boundary of $Z(2)$. Using Lemma 2.18 as in the proof of Theorem 2.20 we
correct $\psi$ to a $C^{1}$-smooth symplectic embedding

$\Phi\colon\bigl{(}\widehat{U},\Sigma\bigr{)}\to\bigl{(}Z(2),\Sigma^{\prime}\bigr{)}$

defined on a neighourhood $\widehat{U}$ of $\Sigma$ such that
$\Phi(\widehat{U})\subset\psi(U)$, whence the closure of $\Phi(\widehat{U})$
is also disjoint from $\partial Z(2)$. Choosing $\widehat{U}$ smaller if
necessary, we can also assume that $\widehat{U}$ retracts onto $\Sigma$, whose
components are topological discs. We therefore find a Liouville form
$\lambda^{\prime\prime}$ on $Z(2)$ that coincides with $\Phi_{*}\lambda$ on
$\Phi(\widehat{U})$ and with $\lambda^{\prime}$ near $\partial Z(2)$. Then the
map

$\begin{array}[]{rcccl}\Psi_{1}&:&{\mathcal{B}}(\Sigma,\lambda)&\longrightarrow&Z(2)\\\
&&p&\longmapsto&\phi_{X_{\lambda^{\prime\prime}}}^{-t^{+}(p)+\varepsilon(p)}\circ\Phi\circ\phi_{X_{\lambda}}^{t^{+}(p)-\varepsilon(p)},\end{array}$

defined for any small enough positive function $\varepsilon(p)$, is a
$C^{1}$-smooth symplectic embedding.

Since the components of $\Sigma$ are discs, the $\phi_{N}$ are
$(\alpha_{{\operatorname{st}}}|_{\Sigma_{N}},\alpha_{{\operatorname{st}}}|_{\Sigma^{\prime}})$-exact,
and so $\Psi_{1}$ is
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact. The
asserted $(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact
symplectic embedding
$\Psi\colon{\mathcal{B}}(\Sigma,\lambda)=\mathbb{R}^{4}\setminus\Gamma\to
Z(2)$ is now obtained from Lemma 2.21. $\square$

###  4.3. Proof of Theorem 3

In this subsection we first give another proof of Theorem 6 for the ball as
domain, that does not use a singular polarization but Biran’s smooth
polarization. We then use this result to prove Theorem 3. We start with a
lemma that is useful if the polarization of the source is smooth.

######  Lemma 4.5.

Let $(\Sigma,d\alpha)$ and $(\Sigma^{\prime},d\alpha^{\prime})$ be compact
connected symplectic surfaces of genus $g$ and $g^{\prime}$, respectively. Let
$b$ be the number of boundary components of $\Sigma$. Assume that

$\operatorname{area}\Sigma<\operatorname{area}\Sigma^{\prime}\quad\mbox{ and
}\quad g+b-1\leq g^{\prime}.$

Then there exists an $(\alpha,\alpha^{\prime})$-exact symplectic embedding
$\varphi\colon\Sigma\to\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip
6.0mu}{\Sigma^{\prime}}$.

Proof. A basis of $H_{1}(\Sigma,\mathbb{Z})$ is given by oriented closed
curves $\gamma_{1},\dots,\gamma_{2g}$ around the $g$ holes and $b-1$ oriented
boundary components $\gamma_{2g+1},\dots,\gamma_{2g+b-1}$, see the red curves
in Figure 4.5. Since $g+b-1\leq g^{\prime}$, there exists an embedding
$f\colon\Sigma\to\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip
6.0mu}{\Sigma^{\prime}}$ as shown in Figure 4.5.

Figure 4.5. The embedding
$f\colon\Sigma\to\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip
6.0mu}{\Sigma^{\prime}}$ (in the figure it is the inclusion).

Notice that $f_{*}\colon H_{1}(\Sigma)\to
H_{1}(\Sigma^{\prime},\partial\Sigma^{\prime})$ is injective, over
$\mathbb{Z}$ and hence over $\mathbb{R}$. In our context of oriented surfaces,
the de Rham isomorphism gives $H_{1}(\Sigma;\mathbb{R})^{*}=H^{1}(\Sigma)$ and
$H_{1}(\Sigma^{\prime},\partial\Sigma^{\prime};\mathbb{R})^{*}=H^{1}_{c}(\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip
6.0mu}{\Sigma^{\prime}})$, where both spaces on the right are the de Rham
cohomology groups. Thus, $f^{*}\colon
H^{1}_{c}(\overset{\raisebox{-1.29167pt}[0.0pt][-1.29167pt]{\mbox{$\scriptscriptstyle\circ$}}\mskip
6.0mu}{\Sigma^{\prime}})\to H^{1}(\Sigma)$ is surjective.

Since the area of $\Sigma^{\prime}$ is larger than the area of $\Sigma$,
Moser’s theorem allows to further assume that $f$ is area-preserving:
$f^{*}d\alpha^{\prime}=d\alpha$. Write $\gamma_{i}^{\prime}=f(\gamma_{i})$.
Since the classes $[\gamma_{i}]$ are linearly independent in $H_{1}(\Sigma)$
and since $f^{*}$ is surjective, there exists a compactly supported closed
$1$-form $\vartheta$ on $\Sigma^{\prime}$ with periods

$\int_{\gamma_{i}^{\prime}}\vartheta\,=\,\int_{\gamma_{i}}\alpha-\int_{\gamma_{i}^{\prime}}\alpha^{\prime},\qquad
i=1,\dots,2g+b-1.$

Let $\phi^{t}_{\vartheta}$ be the flow associated to the vector field
$X_{\vartheta}$ defined by $d\alpha^{\prime}(X_{\vartheta},\cdot)=\vartheta$.
This flow is complete because $\vartheta$ is compactly supported, and it is
symplectic because $\vartheta$ is closed. A classical computation for the flux
shows that

$\int_{\phi_{\vartheta}^{t}(\gamma_{i}^{\prime})}\alpha^{\prime}\,=\,\int_{\gamma_{i}^{\prime}}\alpha^{\prime}+t\int_{\gamma_{i}^{\prime}}\vartheta,\qquad
i=1,\dots,2g+b-1,$

see e.g. [21, $\S$ 10.2]. Thus $\phi_{\vartheta}^{1}\circ f$ is a
$(\alpha,\alpha^{\prime})$-exact symplectic embedding of $\Sigma$ into
$\Sigma^{\prime}$. The proof of the proposition is complete. $\square$

######  Theorem 4.6.

$B^{4}(1)\backslash\Delta_{k}\;\lhook\joinrel\xrightarrow{\;\\!\alpha_{{\operatorname{st}}}\,}\;Z^{4}\bigl{(}\tfrac{2}{k}\bigr{)}$.

