\section{Advances in Distortion-Based Approaches}
\label{sec:advances_in_distortion_based_approaches}
A class of approaches using an intrinsic representation follows at their core a \emph{topology-first approach}, where they construct an initial valid inter-surface map \(\Phi\) between the input surfaces. This initial solution resolves the topological degree of freedom. Only afterwards, they focus on optimizing the geometric aspect of the map, aiming for minimal distortion of the map between the surfaces. The focus of these methods is on \emph{correspondence optimization} rather than \emph{correspondence establishment}.

In the following, we will have a look at key aspects that differentiate algorithms in this class of approaches and report on recent advances. We summarize these findings in \Cref{tab:homeomorphism-comp}.

\begin{table*}[h]
\centering
\begin{threeparttable}
\begin{tabular}{c||c|c|c|c|c}
\textbf{\textbf{Method}} & 
\makecell{Surface Topology\\Class} & 
\makecell{Distortion\\Measure} & 
\makecell{Map \\ Representation} & 
\makecell{Intermediate\\Domain} & 
\makecell{Shape\\Collection} \\
\hline
\hline
\cite{schreiner2004-interSurfaceMapping} & all & to intermediate domain & overlay mesh & \yes (base complex)  & \no \\
\hline
\cite{schmidt2019-injectiveMaps} & \makecell{disk} & end-to-end & overlay mesh & \yes (plane)  & \no  \\
\hline
\cite{schmidt2020in-constantCurvatureMetrics} & all & end-to-end & overlay mesh & (\no)\add{\tnote{\(\dag\)}}  & \no  \\
\hline
\cite{schmidt2023-adaptiveTriangulation} & sphere & end-to-end & intermediate mesh & \yes (unit sphere)  & \yes \\
\hline
\cite{morreale2021in-neuralSurfaceMaps} & disk & end-to-end & neural surface map & \yes (plane) & \yes\\
\hline
\cite{morreale2024in-neuralSemantic} & sphere & end-to-end & neural surface map & \yes (plane) & \chg{\no}\\
\hline
\cite{takayama2022in-compatibleIT} & all & end-to-end & \makecell{compatible intrinsic \\ triangulation} & \no  & \no \\
%\hline
%TODO-method & TODO-topo & TODO-end-to-end & TODO-mapRep & TODO-interDom  & TODO-shapeColl \\
\end{tabular}
\begin{tablenotes}
\item[\add{\(\dag\)}] \add{Uses constant curvature metrics, and constructs an intermediate domain locally.} % CHANGE, maybe weglassen
\end{tablenotes}
\end{threeparttable}
\caption{Summary of distortion-based methods.}
\label{tab:homeomorphism-comp}
\end{table*}

\subsection{Overlay Meshes}
\label{subsec:overlay_meshes}

A standard approach for representing inter-surface maps \(\Phi\) in a piecewise linear fashion is to employ \emph{overlay meshes}. Intuitively, an overlay mesh is the common triangulation of both surfaces \(\surfaceA\) and \(\surfaceB\) between which the inter-surface map \(\mapPhi\) maps. In general, the map cannot be linear in the faces of \(\surfaceA\), but must also map the edges to piecewise linear curves on the other surface. Thus, the connectivity of \(\surfaceA\) must be embedded in \(\surfaceB\), and by subsequently triangulating the resulting polygonal mesh, the overlay mesh can be obtained. The vertex set of this mesh comprises the original vertices of both surfaces as well as additional intersection vertices generated by edge-edge intersections. 

%make clear, that the overlay mesh changes from iteration to iteration, because it depends on the current vertex-to-vertex map

Overlay meshes serve as a foundation for several optimization-based inter-surface map approaches \cite{schmidt2019-injectiveMaps, schmidt2020in-constantCurvatureMetrics}. However, they are associated with significant computational and numerical challenges. First, the complexity of the overlay mesh, \ie, their element count, scales with the product of the input mesh complexities. Consequently, optimization methods operating on overlay meshes tend to be computationally expensive. Further, the geometric quality of these meshes is generally poor: since vertices from \(\surfaceA\) may be mapped arbitrarily close to vertices or edges of \(\surfaceB\), the triangulation often contains \emph{sliver triangles} (commonly referred to as caps and needles), which are known to compromise numerical robustness in subsequent computations. Even overlay meshes constructed from high-quality input surfaces, tessellated with equilateral triangles, can still contain arbitrarily ill-conditioned elements.

