\section{Topological Aspects of Correspondences}
\label{sec:topology}

Despite the fundamental interplay between geometry and topology, only few works have explicitly considered topological aspects of correspondences. We will cover their contributions in more detail in the following. 

\subsection{Homeomorphism Recovery}
Many common representations of surface maps, \ie, element-to-element maps or functional maps, are not continuous nor bijective in a strict sense. It is possible to convert such a representation into an actual surface homeomorphism by applying a method with an \emph{intrinsic map} representation and bijectivity guarantees using the correspondences provided by the input map as guidance. These methods typically employ a \emph{topology-first} strategy, \ie, they start from an initial map and then continuously optimize its geometric quality \emph{within} a fixed homotopy class. Since these methods rely on continuous deformation, they cannot traverse the space of homotopy classes, making initialization and homotopy-aware modeling critical for achieving meaningful correspondences. In the top row of \Cref{fig:topology-first}, the initial map is in an unintended homotopy class, and therefore introducing unnecessary distortion. The initial map in the bottom row is in the expected homotopy class and ready for optimization. 

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.95\linewidth]{fig/Topology/topology-first.png}
    \caption{ If the paths of the cut graph lie in unexpected homotopy classes (top row, left), the corresponding map \add{\(f\)} (top row, right) shows artifacts. When the paths of the cut graph are in the expected homotopy class (bottom row), the resulting map \add{\(f\)} is well suited for optimization. Adapted from \cite{born2021-mapHomologyInterference}.}
    \label{fig:topology-first} 
\end{figure}

To the best of our knowledge, two works have been proposed that address topology-related aspects of map recovery, identifying the \emph{intended} homotopy class of the map. 


\subsubsection*{Layout Embedding}\
A popular approach for obtaining the initial map involves connecting a sparse set of landmarks on both surfaces in a compatible manner, \ie, if two landmarks are connected on one surface, the corresponding landmarks are also connected on the other surface, and the edges around each vertex follow the same cyclic order. The resulting graph structure, called \emph{layout} \(\layout\), is an abstract 2-dimensional cell complex \cite{born2021-layoutEmbedding} without geometric information.

Consider a set of layout vertices on a shape that should be connected according to a prescribed connectivity. Even if only the homotopy of the paths connecting the vertices is considered, there are infinitely many choices for routing a path around the other embedded layout vertices, as shown in \Cref{fig:cube_homotopy}. These choices constitute the discrete degrees of freedom of the layout embedding problem. Additionally, even if the homotopy classes are fixed per path, the geometric shape of a path is a continuous degree of freedom.

To make the problem tractable, a common strategy is to embed edges sequentially as shortest paths between layout vertices. In this setting, the layout embedding is fully determined by the \emph{insertion order} of the layout edges. It is important to note that paths are not allowed to intersect. Thus, earlier insertions may block the optimal path between other layout vertex pairs. As a result, these paths must take detours to avoid intersections with earlier insertions while preserving the cyclic ordering. As a consequence, the insertion order implicitly determines the homotopy class of the embedding.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.9\linewidth]{fig/Topology/cube_homotopy.png}
    \caption{For a fixed layout \(\layout\) with a fixed embedding of the layout vertices in a target mesh \add{\(\mathcal{T}\)}, infinitely many topologically distinct embeddings for the layout edges are available. The overall shortest embedding (d) corresponds to the embedding that is generally considered intuitive. Taken from \cite{born2021-layoutEmbedding}.}
    \label{fig:cube_homotopy}
\end{figure}

While there is no objectively \emph{wrong} homotopy class for the embedding, usually the overall shortest embedding corresponds to the \emph{intended} layout embedding which aligns with human intuition.
Earlier works \cite{praun2001-consistentMeshParam, schreiner2004-interSurfaceMapping} have chosen the insertion order of edges in a greedy fashion based on heuristics, but they did not allow for backtracking. Consequently, a wrong decision early on would result in paths belonging to an \emph{unintuitive} homotopy class, leading to long detours and twists. 

To circumvent this problem, Born \etal \cite{born2021-layoutEmbedding} proposed a \emph{branch-and-bound} approach, optimizing for the insertion order leading to the overall shortest embedding. They proposed a custom branch-and-bound strategy, capable of handling the large search space. They demonstrate that various downstream applications, \eg, quad-meshing and inter-surface mapping, benefit from the robustness of their method. However, their method is only applicable to genus-zero meshes because they assume that every closing cycle encloses a patch with disc topology. For higher genus objects, this is not the case, as a cycle can wrap around one or more handles.