Proof. The polarization of $\operatorname{\mathbb{C}P}^{2}$ of degree $k$ in
Example 2.8 restricts to a tame Liouville polarization
$(\Sigma_{k},\lambda_{k})$ of the ball whose skeleton is $\Delta_{k}\cap
B^{4}(1)$, so

$B^{4}(1)\backslash\Delta_{k}={\mathcal{B}}\bigl{(}\Sigma_{k},\lambda_{k}\bigr{)}.$

One readily checks that this Liouville polarization is extendable. Moreover,
the area, genus, and number of punctures of $\Sigma_{k}$ are well-known and
given by

${\mathcal{A}}_{\omega}(\Sigma_{k})=k,\quad
g(\Sigma_{k})=\frac{(k-1)(k-2)}{2},\quad b(\Sigma_{k})=k.$

On the other side, proceeding as in the proof of Proposition 4.1, we take for
any $A\in\mathbb{N}$ two points $p_{i}$ in $D(\frac{2}{k})$ and $kA$ points
$q_{j}$ in $D(A)$, all with weight $\frac{1}{k}$, and consider the union of
the discs $D(\frac{2}{k})\times q_{j}$ and $p_{i}\times D(A)$. Using Lemma 3.5
and resolving all the $2kA$ intersections as in Proposition 3.8, we obtain a
smooth Liouville polarization $\Sigma_{k}^{\prime}$ of $D(\frac{2}{k})\times
D(A)$ with residue $\frac{1}{k}$. Then

${\mathcal{A}}_{\omega}(\Sigma_{k}^{\prime})=4A\quad\mbox{ and }\quad
g(\Sigma_{k}^{\prime})=kA-1.$

When $A\geq\frac{k}{2}$, we have
${\mathcal{A}}_{\omega}(\Sigma^{\prime}_{k})>{\mathcal{A}}_{\omega}(\Sigma_{k})$
and $g(\Sigma_{k}^{\prime})\geq g(\Sigma_{k})+b(\Sigma_{k})-1$ since $k\geq
2$, so by Lemma 4.5 there exists an
$(\alpha_{{\operatorname{st}}}|_{\Sigma_{k}},\alpha_{{\operatorname{st}}}|_{\Sigma_{k}^{\prime}})$-exact
symplectic embedding of $\widehat{\Sigma}_{k}$ into $\Sigma_{k}^{\prime}$, and
by Theorem 2.20 there exists an
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact symplectic
embedding of
${\mathcal{B}}\bigl{(}\Sigma_{k},\lambda_{k}\bigr{)}=B^{4}(1)\backslash\Delta_{k}$
into $Z^{4}(\frac{2}{k})$. $\square$

######  Remark 4.7.

When $k=2$ we can get a better estimate and symplectically embed
$B^{4}(1)\backslash\Delta_{2}=B^{4}(1)\backslash\mathbb{R}^{2}$ into the
ellipsoid $E(\frac{1}{2},2+\varepsilon)$, like [29]. This is because
$\Sigma_{2}$ is a sphere with two punctures, hence has no genus. Thus a
cylindrical extension of $\Sigma_{2}$ symplectically embeds into a disc of
area $2+\varepsilon$. Since by Example 2.14, $(D(2+\varepsilon),\frac{1}{2})$
is a polarization of $\textnormal{\small SDB}(D(2+\varepsilon),\frac{1}{2})$,
Theorem 2.20 shows that $B^{4}(1)\backslash\Delta_{2}$ embeds into
$\textnormal{\small SDB}(2+\varepsilon,\frac{1}{2})$. And by [23, Lemma 2.1],
$\textnormal{\small SDB}(D(2+\varepsilon),\frac{1}{2})$ is symplectomorphic to
$E(2+\varepsilon,\frac{1}{2})$. $\diamond$

Recall that the open ellipsoid $E(a,b)$ is defined as

$E(a,b)\,=\,\left\\{(z,w)\in\mathbb{C}^{2}\;\bigg{|}\;\frac{|z|^{2}}{a}+\frac{|w|^{2}}{b}<1\right\\}.$

######  Proposition 4.8.

For $d,N\in\mathbb{N}$ coprime there exists for every $m\in\mathbb{N}$ with
$m\geq d$ and $(m,d,N)\neq(2,2,1)$ a symplectic embedding

$B^{4}(1)\setminus\Delta_{mNd}\,\lhook\joinrel\xrightarrow{\;\\!\alpha_{{\operatorname{st}}}\,}\,E\bigl{(}\tfrac{1}{d},d+\tfrac{1}{N}\bigr{)}.$

Proof. As above, consider the extendable Liouville polarization
$(\Sigma_{mNd},\lambda_{mNd})$ of $B^{4}(1)$ of degree $mNd$, whose skeleton
is $\Delta_{mNd}$. We have that ${\mathcal{A}}_{\omega}(\Sigma_{mNd})=mNd$,

$g(\Sigma_{mNd})=\frac{1}{2}(mNd-1)(mNd-2)\quad\mbox{and}\quad
b(\Sigma_{mNd})=mNd.$

In order to produce the required Liouville polarization of the target
ellipsoid, consider the symplectic ramified covering

$\begin{array}[]{rcccl}\Phi&:&B^{4}(N(Nd^{2}+d))&\longrightarrow&E(N,Nd^{2}+d)\\\
&&(R_{1},\theta_{1},R_{2},\theta_{2})&\longmapsto&\big{(}\frac{R_{1}}{Nd^{2}+d},(Nd^{2}+d)\theta_{1},\frac{R_{2}}{N},N\theta_{2}\big{)}.\end{array}$

One checks without difficulty that the Liouville polarization
$(\Sigma_{m},\lambda_{m})$ of $B^{4}(N(Nd^{2}+d))$ of degree $m$ projects
under $\Phi$ to a smooth Liouville polarization of degree $m$ of
$E(N,Nd^{2}+d)$ (smoothness requires that $N$ and $Nd^{2}+d$ are coprime,
which is equivalent to our assumption that $N$ and $d$ are coprime). This
polarization can therefore be seen as a Liouville polarization
$(\Sigma^{\prime},\lambda^{\prime})$ of degree $mNd$ of
$E(\frac{1}{d},d+\frac{1}{N})$. The polarizing curve $\Sigma^{\prime}$ has
area

${\mathcal{A}}_{\omega}(\Sigma^{\prime})\,=\,m(Nd+1)\,>\,{\mathcal{A}}_{\omega}(\Sigma_{mNd}),$

and by the Riemann–Hurwitz formula its genus is

$g(\Sigma^{\prime})\,=\,\frac{1}{2}\bigl{(}(mN-1)(mNd^{2}+md-1)-m+1\bigr{)}$

which is $\geq g\bigl{(}\Sigma_{mNd}\bigr{)}+b\bigl{(}\Sigma_{mNd}\bigr{)}-1$
by our assumption on $(m,d,N)$. By Lemma 4.5, there exists an
$(\alpha_{{\operatorname{st}}}|_{\Sigma_{mNd}},\alpha_{{\operatorname{st}}}|_{\Sigma^{\prime}})$-exact
symplectic embedding of $\widehat{\Sigma}_{mNd}$ into $\Sigma^{\prime}$, and
by Theorem 2.20 an
$(\alpha_{{\operatorname{st}}},\alpha_{{\operatorname{st}}})$-exact symplectic
embedding of
${\mathcal{B}}(\Sigma_{mNd},\lambda_{mNd})=B^{4}(1)\backslash\Delta_{mNd}$
into $E(\frac{1}{d},d+\frac{1}{N})$. $\square$

Proof of Theorem 3. It is shown in [30, $\S$ 6.2] that for every
$\varepsilon>0$ there exists $s_{0}\in\mathbb{R}$ such that for every $s\geq
s_{0}$ the ellipsoid $\lambda_{s}E(\frac{1}{s},s)$ of volume
$B^{4}(a-\frac{\varepsilon}{2})$ symplectically embeds into $(M,\omega)$. The
claim now follows by precomposing this embedding with a scaling of a suitable
embedding from Proposition 4.8. $\square$

The above proof gives no upper bound for $k$. However, if $(M,\omega)$ is a
rational symplectic manifold, or an affine part $M\setminus\Sigma$ therein,
then by [23] there exists an explicit volume filling ellipsoid in $M$ that can
be chosen to lie in the complement of $\Sigma$. Together with Proposition 4.8
one obtains an upper bound for $k$.

