\section{Evaluation of Shape Correspondence}
\label{sec:eval}

From a theoretical standpoint, there exist infinitely many possible correspondences between two shapes. The goal of shape correspondence establishment is therefore to identify the most meaningful correspondence within this vast space.

This naturally leads to the need for a \emph{quantitative assessment} of correspondence quality. Beyond serving as a benchmark for algorithm performance, such assessments can also act as proxies for \emph{shape similarity}, which is particularly relevant in downstream tasks such as shape retrieval \cite{vanKaick2011-survey}. However, defining and evaluating correspondence quality quantitatively remains challenging. In fact, identifying a suitable evaluation metric is often as difficult as solving the correspondence problem itself, because the metric could be used as objective in optimization based approaches. Ideally, a correspondence should capture both geometric and semantic aspects, but semantics are especially hard to formalize and quantify. As a result, the community has developed a range of visualization techniques that support \emph{qualitative analysis} and facilitate the comparison of correspondence algorithms.

In this section, we review the available evaluation tools, starting with quantitative metrics and then discussing visualization techniques for qualitative analysis.

\subsection{Quality Metrics}
\subsubsection*{Geodesic Error}\
Whenever \emph{ground truth} correspondences are available, the most straightforward way to evaluate map quality is by computing the sum of the \emph{geodesic distances} between the mapped points and their ground truth counterparts, typically normalized by the square root of shape area, commonly referred to as Princeton protocol \cite{kim2011-evalGeodesicError}. For this purpose, indirect map representations, \eg, functional maps, must first be converted into an explicit map representation, \eg, a vertex-to-vertex map.
When working with point clouds, geodesic distances may not be directly accessible. The euclidean distance can serve as a reasonable approximation.

To quantitatively compare two maps, a common measure is the \emph{cumulative geodesic error}, defined as the fraction of points (or surface area) whose geodesic error falls below a given threshold \cite{kim2011-evalGeodesicError}. By varying the threshold, one obtains a cumulative curve that concisely summarizes the map's accuracy across different error tolerances.

%\begin{itemize}
%    \item can be used to evaluate a correspondence (e.g. for evaluation or supervised learning)
%    \item CQC: correspondence quality characteristics from name from \cite{li2024fm-operatorPreservation}): percentage of points that have less than a certain threshold error
%\end{itemize}

\subsubsection*{Feature Matching}\
Since a meaningful correspondence relates similar features on the surfaces, a natural quality measure is to compare and quantify the difference between features associated with corresponding points. Such features can range from simple, locally derived geometric quantities, \eg, mean or Gaussian curvature, principal curvatures, or local feature size (\eg, shape diameter function), to more sophisticated descriptors.

A particularly important class comprises \emph{shape signatures}, which are constructed to be characteristic or invariant under certain transformations, such as rigid motions, scaling, and isometries. By design, these signatures aim to be unique (or nearly so) within a given object class, thereby enabling identification of corresponding structures across shapes. The choice of signature effectively defines the semantic notion of similarity. Popular examples include the Pose-Oblivious Shape Signature \cite{gal2007-poseObliSignature}, Global Point Signature \cite{rustamov2007signatures-gps}, SHOT \cite{tombari2010-SHOT}, Heat Kernel Signature \del{\hbox{\cite{bronstein2010-evalHKS-wrong}}}\add{\cite{sun2009signatures-hks}}, Wave Kernel Signature \cite{aubry2011signature-wks}, and Biharmonic Kernel Signature \cite{rustamov2011signature-bks}. For a more exhaustive list of available spectral descriptors, we refer to the survey \cite{liu2024fm-survey}.

Finally, features can also be estimated in a data-driven manner via neural networks \cite{qi2017pointnet, qi2017pointnet++, sharp2022fm-diffusionnet}.

\subsubsection*{Surface Approximation Error}\
For methods that work with an extrinsic map representation, the surface approximation error evaluates how well the mapped source surface aligns with the target surface. Possible candidates are the \emph{Chamfer distance}, capturing the average deviation between two surfaces by integrating point-to-surface distances, and the \emph{Hausdorff distance}, measuring the maximum deviation, \ie, the largest pointwise mismatch between the two shapes.

\subsubsection*{Distortion}\
\chg{Analogously to classical surface parametrization tasks, correspondences, in particular inter-surface maps, are often assessed by measuring overall distortion based on the map Jacobian \(J_{\mapPhi}\)}.

