\section{Practical applications}
\label{sec:applications}

\begin{figure*}[t]
  \centering
\begin{subfigure}{0.49\linewidth}
    \centering
    \setlength{\fboxsep}{0pt}
    \includegraphics[width=0.9\linewidth]{figures/applications/segmentation_large.jpg}
    \caption{Segmentation transfer}
\end{subfigure}
  \begin{subfigure}{0.49\linewidth}\centering
    \includegraphics[width=0.65\linewidth]{figures/applications/deformation_compressed}
    \vfill
    \caption{Shape interpolation}
  \end{subfigure}\hfill
\begin{subfigure}{0.49\linewidth}
  \centering
  \vfill
  \includegraphics[width=0.9\linewidth]{figures/applications/medical_applications}
  \vfill
  \caption{Statistical shape modelling}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
  \centering
  \includegraphics[width=0.95\linewidth]{figures/applications/animation_transfer_notext}
  \vfill
  \caption{Animation transfer}
\end{subfigure}
  \caption{
        \textbf{Practical applications} of shape correspondence span numerous tasks in graphics and geometry processing.
        \textbf{(a)} Segmentation transfer propagates semantic part annotations across a collection of registered shapes.
        \textbf{(b)} Shape interpolation uses correspondences to generate smooth transitions between source and target shapes.
        \textbf{(c)} Statistical shape modelling uses correspondences to build low-dimensional shape spaces, including anatomical models of organs and tissues.
        \textbf{(d)} Animation transfer allows to propagate motion sequences across registered shapes.
        Figures adapted from \cite{cao_spectral_2024,abdelreheem2023zero,el_amrani_universal_2024,Musoni2021ReposingAR}.
        }  
  \label{fig:practical_applications}
\end{figure*}

Shape correspondence lies at the core of numerous applications involving 3D meshes.
In this section, we highlight some of the most prominent use cases.
Examples of these applications are illustrated in Figure~\ref{fig:practical_applications}.

\paragraph{Information transfer.}

A well-known application of shape correspondence is the transfer of annotations between shapes, which enables numerous tasks in graphics, animation, and robotics.
A classic example is texture transfer, where a texture defined on a source shape is projected onto a target shape, which is often used to visually inspect the quality of correspondences.
Segmentation transfer enables the propagation of part labels across a dataset of registered shapes, allowing a small number of manual annotations to be scaled to a shape collection~\cite{abdelreheem_satr_2023}. 
Deformation transfer propagates articulations, such as pose changes in human bodies, and can extend to complete deformation systems, including rigging properties~\cite{avril2016animation,musoni2021functional} or the mesh structure itself~\cite{melzi2020intrinsic, sumner2004deformation, lakshmipathy2025kinematic}.
Beyond graphics, shape correspondence finds applications in robotics for transferring contact point annotations between objects to guide robotic arm grasping~\cite {zhu_densematcher_2024}.

\paragraph{Shape interpolation.}

Shape interpolation (or morphing) aims to generate a continuous sequence of intermediate shapes that smoothly deform a source geometry into a target.
Accurate correspondences help preserve semantic structure and geometric detail throughout the interpolation process. 
Recent approaches vary in how they incorporate correspondence: some treat it as external supervision or initialization~\cite{sang20254deform, sang2025twosquared}, others solve for correspondence and interpolation jointly~\cite{eisenberger2021neuromorph}, or couple spectral and spatial representations to improve consistency~\cite{cao_spectral_2024}.

\paragraph{Statistical shape modelling and medical applications.}

Statistical Shape Models (SSMs) capture shape variability within a category by learning low-dimensional representations from populations of 3D shapes. 
This process relies on dense correspondences to ensure consistent geometry alignment across the population.


Two broad modelling paradigms exist for building SSMs.
Template-based methods assume shared mesh topology and model deformation within a fixed coordinate system, with prominent examples including SMPL~\cite{loper2015smpl}, MANO~\cite{romero2017mano}, and 3DMM~\cite{egger20203d}. 
Recent extensions adapt this paradigm to animals~\cite{li2024learning} or anatomical skeletons~\cite{Keller_OSSO_CVPR_2022,keller2023skel} by combining template correspondences with learned statistical priors. 
In contrast, groupwise methods jointly estimate correspondences and a shape model to avoid template bias, using encoder–decoder architectures~\cite{iyer2023mesh2ssm} or functional maps~\cite{greenspan_s3m_2023,el_amrani_universal_2024}. 
In both cases, correspondence is the key component enabling statistical analysis.


A major application domain of SSMs is medical data, where correspondence establishes anatomical mappings for disease classification and population-level studies~\cite{adams_point2ssm_2023,iyer2025mesh2ssm++}. 
Functional map-based frameworks have proven effective for unsupervised correspondence estimation when mesh representations are available~\cite{klatzow2022mumatch,greenspan_s3m_2023,el_amrani_universal_2024}. 
In clinically assisted interventions, however, data may arise from partial point clouds acquired via laparoscopic imaging, ultrasound, X-ray~\cite{yang2023learning,gafencu2024shape,gafencu2025us}, or time-of-flight (ToF) depth sensors~\cite{bastian2023disguisor,wang2025beyond}. 
Template-based deformation approaches have improved robustness in applications including abdominal organs~\cite{bongratz2023abdominal}, cortical surfaces~\cite{bongratz2024neural}, and orthopaedic fracture reduction planning~\cite{YibSut_SimtoReal_MICCAI2025}. 
Atlas construction remains another important research direction~\cite{kalaie2025mdeicalAtlas}. 
For a broader overview, we refer to a survey by Zhang et al.~\cite{zhang2025survey} and the extensive MedShapeNet dataset~\cite{li2025medshapenet}.




\paragraph{Shape retrieval.}

Shape retrieval seeks to identify shapes in a database that best match a query despite variations in pose, deformation, or partiality. 
Recent methods increasingly integrate correspondence estimation into the retrieval process. 
Dense maps can define similarity metrics directly between query and candidates~\cite{di2023u}, while keypoint or part-level correspondences support retrieval with subsequent deformation~\cite{zhang2024kp,di2024shapematcher}. 
Other approaches extend correspondence-based metrics to multimodal retrieval settings~\cite{gumeli2022roca,gao2024diffcad}.  
Overall, correspondence helps retrieval focus on geometric rather than purely visual similarity.

\paragraph{Dynamic reconstruction.} 
When reconstructing a dynamic scene into a sequence of 3D frames, the underlying problem implicitly assumes dense correspondence between the frames. 
Instead of finding correspondences between image keypoints, most recent methods rely on a deformation field between a reconstructed template or an initial frame and subsequent frames~\cite{shao2023tensor4d,ebbed2025moangelo}. 
While most methods restrict correspondence to be implicit through the deformation model, it can also be used directly to impose geometric constraints, such as maintaining curvature consistency~\cite {chen2025adaptive}. 

\paragraph{Data alignment.} 
Correspondence is of significant importance for annotating 3D data. For example, it is common to decorate acquisition with an aligned template, providing a semantic meaning to otherwise unordered points~\cite{bhatnagar2022behave, wang20244d}. Correspondence can also be used to unify different datasets in a common semantic frame~\cite{melzi2019shrec}.

\paragraph{Measures and shape analysis.} 
By bringing shapes into correspondence, it is also possible to compare intrinsic properties of objects, especially in terms of measures. For example, correspondence methods are adopted in anthropometry, the science that studies the measurements and proportions of the human body \cite{huang2025shape, zhao2022skin3d}.