##  5\. Lagrangian rigidity

Proof of Theorem 4. Let $\iota\colon L\hookrightarrow D(A)\times D(B)$ be a
Lagrangian embedding of a closed surface. Assume that there exists a
Hamiltonian diffeomorphism $\phi$ of $\mathbb{R}^{4}$ such that

$\phi(L)\subset\bigl{(}D(A)\times
D(B)\bigr{)}\backslash\bigl{(}\Gamma_{\\!\leq a}\\!\times\Gamma_{\\!\leq
b}\bigr{)}$

Since $\phi(L)$ is compact and disjoint from $\bigl{(}\Gamma_{\\!\leq
a}\cup\partial D(A)\bigr{)}\times\bigr{(}\Gamma_{\\!\leq b}\cup\partial
D(B)\bigr{)}$, we can modify $\Gamma_{\\!\leq a}$ and $\Gamma_{\\!\leq b}$
near the vertices of their closure to regular grids $\Gamma^{\prime}_{\\!\leq
a}$ and $\Gamma^{\prime}_{\\!\leq b}$ that still divide $D(A)$ and $D(B)$ into
topological discs of area $\leq a$ and $\leq b$, and such that still

$\phi(L)\subset\bigl{(}D(A)\times
D(B)\bigr{)}\backslash\bigl{(}\Gamma^{\prime}_{\\!\leq
a}\times\Gamma^{\prime}_{\\!\leq b}\bigr{)}.$

By Theorem 6 there exists an exact symplectic embedding

$\psi\colon\bigl{(}D(A)\times
D(B)\bigr{)}\backslash\bigl{(}\Gamma^{\prime}_{\\!\leq
a}\times\Gamma^{\prime}_{\\!\leq b}\bigr{)}\,\to\,Z^{4}(a+b).$

Then $\psi\circ\phi\circ\iota\colon L\to L^{\prime}\subset Z^{4}(a+b)$ is a
Lagragian embedding. Since $\phi$ and $\psi$ are exact, the action classes
$\iota^{*}[\alpha_{{\operatorname{st}}}]$ and
$(\psi\circ\phi\circ\iota)^{*}[\alpha_{{\operatorname{st}}}]$ in
$H^{1}(L;\mathbb{R})$ coincide. Since by Stokes’ theorem the minimal area
(1.1) is also the minimal action

$A_{\min}(L)\,=\,\inf\left\\{\int_{\gamma}\alpha_{{\operatorname{st}}}\mid[\gamma]\in\pi_{1}(L),\,\int_{\gamma}\alpha_{{\operatorname{st}}}>0\right\\},$

we obtain that

$A_{\min}(L^{\prime})=A_{\min}(L).$

And since $L^{\prime}\subset Z^{4}(a+b)$, its displacement energy $e(L)$ in
$\mathbb{C}^{2}$ is $<a+b$, so by Chekanov’s result from [8],
$A_{\min}(L^{\prime})\leq e(L)<a+b$. Altogether, $A_{\min}(L)<a+b$, as we
wished to prove. $\square$

##  6\. Legendrian barriers

Theorem 5 on the existence of short Reeb chords between Legendrian curves is
obtained from the Lagrangian rigidity result of the previous paragraph via a
construction of Mohnke [22], that associates to the Reeb trajectory of a
Legendrian knot a Lagrangian torus. We first review this construction.

######  Lemma 6.1.

Let $(M^{3},\xi,\alpha)$ be a contact manifold,
$\bigl{(}SM=M\times\mathbb{R}_{>0},d(R\alpha)\bigr{)}$ its symplectization,
$\Lambda\subset M$ a Legendrian knot, and $X\subset M$ any subset. Assume that
there is no Reeb chord of length $\leq T$ from $\Lambda$ to $\Lambda\cup X$.

Then there exists a Lagrangian torus $L$ in $(M\backslash X)\times(0,1]\subset
SM$ with ${\mathcal{A}}_{\min}(L,SM)=T$.

Proof. Consider the map

$\iota\colon\Lambda\times(0,1]\times[0,\infty)\,\to\,SM,\quad(p,\tau,t)\mapsto\left(\Phi_{\alpha}^{t}(p),\tau\right),$

where $\Phi^{t}_{\alpha}$ is the Reeb-flow on $(M,\alpha)$. The restriction of
$\iota$ to every band $\\{p\\}\times(0,1]\times[0,\infty)$, $p\in\Lambda$, is
symplectic for the the forms $d\tau\wedge dt$ and $d(R\alpha)$ on the domain
and the target, respectively. Since there is no Reeb chord of length $\leq T$
from $\Lambda$ to $\Lambda\cup X$, we find $\varepsilon>0$ such that $\iota$
is an embedding of $\Lambda\times(0,1]\times[0,T+\varepsilon)$ into

$V\,:=\,(M\backslash X)\times(0,1]\,\subset\,SM.$

Take a closed disc $D_{\gamma}$ in $(0,1]\times[0,T+\varepsilon]$ of area $T$
and smooth oriented boundary $\gamma$. Then $\iota(\Lambda\times\gamma)$ is an
embedded Lagrangian torus in $V$. The actions of the generators $[\Lambda]$
and $[\gamma]$ of $\pi_{1}(L)$ are $\int_{\Lambda}\alpha=0$ and
$\int_{\gamma}\alpha=T$, so ${\mathcal{A}}_{\min}(L,SM)=T$. Moreover, the
symplectic disc $\iota(D_{\gamma})$ lies in $V$, has boundary on $L$, and has
area $T$. $\square$

Returning to the setting of Theorem 5, we notice that when $S$ is the smooth
boundary of a starshaped domain $U\subset\mathbb{R}^{4}$, with contact form
$\lambda_{S}=\alpha_{{\operatorname{st}}}|_{S}$, then the exact
symplectomorphism

$\begin{array}[]{lcl}\bigl{(}S\times\mathbb{R}_{>0},R\;\\!\lambda_{S}\bigr{)}&\longrightarrow&\bigl{(}\mathbb{R}^{4}\backslash\\{0\\},\alpha_{{\operatorname{st}}}\bigr{)}\\\\[1.99997pt]
(s,R)&\longmapsto&R\,s\end{array}$

identifies $S\times(0,1]$ with $U\backslash\\{0\\}$ and $X\times(0,1]$ with
the part in $U\backslash\\{0\\}$ of the cone over $X$ centered at the origin
of $\mathbb{R}^{4}$.

