\section{Outlook and Future Work}
\label{sec:outlook}

\chg{We provide an organized view of shape correspondence problems and discuss common paradigms for establishing and refining correspondences between surfaces. We highlight topological considerations in the context of shape correspondence and summarize approaches for evaluation alongside an overview of available datasets. We hope that this organized overview facilitates future research.}

\chg{In conclusion, methods based on data-driven features and learning-based approaches achieve strong performance across a range of settings, from near-isometric to more challenging non-isometric cases. Each of the discussed paradigms has distinct strengths and limitations. Combinatorial methods can offer strong guarantees of geometric consistency and global optimality, but they usually struggle with scalability, for instance when input geometries have triangle counts exceeding an order of magnitude of \(10^3\). Accordingly, a common use case for these methods is to serve as an initialization for subsequent correspondence refinement. Functional map methods are appealing due to their compact formulation in function space and are frequently used to extend sparse correspondences to dense ones. However, for a broad range of applications, an additional conversion step is required to obtain an element-to-element map. Deformation-based approaches enable correspondence across shape collections by embedding shapes in a latent space. Distortion-based methods relying on intrinsic representations can convert sparse or dense element-to-element maps into continuous point-to-point maps, providing strong bijectivity guarantees. Nevertheless, they face scalability challenges, with runtimes reaching up to several days, are largely restricted to well-defined 2-manifold inputs, and tend to struggle in highly non-isometric settings.}

\chg{In the full-to-full setting, and in line with the benchmarks in BeCoS~\cite{ehm2024partialBenchmark}, one of the remaining challenges is \emph{semantic} incompleteness, where corresponding parts are meaningfully different or missing despite both shapes being \emph{geometrically complete}. Typical examples include the presence or absence of horns on animals or armrests on chairs. More comprehensive and standardized benchmarking, particularly across varying degrees of non-isometry, for example using datasets such as SHREC'20~\cite{dyke2020dataset-shrec20} with sub-challenges of increasing non-isometry, would help the community to better identify the limitations of current methods and allow for better comparison across methods. Overall, we argue that strongly non-isometric correspondence remains challenging even in the full-to-full setting. Benchmarking accounting for noise, topological artifacts, and outliers would also provide a clearer picture of current capabilities.}

\chg{While the full-to-full setting is comparatively well studied, challenges become more pronounced once one moves beyond full-to-full correspondences. The performance of state-of-the-art algorithms degrades in full-to-partial and, in particular, in partial-to-partial scenarios. As demonstrated by the BeCoS benchmark \cite{ehm2024partialBenchmark}, existing methods struggle in these settings, often producing noisy or unreliable correspondences. This underscores the need for further methodological advances tailored specifically to partial shape matching. At the same time, the dataset introduced in BeCoS provides a valuable foundation for more rigorous quantitative evaluation and for training learning-based approaches that may better generalize to these challenging scenarios.}

\chg{Current research and publicly available datasets commonly focus on organic, low-genus shapes, such as humans and animals, where feature misalignment is not always visually apparent. In contrast, many industrial applications require precise alignment of hard features in man-made objects, which frequently exhibit higher genus. This regime remains underrepresented in existing datasets. Future work could explicitly account for the topological degrees of freedom in such cases, which is particularly important for methods based on intrinsic representations. These approaches, commonly used in distortion-driven frameworks, typically rely on continuous updates of the map and are therefore restricted to optimization within a fixed homotopy class. However, for man-made objects in particular, it may be necessary to optimize across different homotopy classes and consider maps that differ by topological twists in order to find optimal correspondences.}

\chg{Finally, different applications impose specific requirements that go beyond those of the correspondence problem itself. For instance, tasks such as texture transfer or the transfer of physical simulation results require maps that are continuous and sufficiently smooth to avoid visual artifacts or implausible physical predictions on the target surface. Mesh processing applications, such as high-quality mesh transfer, benefit from correspondences that induce high-quality element maps, even if this sometimes comes at the expense of strict correspondence accuracy. Moreover, in this setting, all geometric features must be captured by the map, which can be challenging for explicit, element-based representations: If features are not properly sampled by the mesh, they may not be represented in the correspondence at all. For example, when transferring a hand mesh to another hand, vertices might be placed slightly too low on a finger such that the fingertip feature is no longer captured by any mesh element, and thus cannot be transferred reliably. Similarly, shape interpolation, morphing, and averaging require correspondences that produce meaningful intermediate shapes and, in particular, avoid self-intersections or other geometric degeneracies. These diverse and often application-specific demands call for more tailored correspondence frameworks that can adapt to the needs of the task at hand rather than relying on a one-size-fits-all formulation.}

\section*{Acknowledgments}
\add{This work was partially funded by the German Research Foundation within the Gottfried-Wilhelm-Leibniz program and by the Excellence Strategy of the Federal Government and the States of Germany. Open Access funding enabled and organized by Projekt DEAL.}

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