Let $N$ be a smooth, connected, compact, oriented Riemannian manifold. Define the cone over $N$ via $\mathcal{C}(N):= (N\times \R^{\geq0})/(N\times\{0\})$. If we let $d_N$ denote the geodesic distance on $N$ and fix some $\delta \in (0,\infty)$, then we can define a metric on $\mathcal{C}(N)$ via \[d_{\mathcal{C}(N)}((n_1,r_1),(n_2,r_2))^2=4\delta^2 \left(r_1^2+r_2^2-2r_1r_2 \overline{\cos}(d_N(n_1,n_2)/2\delta)\right).\]
Let $M$ be another smooth, connected, compact, oriented Riemannian manifold with dimension $\geq2$. Any function $q:M\to \mathcal{C}(N)$ can be decomposed into component functions by $q(x)=(\overline{q}(x),\hat{q}(x))$ where $\overline{q}:M\to N$ and $\hat{q}:M\to \R^{\geq0}$.
Define the space of $L^2$-functions from $M$ to $\mathcal{C}(N)$ as \[L^2(M,\mathcal{C}(N)):=\{q:M\to \mathcal{C}(N) \text{ s.t. } \int_M \hat{q}(m)^2 dm<\infty\}\]
Given $q_1,q_2:M\to\mathcal{C}(N)$. The $L^2_\delta$ distance between $q_1$ and $q_2$ is given by \[d_{L^2_\delta}(q_1,q_2)^2=\int_M d_{\mathcal{C}(N)}(q_1(x),q_2(x))^2 dm.\] By decomposing $q_1$ and $q_2$, we can alternatively write \[d_{L^2_\delta}(q_1,q_2)^2= 4\delta^2 \left(\int_M \hat{q_1}(x)^2 dm +\int_M \hat{q_2}(x)^2 dm -2\int_M \hat{q_1}(x)\hat{q_2}(x)\overline{\cos}(d(\overline{q_1}(x),\overline{q_2}(x))/2\delta)dm\right)\]
We define the right action of the diffeomorphisms of $M$ on $L^2(M,\mathcal{C}(N))$ component-wise. We treat $\hat{q}$ as a half density and define the action of $\Diff(M)$ on this component as the action on half-densities. Thus, we define the action of $\Diff(M)$ on $L^2(M,\mathcal{C}(N))$ given by
\begin{align*}L^2(M,\mathcal{C}(N))\times\Diff(M)&\to L^2(M,\mathcal{C}(N)) \text{ via }\\ (\overline{q},\hat{q}),\gamma&\mapsto q*\gamma:=\left(\overline{q}\circ \gamma,\hat{q}\circ\gamma \cdot \sqrt{|D\gamma|}\right)\end{align*}
Let $q\in L^2(M,\mathcal{C}(N))$. We can define a Borel measure $\mu_q$ on $N$ where for all $U\subseteq N$ open \[\mu_q(U)=\int_{\overline{q}^{-1}(U)} \hat{q}(x)^2dm.\]
We will now prove some facts about the map $\Psi:L^2(M,\mathcal{C}(N)) \to \mathcal{M}(N)$ given by $\Psi(q):=\mu_q $
Let $q\in L^2(M,\mathcal{C}(N))$. Then for all $\gamma\in \Diff(M),\,\mu_{q}=\mu_{q*\gamma}.$
Let $U\subseteq N$ open.
\begin{align*}\mu_{q*\gamma}(U)&=\int_{\gamma^{-1}(\overline{q}^{-1}(U))} (\hat{q}\circ{\gamma}(x)\cdot\sqrt{|D\gamma|})^2dm\\&=\int_{\gamma^{-1}(\overline{q}^{-1}(U))} \hat{q}\circ{\gamma}(x)^2\cdot|D\gamma|dm=\int_{\overline{q}^{-1}(U)} \hat{q}(x)^2dm=\mu_{q}(U).\end{align*}
The map $\Psi:(L^2(M,\mathcal{C}(N)),d_{L^2_\delta})\to (\mathcal{M}(N),\operatorname{WFR}_\delta)$ is Lipschitz continuous with constant $K=1$.
Let $q_1,q_2\in L^2(M,\mathcal{C}(N))$ with $\mu_{q_1}=\Psi(q_1)$ and $\mu_{q_2}=\Psi(q_2)$.
For any semi-coupling $(\pi_1,\pi_2)\in\overline{\Gamma}(\mu_{q_1},\mu_{q_2})$,\[\operatorname{WFR}_\delta(\mu_{q_1},\mu_{q_2})^2\leq J_\delta(\pi_1,\pi_2).\]
We prove the lemma by constructing $(\pi_1,\pi_2)\in \overline{\Gamma}(\mu_{q_1},\mu_{q_2})$ such that $J_\delta(\pi_1,\pi_2)=d_{L^2_\delta}(q_1,q_2)^2$.
Let $N$ be a smooth connected compact Riemannian manifold and $M$ be a smooth connected compact Riemannian manifold of dimension 2 or higher.
a.) For all $\mu_1,\mu_2\in\mathcal{M}(N)$ and $q_1,q_2\in L^2(M,\mathcal{C}(N))$ such that $\mu_1=\Psi(q_1)$ and $\mu_2=\Psi(q_2)$ we have \begin{align*}\operatorname{WFR}_\delta(\mu_1,\mu_2)= \inf\limits_{\gamma\in\Diff(M)}d_{L^2_\delta}(q_1,q_2*\gamma).\end{align*}
b.) Moreover, for all $\mu \in \mathcal{M}(N)$ there exists $q\in L^2(M,\mathcal{C}(N))$ such that $\mu=\Psi(q)$. If $\mu$ is a finitely supported measure given by $\mu=\sum_{i=1}^{n}a_i\delta_{u_i}$, then one can choose $q$ piecewise constant.