Proof of Theorem 5: Let $S$ be the smooth boundary of a starshaped domain
$U\subset C^{4}(1)\subset\mathbb{R}^{4}$. Arguing by contradiction, assume
that $\Lambda$ is a Legendrian knot in $S$ with no Reeb chord of length
$\leq\delta_{1}+\delta_{2}$ from $\Lambda$ to $\Lambda\cup\Lambda_{\delta}$,
where
$\Lambda_{\delta}=(\Gamma_{\\!\delta_{1}}\\!\times\Gamma_{\\!\delta_{2}})\cap
S$. Since $\Gamma_{\\!\delta_{1}}$ and $\Gamma_{\\!\delta_{2}}$ are radial,
$\Gamma_{\\!\delta_{1}}\\!\times\Gamma_{\\!\delta_{2}}$ lies in the cone over
$\Lambda_{\delta}$. Hence Lemma 6.1 provides a Lagrangian torus

$L\,\subset\,U\setminus\left(\Gamma_{\\!\delta_{1}}\\!\times\Gamma_{\\!\delta_{2}}\right)\,\subset\,C^{4}(1)\setminus\left(\Gamma_{\\!\delta_{1}}\\!\times\Gamma_{\\!\delta_{2}}\right)$

with ${\mathcal{A}}_{\min}(L)=\delta_{1}+\delta_{2}$. This is a contradiction
to Theorem 4. $\square$

##### An illustration.

To get a feeling for the phenomenon of Legendrian barriers, we explictely work
out the case where $S$ is the round sphere $S^{3}(1)=\partial B^{4}(1)$ and
$\delta_{1}=\delta_{2}=\frac{1}{k}$. Recall that

$\Delta_{k}\,=\,\bigcup_{0\leq i,j\leq k-1}\xi^{i}\mathbb{R}_{\geq
0}\times\xi^{j}\mathbb{R}_{\geq 0}$

where $\xi$ is the $k$ th root of unity $e^{2\pi i/k}$. Hence the Legendrian
graph $\Lambda_{k}:=\Delta_{k}\cap S^{3}(1)$ is the union of the $k^{2}$
Legendrian quarter-circles

$Q_{i,j}\,:=\,\left(\xi^{i}\mathbb{R}_{\geq 0}\times\xi^{j}\mathbb{R}_{\geq
0}\right)\cap S^{3}(1).$

Note that $Q_{i,j}$ and $Q_{i^{\prime},j^{\prime}}$ are disjoint if $i\neq
i^{\prime}$ and $j\neq j^{\prime}$, and intersect at one end-point if either
$i\neq i^{\prime}$ or $j\neq j^{\prime}$. We group these $k^{2}$ quarter-
circles in $k$ sets

$Q_{j}\,:=\,\coprod_{i=0}^{k-1}Q_{i,i+j}.$

Since the Reeb flow on $S^{3}(1)$ is the Hopf flow

$\Phi_{R}^{t}(z_{1},z_{2})\,=\,e^{2\pi it}(z_{1},z_{2})\,=\,\left(e^{2\pi
it}z_{1},e^{2\pi it}z_{2}\right),$

we have $\Phi_{R}^{\frac{i}{k}}(Q_{0,j})=\xi^{i}\,Q_{0,j}=Q_{i,i+j}$, that is,
$\Phi_{R}^{\frac{1}{k}}$ cyclically acts on the components of $Q_{j}$.

The full sweep out of $Q_{0,j}$ under the (backward) Reeb flow is the
Lagrangian surface

$L_{j}\,:=\,\bigcup_{t\in\mathbb{R}}e^{-2\pi
it}\,Q_{0,j}\,=\,\bigcup_{t\in[0,1]}e^{-2\pi
it}\,Q_{0,j}\,=\,\bigcup_{t\in[0,\frac{1}{k}]}e^{-2\pi it}\,Q_{j}.$

Therefore, if $\pi\colon S^{3}\to S^{2}$ denotes the Hopf fibration (whose
fibers are the Reeb trajectories), then

$L_{j}\,=\,\pi^{-1}(h_{j}),\quad\text{where
}h_{j}=\pi(Q_{0,j})\,=\,\pi(Q_{j}).$

A computation shows that $h_{j}$ is a half-great circle joining the north pole
to the south pole of $S^{2}$. Since the area of the reduced space $S^{2}$ is
$1$, it follows that the pairs $(h_{j},h_{j+1})$ each bound an open disc
$D_{j}$ of area $\frac{1}{k}$. It is also not hard to see that there exists a
diffeomorphism $\pi^{-1}(D_{j})\to D(\frac{1}{k})\times S^{1}$ that takes
$\lambda_{S^{3}(1)}$ to $\alpha_{{\operatorname{st}}}\oplus dt$. Altogether,
we see that

$L\,:=\,\bigcup_{j}L_{j}\,=\,\bigcup_{t\in[0,\frac{1}{k}]}\Phi^{-t}_{R}(\Lambda_{k})$

cuts $S^{3}(1)$ into $k$ connected components, each of which can be identified
with $\bigl{(}D(\frac{1}{k})\times
S^{1},\alpha_{\operatorname{st}}+dt\bigr{)}$, where
$S^{1}=\mathbb{R}/\mathbb{Z}$.

Now take a Legendrian knot $\Lambda$ in $S^{3}(1)$ such that there is no Reeb
chord of length $\leq\frac{1}{k}$ from $\Lambda$ to $\Lambda_{k}$.
Equivalently, $\Lambda$ is disjoint from
$\bigcup_{t\in[0,\frac{1}{k}]}\Phi^{-t}_{R}(\Lambda_{k})$, which is $L$. So
$\Lambda$ lies in one of the connected components of $S^{3}(1)\backslash L$
and can be seen as a Legendrian knot in $\bigl{(}D(\frac{1}{k})\times
S^{1},\alpha_{\operatorname{st}}+dt\bigr{)}$. But a very classical and
elementary argument (see e.g. [1, p. 192]) shows that such a knot has a Reeb
chord of length $\leq\frac{1}{k}$ (in fact, of length $<\frac{1}{2k}$).

Summing up this discussion, in the case of the round sphere $S=S^{3}(1)$,
where the Reeb flow is explicit, the barrier property of $\Lambda_{k}$
directly follows from the fact that the $\frac{1}{k}$-negative Reeb chords
starting at $\Lambda_{k}$ disconnect $S$ into pieces all of whose Legendrian
knots have small Reeb chords. There is no reason, however, that this
disconnectedness property remain true when the Reeb flow is modified by taking
the contact form on an arbitrary starshaped domain in $C^{4}(1)$. It is
therefore remarkable that the Legendrian rigidity result holds true.

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# Detecting Temporal shape changes with the Euler Characteristic Transform

Lewis Marsh1,2, Felix Y Zhou2,3, Xiao Qin2,4, Xin Lu2, Helen M Byrne1,2,
Heather A Harrington1,5 1Mathematical Institute, University of Oxford, Oxford,
UK
2Ludwig Institute for Cancer Research, University of Oxford, Oxford, UK
3Lyda Hill Department of Bioinformatics, UT Southwestern Medical Center,
Dallas, Texas, USA
4Department of Oncology, University College London, London, UK 5Wellcome
Centre for Human Genetics, University of Oxford<EMAIL_ADDRESS>
(Date: June 2022)

###### Abstract.

Organoids are multi-cellular structures which are cultured _in vitro_ from
stem cells to resemble specific organs (e.g., brain, liver) in their three-
dimensional composition. Dynamic changes in the shape and composition of these
model systems can be used to understand the effect of mutations and treatments
in health and disease. In this paper, we propose a new technique in the field
of topological data analysis for DEtecting Temporal shape changes with the
Euler Characteristic Transform (DETECT). DETECT is a rotationally invariant
signature of dynamically changing shapes. We demonstrate our method on a data
set of segmented videos of mouse small intestine organoid experiments and show
that it outperforms classical shape descriptors. We verify our method on a
synthetic organoid data set and illustrate how it generalises to 3D. We
conclude that DETECT offers rigorous quantification of organoids and opens up
computationally scalable methods for distinguishing different growth regimes
and assessing treatment effects.

