
\section{Deformation-based methods}
\label{sec:deformation}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/deformation/deformation}
    \caption{\textbf{Deformation methods} recover the correspondence between shapes by finding a deformation that aligns them spatially.}
    \label{fig:deform_2}
\end{figure}


\begin{table*}[ht]
    \footnotesize
    \centering
    \begin{tabular}{l@{\hspace{0.3em}}c@{\hspace{0.5em}}c@{\hspace{0.5em}}c@{\hspace{0.5em}}c@{\hspace{0.5em}}c}
        \toprule
         \textbf{Method} & \textbf{Category} & \textbf{Correspondence} & \textbf{Embedding Space} & \textbf{Deformation Model} & \textbf{Training} \\ 
         \midrule
        Revisited-CPD \cite{fan2022coherent} &  \textbf{O} & Soft & Euclidean & Coherent & \NONE \\ 
        Geo-CPD \cite{hirose2022geodesic} &  \textbf{O} & Soft & Manifold & Coherent & \NONE \\
        Tajdari et al. \cite{tajdari2022feature} &  \textbf{O} & Hard & Euclidean & Affine & \NONE \\
        Yao et al. \cite{yao2023fast} &  \textbf{O} & Hard & Euclidean & Locally-affine & \NONE \\
        Zhao et al. \cite{zhao2024correspondence} &  \textbf{O} & Soft & Euclidean & Kernel-based & \NONE \\
        Merrouche et al. \cite{merrouche2025matching} &  \textbf{O} & Hard & Neural & ARAP & \NONE \\
        SPARE \cite{yao2025spare} &  \textbf{O} & Hard & Euclidean & ARAP & \NONE \\
        \midrule
        Jiang et al. \cite{jiang2023non} &  \textbf{H} & Neural & Neural & ARAP & \SUP \\
        Chen et al. \cite{chen2023rethinking} &  \textbf{H} & Global & Neural & Parametric & \SUP \\ %
        Qin et al. \cite{qin2023deep} &  \textbf{H} & Neural & Neural & Parametric & \SUP \\
        ArtEq \cite{Feng_2023_ICCV} &  \textbf{T} & Neural & Semantic & Parametric & \SUP \\ 
        Guo et al. \cite{guo2024diffusion} &  \textbf{H} & Hard & Neural & Coherent & \SUP \\ %
        ETCH \cite{li2025etchgeneralizingbodyfitting} &  \textbf{T} & Hard & Manifold & Parametric & \SUP \\
        Jung et al. \cite{jung2025variable} &  \textbf{T} & Hard & Euclidean & Kernel-based & \NONE \\
        \midrule
        NDP \cite{li2022non} &  \textbf{L} & Neural & Neural & Neural Field & \SUP \\
        IFS \cite{avidan_implicit_2022} &  \textbf{L} & Neural & Implicit & Neural Field & \UNSUP  \\
        LVD \cite{corona2022learned} &  \textbf{H} & Neural & Neural & Neural Field & \SUP \\ %
        Sundararaman \cite{sundararaman_deformation_2024} &  \textbf{L} & Neural & Spectral & Jacobian-based & \SUP\\
        Merrouche \cite{Merrouche_2023_BMVC} &  \textbf{L} & Neural & Neural & Hierarchical & \UNSUP \\
        Lin et al. \cite{lin2023leveraging} &  \textbf{L} & Neural & Manifold & Neural Field & \UNSUP \\
        Feng et al. \cite{feng2023differentiable} &  \textbf{L} & Neural & Neural & Parametric & \SUP \\
        NICP \cite{marin2024nicp} &  \textbf{H} & Neural & Neural & Neural Field & \SUP \\
         \bottomrule
    \end{tabular}
    
    \caption{Overview of \textbf{deformation-based} matching methods. Methods are categorized by their approach (\textbf{O}ptimisation, \textbf{T}emplate, \textbf{L}earning, or \textbf{H}ybrid), correspondence type (Hard, Soft, Neural, Global), feature space, deformation model, and training requirements (\NONE one, \SUP upervised, or \UNSUP nsupervised).
    }
    \label{tab:deformation_methods}
\end{table*}






So far, the methods that we considered retrieve the correspondence without modifying the input shapes, mainly by relying on intrinsic properties of the geometry. 
In this section, we instead present methods that recover correspondence by computing a deformation field that aligns a source surface with a target surface in the ambient 3D coordinate space.
Unlike \textit{intrinsic} methods, which rely on surface-aware metrics like geodesic distances, this \textit{extrinsic} paradigm fundamentally considers Euclidean metrics and is thus broadly applicable to various 3D data representations, including unstructured point clouds, without requiring manifold connectivity or consistent topology between the shapes being registered.
It is worth mentioning that while the majority of such works define the problem in terms of point sets, they are commonly applied to mesh vertices.
Surface properties are often used both for regularisation and as a domain for point sampling. 

