Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework

$\renewcommand{\Diff}{\mathcal{D}}\newcommand{\dist}{\mathrm{dist}}\renewcommand{\Imm}{\mathcal{I}}\newcommand{\Shape}{\mathcal{S}}\newcommand{\R}{\mathbb{R}}\newcommand{\vol}{\operatorname{vol}}\newcommand{\Vol}{\mathrm{Vol}}\newcommand{\Var}{V}$

<h4 align="left">Elastic Shape Analysis of Surfaces</h4><hr>
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				<p>
				The goal of elastic shape analysis is to define a Riemannian metric on the space of unparameterized surfaces.
				</p>
				<hr><p>Let $M$ be a smooth oriented, Riemannian 2-manifold.</p>
				<p>
				<b>Parameterized Shapes:</b><br>
				$\qquad \qquad \qquad \Imm =\operatorname{Imm}(M,\R^3)/\text{translations}$<br>
				<b>Reparameterization Group:</b><br>
				$\qquad \qquad \qquad \Diff = \operatorname{Diff}_+(M)$<br>
				<b>Unparameterized Shapes:</b><br>
				$\qquad \qquad \qquad \Shape = \Imm/\Diff\qquad [q]=\{q\circ\phi|\phi\in\Diff\}$
				</p>
				<hr>
				<p>We equip $\Imm$ with a metric $G$ that is invariant under the action of $\Diff$ and equip $\Shape$ with the quotient metric.<br>
				</p><hr><p>
				Traditionally, to solve geodesic boundary value problems between $[q_0],[q_1]\in\Shape$, we simultaneously find an optimal reparameterization $\phi$ and an optimal path between $q_0$ and $q_1\circ\phi$ which minimizes the path energy functional. </p>
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				<img src="figs/OptOrbits1.png" width="80%" /><p>This requires implementing the action of $\Diff$ on $\Imm$ which requires restrictive parameterizations of the objects.</p>
				</div>
				</div>
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<h4 align="left">Second Order Sobolev Metrics</h4><hr>
				<p align="left">
				Let $q\in \Imm$ and $h,k\in T_q\Imm$. We let $g_q$ be the pullback metric of the Euclidean metric on $\R^3$. A second order Sobolev metric is then given by
				\begin{equation}
				\label{eq:H1metric}
				G_q(h,k)=\int_M \langle h,k \rangle +g_q^{-1}(dh,dk)+ \langle\Delta_q h,\Delta_q k\rangle\vol_q.
				\end{equation}
				where <br>
				+ $dh$ and $dk$ are viewed as a vector valued one forms, <br>
				+ and $\Delta_q$ is the surface Laplacian.<br>
				</p>
				<div class="b"><div class="btitle" align="left"><h4>Note</h4></div>
				<p align="left">
				For fixed coordinate  we can view $g_q$, $dh$, and $dk$ as matrix fields and \[g_q^{-1}(dh,dk)=\operatorname{tr}(dh\cdot g_q^{-1}\cdot dk^T)\]
				</p>
				</div>
				<p class="fragment fade-in" align="left">
					Following the construction of Su et. al.$^1$ we will further decompose the first order term, in four different terms which each have a geometric interpretation. Therefore, we write
					\begin{equation}
					dh= dh_m+dh_++dh_\perp+dh_0.
					\end{equation}
				</p>
				<p align="left"><sub>$^1$. Su, Bauer, Preston, Laga, and Klassen, “Shape analysis of surfaces using general elastic metrics”.</sub><br></p>