In \cite{schmidt2023-adaptiveTriangulation}, this problem is addressed: They compute an inter-surface map represented as a piecewise linear function, while maintaining numerical stability and achieving good runtimes. The core idea is to introduce a third triangulation with high-quality elements that approximates the map. This intermediate triangulation is updated throughout the optimization, while implicitly tracking the true inter-surface map. The explicit overlay computation is deferred until the final map is extracted. Since the map resolution is decoupled from the input surfaces, this method can be used to compute shape correspondences between entire shape collections in a reasonable time. However, the method relies on spherical embeddings for the map representation, which restricts its applicability to genus-zero meshes.

\subsection{With Intermediate Domain}
The computation of maps from a surface to a specific domain, often canonical domains such as the plane, is a well-studied topic within the geometry processing community. Thus, it comes to no surprise that many inter-surface map approaches build on this work by reformulating the problem and introducing such a canonical domain as \emph{intermediate domain}. A map \(\mapPhi\) from a source surface \(\surfaceA\) to a target surface \(\surfaceB\) is then expressed as the composition of two maps, \(\mapPhi := f_{\domain \rightarrow \surfaceB} \circ f_{\surfaceA \rightarrow \domain}\), where the first map \(f_{\surfaceA \rightarrow \domain}\) maps from \(\surfaceA\) to a well-studied intermediate domain \(\domain\), and the second map \(f_{\domain \rightarrow \surfaceB}\) maps from the intermediate domain  \(\domain\) to \(\surfaceB\).

A popular choice for the intermediate domain is the Euclidean plane \cite{schmidt2019-injectiveMaps, morreale2021in-neuralSurfaceMaps, morreale2024in-neuralSemantic}. This preference is largely motivated by its foundational role in surface parameterization. Alternative intermediate domains have also been considered, most notably the sphere \cite{schmidt2023-adaptiveTriangulation}. However, the use of a fixed canonical domain introduces inherent challenges due to topological constraints. For example, for a planar domain, the Uniformization Theorem states that only surfaces with disk topology are homeomorphic to it. As a result, applying such methods to closed surfaces is non-trivial and typically requires the introduction of cuts and transition functions to manage the topology. Apart from canonical domains, it is also possible to compute a base complex suited for both shapes and use this as an intermediate domain \cite{schreiner2004-interSurfaceMapping}.

In the spirit of surface parameterization, older methods \cite{schreiner2004-interSurfaceMapping} aim to minimize distortion between the input surfaces and the intermediate domain. However, this strategy generally does not produce high-quality inter-surface maps with low \emph{end-to-end} distortion. In fact, the distortion to the intermediate domain provides limited insight into the distortion of the composite map \(\mapPhi\), as the distortion introduced by the map to the intermediate domain can be canceled as well as amplified by the second map to the target surface.

To address this limitation, recent methods \cite{schmidt2019-injectiveMaps, schmidt2020in-constantCurvatureMetrics,morreale2021in-neuralSurfaceMaps,takayama2022in-compatibleIT, schmidt2023-adaptiveTriangulation} explicitly optimize the end-to-end distortion of the full inter-surface map \(\Phi\). In these approaches, the intermediate domain is merely used to express the degrees of freedom of the map, while distortion is evaluated directly between the source surface and its image induced by \(\Phi\). 

In \cite{morreale2021in-neuralSurfaceMaps}, \emph{overfitted} neural surface maps \(\phi: \Omega \subset \reals^2 \rightarrow \reals^3\) are used to represent the parametrization of a surface with disk-topology. An inter-surface map can be directly obtained by map composition. Since these neural surface maps represent the input surfaces, they should not be changed. Thus, to optimize the map between the surfaces, a third neural surface map \(h: \Omega \rightarrow \Omega\) is introduced and optimized. The approach is easily extendable to shape collections by introducing multiple intermediate maps \(h_i\). \cite{morreale2024in-neuralSemantic} extend this framework to genus-zero surfaces and incorporate semantic features. To accommodate for semantic features, they do not require the map to have low distortion, but instead require the Jacobian of the map to change slowly. 


\subsection{Without Intermediate Domain}
Few methods avoid relying on an intermediate domain for their map representation. In \cite{schmidt2020in-constantCurvatureMetrics}, the vertices of one mesh are directly mapped to the other surface. The image of the edges is defined as a geodesic under a constant curvature metric. In this representation, the surface is locally embedded on a model manifold with constant curvature, but no global embedding is needed. Although different metrics are needed for genus 0, 1, and \(\geq 2\) surfaces, their approach still yields a unified formulation across all topologies.

The method of Takayama \cite{takayama2022in-compatibleIT} \emph{entirely} avoids an intermediate domain. Instead, it leverages intrinsic triangulations combined with a heuristic edge-flipping strategy to construct two compatible triangulations, which are then optimized for low distortion.

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