
\section{Open challenges}
\label{sec:challenges}

Despite the extensive progress made in non-rigid shape correspondence in recent years, several fundamental challenges remain unresolved and limit its applicability to arbitrary real-world data. 

First, existing correspondence methods still struggle with non-trivial partiality and topological changes in the geometry. 
Both of these are commonly encountered in 3D scans that contain occlusions or self-intersections. 
While a few approaches explicitly tackle the partial case (see \Cref{sub:partial-to-full} and \Cref{sub:partial-to-partial}), they require extensive training data or rely on hard assumptions to work. 
Even more challenging is the case of topological changes. 
The spectral properties can change significantly under different topologies, even if the local geometry is similar~\cite{laehner2016topkids}. 
This makes the line of spectral methods incredibly sensitive to topological noise, and most papers do not even show results in this direction. 
In addition, not many datasets, and only very small ones, contain topologically noisy shapes with ground-truth correspondences. 
For this reason, supervised learning methods cannot be applied in this domain, while unsupervised methods struggle due to violations of structural assumptions arising from changes shape topology.

Another major challenge is the lack of large, diverse datasets. 
The most famous and commonly used isometric datasets, FAUST~\cite{bogo_faust_2014} and SCAPE~\cite{anguelov_scape_2005}, are synthetic; they consist only of human shapes and are generated from a fixed template. 
This leads to near-isometric and very clean meshes, resulting in benchmarks that recent methods have begun to saturate. 
On the other hand, there is no large collection of alternatives. 
This is in part due to the challenge of annotating datasets at scale; the largest existing datasets, therefore, still consist of synthetic data. 
As a result, methods are not sufficiently challenged by the lack of realism in noise and data distributions.
The implication that almost all existing data consist of clean shapes, with mostly smooth surfaces and minimal geometric irregularities, is that generalisation to real-world settings is limited. 
The recent dataset BeCoS~\cite{ehm2025beyond} made a step towards realism by simulating extreme partiality using ray casting, but also found that existing methods already struggle under much simpler transformations, such as arbitrary rotations. 
While no standardised benchmarks exist to test such cases, the findings suggest that the current generation of correspondence methods is unlikely to be widely generalisable in real-world scenarios.

\paragraph{Connecting quantitative metrics to downstream applications.}
A further challenge that cuts across all three methodological families is the disconnect between what current benchmarks measure and downstream application requirements.
Most methods are primarily evaluated on geodesic accuracy, yet applications such as deformation transfer~\cite{sumner2004deformation,vigano2025nam}, texture mapping, and statistical shape modelling~\cite{loper2015smpl,magnet2023assessing} also require smooth and neighbourhood-preserving maps.
While recent developments in unsupervised spectral methods enable the curation of datasets with geometric descriptors, the resulting pointwise maps lack such guarantees (see \cref{sec:spectral}).
Combinatorial methods offer a promising way of estimating smooth maps from these descriptors, but remain limited in scalability~\cite{roetzer2025geco,amrani2025highres}.
Closing this gap, either through scalable formulations with geometric guarantees or by employing metrics that go beyond geodesic error, remains an important open problem.
In domain-specific settings such as medical image analysis, a further issue is that purely geometric descriptors may be insufficient or prohibitively difficult to obtain. 
Incorporating prior knowledge from domain experts, such as anatomical landmarks or physiological priors, can improve the ability to produce meaningful correspondences and ultimately drive application impact~\cite{greenspan_s3m_2023,el_amrani_universal_2024,bongratz2023abdominal}.

\paragraph{Potential future directions.}
Problems with partiality, topological changes, and other noise are closely related to implicit assumptions that most methods make about the solution, such as injectivity.
It may be necessary to make a fundamental change in the problem's modelling to overcome these challenges reliably.
This holds for both optimisation- and learning-based approaches, which fall in the same categories and rely on the same basic frameworks. 
To evaluate progress in this direction, new datasets, especially large-scale ones based on real-world data, are needed to capture the diversity of possible data and avoid overfitting to the small data pool.
A potential solution might be to incorporate foundation models from other modalities, leveraging their extensive knowledge of the world to compensate for ill-posed settings and a lack of diverse training data. 
When correspondence methods become robust in real-world scenarios, they will enable a wide range of different applications with noisy data, for example, in medical settings or dynamic reconstruction, that would greatly benefit from correspondence information but cannot provide the clean meshes expected by most methods at the moment. 


\section{Conclusions}
\label{sec:conclusions}
The report provides a broad overview of recent advances in non-rigid 3D shape correspondence. 
We discuss three recent branches of techniques which have been emphasised over the last few years: spectral methods based on functional maps, combinatorial formulations with discrete constraints, and deformation-based techniques. 
While the field of non-rigid 3D shape correspondence estimation has seen considerable advances in long-standing paradigms, it is now also fuelled by the recent progress in related computer vision fields and machine learning, including research on foundation models and quantum computing. 

We conclude that further progress depends substantially on the availability of new datasets that cover a wide variety of scenarios, especially for learning-based techniques that rely on large training datasets. 
We also foresee that the outlined open challenges, e.g., ~partial-to-partial shape matching, multi-shape matching, and handling different kinds of noise in the input data, such as topological changes, will keep the current community and the new generation of researchers curious and excited. 
The field is continuing to grow, and we are excited for the next generation of robust techniques and tools, broadly applicable across 3D computer vision, graphics, and related fields in diverse scenarios and applications. 


\section{Acknowledgements}
This work is supported by the ERC starting grant no. 101160648 (Harmony); the DFG grant LA 5191/2-1; the UK Royal Society grant NIF/R1/254128; the DFG (German Research Foundation) project number 534951134; the ERC advanced grant no. 884679 (SIMULACRON); JSPS KAKENHI No. 21H05054, 24H00742, and 24H00748; the Federal Ministry of Research, Technology and Space of Germany and the state of North Rhine-Westphalia as part of the Lamarr Institute for Machine Learning and Artificial Intelligence.