##  1\. Introduction

Organoids are three-dimensional multi-cellular _in vitro_ cell cultures that
can be generated from stem cells. Organoids are also known as _mini organs_
because their constituent cells can differentiate into various cell lineages
with defined cellular functions. At early times, their growth is approximately
radially symmetric. Growth factors in the culture medium surrounding the
organoids, drive the stem cells to proliferate and the organoids to increase
in size. On longer times, the progeny of the stem cells differentiate or
specialise, into different cell types that self-organise to produce tissues
whose composition resembles the original organ. Excessive cell proliferation
combined with cell differentiation can lead to complex shape changes, with the
outer boundary of the organoids adopting intricate and asymmetric structures.
The increasing use of organoids for studying tissue development and disease
progression tissue responses to genetic and environmental perturbations can be
attributed to their ability to recapitulate the 3D cellular architecture and
function of their tissue of origin than 2D cell cultures [26]. Organoids are
also being used to investigate tissue responses to genetic and environmental
perturbations and for drug testing and development [6]. While organoids can
now be generated to study multiple organs, including the brain, kidney and
liver, the most widely studied organoids are those that derive from the
intestinal epithelium, one of the fastest-renewing mammalian tissues.

Differentiated cells with specific cellular functions often exhibit defined
cellular morphology. Descriptors of morphology can be used as a phenotypic
read-out of the organoids, which may reflect the underlying genetic
composition and its response to environmental perturbations. Therefore,
several studies have focused on quantifying and analysing organoid morphology.
Simple measures, such as cell numbers, organoid volume and surface area,
diameter, shape factor (ratio of surface area to volume), and growth rate have
been used to relate morphology, genotype and drug responses [5, 18, 23, 40].
Furthermore, deep learning methods can segment organoid images and extract
morphological features including organoid perimeter and eccentricity [17, 22].
At the same time, mechanistic models have been developed to investigate the
relationship between stem cell proliferation, cell fate specification,
organoid growth and morphology. These agent-based and continuum-based models
have been compared with experimental data using simple growth curves [18, 35,
40]. The complexity of organoid morphology lends itself to more sophisticated
analysis. For example, genus and average curvature [19], measures from
geometry and topology, have been used to distinguish shape changes in
organoids. Here we propose studying the geometry and topology of organoids
with topological data analysis.

Topological data analysis (TDA) is a collection of data science methods for
quantifying shape in data sets by using techniques from topology. TDA has been
successfully applied to a variety of problems in applied mathematics, which
include sensor network coverage [10], image signalling [3, 7], skeletonising
of images [12], discovering periodicities in time-series [33], material
science [24, 25], and financial mathematics [16, 28]. A field of applied
mathematics in which TDA has been particularly successful is mathematical
biology, for example in studying enzyme kinetics [29], neuron morphology [21,
2] and protein structures [13, 14, 30].

One technique in TDA, introduced by Turner, Mukherjee and Boyer [36], is the
Euler Characteristic Transform (ECT). The ECT gives a signature for shapes
embedded in Euclidean space $\operatorname{\mathbb{R}}^{d}$. The ECT and its
extensions have been applied to a range of problems including MRT scans of
glioblastoma patients [8, 20], to classification of 3D shapes (e.g. mammalian
teeth [37], protein structures [34] and barley seeds [1]). The above studies
highlight the insight that ECT can generate when applied to static data. Here
we show the additional information that can be gained from generalising the
ECT to dynamic shape changes (i.e., spatio-temporal data).

We propose an extension to the smooth ECT (SECT), _DEtecting Temporal shape
changes with the Euler Characteristic Transform_ (DETECT) which is the first
application of a rotationally invariant and dynamic ECT. The output from
DETECT is a signature in a Hilbert space that can be thought of as a surface
in $\operatorname{\mathbb{R}}^{3}$. To aid classification of the resulting
signature, we apply a non-linear transformation using kernel methods. Other
studies have successfully used kernels in conjunction with the ECT [8, 34,
37]; we are the first to use approximate feature embeddings with the Nystroem
method [39]. In practice, Nystroem approximations typically scale better in
runtime and memory than kernel methods [39], including Gaussian process
approaches with the ECT [8, 34, 37]. Compared with standard kernel methods,
these approximations also allow a wider variety of methods to be used with
DETECT, including random forests. We apply DETECT to dynamic data describing
the boundaries of experimental and synthetic organoids in 2D and 3D. We
demonstrate the proposed method on the applicability of both synthetic and
experimental on 2D and 3D data to distinguish morphology of organoids. First,
we show that the smooth Euler Characteristic Transform super-cedes standard
measures used to study organoid morphology. Next, we show that the temporal
method proposed, DETECT, improves the classifying of different organoid
treatment groups. Our paper is organised as follows. We first introduce
relevant background on topological data analysis and kernel methods. We then
introduce a novel signature, DETECT, which quantifies the evolution of a shape
over time. By applying DETECT to a data set of 2D organoid boundaries
segmented from videos of mouse small intestine organoid experiments, we
demonstrate that DETECT can distinguish between treated and untreated
organoids. Our results thereby give insights into how cancer treatments affect
the morphology of organoids. We highlight that DETECT outperforms classical
shape descriptors at this classification task. Finally, we demonstrate on a
data set of 3D organoid boundaries generated by a mechanistic model of Yan et
al. [40] that our method generalises to 3D boundaries and verify that it can
extract biologically meaningful information in a setting in which we have
precise control over all biologically meaningful factors.

##  2\. Mathematical methods

###  2.1. Preliminaries

####  2.1.1. Simplicial complexes and filtrations

To undertake our work, we require a mathematical definition of a shape. In
topological data analysis (TDA) and computational geometry, simplicial
complexes are widely used for this purpose:

######  Definition 1.

Given a finite set of vertices $X$, an _abstract simplicial complex_ is a set
of subsets of $X$, denoted $\operatorname{\mathcal{K}}$, such that for any
$\tau\in\operatorname{\mathcal{K}}$ and $\sigma\subseteq\tau$, we have
$\sigma\in\operatorname{\mathcal{K}}$. We call
$\sigma\in\operatorname{\mathcal{K}}$ a _simplex_ of
$\operatorname{\mathcal{K}}$. Moreover, for any simplex $\sigma$, we define
$\dim(\sigma)=|\sigma|-1$ and
$\operatorname{\mathcal{K}}_{i}=\\{\sigma\in\operatorname{\mathcal{K}}\,|\,\dim(\sigma)=i\\}$.