\subsubsection*{Map Homology Inference}\
Most shape correspondence methods either provide only sparse correspondences or work with representations that are not trivially extendable to an embedded point-to-point map. In practice, maps may also contain defects such as noise, outliers, or overlaps, making them inherently ambiguous \wrt map topology. Converting such a map to a true homeomorphism requires resolving these ambiguities and defects in a globally consistent manner and committing to a specific, intended map topology.

\cite{born2021-mapHomologyInterference} propose a method for map homology inference from imperfect input correspondences. Specifically, they compute a \emph{homology map}, \ie, a one-to-one correspondence between the homology classes of cycles on the source and target surfaces, derived from the input correspondences. While the homology map does not fully determine the homotopy of a homeomorphism, it captures key topological features by identifying corresponding handles and tunnels. 

The method takes as input two closed surfaces (triangle meshes) together with an initial, potentially low-quality correspondence relation. For each surface with genus \(g\), a homology basis is computed, represented by \(2g\) cycles of oriented edges per mesh. From this, the homology map is inferred and represented as a matrix \( M \in \mathbb{Z}^{2g \times 2g}\). Computationally, this amounts to solving an integer quadratic program. The key challenge lies in computing the image of the homology basis under an unreliable map. Instead of mapping them directly, the authors \cite{born2021-mapHomologyInterference} map the low-frequency cohomology basis, which is maximally smooth and comparable across the entire surface. They further incorporate a global topological consistency constraint, restricting the search to the subspace of homology maps compatible with homeomorphisms. These two ingredients make the inference robust, enabling recovery of the intended homology map even in the presence of noise, outliers, and sparsity.

Still, their method cannot distinguish paths that are homologous but non-homotopic, \eg, those related by Dehn twists along separating cycles.

\subsection{Correspondence between Surfaces with Different Topologies}
Even though a homeomorphism exists only between topologically identical surfaces, correspondences between topologically different surfaces can still be of interest, \eg, between chairs whose backrests contain a different number of spindles. While several approaches can operate on surfaces with differing topology, they do not explicitly account for topological differences. Park \etal \cite{park2016-eulerianApproachTopo} propose a method that establishes correspondences between surfaces with different topology by removing an appropriate set of points from each original surface, yielding modified surfaces that admit a homeomorphism.

At the core of their approach is a deformation process that aligns the source mesh with the target mesh. Unlike most methods, they adopt an Eulerian viewpoint: Operating on an implicit surface representation, they compute the morph from source to target by iteratively tracking the deformation of the implicit surface and applying the same deformation to the explicit mesh. A key advantage of this formulation is that it produces a velocity field defined over the entire domain, which both drives the deformation and identifies suitable locations for topological surgery, namely, the non-degenerate saddle points of the velocity field.

\subsection{Adapted Objective Functions}
Numerous formulations for isometric maps rely on a notion of shortest distance, \eg, the geodesic distance, between pairs of surface points, typically vertices. However, this becomes problematic on meshes with topological artifacts, such as holes, accidental shortcut tunnels (\eg, nearly degenerate triangles between separate features), or merged features like touching hands, as the shortest distance between nearby vertex pairs can change drastically depending on whether an artifact is present or not.

\cite{sahilliouglu2022-augmentedPathsAndReodesics} introduce reodesics, a robust counterpart to minimal geodesics in the presence of topological noise, such as shortcut tunnels. A reodesic between two vertices is constrained to pass through a third vertex, such that they are locally shortest but not necessarily globally shortest. To make reodesics comparable across two surfaces, one anchor correspondence must already be known: This correspondence is used to reroute paths between other vertex pairs. Given a \emph{brute-force} map, computed without accounting for topological noise, the reodesics can be used to validate candidate matches: A prediction is accepted if the corresponding reodesics agree. For additional robustness, the paths are augmented with descriptors, \ie, heat vectors, for comparison.

The Wormhole loss \cite{bracha2024ex-wormholeLoss} circumvents the problem of partial shapes containing holes by identifying \emph{consistent} pairs, \ie, pairs whose geodesic distance remains unchanged between the complete and partial shape. To detect such pairs, it employs a conservative heuristic: for each candidate pair, it computes both the geodesic distance between the points, and a lower bound given by the sum of their distances to the nearest boundary plus the Euclidean distance between those boundary points. If the geodesic distance is smaller than this bound, the connection is guaranteed to not be influenced by a hole, and the pair is deemed consistent. Intuitively, the heuristic investigates if a pair of vertices is intrinsically closer to each other than to the closest boundary. If this is the case, their geodesic distance is unaffected by holes. Notably, this formulation requires no access to the full shape and is thus suitable for unsupervised training. 


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