In revisiting the literature of recent years, we categorise deformation works into three main branches: optimisation-, template-, and learning-based. 
Drawing a clear distinction among them is impossible, as many methods are hybrid, and our categorisation follows what we believe are the main principles behind the paper.
We additionally emphasise the type of optimisation that is conducted, feature embedding space, correspondence type, and deformation model (see ~\cref{tab:deformation_methods}).
We begin with background on seminal techniques that continue to serve as inspiration. 


\subsection{Background}
While intrinsic methods come with a number of advantages, such as invariance of the shape's embedding in the ambient Euclidean domain and strong structural priors, their theoretical premises are also a source of drawbacks. 
Noise, topological changes, as well as incomplete or cluttered shapes, often introduce enough instability to make purely intrinsic methods inapplicable. 
On the contrary, we have already seen that considering the extrinsic information of shapes can play an important role, for example, in distinguishing symmetries, and provide semantical nuances difficult to catch on a pure surface level.
Along this line, several methods rely on the assumption that the correspondence is equivalent to a certain, generally unknown, deformation. For the sake of simplicity, we formalise the problem in an idealised bijective setting. 
Considering two sets of points $X_{\mathcal{M}}$ and $X_{\mathcal{N}}$, and assuming there exist a permutation $\Pi: X_{\mathcal{N}} \rightarrow \widehat{X}_{\mathcal{N}} \in \mathbb{R}^{n_{\mathcal{M}} \times 3}$ that reorganizes their vertices under the desired semanticity, the idea is to recover an equivalent deformation $\mathcal{F}:X_{\mathcal{M}} \rightarrow \widehat{X}_{\mathcal{M}} \in \mathbb{R}^{n_{\mathcal{M}} \times 3}$ such that:
\begin{equation}
    \mathcal{F}(\mathbf{X}_{\mathcal{M}}) \approx {\Pi}(\mathbf{X}_{\mathcal{N}}).
    \label{eq:deformation}
\end{equation}
It is important to outline a few observations, which will help us anticipate the challenges addressed in the literature. First, perfect alignment can only occur when we have access to a perfect deformation model, and the two shapes share the same discretisation; this generally holds only for the synthetic settings. Second, given the permutation ${\Pi}$ or the deformation $\mathcal{F}$, the other can be recovered in a closed form \cite{besl1992method}; however, the standard case (and the one we will focus on) assumes that both are unknown, an ill-posed problem with many plausible solutions $\mathcal{F}$ and $\Pi$. 
Finally, it is often the case that only subsets of the points can be correctly aligned; clutter, noise, and outliers often require careful consideration and ad-hoc techniques. 
For these reasons, the problem is classically phrased as an energy minimisation.

\paragraph{Energy-based formulation.}
Solving \cref{eq:deformation} via optimisation requires defining an energy, which aims to align the shapes, but also recovers a plausible deformation. 
Hence, the energy is generally composed of data ($\Energy_\text{align}$) and regularization ($\Energy_{\text{reg}}$) terms, with a weighting factor $\alpha$:
\begin{align}
\Energy = \Energy_{\text{align}} + \alpha \Energy_{\text{reg}}
\end{align}

