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">Shape Analysis of Surfaces</h4><hr>

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          The goal of shape analysis is to define a distance (or better yet a Riemannian metric) on the space of unparameterized surfaces. By the space of unparamerteized surfaces we will consider the quotient given by the space of parameterized surfaces modulo the action of the appropriate reparameterization group. Using the metric we define, one can perform statistical analysis on populations of shape data.</p>
          <hr><p>Let $M$ be a smooth oriented, Riemannian 2-manifold.</p>
						<p>
						\begin{align*}
						&\textbf{Parameterized Shapes:}\\
						&\qquad \Imm =\operatorname{Imm}(M,\R^3)/\text{translations}\\
						&\textbf{Reparameterization Group:}\\
						&\qquad \Diff = \operatorname{Diff}_+(M)\\
						&\textbf{Unparameterized Shapes:}\\
						&\qquad \Shape = \Imm/\Diff\qquad [f]=\{f\circ\gamma|\gamma\in\Diff\}
						\end{align*}
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					<img src="figs/OptOrbits1.png" height="100%" />
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<h4 align="left">Shape Analysis of Surfaces</h4><hr>
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						Our goal now is to equip the manifold $\Imm$ with a Riemannian metric $G$ that is invariant to the the action of $\Diff$ i.e.
						\begin{multline}
						\text{for }  q\in \Imm, \; h,k\in T_q\Imm, \; \phi\in\Diff,\\
						G_q(h,k)=G_{q\circ \phi}(h\circ\phi,k\circ\phi). \qquad (1)
						\end{multline}<br>
						The geodesic (psuedo) distance associated with $G$ on $\Imm$ is given by
						\begin{multline*}
						d^{\Imm}(q_0,q_1)=\inf\limits_{q\in \mathcal{P}_{q_0}^{q_1}} \int_0^1\sqrt{G_{q(t)}(\partial_t q(t),\partial_t q(t))} dt\\
						\text{ where }
						\mathcal P_{q_0}^{q_1}:= \left\{C^{\infty}([0, 1], \Imm): q(0) = q_0, q(1) = q_1\right\}.
						\end{multline*}<br>
						Given a metric satisfying (1), the projection $\pi:\Imm\to \Shape$ is a Riemannian submersion and the geodesic distance corresponds to a distance on $\Shape$ via
						\begin{equation*}
						d^{\Shape}([q_0],[q_1])=\inf_{\phi\in \Diff} d^{\Imm}(q_0,q_1\circ\phi).
						\end{equation*}
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						<img src="figs/OptOrbits1.png" width="100%" />
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<h4 align="left">Approaches of Laga, et al. $^1$ and Su, et al.$^2$</h4><hr>
						<p align="left">Under the assumption $M=S^2$, Laga, et al. (2017) and Su, et al. (2020) discretize $\Diff$ via spherical harmonics and solve the matching problem over this discretization with respect to different metrics on $\Imm$.
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						<p class="fragment fade-in" align="left">
						This approach:
						</p>
						<p class="fragment fade-left">+ provides a stable method for solving the matching of two surfaces.</p>
						<p class="fragment fade-left">+ requires genus-zero surfaces given by their spherical parameterizations.</p>
						<hr><p class="fragment fade-in">The difference between these frameworks is the choice of distance.</p>
						<p class="fragment fade-left">+ Laga et al. use the square root normal field (SRNF) psuedo-distance.</p>
						<p class="fragment fade-left">+ Su et al. use the geodesic distance of a novel Riemannian metric.</p>
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					  <img src="figs/su_armadillo.png" width="100%" />
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						<hr>
					 	<p align="left">
					 		<sub>$^1$ Laga, Xie, Jermyn, and Srivastava, “Numerical inversion of SRNF maps for elastic shape analysis of genus-zero surfaces”.</sub><br>
							<sub>$^2$ Su, Bauer, Preston, Laga, and Klassen, “Shape analysis of surfaces using general elastic metrics”.</sub></p>

<h4 align="left">Existing Methods for Surface Matching Contd.</h4><hr>
						<p align="left">Finally, Bauer, et al.$^3$ (2020) proposed an varifold matching framework to register surfaces with respect to the SRNF pseudo-distance.
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						<p class="fragment fade-in">Using the varifold distance for elastic matching provides several benefits:</p>
						<p class="fragment fade-left">+ the reparametrization group $\Diff$ does not need to be discretized and its action on $\Imm$ does not need to be implemented.</p>
						<p class="fragment fade-left">+ the framework analyzes simplicial meshes without having to produce spherical parametrizations first</p>
						<p class="fragment fade-left">+ the framework can easily analyze surfaces with more general topologies with or without boundary</p>
						<hr>
						<p class="fragment fade-in">This framework has drawbacks as well:</p>
						<p class="fragment fade-left">+ finding horizontal geodesics requires a separate step from registration.</p>
						<p class="fragment fade-left">+ the framework has numerical instabilities in some cases.</p>
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						<tr><img src="figs/varifold_multires_match.png" width="100%" /></tr>
						<tr><img src="figs/varifold_skull.png" width="100%" /></tr>
						<tr><td><img src="figs/skull2_multires_matchedsource_high.png" width="100%" /></td>
						<td><img src="figs/skull2_multires_matchedtarget_high.png" width="100%" /></td></tr>
						</table>
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						<hr>
					 	<p align="left">
					 		<sub> $^3$ Bauer, Charon, Harms, and Hsieh, “A numerical framework for elastic surface matching, comparison, and interpolation”.<br>