A _geometric (or embedded) simplicial complex_ $\operatorname{\mathcal{K}}$ is
an abstract simplicial complex that is endowed with an embedding in
$\operatorname{\mathbb{R}}^{d}$. That is,
$X\subset\operatorname{\mathbb{R}}^{d}$ and for all
$\sigma,\tau\in\operatorname{\mathcal{K}}$ with $\sigma\neq\tau$ we have
$\mathrm{relint}(\mathrm{cvx}(\sigma))\cap\mathrm{cvx}(\tau)=\varnothing$.
Here, $\mathrm{relint}$ is the relative interior111For a convex set
$C\subset\operatorname{\mathbb{R}}^{d}$, that is $\\{x\in C\,|\,\forall y\in
C:\exists\lambda>1:\lambda x+(1-\lambda)y\in C\\}$. and $\mathrm{cvx}$ the
convex hull.222That is, the intersection of all convex subsets of
$\operatorname{\mathbb{R}}^{d}$ that contain the given set of points. We can
then think of $\operatorname{\mathcal{K}}$ equivalently as the union of the
convex hulls of all of its simplices, which is a topological subspace of
$\operatorname{\mathbb{R}}^{d}$.

Given two simplicial complexes $\operatorname{\mathcal{K}}$ and
$\operatorname{\mathcal{K}}^{\prime}$, a _simplicial map_ $f$ is a function
$f:X\to X^{\prime}$, extending to a map
$f:\operatorname{\mathcal{K}}\to\operatorname{\mathcal{K}}^{\prime}$ by
$f(\sigma)=\left\\{f(x)\,|\,x\in\sigma\right\\}$.

Figure 1. Example of two simplicial complexes, $\mathcal{K}_{1}$ and
$\mathcal{K}_{2}$, embedded in $\operatorname{\mathbb{R}}^{2}$. Vertices as
blue dots, 1-simplices as red lines.

Note that any abstract simplicial complex can be viewed as a geometric
simplicial complex by considering $V$ to be a subset of the free vector space
generated by $V$. This geometric simplicial complex is called the _geometric
realisation_ of $\operatorname{\mathcal{K}}$.

In most settings, abstract simplicial complexes are more amenable to
computations while geometric simplicial complexes, perhaps unsurprisingly,
contain geometric information. The organoid boundaries extensively studied in
the following section are all equivalent when modelled as abstract simplicial
complexes. We use filtrations to study and compare geometric simplicial
complexes representing organoid boundaries.

######  Definition 2.

A _filtration_ of a simplicial complex $\operatorname{\mathcal{K}}$ is a
function $f:\operatorname{\mathcal{K}}\to\operatorname{\mathbb{R}}$ such that
$f(\sigma)\leq f(\tau)$ for $\sigma\subseteq\tau$. For
$j\in\operatorname{\mathbb{R}}$, we then define

$\operatorname{\mathcal{K}}^{j}=\left\\{\sigma\in\operatorname{\mathcal{K}}\,|\,f(\sigma)\leq
j\right\\}.$

All $\operatorname{\mathcal{K}}^{j}$ are simplicial complexes in their own
right and
$\operatorname{\mathcal{K}}^{j}\subseteq\operatorname{\mathcal{K}}^{i}$ for
$j\leq i$.

####  2.1.2. Kernels and kernel approximations

Before introducing further topology, we will briefly discuss kernels and their
associated Hilbert spaces - a theory we will use when defining key topological
signatures. Kernels are generalisations of inner-products. The motivation for
generalising inner-products is twofold. First, we may want to use data science
methods requiring an inner-product (e.g. support vector classification,
principal component analysis or k-means) on data in spaces not endowed with an
inner-product. Second, even if it is possible to define an inner-product in
data space, features in the data may not be linear. For example, not every
labelled data set can be separated by a plane, resulting in inaccurate
classification. In such an instance, SVC in combination with kernels could, by
contrast, allow a separation.

######  Definition 3.

Let $\mathcal{X}$ be a non-empty set. A function
$k:\mathcal{X}\times\mathcal{X}\to\operatorname{\mathbb{R}}$ is called a
_kernel_ if it is symmetric and positive definite. That is, for any
$x_{1},...,x_{n}\in\mathcal{X}$ and
$a_{1},...,a_{n}\in\operatorname{\mathbb{R}}$, we get

$\sum_{i,j=1}^{n}a_{i}a_{j}k\left(x_{i},x_{j}\right)\geq 0.$

For general $\mathcal{X}$, the Kronecker delta gives a kernel. If
$\mathcal{X}$ is an inner-product space, the inner-product is a kernel. If
$\mathcal{X}$ is a metric space with metric $d$, then the function

$k(x,y):=\exp\left(-\frac{d(x,y)^{2}}{\lambda}\right),$

where $\lambda>0$ is a hyperparameter, is a kernel called the _Gaussian
kernel_ , which we employ in this work.

######  Definition 4.

Let $k$ be a kernel on some set $\mathcal{X}$. If we define
$\mathcal{H}_{0}=\mathrm{span}_{\operatorname{\mathbb{R}}}\\{k(x,\,\cdot\,)\,|\,x\in\mathcal{X}\\},$
then $\mathcal{H}_{0}$ is a vector space of functions from $\mathcal{X}$ to
$\operatorname{\mathbb{R}}$. Note that

$\langle k(x,\,\cdot\,),k(y,\,\cdot\,)\rangle_{\mathcal{H}_{0}}:=k(x,y)$

defines an inner-product on $\mathcal{H}_{0}$ by bi-linear extension. We
define $\mathcal{H}=\overline{\mathcal{H}_{0}}$, the completion of
$\mathcal{H}_{0}$. Then $\mathcal{H}$ is a Hilbert space, called the
_Reproducing Kernel Hilbert Space_ (RKHS) of $k$.

The name RKHS derives from the fact that for any $f\in\mathcal{H}$ and
$x\in\mathcal{X}$, we have $\langle
k(x,\,\cdot\,),f\rangle_{\mathcal{H}}=f(x)$, which is called the reproducing
property of $k$. In particular, the reproducing property of an RKHS together
with the Cauchy-Schwarz inequality gives that the linear functionals
$\delta_{x}:\mathcal{H}\to\operatorname{\mathbb{R}}$ defined by
$\delta_{x}(f)=f(x)$ at $f\in\mathcal{H}$ are continuous for all
$x\in\mathcal{X}$.

The main point - in the context of this work - of defining an RKHS is to
illustrate that applying a kernel to elements of $\mathcal{X}$ can be viewed
as first embedding $\mathcal{X}$ into some Hilbert space of functions
$\mathcal{H}$ (by $x\mapsto k(x,\,\cdot\,)$) and then taking an inner-product
of such embedded elements. While $\mathcal{H}$ may be infinite-dimensional and
thus the embedding of $\mathcal{X}$ into $\mathcal{H}$ is intractable in
general, we never need to compute the (exact) embedding itself - computing
$k(x,y)$ for all $x,y\in\mathcal{X}$ is sufficient for any downstream method
relying on the inner-product of $\mathcal{H}$ only. This insight is called the
_kernel trick_.

If $\mathcal{X}$ is of finite size $n$, then computing $k(x,y)$ for all pairs
$k(x,y)$ may require only finitely many computations and will scale as
$\mathcal{O}(n^{2})$. To enable computations for large $n$, a number of
approximation methods of lower computational complexity have been developed.
One such method is the _Nystroem approximation_ :

######  Definition 5.

Let $\mathcal{X}=\\{x_{1},...,x_{n}\\}$ be of finite size $n$ and let $m<n$,
where $m$ is an integer. Denote by $K$ the $n\times n$-matrix with $i,j$-entry
$k(x_{i},x_{j})$. Then $K$ can be written in block-form as

$K=\begin{bmatrix}K_{11}&K_{12}\\\ K_{21}&K_{22}\end{bmatrix}.$

For $C:=[K_{11},K_{12}]^{T}$, the $m$-th Nystroem approximation of $K$ is the
matrix

$\tilde{K}=CK_{11}^{\dagger}C^{T},$

where † denotes the Moore-Penrose pseudo-inverse.