\paragraph{Seminal registration methods.} %
A survey on non-rigid registration would be incomplete without discussing two seminal works, which still serve as inspiration for many recent methods: Iterative Closest Point (ICP) ~\cite{besl1992method} and Coherent Point Drift (CPD) \cite{myronenko2010point}.
Since both the correspondence and the deformation are unknown, ICP proposes iterating between the two, fixing one and optimising the other. Starting from an initial configuration of the two shapes (i.e., $\widetilde{\mathcal{F}}(X_{\mathcal{M}}) = X_{\mathcal{M}}$), ICP obtains an estimated correspondence $\widetilde{\Pi}$ by Euclidean nearest-neighbour pairing:
\begin{equation}
    \widetilde{\Pi} = \arg\min\limits_{\Pi} \|\widetilde{\mathcal{F}}(X_{\mathcal{M}}) - \Pi(X_{\mathcal{N}})\|_2^2.
    \label{eq:corr}
\end{equation}
Next, the registration is updated $\widetilde{F}$ by minimizing the point-to-point distance with a least-squares objective:
\begin{equation}
    \widetilde{\mathcal{F}} = \arg\min\limits_{\mathcal{F}} \sum\limits_{i = 1}^{m}\|\mathcal{F}(X_{\mathcal{M}_i}) - \widetilde{\Pi}(X_{\mathcal{N}})_i\|_2^2.
    \label{eq:def}
\end{equation}
The two steps are iterated until convergence. 
Notice that to avoid trivial solutions, $\mathcal{F}$ should be restricted to a specific family, which originally was the class of rigid transformations~\cite{besl1992method}. 
Over the past several decades, numerous non-rigid variations have been proposed. 
The first non-rigid extension was proposed by Amberg et al. \cite{amberg2007optimal}, which revises the transformation in terms of local affine transformations, ensuring that changes are minimal between nearby vertices. 
By fixing the amount of desired stiffness, the authors derive a cost function that still admits a closed-form to recover an optimal iterate. 

As opposed to the discrete, nearest-neighbour-based approaches pioneered by ICP, probabilistic methods generate soft matches where each point in the source set $X_{\mathcal{M}}$ has a certain probability of corresponding to every point in the target set $X_{\mathcal{N}}$. 
The seminal Coherent Point Drift (CPD) algorithm by Myronenko and Song \cite{myronenko2010point} reformulates the geometric alignment problem through statistical density estimation. 
CPD treats the source points as centroids of a Gaussian Mixture Model (GMM), while the target points are considered as observations generated from this GMM. 
To handle outliers and noise, CPD augments the mixture model with a uniform distribution component, yielding the probability density:
\begin{equation}
p(\mathbf{x}) = w \cdot \frac{1}{n_{\mathcal{N}}} + (1-w) \sum_{i=1}^{n_{\mathcal{M}}} \frac{1}{n_{\mathcal{M}}} \mathcal{N}(\mathbf{x} | \mathcal{F}(x_i), \sigma^2 I)
\end{equation}
where $w \in [0,1]$ represents the assumed proportion of outliers, $\mathcal{F}$ denotes the transformation function, and $\sigma^2$ is the isotropic variance of the Gaussian components.
The registration problem can then be approximated through maximum likelihood estimation (MLE).


Since the two streams of work rely on different definitions of data fitting, the resulting regularisation techniques also differ. For ICP, the deformation is often chosen within a specific class, such as $\mathrm{SE}(3)$~\cite{besl1992method} or local Affine transformations \cite{amberg2007optimal}, which naturally restrict the valid solutions. 
CPD instead admits free displacements and thus relies on Motion Coherence Theory (MCT), penalising derivatives of all orders of the displacement field and ensuring neighbouring points are deformed consistently (this results in neighbouring points moving \textit{coherently}). 


\subsection{Revisiting classical methods}
\begin{figure*}
    \centering
    \includegraphics[width=\linewidth]{figures/deformation/GeoCPD}
    \caption{\textbf{An example of deformation} from GeoCPD \cite{hirose2022geodesic}. The source shape (black) is deformed toward the target one (blue), toward multiple steps till final convergence.}
    \label{fig:deform_1}
\end{figure*}



\paragraph{Optimisation-based.}

The standard point-to-plane metric as used in ICP only considers the normals of the static target surface. 
The recent SPARE \cite{yao2025spare} method thus proposes a symmetrised point-to-plane distance metric. 
This formulation incorporates normal information from both the deforming source and the target surfaces, providing a more accurate, second-order approximation of the local geometry, which can accelerate convergence and improve final accuracy \cite{yao2025spare}.
Alternatively, the problem of aligning distribution can be rephrased as a fuzzy clustering problem \cite{zhao2024correspondence}.
Beyond penalising smoothness through motion coherence or stiffness terms, geometrically derived features can provide additional constraints \cite{tajdari2022feature,hirose2022geodesic}.