Specifically, the function $q$ given by \[q(x)=\begin{cases}\left(u_j,\sqrt{\frac{a_j}{\text{area}(\sigma_j)}}\right)&\text{ if } 1\leq j\leq n\\ (u_1,0)& \text{ if } n< j\leq m\end{cases},\] where $\{\sigma_j\}_{j=1}^m$ is a subdivision of the canonical triangulation of $M$ with $m\geq n$, satisfies $\mu=\Psi(q)$.
For $\mu\in\mathcal{M}_0(N)$ we construct an explicit piecewise constant $q\in L^2(M,\mathcal{C}(N))$ such that $\mu=\Psi(q)$.
We then show for $\mu_1,\mu_2\in\mathcal{M}_0(N)$ and $q_1,q_2\in L^2(M,\mathcal{C}(N))$ such that $\mu_1=\Psi(q_1)$ and $\mu_2=\Psi(q_2)$ we have \begin{align*}\operatorname{WFR}_\delta(\mu_1,\mu_2)= \inf\limits_{\gamma\in\Diff(M)}d_{L^2_\delta}(q_1,q_2*\gamma).\end{align*}
We then conclude the result for general Radon measures by a density argument leveraging the Lipschitz continuity result from above.
Consider $N=S^2$ with $d_N(n_1,n_2):= \cos^{-1}(n_1\cdot n_2)$ and $\delta=\frac{1}{2}$. Thus, $\mathcal{C}(N)=\R^3$ and
\begin{align*}d_{\mathcal{C}(N)}((n_1,r_1),(n_2,r_2))^2&=4\delta^2 \left(r_1^2+r_2^2-2r_1r_2 \overline{\cos}(d_N(n_1,n_2)/2\delta)\right)\\&=r_1^2+r_2^2-2r_1r_2 (n_1\cdot n_2)\\&=\|r_1n_1 - r_2n_2\|^2\end{align*}
In particular, if $M$ is a smooth, connected, compact, oriented Riemannian 2-manifold and $q_1,q_2\in L^2(M,\mathcal{C}(N))$, \[\inf\limits_{\gamma\in\Diff(M)}d_{L^2_\delta}(q_1,q_2*\gamma)^2=\inf\limits_{\gamma\in \Diff(M)}\int_M\left\|q_1-q_2\circ\gamma\sqrt{|D\gamma|}\right\|^2dm\]
Therefore, if $f_1,f_2\in \Imm(M,\R^3)$, then \[d_{\Shape}([f_1],[f_2])^2=\inf\limits_{\gamma\in \Diff(M)}\int_M\left\|\phi(f_1)-\phi(f_2)\circ\gamma\sqrt{|D\gamma|}\right\|^2dm=\operatorname{WFR}_\delta^2(\mu_{\phi(f_1)},\mu_{\phi(f_2)})\]
Let $\mu\in \mathcal{M}(S^2)$ such that the support of $\mu$ is not concentrated on a great circle and \begin{equation*}\int_{S^2}x\,d\mu(x)=0.\end{equation*} Then, there exists a unique (up to translation) convex body whose surface area measure is $\mu$. Moreover, if $\mu$ is finitely supported then the convex body is a polytope.
The WFR is a distance on the Radon measures on $S^2$. So in particular, $\operatorname{WFR}(\mu,\nu)=0$ if and only if $\mu=\nu$.
Let $f\in \Imm(S^2,\R^3)$ such that the image of the Gauss map spans $\R^3$. Then there exists a unique (up to translations) convex body $f_1$ that is indistinguishable from $f$ by the SRNF shape distance, i.e, $ d_{\mathcal S}([f],[f_1])=0$.
Let $f\in\Imm(S^2,\mathbb R^3)$ and let $q=\phi(f)\in L^2(S^2,\R^3)$. Then $q$ satisfies the closure condition $\int_{S^2}q(x)|q(x)|dm=0$. Moreover, the closure of the image of $\phi$ is given by the set
\[\mathcal{U}:=\left\{ q\in L^2(S^2,\R^3) \text{ such that } \int_{S^2}q(x)|q(x)|dm=0\right\}.\]
We present a novel algorithm for computing the WFR. Additionally, we propose a formulation that characterizes the WFR as a diffeomorphic optimization inspired by problems from elastic shape analysis and leverage this formulation to answer some open questions about the SRNF shape distance.
+ What happens to the diffeomorphic formulation of the WFR if we optimize over some subset of $\Diff(M)$.
+ Can we regularize the transport by including a penalty on the diffeomorphisms of $M$?
This talk is based on:
https://arxiv.org/pdf/2105.06510.pdf
and
https://arxiv.org/pdf/2301.00284.pdf
The code for our algorithm with $N=S^2$ available at:
https://github.com/emmanuel-hartman/WassersteinFisherRaoDistance
The code for the general version of our algorithm is coming soon.
These slides are available at:
www.math.fsu.edu/~ehartman/Slides/SRNF_Slides/talk.html
Thank you for your attention!