\section{Advances in Shape Alignment}
\label{sec:advances_in_shape_alignment}
Recent years have witnessed significant progress in shape alignment. In this section, we review these advances from several complementary perspectives, including template-based versus template-free approaches, deformation representations, hierarchical strategies, and regularization techniques. These categories capture the main underlying design choices and highlight the respective advantages and limitations of different methods. A summary of key approaches and their properties is provided in \Cref{tab:shape-alignment-comp}.

\begin{table*}[htb]
\centering
\begin{threeparttable}
\begin{tabular}{c||c|c|c|c|c|c|c}
\textbf{Method} & 
\makecell{Objective}& % \\ Intrinsic \\ or Extrinsic} & 
\makecell{Regularization}& 
\makecell{Template-\\based}&
\makecell{Super-\\vision}& 
\makecell{Initialization}& 
\makecell{Partial}& 
\makecell{Shape\\Collection} \\
\hline
\hline
% SMRO % can also do unsupervised training, eval for correspondence only supervised, unsup only for segmentation transfer CHANGE
\makecell{\cite{sundararaman2022ex-reducedRepresentationDF}} & \makecell{GT or CD} & \makecell{volume preservation,\\
ARAP} & \makecell{\yes} & \makecell{\del{(}\yes\del{)}\add{\tnote{\(\dag\)}}} & \makecell{\no} & \makecell{\no} & \makecell{\yes} \\
\hline
% Deep Implicit Templates
\makecell{Deep Implicit \\ Templates \cite{zheng2021-deepImplicitTemplates}} & \makecell{reconstruction\\loss} & \makecell{smoothness} & \makecell{\yes} & \makecell{weakly} & \makecell{\no} & \makecell{\no} & \makecell{\yes} \\
\hline
% ShapeFlow % eigentlich unsupervised CHANGE
\makecell{ShapeFlow\\ \cite{jiang2020-shapeflow}} & \makecell{CD} & \makecell{invertibility} & \makecell{\yes} & \makecell{\del{(}\no\del{)}} & \makecell{\no} & \makecell{\no} & \makecell{\yes} \\
\hline
% DIF correspondence is unsupervised, some other step is not
\makecell{DIF \\ \cite{Deng2021-deformedImplicitField}} & \makecell{normal\\consistency} & \makecell{smoothness} & \makecell{\yes} & \makecell{\no} & \makecell{\no} & \makecell{\no} & \makecell{\yes} \\
\hline
% Gencorres
\makecell{GenCorres \\ \cite{yang2023ex-gencorres}} & \makecell{VAE loss} & \makecell{cycle-consistency,\\ARAP, ACAP} & \makecell{\yes} & \makecell{\no} & \makecell{\no} & \makecell{\no} & \makecell{\yes} \\
\hline



% AGK
\makecell{\cite{aigerman2022-neuralJacoianFields}} & \makecell{GT vertex\\positions} & \makecell{GT Jacobians} & \makecell{\no} & \makecell{\yes} & \makecell{\no} & \makecell{\yes} & \makecell{\no} \\
\hline
% Neuromorph
\makecell{NeuroMorph \\ \cite{eisenberger2021ex-neuromorph}} & \makecell{ED} & \makecell{ARAP, GD} & \makecell{\no} & \makecell{weakly} & \makecell{\no} & \makecell{\yes\tnote{\(\ast\)}} & \makecell{\no} \\
\hline
% CorrNet3D % eig unsupervised, but also say how supervised can be done CHANGE
\makecell{CorrNet3D\\\cite{zeng2021ex-corrnet3d}} & \makecell{reconstruction\\loss} & \makecell{neighborhood\\preservation} & \makecell{\no} & \makecell{\del{(}\no\del{)}} & \makecell{\no} & \makecell{\no} & \makecell{\no} \\
\hline
% ANVL
\makecell{\cite{atzmon2021-augmentingNeuralDefFields}} & \makecell{VAE loss} & \makecell{AKAP} & \makecell{\no} & \makecell{\no} & \makecell{\no} & \makecell{\no} & \makecell{\yes} \\
\hline
% Mesh Draping
\makecell{Mesh Draping \\ \cite{hertz2023-meshDraping}} & \makecell{CD} & \makecell{angle distortion,\\triangle quality} & \makecell{\no} & \makecell{\no} & \makecell{landmarks} & \makecell{\yes} & \makecell{\no} \\
\hline
% deformation guided unsupervised
\makecell{\cite{merrouche2023ex-deformationGuidedUnsup}} & \makecell{GD, ED} & \makecell{cycle-consistency,\\ARAP} & \makecell{\no} & \makecell{\no} & \makecell{\no} & \makecell{\no} & \makecell{\no} \\
\hline
% GenAnalysis
\makecell{GenAnalysis\\\cite{yang2025ex-genanalysis}} & \makecell{reconstrucion\\loss} & \makecell{AAAP} & \makecell{\no} & \makecell{\no} & \makecell{\no} & \makecell{\yes\tnote{\(\ast\)}} & \makecell{\yes} \\
\hline
% divergence-free
\makecell{\cite{eisenberger2019ex-divergenceFree}} & \makecell{feature similarity,\\ED} & \makecell{implicit volume\\preservation} & \makecell{\no} & \makecell{ -\add{\tnote{\(\ddag\)}} } & \makecell{\no} & \makecell{\no} & \makecell{\no} \\
%\hline
%\makecell{TODOmethod} & \makecell{TODOobj} & \makecell{TODOreg} & \makecell{TODOtemp} & \makecell{TODOsuper} & \makecell{TODOinit} & \makecell{TODOpartial} & \makecell{TODOgroup} \\
%\hline
\end{tabular}
\begin{tablenotes}
\item[\add{\(\dag\)}] \add{Presents unsupervised training used for segmentation transfer.} % CHANGE, maybe weglassen
\item[\(\ast\)] Also capable of partial-to-partial correspondences.
\item[\add{\(\ddag\)}] \add{Non data-driven method.}
\end{tablenotes}
\end{threeparttable}
\caption{Summary of recent shape alignment methods. Note that the regularization column \add{only} contains the most important regularization terms. We use the following abbreviations: GT: Ground truth, CD: Chamfer distance, ED: Euclidean distance, GD: Geodesic distance, ARAP: As-Rigid-As-Possible, ACAP: As-Conformal-As-Possible, AKAP: As-Killing-As-Possible, \add{AAAP: As-Affine-As-Possible.}}
\label{tab:shape-alignment-comp}
\end{table*}