For piecewise-linear map definitions, the Jacobian is constant per face: Consider a face \(f\) in the common triangulation of both input surfaces \(\surfaceA\) and \(\surfaceB\), with vertex positions \(a_{\surfaceA}, b_{\surfaceA}, c_{\surfaceA}\) on \(\surfaceA\), and  corresponding  positions \(a_{\surfaceB}, b_{\surfaceB}, c_{\surfaceB}\) on \(\surfaceB\). A local coordinate system is constructed by defining the edges 
\(e_0 = b_{\surfaceA} - a_{\surfaceA}\) and \(e_1 = c_{\surfaceA} - a_{\surfaceA}\) on \(\surfaceA\), and analogously \(e'_0 = b_{\surfaceB} - a_{\surfaceB}\) and \(e'_1 = c_{\surfaceB} - a_{\surfaceB}\) on \(\surfaceB\). The local Jacobian of the map \(\mapPhi\) is then given by:
%
\[
J_{\mapPhi}(f) = [e'_0 \ e'_1] [e_0 \ e_1]^{-1}.
\]
%
\chg{The singular value decomposition of \(J_{\mapPhi}\) yields the singular values \(\sigma_{1}\) and \(\sigma_{2}\), which characterize local stretching along the principal directions. Based on the singular values, several distortion measures are commonly used~\cite{smith2015-evalBijectiveParam,schmidt2019-injectiveMaps,schmidt2020in-constantCurvatureMetrics, Abulnaga2023-symmetricEnergies}. Here, we present two examples and express them as pointwise measures.}
%
\chg{
\begin{itemize}
    \item \textbf{Symmetric Dirichlet energy:} 
    \[
    E_{\text{SDE}}(J_{\mapPhi}) = \sigma_{1}^2 + \sigma_{2}^2 + \sigma_{1}^{-2} + \sigma_{2}^{-2},
    \]
    \item \textbf{As-rigid-as-possible:}
    \[
    E_{\text{ARAP}}(J_{\mapPhi}) = \bigl(\sigma_{1} - 1)^2 + (\sigma_{2} - 1)^2.
    \]
\end{itemize}
}
\chg{The overall distortion induced by a map \(\mapPhi\) is obtained by integrating the pointwise measure \(E_\star(J_{\mapPhi})\) over the input surfaces. 
In the discrete setting, this corresponds to integrating the pointwise measure per face \(f\), and summing this over all triangles (with piecewise constant Jacobian \(J_{\mapPhi}(f)\)) of the common triangulation:
\[
E_\star(\Phi) = \sum_{f \in F} |f| \; E_\star(J_{\mapPhi}(f)),
\]
where \(|f|\) is the area of face \(f\) on \(\surfaceA\).
}

\add{In the correspondence setting, neither mapping direction is favored over the other, so we are interested in symmetric objectives satisfying \(E_\star(\mapPhi) = E_\star(\mapPhi^{-1} )\). Not every distortion measure \(E_\star\) is symmetric, but it can easily be symmetrized by computing~\cite{Abulnaga2023-symmetricEnergies}:
\[
E{^\text{total}_\star}(\mapPhi) = E^{\text{sym}}_\star(\mapPhi)
= \frac{1}{2}\,E_\star\bigl(\Phi\bigr)
+ \frac{1}{2}\,E_\star\bigl(\Phi^{-1}\bigr).
\]
Expressed in terms of singular values for a pointwise measure, this leads to~\cite{Abulnaga2023-symmetricEnergies}: 
\[
E^{\text{sym}}_\star(J_{\mapPhi})
= \frac{1}{2}\, E_\star(J_{\mapPhi})
+ \frac{1}{2} |\sigma_1\sigma_2|\,E_\star\bigl(J^{-1}_{\mapPhi}).
\]
For an in-depth discussion, we refer to \cite{Abulnaga2023-symmetricEnergies}. Here, we focus on the key observation: %While a distortion measure promotes isometry with its minimum value for \((\sigma_1, \sigma_2) = (1,1)\), this may no longer be true for the symmetric version, see~\Cref{fig:symmetric_distortion_figures}.%
An asymmetric distortion measure may promote isometry (\ie, its minimum is at \((\sigma_1, \sigma_2) = (1,1)\)), while its symmetric version does not.}
%
\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth]{fig/distortion_measures.pdf}
  \caption{Following the visualization style of \cite{Abulnaga2023-symmetricEnergies}, we plot the energy landscape (dark colors correspond to small values) for different distortion measures and their symmetrized versions on \([0,2]^2\). The minimum of the function is indicated by a circle, and a dot at \((1,1)\) marks the expected location of the minimum for an isometry-promoting function. Note that the symmetrized version of the standard SDE does not promote isometry.}
  \label{fig:symmetric_distortion_figures}
\end{figure}
%
\add{In~\Cref{fig:symmetric_distortion_figures}, it can be seen that the symmetric version of ARAP promotes isometry, while the symmetric version of the SDE does not. However, this property can be restored by a slight modification of the energy: \(E_{SDE'} = E_{SDE} - 4\).}

%While various alternatives exist, symmetric variants are particularly relevant in shape matching, as they satisfy \(E(J_{\mapPhi}) = E(J^{-1}_{\mapPhi})\) and therefore do not favor one mapping direction over the other.

