\section{Introduction}
\label{sec:intro}
Shape correspondence seeks to establish an intuitive and meaningful relationship between the points of two or more similar input shapes. Here, intuitive and meaningful refer to correspondences that respect both \textit{geometric structure} and \textit{semantic similarity} \cite{li2014-survey}. The intricate interplay between geometry and semantics across multiple surfaces makes shape correspondence a fundamentally challenging problem. Nevertheless, it is a core component of shape analysis, enabling a wide range of downstream applications. The most immediate use case for correspondences is the transfer of information between shapes, such as geometric detail, textures, segmentations, and parameterizations. Correspondences further enable joint shape processing tasks including co-segmentation, co-field synthesis and compatible remeshing. Further applications include shape interpolation and morphing, shape averaging, and statistical shape modeling and analysis.

% example tasks:
% - generative shape modelling https://dl.acm.org/doi/abs/10.1145/3306346.3322956
% - 

The diversity of applications imposes varying qualitative and quantitative requirements and causes significant challenges. Existing methods have achieved notable success in computing meaningful correspondences, \eg, through deformation-based optimization or data-driven approaches. Some methods emphasize stricter guarantees, \eg, geometric consistency, bijectivity, and smoothness of the map describing the correspondence between a source and a target surface. Despite substantial efforts by the research community and numerous contributions, critical limitations persist. Current methods often lack robustness, struggle with computational scalability, especially for larger shape collections, and rarely combine semantic meaningfulness with strong guarantees for the quality of the map. Addressing these challenges is essential for broadening the applicability and reliability of shape correspondence techniques.

In this work, we aim to provide a comprehensive overview of the field, covering the full range of shape correspondence formulations. While several surveys have previously addressed the topic \cite{vanKaick2011-survey, li2014-survey, sahilliouglu2019-survey}, the rapid pace of progress calls for a new synthesis that consolidates recent advances and offers a structured perspective on the diverse problem settings, paradigms, and open challenges. By organizing and contextualizing the literature, we seek to make the field more accessible, thus supporting continued progress.

We begin by revisiting a unified formalization of shape correspondence (\Cref{sec:problem_setting}) and highlight how specific problem instances may differ. We clarify related terminology, including concepts such as geometric consistency and surface homeomorphism. Furthermore, we structure the vast body of literature, identifying key paradigms (\Cref{sec:paradigm_overview}) used to establish and refine correspondences and summarizing recent progress within each of them (\Cref{sec:advances_in_shape_alignment,sec:advances_in_combinatorial_approaches,sec:advances_in_distortion_based_approaches,sec:advances_in_the_functional_maps_framework}). Methods based on more niche paradigms are presented in \Cref{sec:machine_learning_approaches}. We also cover topological aspects of correspondences between surfaces~(\Cref{sec:topology}), as well as approaches for evaluation and visualization~(\Cref{sec:eval}). In addition, we review datasets commonly used for training and benchmarking (\Cref{sec:datasets}). Finally, we discuss open challenges and outline promising directions for future research (\Cref{sec:outlook}).

\subsection{Scope}
To keep this report focused, we restrict our attention to methods that compute correspondences between shapes undergoing general and potentially large non-rigid deformations. A closely related topic is 3D registration, which typically assumes rigid or near-isometric deformations and may involve incomplete or noisy data, typically in the context of surface acquisition, where an object is scanned from different directions. For a comprehensive overview of that area, we refer the reader to the survey by Deng \etal \cite{deng2022-registrationSurvey}. Our scope is further limited to techniques that establish surface-to-surface correspondences. Thus, we do not cover volumetric (interior) mappings. Likewise, curve-to-surface and curve-to-curve correspondences are beyond the scope of this survey. Finally, methods for (partial) symmetry detection are not covered, although some of the discussed techniques may incidentally support such tasks.