If $K$ is of approximate rank $m$ (or less), then $\tilde{K}\approx K$ [27].
Typically, the approximate rank of a Gram matrix is much less than $n$ if $n$
is large. Computing $\tilde{K}$ only requires $\mathcal{O}(mn)$ evaluations of
$k$, where $m$ is typically fixed. In practice, we only compute
$C(K_{11}^{\dagger})^{1/2}$ (to save computer memory), as the standard inner-
product of the $i$-th and $j$-th rows of $C(K_{11}^{\dagger})^{1/2}$
approximately gives $k(x_{i},x_{j})$. This matrix $C(K_{11}^{\dagger})^{1/2}$
can be used as a non-linear transformation and can be further analysed. In
particular, we take the row vectors and feed them to a method using inner-
products. Computing $C(K_{11}^{\dagger})^{1/2}$ has a runtime complexity of
$\mathcal{O}(nm^{2}+m^{3})$. There exist sampling heuristics for picking an
optimal set of $m$ points from $\mathcal{X}$ [27].

####  2.1.3. The Euler Characteristic and the Euler Characteristic Transform

The _Euler characteristic_ is a topological invariant of simplicial complexes.
Any two simplicial complexes that are (homotopy) equivalent as topological
spaces have the same Euler characteristic. Conversely, if two simplicial
complexes have different Euler characteristics, we can conclude that they are
topologically different (i.e. not homotopy equivalent). We can compute the
Euler characteristic entirely from the combinatorial information of an
abstract simplicial complex.

######  Definition 6.

Let $\operatorname{\mathcal{K}}$ be a simplicial complex. Then its Euler
characteristic is

$\chi(\operatorname{\mathcal{K}})=\sum_{i=0}^{\infty}(-1)^{i}\cdot|\operatorname{\mathcal{K}}_{i}|.$

Given a simplicial complex $\operatorname{\mathcal{K}}$ embedded in
$\operatorname{\mathbb{R}}^{d}$, using a sequence of Euler characteristics
induced by a filtration, yields additional discriminative information to
$\chi(\operatorname{\mathcal{K}})$: Let $v\in
S^{d-1}=\left\\{x\in\operatorname{\mathbb{R}}^{d}:\|x\|_{2}=1\right\\}$ be a
fixed direction in $\operatorname{\mathbb{R}}^{d}$. We then call the
filtration on $\operatorname{\mathcal{K}}$ induced by

$f_{v}:\operatorname{\mathcal{K}}\to\operatorname{\mathbb{R}},\quad\sigma\mapsto\min_{x\in\sigma}\left\\{\langle
x,v\rangle\right\\}$

the sub-level set filtration of $\operatorname{\mathcal{K}}$ in direction $v$,
where $\langle\,\cdot\,,\,\cdot\,\rangle$ is the standard inner-product in
$\operatorname{\mathbb{R}}^{d}$. We denote the above filtration
$\operatorname{\mathcal{K}}^{\langle\,\cdot\,,v\rangle}$ and each sub-level-
set at $t\in\operatorname{\mathbb{R}}$ as
$\operatorname{\mathcal{K}}^{\langle\,\cdot\,,v\rangle\leq t}$.

Assume that $a\in\operatorname{\mathbb{R}}$ is larger than the diameter of
$\operatorname{\mathcal{K}}$ in $\operatorname{\mathbb{R}}^{d}$. Then
$f_{v}^{-1}((-\infty,-a))=\varnothing$ and
$f_{v}^{-1}((-\infty,a))=\operatorname{\mathcal{K}}$ for all $v\in S^{d-1}$.
We can now define the Smooth Euler Characteristic transform and related
constructions [36]:

######  Definition 7.

First, let the _Euler characteristic curve_ in a fixed direction $v$ be

$\mathrm{ECC}_{\operatorname{\mathcal{K}}}^{v}:[-a,a]\to\operatorname{\mathbb{Z}},\qquad
t\mapsto\chi\left(\operatorname{\mathcal{K}}^{\langle v,\,\cdot\,\rangle\leq
t}\right).$

Secondly, this curve is smoothed by defining the _smooth Euler characteristic
curve_ (SEC) as follows:

$\mathrm{SEC}_{\operatorname{\mathcal{K}}}^{v}:[-a,a]\to\operatorname{\mathbb{R}},\qquad
t\mapsto\int_{-a}^{t}\left(\mathrm{ECC}_{\operatorname{\mathcal{K}}}^{v}(x)-\overline{\mathrm{ECC}_{\operatorname{\mathcal{K}}}^{v}}\right)\,\mathrm{d}x,$

where $\overline{\mathrm{ECC}_{\operatorname{\mathcal{K}}}^{v}}$ is the mean
of the function $\mathrm{ECC}_{\operatorname{\mathcal{K}}}^{v}$ over the
interval $[-a,a]$.

Thirdly, the ECT of a shape $\operatorname{\mathcal{K}}$ is then the map

$\mathrm{ECT}(\operatorname{\mathcal{K}}):S^{d-1}\to CF([-a,a]),\qquad
v\mapsto\mathrm{ECC}_{\operatorname{\mathcal{K}}}^{v},$

where $CF([-a,a]$ is the set of constructible integer-valued functions on
$[-a,a]$. For a more comprehensive discussion of these constructible functions
and ECT, see [9, 5]. Finally, we can define the _smooth Euler characteristic
transform_ (SECT):

$\mathrm{SECT}(\operatorname{\mathcal{K}}):S^{d-1}\to{L}^{2}([-a,a]),\qquad
v\mapsto\mathrm{SEC}_{\operatorname{\mathcal{K}}}^{v}.$

Here, $L^{2}([-a,a])$ is the Hilbert space of all functions on $[-a,a]$ that
can be generated by Fourier series.

Turner and colleagues have shown that both the ECT and SECT are injective,
making them sufficient statistics for comparing shapes embedded in
$\operatorname{\mathbb{R}}^{2}$ and $\operatorname{\mathbb{R}}^{3}$ [36].
Ghrist et al. [15] and Curry et al. [9] independently extended this
injectivity result to general $\operatorname{\mathbb{R}}^{d}$. The ECT or SECT
therefore also discriminate between shapes that are equivalent up to
translation, rotation, reflection, and combinations thereof. The issue of
discriminating between shapes equivalent up to translation can be overcome by
re-centring simplicial complexes by subtracting the mean of all vertices from
each vertex in the simplicial complex. However, resolving rotation and
reflection requires more care. Fortunately, Curry et al. [9] present a variant
of the ECT that is injective on the space of shapes modulo actions of the
orthogonal group $O(d)$.

Before introducing their result, we recall the notion of a pushforward
measure:

######  Definition 8.

Let $(X_{1},\Sigma_{1})$ and $(X_{2},\Sigma_{2})$ be measurable spaces,
$f:X_{1}\to X_{2}$ be a measurable function, and $\mu:\Sigma_{1}\to[0,\infty]$
be a measure on $X_{1}$. Then $f_{*}\mu$, the pushforward of $\mu$ along $f$,
is the measure on $X_{2}$ defined by $(f_{*}\mu)(U)=\mu(f^{-1}(U))$ for each
$U\in\Sigma_{2}$.

