This can be evaluated by \textit{inverse consistency}, which computes correspondences in both directions,
$\Pi_{\mathcal{M}\mathcal{N}}$ and $\Pi_{\mathcal{N}\mathcal{M}}$, by applying the same model
with swapped inputs.
The forward and backward maps are then composed (i.e., $\Pi_{\mathcal{N}\mathcal{M}} \circ \Pi_{\mathcal{M}\mathcal{N}}$ and $\Pi_{\mathcal{M}\mathcal{N}} \circ \Pi_{\mathcal{N}\mathcal{M}}$), and the average geodesic distance between the composed maps and the identity map $I$ is measured.
Low values indicate approximately bijective and mutually consistent correspondences.
In addition to inverse-consistency, bijectivity can be further assessed using \textit{orientation preservation}
criteria, such as the triangle inversion rate, which measures the fraction of faces whose orientation is
reversed after mapping and indicates locally inconsistent correspondences.

\paragraph{Visualisation.}
In addition to quantitative metrics, correspondences are commonly assessed qualitatively through visualisation.
A standard approach is \textit{colour transfer} between shapes, which provides an illustration of the overall correspondence structure but may fail to reveal subtle local errors.
More diagnostic visualisations can be obtained by transferring \textit{high-frequency textures}, which are much more sensitive to small-scale errors.
Similarly, \textit{deformation transfer}~\cite{sumner2004deformation,vigano2025nam} assesses correspondence quality by propagating the deformation field defined on a source shape to the target, making local mismatches appear as visible topological artefacts.

\paragraph{Accuracy to downstream application.}
While the metrics above provide a comprehensive quantitative assessment of correspondence quality, many methods in the literature optimise primarily for geodesic accuracy or other quantifiable metrics.
In practice, however, downstream applications may have additional requirements that pure vertex-to-vertex or spectral maps may not satisfy.
For instance, deformation transfer~\cite{sumner2004deformation} and topology transfer require correspondences that are not only accurate but also smooth and bijective; without such guarantees, transferred deformations can exhibit artefacts (see \cref{fig:vis_deformation}).
Similarly, statistical shape modelling demands geometrically consistent maps across shape collections to produce meaningful modes of variation~\cite{loper2015smpl,magnet2023assessing}. 
In domain-specific settings such as medical shape analysis, purely geometric descriptors may be insufficient or prohibitively challenging to obtain. Incorporating prior knowledge from domain experts, such as anatomical landmarks or physiological constraints, can thus be essential for incorporating such methods into clinical pipelines~\cite{el_amrani_universal_2024,bongratz2023abdominal}.
These application-specific requirements highlight that correspondence accuracy, while necessary, can be insufficient on its own, motivating the development of methods that jointly account for the demands of the intended downstream application.
We revisit this gap between benchmark performance and practical applicability in \cref{sec:challenges}.