To address noise, outliers, and acquisition artefacts, optimisation can be made more resilient through robust norms such as the $L_p$ norm with $p<2$ (e.g., $L_1$) for both alignment and regularisation terms \cite{yao2020quasi,zhao2024correspondence}.
Unlike the $L_2$ norm, which quadratically penalises large errors and is thus highly sensitive to outliers, these norms apply a less severe penalty, making the overall cost function more resilient to gross errors.
However, many robust norms are non-smooth and non-differentiable, complicating optimisation and inhibiting standard gradient-based solvers.
Recent work by Yao et al. \cite{yao2023fast} addresses this through the Majorisation-Minimisation (MM) algorithm with Welsch's function \cite{holland1977robust}, enabling closed-form solutions that prove computationally superior to non-smooth proximal methods (such as ADMM \cite{boyd2011distributed}) while maintaining robustness.

Despite the use of the Fast Gaussian Transform, the speed of the CPD algorithm remains a crucial drawback \cite{golyanik2016extended}. 
Subsequent works have significantly advanced the regularization framework: Hirose proposed an accelerated version using downsampling \cite{Osamu2021Acceleration} and later introduced a Bayesian formulation that generalizes both CPD and ICP by rephrasing coherence as a distribution prior \cite{Osamu2021Bayesian}, while also extending the regularization to use geodesic distances rather than Euclidean ones to better preserve the intrinsic geometry of the surface during deformation \cite{hirose2022geodesic}. 
Other extensions have adapted the coherence constraint to non-Euclidean settings \cite{fan2022coherent} and incorporated topological preservation through joint optimisation of explicit and implicit representations \cite{merrouche2025matching}.

\paragraph{Use of learned priors.}

Learned features can be used to complement non-rigid deformation-based approaches.
For instance, Jiang et al. learn deep features via functional maps (see \cref{sec:spectral}), which induces an initial set of point-to-point correspondences $\hat{\Pi}$ \cite{jiang2023non}. 
This permutation estimate can then be filtered based on geometric consistency and used to recover the deformation function $\mathcal{F}$. 
Guo et al. \cite{guo2024diffusion} address cloth deformation by learning a diffusion-based prior from real-world clothing deformations. 
Learned methods can be used to prune correspondences in a manner more suitable for non-rigid deformations \cite{qin2023deep}.
As with some functional map-based methods \cite{attaiki2021dpfm}, transformer-based modules can be useful in guiding overlap prediction to handle deformations between partial shapes \cite{mei2023overlap}.
Auxiliary tasks can also yield learned feature representations useful for registration \cite{chen2023rethinking}.

\subsection{Template fitting}

For certain classes, we might have access to an archetypal shape that describes the structure, and often comes with information on how shapes of that class are expected to deform. 
Such a model is referred to as a template, and when it is available, it provides a strong regularisation. 
Such a template is generally crafted by an artist or domain experts \cite{gao2023automated, dale1999cortical}.
The template can also be enriched with a learned parametric deformation space by aligning it to a collection of shapes. A seminal work in this sense is that of Blanz and Vetter \cite{blanz2023morphable}, which aligned a few hundred human faces to learn PCA, both to model the subject's geometry and texture. 
More recent shape models are the SMPL models for humans \cite{loper2015smpl, pavlakos2019smplx}, SMAL for the animals \cite{zuffi2017small}, and MANO for the hands \cite{romero2017mano}, but learning such morphable shapes is also popular in medicine \cite{MacFra_Unraveling_MICCAI2025, magnet2023assessing, posner2025statistical}. 

While generally such a procedure requires a large amount of curated data \cite{achenbach2017fast, loper2015smpl, pons2015dyna, zuffi2024varen}, it is worth mentioning that recent approaches also leverage data in the wild, such as images and videos from the web \cite{li2024learning}.