%The overall distortion induced by a map is obtained by integrating the pointwise measure over the input surfaces. 
%In the discrete setting, this corresponds to integrating the pointwise measure per face, yielding \(E(f)\), and summing this over all triangles (with piecewise constant Jacobian \(J\)) of the common triangulation:
%\[
%E(\Phi) = \sum_{f \in F} E(f).
%\]
%
In addition to reporting \chg{the} global value \add{\(E{^\text{total}_\star}\)}, one can also consider the \emph{cumulative distortion distribution}, defined as the percentage of surface area where the distortion does not exceed a given threshold. Varying the threshold yields a cumulative curve that highlights how distortion is distributed across the surface.
%
\subsubsection*{Elastic Energy}\
\add{The previous formulations for measuring distortion are usually used for near-isometric shape matching. For matching shapes that are not approximately isometric, a common alternative is to use \emph{elastic energies}~\cite{ezuz2019-elasticEnergy, Edelstein2020-ENIGMA-elastic}: 
\[
E_\text{elastic}(\surfaceA, \surfaceB) = E_\text{mem}(\surfaceA, \surfaceB) + \omega_\text{bend}E_\text{bend}(\surfaceA, \surfaceB), 
\]
where \(E_\text{mem}(\surfaceA, \surfaceB)\) is a membrane energy based on Jacobian of the map, penalizing area distortion. \(E_\text{bend}(\surfaceA, \surfaceB)\) is a bending energy penalizing misalignment of curved features, and \(\omega_\text{bend}\) is the corresponding weighting factor balancing the contribution of the bending term.} 
%
\subsection{Visualization Techniques for Shape Correspondence}
Beyond quantitative analysis, visual inspection remains a crucial component in evaluating shape correspondence. It enables assessment of whether a computed map matches human intuition, which is inherently difficult to formalize numerically.

A range of visualization techniques has been developed, each emphasizing different aspects of correspondences:

\begin{itemize}
    \item \textbf{Connecting lines between corresponding points} offers a direct and intuitive visualization of discrete point-to-point correspondences. To maintain visual clarity, this visualization is typically applied to a sparse subset of points rather than the entire set. It is particularly effective for verifying whether prominent geometric or semantic landmarks are consistently and correctly matched.
    %
    \item \textbf{Segmentation transfer} consists of partitioning the source surface into meaningful regions and propagating this segmentation to the target surface via the correspondence. This approach enables assessment of whether large regions are mapped coherently. When the segmentation encodes semantic structure, it directly reveals whether the correspondence preserves semantics.
    %
    \item \textbf{Texture transfer}, typically using a checkerboard pattern with annotated quadrilaterals, reveals qualities of the map such as smoothness, local distortion, and continuity. Moreover, they can expose local fold-overs or inversions that may be overlooked by purely quantitative metrics.
    %
    \item \textbf{Pushforward (or Pullback) of surface normals} is an effective way to show higher-frequency details of a correspondence. The idea is to compute the normals on the source surface and transfer these normals to the target surface. The pushforward can be visualized directly on the target surface using a color map on the unit sphere \cite{hartwig2023fm-elasticBasis} or they can be used to shade the target surface \cite{melzi2019fm-visualAssessments}. This approach is particularly effective in highlighting correspondence quality of high curvature regions such as creases. 
    %
    \item \textbf{Local distortion} per face allows the analysis of where and to what extent the map introduces distortion. While the integration of distortion provides a global measure of quality, visualizing the local distortion offers more detailed insights into the spatial distribution of mapping artifacts.
    %
    \item \textbf{Triangulation transfer} from source to target allows to see if the matching is geometrically consistent. If it is not, artifacts such as large triangles spanning the space between distant features may appear.
    %
    \item \textbf{Morphing animations} that interpolate between source and target shapes facilitate qualitative assessment of the overall continuity and plausibility of the correspondence by visualizing the deformation from one shape to the other.
\end{itemize}

Together, these visualization strategies complement quantitative metrics by offering qualitative insights that are crucial for practical assessment and algorithm development. 
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