\subsection{Template-Based vs. Template-Free Approaches}
Template-based methods use a canonical representation to which all shapes of a class are mapped, thereby inducing a star-shaped correspondence topology. Such templates can take the form of explicit meshes or point clouds \cite{sundararaman2022ex-reducedRepresentationDF}, implicit fields \cite{zheng2021-deepImplicitTemplates, Deng2021-deformedImplicitField}, or latent-space templates \cite{jiang2020-shapeflow, yang2023ex-gencorres}.

Most commonly, the template is either picked from the shape collection \cite{sundararaman2022ex-reducedRepresentationDF} or it represents some average shape \cite{zheng2021-deepImplicitTemplates}. Deng \etal \cite{Deng2021-deformedImplicitField} learn a fusion of signed distance function (SDF) values across the collection rather than a valid SDF of an average shape. This superposition, combined with a learned correction field, enables the explicit handling of topological differences and regions where simple deformation fails. GenCorres \cite{yang2023ex-gencorres} combines an implicit generator trained using the VAE-loss proposed in SALD \cite{atzmon2020-SALD} with an explicit mesh template, allowing correspondences to be first established in an implicit representation and then refined using an explicit mesh.

Template-free approaches establish correspondences without relying on a canonical template. They either compute deformations directly between shapes \cite{eisenberger2021ex-neuromorph, hertz2023-meshDraping} or learn a latent space \cite{jiang2020-shapeflow, atzmon2021-augmentingNeuralDefFields} in which a latent code represents each shape. Note that ShapeFlow \cite{jiang2020-shapeflow} uses a template for shape matching but not for establishing the latent space.

\subsection{Deformation Representation}
The representation of shape deformations lies at the core of shape alignment methods. The most straightforward way is to prescribe the final position or a deformation field that produces the deformed shape in one go \cite{hertz2023-meshDraping}. Alternatively, a continuous deformation process \cite{eisenberger2019ex-divergenceFree, jiang2020-shapeflow, eisenberger2021ex-neuromorph} can be used to obtain the final deformation, which also considers intermediate shapes. Instead of predicting vertex positions or a continuous deformation directly, \cite{aigerman2022-neuralJacoianFields} predict the Jacobians of the deformation and reconstruct the deformed surface from the predicted Jacobians by solving a Poisson equation.

More structured representations include handle-based \cite{shechter2022ex-neuralmls} or cage-based representations \cite{yifan2020ex-neuralCages}, which manipulate shapes via a set of control points, and learned latent embeddings \cite{jiang2020-shapeflow}. Reduced representations are also widely used, realized via basis representations \cite{eisenberger2019ex-divergenceFree}, patch-based deformation models \cite{merrouche2023ex-deformationGuidedUnsup}, or mesh-free function approximation methods \cite{sundararaman2022ex-reducedRepresentationDF}. Sundararaman \etal \cite{sundararaman2022ex-reducedRepresentationDF} show that using a reduced representation proves beneficial in data-driven approaches, as it allows high-quality results to be obtained with fewer training samples.

\subsection{Hierarchical Approaches}
Deformation-based methods are prone to undesirable local optima. Hierarchical methods address these instabilities. Mesh Draping \cite{hertz2023-meshDraping} gradually maps source mesh vertices using positional encodings of increasing frequency, biasing the optimization towards smooth deformations at first and progressively enabling finer geometric detail. 