<h4 align="left">Varifold Representations$^{1,2,3,4}$</h4><hr>
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		<div class="col-md-8" markdown="1" align="left"><p>
		For $[q]\in \Shape$ we can associate a varifold $\mu_q\in\mathcal{M}(\mathbb R^3\times S^{2})$.<br><br>
	  In particular, $\mu_q:= (q,n_q)_*\operatorname{vol}_q$ where $n_q$ is the unit normal map of $q$</p><br><br>
		<p class="fragment" data-fragment-index="2">
		We take  a norm $\|\cdot\|_{V}$ on $\mathcal{M}(\mathbb R^3\times S^{2})$ where for $[q],[p]\in\Shape$ the scalar product between the associated varifolds $\mu_{q}$ and $\mu_{p}$ can be written as: \begin{equation*}\label{equ:norm_var}
		\langle \mu_{q},\mu_{p}\rangle_{V}=\iint_{M \times M}e^{-\alpha||q(x)-p(y)||^2}\langle n_q(x), n_p(y)\rangle^2\vol_{q}(x) \vol_{p}(y).
		\end{equation*}</p> <br>
		<span class="fragment" data-fragment-index="3" align="left">
			<p>Thus we define our choice of relaxation term as,\[\Gamma([q],[p]):=||\mu_p-\mu_q||^2_{V}=\langle \mu_{p},\mu_{p}\rangle_{V}+\langle \mu_{q},\mu_{q}\rangle_{V}-2\langle \mu_{p},\mu_{q}\rangle_{V}.\]</p>
		</span>
	</div>
		<div class="col-md-4" markdown="1" align="right"><img src="figs/Face_fullsample.png" width="50%" /><img src="figs/Face_downsampled.png" width="50%" /><hr><img src="figs/varifold_similar.png" width="100%" /></div>
		<p align="left"><sub>$^1$Charon & Trouvé. "The varifold representation of nonoriented shapes for diffeomorphic registration."</sub><br>
		<sub>$^2$Kaltenmark, et al. "A general framework for curve and surface comparison and registration with oriented varifolds."</sub><br>
	  <sub>$^3$Feydy et al. "Optimal transport for diffeomorphic registration"</sub><br>
    <sub>$^4$ Roussillon & Glaunès. "Representation of surfaces with normal cycles and application to surface registration."</sub><br></p>
		</div>

<h4 align="left">The Varifold Norm and Partial Matching</h4><hr>
		<p align="left">What to do when there are parts of $q_0$ that are not matched to $q_1$?<br><img src="figs/hands_code.png" width="75%" /></p>
		<table><tr>
		<td><img src="figs/partial_hand_2_source.png" height="100%" /></td><td><img class="fragment" data-fragment-index="1" src="figs/partial_hand_2.png" height="100%" /></td><td><img src="figs/partial_hand_2_target.png" height="100%" />
		</td></tr></table><p class="fragment" data-fragment-index="2" align="left">
		Consider $\omega:M\to [0,1]$ interpreted as the probability that $q_0(x)$ is matched to $q_1$ and use a $\omega-$weighted varifold representation of $[q(1)]$ denoted $\mu_{q(1),\omega}$.<br>
		Thus, our relaxed partial matching problem becomes
		$$\inf\limits_{\{\omega:M\to [0,1]\}}\inf\limits_{\{q\in C^\infty([0,1],\Imm)\}} \displaystyle\int_0^1 G_{q(t)}(\partial_t q(t),\partial_t q(t)) dt +\lambda_0||\mu_{q_1}-\mu_{q(0)}||^2_{V}+ \lambda_1 ||\mu_{q_1}-\mu_{q(1),\omega}||^2_{V}$$
		<img src="figs/partial_hands_code.png" width="50%" />
		<img class="fragment" data-fragment-index="3" src="figs/partial_hand_1.png" height="100%" />
		</p>

<h4 align="left">Conclusion</h4><hr>
		<p align="left">We propose a numerical framework for elastic shape analysis of 3D surfaces equipped with a family of elastic second-order Sobolev metrics. Furthermore we demonstrate our software's ability to:<br><br>
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			<p class="fragment fade-left">+ perform statistical analysis under the proposed family of metrics.</p>
			<p class="fragment fade-left">+ solve boundary value geodesic problems and other analysis tasks in the unregistered setting.</p>
			<p class="fragment fade-left">+ handle to partial matching tasks between incomplete or topologically inconsistent representations of shapes.</p>
			<hr>
			<p><sub>
			This talk is based on:<br><a href="https://link.springer.com/article/10.1007/s11263-022-01743-0">"Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics", IJCV. </a><br>
			The code for this project is available at:<br> <a href="https://github.com/emmanuel-hartman/H2_SurfaceMatch">github.com/emmanuel-hartman/H2_SurfaceMatch</a><br>
			These slides are available at:<br><a href="https://www.math.fsu.edu/~ehartman/Slides/CIRM2024/talk.html">www.math.fsu.edu/~ehartman/Slides/CIRM2024/talk.html</a><br>
		</sub></p>
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		<img src="figs/skulls.gif" width="80%" />
		<br><br>
		<center><p class="fragment fade-left">Thank you to the organizers for putting together this workshop and thank you for your attention!</p></center>