<h4 align="left">Our Framework</h4><hr>
						<p align="left">We propose a novel numerical framework for performing shape analysis of surfaces. This framework uses varifold matching to solve the registration problem with respect to a family of split second order Sobolev metrics.
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							<p class="fragment fade-in">This framework has several benefits:</p>
							<p class="fragment fade-left">+ produces horizontal geodesics as a part of the distance computation.</p>
							<p class="fragment fade-left">+ works without having to produce spherical parametrizations first.</p>
							<p class="fragment fade-left">+ can easily analyze surfaces with more general topologies with or without boundary as well as shape complexes.</p> <hr>
							<p class="fragment fade-in">Additionally, our numerical framework has been expanded to</p>
							<p class="fragment fade-left">+ include weights on the varifolds to produce partial matching results. </p>
							<p class="fragment fade-left">+ solve an initial value problem to compute the exponential map on $\Imm$</p>
							<p class="fragment fade-left">+ compute Karcher means, Tangent Space PCA, and Parallel Transport.</p>
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						<tr><img src="figs/main.png" width="100%" /></tr>
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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 \[g_q^{-1}(dh,dk)=\operatorname{tr}(dh\cdot g_q^{-1}\cdot dk^T)\]
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							<p class="fragment fade-in" align="left">
								Following the construction of Su et. al. 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>

<h4 align="left">Relaxed matching and Varifolds</h4><hr>
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							For $q\in \Imm$ we can associate a varifold $\mu_q\in\mathcal{M}(\mathbb R^3\times S^{2})$ and endow $\mathcal{M}(\mathbb R^3\times S^{2})$ with a norm $\|\cdot\|_{V}$.
							In our standard matching framework, we consider the class of norms where the scalar product between varifolds $\mu_{q_0}$ and $\mu_{q_1}$ can be expressed as:\begin{equation*}\label{equ:norm_var}
							\langle \mu_{q_0},\mu_{q_1}\rangle_{V}=\iint_{M \times M}k(x_0,n_0,x_1,n_1)\vol_{q_0} \vol_{q_1}
							\end{equation*}</p>
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							<span class="fragment" data-fragment-index="1">
							<div class="b" align="left"><div class="btitle"><h4 align="left">Lemma</h4></div><p align="left">
							For all $q\in \Imm^*$ and $\phi\in \Diff$, $\mu_{q}=\mu_{q\circ\phi}$. Thus, for all $q_0,q_1\in \Imm$ and $\phi_0,\phi_1\in \Diff$,
							\[\|\mu_{q_0}-\mu_{q_1}\|_{V}=\|\mu_{q_0\circ\phi_0}-\mu_{q_1\phi_1}\|_{V}\] and the norm vanishes if $q_0$ is in the same orbit as $q_1$ under the action of $\Diff$.
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							<span class="fragment" data-fragment-index="2" align="left">
								<p>Thus, $\Gamma(q,p):=||\mu_p-\mu_q||^2_{V}$ is a natural choice for our relaxtion term.</p>
							</span>

<h4 align="left">The Varifold Norm and Partial Matching</h4><hr>
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								<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">
								In our partial matching model, we define a weight function $\rho:M\to [0,1]$ that quantifies whether a point of the source shape should be matched to the target. The weighted scalar product between varifolds $\mu_{q_0}$ and $\mu_{q_1}$ can be expressed as:
								\begin{equation*}
								\langle \mu_{q_0},\mu_{q_1}\rangle_{V,\rho}=\iint_{M \times M}k(x_0,n_0,x_1,n_1)\rho(x_0)\vol_{q_0} \vol_{q_1}
								\end{equation*}
								We redefine our our relaxed energy using this norm and optimize with respect to both the path $q$ and the weighting function $\rho$.<br>
								<img class="fragment" data-fragment-index="3" src="figs/partial_hand_1.png" height="100%" />
								</p>

<h4 align="left">Conclusion And Future Work</h4><hr>
						<p align="left">We propose a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of elastic second-order Sobolev metrics. Building on this, we develop tools for the statistical shape analysis of sets of surfaces and extend the framework for partial matching. <br> <br>
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							<p class="fragment fade-left">+ Use this framework for statistical shape analysis applications.</p>
							<p class="fragment fade-left">+ Consider applications where we want to restrict the basis of deformations of a certain space.</p>
							<hr>
							<p>
							This talk is based on:<br><a href="https://arxiv.org/abs/2204.04238">arxiv.org/abs/2204.04238</a><br>
							The code 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/H2_Slides/talk.html">www.math.fsu.edu/~ehartman/Slides/H2_Slides/talk.html</a>
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						<center><p>Thank you for your attention!</p></center>