\paragraph {Fitting.} To take advantage of the available template, its structural properties can be used during the optimisation. For example, humans can be represented as piece-wise rigid deformations, as done in the Stitched Puppet formulation \cite{zuffi2015stitched}. 
Recent methods make use of local $\mathrm{SO}(3)$-equivariance features to represent the local rigidity \cite{li2025etchgeneralizingbodyfitting,Feng_2023_ICCV}, though in practice, such equivariance assumes little or no noise. 
Since templates provide a smooth and complete geometry, they can be integrated with intrinsic methods, enhancing their robustness \cite{marin2020farm}. 
It is likely that a template cannot catch all the nuances of a specific class. Hence, it is often the case that to align it to the data, the local details are delegated to free-form offsets. 
Examples are for skin details \cite{marin2019high}, cloths \cite{ma2020learning}, and soft-tissues \cite{pons2015dyna}. 
Jung et al. take an approach of jointly optimising a shared template with respect to a collection of input shapes \cite{jung2025variable}.
The template fitting can also be enriched by visual semantic information present on the surface, such as textures \cite{antic2024close, lazova2019360}.
Recent developments in capturing technologies have also enabled the study of fine-grained template deformation, such as that in fabrics. 
Modelling wrinkles can be achieved by physics modelling \cite{yu2025phydeformer} or learned-priors \cite{guo2024diffusion}.


\subsection{Learning-based methods}
With the advent of deep learning, many methods leverage deep neural networks trained to learn a deformation model from a collection of data.  
3DCoded \cite{groueix_3d-coded_2018} popularised this line of work, training an autoencoder to deform a template into all the shapes of the training set. 
In case ground truth registration is available, such a network can be trained in a supervised way using a vertex-to-vertex loss. 
On the contrary, it can also be trained without supervision, relying on Chamfer Distance. 
In the follow-up \cite{deprelle2019learning}, the same authors also investigated the possibility of learning primitive structures together with the deformation module. 
A revision of the 3DCoded formulation has also been proposed with Transformer architecture \cite{trappolini_shape_2021}, where the network relies on an attention mechanism that weights points by their area.

The deformation can be represented globally, for instance, as a set of hierarchical MLPs \cite{li2022non}.
Merrouche et al. \cite{Merrouche_2023_BMVC} suggest training both an association and a deformation network relying on hierarchical features.
Temporal network modules such as GRU can be used to represent the deformation incrementally, enhancing robustness \cite{feng2023differentiable}.
A more nuanced geometric view is achieved through Neural Jacobian Fields (NJF) \cite{aigerman_neural_2022}, which first learns a matrix field in the ambient space and then derives the Jacobian for each triangle via an intrinsic projection. 
A final global deformation can then be recovered by solving a Poisson equation based on the locally learned Jacobians.
Sundararaman et al. build upon NJF by abandoning the need for a global shape encoder, representing shapes' vertices with their local Jacobians, learning a coarse-to-fine deformation entirely based on local properties \cite{sundararaman_deformation_2024}.


Particularly convenient is relying on implicit representations, since they provide spatial and differentiable information useful for the fitting. 
Additionally, they offer a flexible representation that is significantly more robust to noise and partial information. Sundararaman et al. propose representing shapes as neural fields and utilising implicit representations to regularise deformation \cite{avidan_implicit_2022}. 
On a similar principle, they propose to learn the deformation as local fields \cite{sundararaman_reduced_2022}. 
In IPNET  \cite{bhatnagar2020combining}, the authors combine a deformable template with a network to predict the occupancy of local body parts, which is then used to guide the SMPL registration. 
Wang et al. \cite{wang2021locally} propose PTF, a similar framework that relies on piece-wise implicit representations. LoopReg \cite{bhatnagar2020loopreg} also proposes a weakly-supervised schema for shape registration. 

While these works rely on implicit representations to represent the shapes, a more recent trend is to leverage neural fields to also represent deformations as a velocity field defined all over $\mathbb{R}^3$, as proposed in Learned Vertex Descent \cite{corona2022learned}. 
The motivation is that they provide a richer representation for the deformation, on which further supervision can be provided. While the original paper utilises the neural field primarily during training, Marin et al. \cite{marin2024nicp} demonstrate that this formulation can also be beneficial at inference time. 
In practice, they propose Neural ICP (NICP), a procedure to iteratively refine the neural deformation field by retrieving its induced correspondence. Neural deformation fields have also been defined intrinsically, in the clothing deformation case \cite{lin2023leveraging}. Lastly, in the majority of previous works, the correspondence step is delegated to features coming from neural backbones. 
















    