Merrouche \etal \cite{merrouche2023ex-deformationGuidedUnsup} adopt a coarse-to-fine strategy, beginning with patch-level matchings, which are further refined to vertex-level correspondences. This is achieved by a patch-based deformation model, which represents shape deformation as a collection of near-rigid patch transformations, reducing the number of optimization variables. Such hierarchical formulations improve robustness to noise while also reducing the dimensionality of the problem, thereby making unsupervised learning feasible.

\subsection{Regularization}
To ensure that estimated deformations remain geometrically plausible, regularization is commonly applied. Two complementary forms of regularization are typically employed:
\begin{itemize}
    \item \textbf{Implicit prior}: Neural network parameterizations inherently constrain the deformation space, since the architecture limits the class of admissible transformations. For instance, Mesh Draping \cite{hertz2023-meshDraping} optimizes a fully-connected network that directly predicts vertex positions, acting as an implicit prior on smooth and coherent deformations. Similarly, \cite{zeng2021ex-corrnet3d} use a neural network to predict the deformation of the point cloud of \(\surfaceA\) such that the point cloud of \(\surfaceB\) is well approximated and vice versa.
    Eisenberger \etal \cite{eisenberger2019ex-divergenceFree} construct deformation fields using a coarse-to-fine basis derived via the Karhunen–Loève expansion, which inherently ensures volume preservation.
    \item \textbf{Explicit prior}: 
    Hand-crafted regularization terms enforce desirable properties either of the deformed surfaces, the deformation field, or the correspondence itself. Well-known regularization terms include ARAP (as-rigid-as-possible) \cite{eisenberger2021ex-neuromorph} and ACAP (as-conformal-as-possible) \cite{yang2023ex-gencorres,yang2023ex-geolatent}, which penalize distortion introduced by the deformation process.
    The deformation field itself can be regularized by AAAP (as-affine-as-possible) \cite{yang2025ex-genanalysis} and AKAP (as-killing-as-possible) \cite{atzmon2021-augmentingNeuralDefFields} formulations. Further, promoting cycle consistency \cite{merrouche2023ex-deformationGuidedUnsup} or invertible deformations \cite{jiang2020-shapeflow} encourages bijective correspondences.
\end{itemize}
The regularization terms are incorporated in frameworks differently.
NeuroMorph \cite{eisenberger2021ex-neuromorph} integrates correspondence and interpolation in a unified neural framework. The network computes a stochastic assignment between two input shapes and produces a continuous interpolation, expressed as a displacement field, in a single forward pass.  Querying the network for intermediate times \(t \in [0,1]\) allows the application of an ARAP loss to ensure meaningful interpolation trajectories.

Methods relying on latent representations can control the construction of the latent space by enforcing that the interpolation of latent codes produces a sequence of interpolated shapes that undergo plausible deformations using regularization terms \cite{jiang2020-shapeflow,sundararaman2022ex-reducedRepresentationDF, yang2023ex-gencorres,yang2025ex-genanalysis}. More precisely, given the latent embedding \(Z_{\surfaceA}\) and \(Z_{\surfaceB} \in \reals^D\) of two shapes, the linear interpolation between these latent codes produces a sequence of interpolated latent codes \(Z_{{\surfaceA}{\surfaceB}}(\alpha) = \alpha\, Z_{\surfaceA}  + (1  - \alpha) Z_{\surfaceB}\). The decoded shapes corresponding to these interpolated latent codes should undergo plausible deformations, which can, for example, be achieved by promoting rigidity. 

For implicit surface representations, imposing rigidity constraints directly is challenging. Therefore, in \cite{atzmon2021-augmentingNeuralDefFields}, the network simultaneously estimates an auxiliary piecewise-linear deformation field which is used to regularize the latent space. 


\color{black}
%\color{red}
%\todo{include}
%Encoder-only Transformers-based architecture for explicit point wise correspondence
%\cite{riva2024ex-localizedGaussians} - attention heads, particularly resembling Gaussian functions centered on each point of the input shape - fixing them improve efficiency and robustness of training
%Implicit field supervision for robust non-rigid shape matching
%SALD: Sign Agnostic Learning with Derivatives,  Matan Atzmon, Yaron Lipman --> VAE loss used by GenCorres
%Geolatent: A geometric approach to latent space design for deformable shape generators
%A morphable model for the synthesis of 3D faces
%Neuralmls: Geometry-aware control point deformation
%Neural cages for detail-preserving 3d deformations
%3dn: 3d deformation network
%Divergence-Free Shape Correspondence by Deformation (already in the 2019 STAR)
%ResNet-LDDMM: Advancing the LDDMM Framework Using Deep Residual Networks (registration but relevant, diffeomorphisms)
%Learnable deformations among 3d shapes Geometry processing with neural fields
%\color{black}