Then Theorem 6.6 in [9] states:

######  Theorem 9.

Let $\operatorname{\mathcal{K}}$ and $\operatorname{\mathcal{K}}^{\prime}$ be
generic simplicial complexes embedded in $\operatorname{\mathbb{R}}^{d}$. Let
$\mu$ be the Lebesgue measure on $S^{d-1}$. If
$\mathrm{ECT}\left(\operatorname{\mathcal{K}}\right)_{*}(\mu)=\mathrm{ECT}\left(\operatorname{\mathcal{K}}^{\prime}\right)_{*}(\mu)$,
then there exists a $\phi\in O(d)$ such that
$\operatorname{\mathcal{K}}=\phi\left(\operatorname{\mathcal{K}}^{\prime}\right)$.

Note that the converse implication of the above theorem is trivial, as the
Lebesgue measure on $S^{d-1}$ is invariant under the action of $O(d)$. Note
that the function mapping ECC to SEC is injective. Therefore, Theorem 9
generalises to the SECT.

For any $x\in[-a,a]$, define
$\delta_{x}:{L}^{2}([-a,a])\to\operatorname{\mathbb{R}}$ by
$\delta_{x}(f)=f(x)$ for all $f\in{L}^{2}([-a,a])$. These functionals, called
evaluation functionals, are continuous and, hence, measurable as
$L^{2}([-a,a])$ is an RKHS. Then if two embedded simplicial complexes,
$\operatorname{\mathcal{K}}$ and $\operatorname{\mathcal{K}}^{\prime}$ say,
satisfy $K=\phi(\operatorname{\mathcal{K}}^{\prime})$ for some $\phi\in O(d)$,
we get

$\int_{S^{d-1}}\delta_{x}\circ\mathrm{SECT}(\operatorname{\mathcal{K}})\,\mathrm{d}\mu=\int_{{L}^{2}([-a,a])}\delta_{x}\,\mathrm{d}(\mathrm{SECT}(\operatorname{\mathcal{K}})_{*}(\mu))$
$=\int_{{L}^{2}([-a,a])}\delta_{x}\,\mathrm{d}(\mathrm{SECT}(\operatorname{\mathcal{K}}^{\prime})_{*}(\mu))=\int_{S^{d-1}}\delta_{x}\circ\mathrm{SECT}(\operatorname{\mathcal{K}}^{\prime})\,\mathrm{d}\mu$

for all $x\in[-a,a]$ by the change of variable formula for integrals with
measure pushforwards and Theorem 9. Hence, the mean of such SECT evaluations,
or collections thereof, form a statistic that can be used for distinguishing
shapes modulo $O(d)$ actions.

Figure 2. Standard ECT pipeline visualised on segmented organoid boundaries.
1) Segmented input data. 2) Illustration of the sub-level-set filtration in
the direction given by the arrow. 3) The ECT in the direction given in 2). 4)
The SECT in the direction given in 2).

####  2.1.4. Previous studies using ECTs and kernel methods

Kernel methods have been used successfully in conjunction with the SECT for
both shape regression and classification problems [8, 37, 34]. Both of these
studies use Gaussian process models. Gaussian processes include the inversion
of a Gram matrix and thus have a runtime of $\mathcal{O}(n^{3})$. Hence, the
Nystroem method we employ scales better to large data sets. These models are
also conceptually more complex than linear regression and SVC, which we use
for regression and classification. To the best of our knowledge, neither the
ECT nor SECT have previously been used in combination with kernel
approximation methods.

###  2.2. Temporal shape detection

The SECT transforms a fixed, static shape. We now extend the definition of the
SECT to get a rotationally invariant temporal signature of a sequence of
shapes:

######  Definition 10.

Let $T=[0,c]$ or $T=\\{0,1,...,n-1\\}$, where $c\in\operatorname{\mathbb{R}}$
and $n\in\operatorname{\mathbb{N}}$, be a set of time points. Let
$\\{\operatorname{\mathcal{K}}(t)\\}_{t\in T}$ be sequence of shapes. Then the
_DETECT_ (DEtecting Temporal shape changes with the Euler Characteristic
Transform) of this sequence is the transform

$\mathrm{DETECT}\left(\\{\operatorname{\mathcal{K}}(t)\\}\right):T\to
L^{2}([-a,a]),\qquad
t\mapsto\left(x\mapsto\int_{S^{d-1}}\delta_{x}\circ\mathrm{SEC}^{v}_{\operatorname{\mathcal{K}}(t)}\,\mathrm{d}v\right).$

Here, we will use DETECT with $T=\\{0,...,n-1\\}$. In practice, after
integrating out the dependence on the direction, DETECT is a continuous
function from $T$ to $L^{2}([-a,a])$ (using the continuity of $\delta_{x}$ and
that $L^{2}([-a,a])$ is an RKHS). Such functions form an infinite dimensional
vector space and thus a finite presentation is not possible in general. We,
therefore, evaluate $\mathrm{DETECT}$ at any fixed $t\in T$ on a finite number
of evenly spaced points $P$ in $[-a,a]$. $\mathrm{DETECT}$ is then represented
approximately by a $|T|\times|P|$-matrix. In this paper, we apply
$\mathrm{DETECT}$ to two time-course data sets of organoid boundaries. We will
consider the space of such matrices to be endowed with the
$\|\,\cdot\,\|_{2}$-norm.

##  3\. Data sets

We analyse experimental 2D video data as well as synthetic 3D spatio-temporal
data.

###  3.1. Intestinal organoids from 2D video experiments

We first acquired a set of imaging data derived from time-lapse imaging of
mouse small intestine organoids. In total, we have 176 organoids and 320 video
frames for each organoid. The data set comprises of 74 wild-type (WT) and 102
p53 knock-out (KO/mutant) genetics organoids. Both groups of organoids further
split into untreated (CNT) and organoids treated with valproic acid and GSK3
inhibitor CHIR99021 (VC). These organoids have been filmed throughout their
growth and the resulting videos have been segmented. After segmentation, we
have 100 points summarising the boundary of each organoid at each video frame
(frames starting at beginning of the experiment and are taken every 15 min.
henceforth). We discard some videos which for technical reasons have fewer
than 320 video frames (as most videos below this threshold still seem to
change their morphology at the end of the video).

The organoid boundaries in this data set are, in the 2D video view, close to
being perfectly circular in the early stages of the videos across all
experimental conditions. This simple geometry is a result of the cellular
homogeneity in the early phases. As time progresses, cells proliferate and
differentiate. Through proliferation, organoids grow in size, and through
stem-cell differentiation, the cellular composition of organoids changes. As
different cell types have different mechanistic properties and differentiation
is not spatially uniform, organoids cease to be spherical. Most notably, they
elongate and their boundary buckles, possibly leading to the growth of finger-
like protrusions. Such growth behaviour is illustrated by the examples of
final video frames presented in Figure 3.

Figure 3. Phenotype effect of VC treatment to intestinal organoids. Static
video snapshots of the final frame. One example is shown for each condition.

Each collection of boundary points is transformed into a simplicial complex
representing the organoid boundaries, yielding a sequence of simplicial
complexes indexed by $t=0,...,319$ for each organoid. We re-centre each
simplicial complex such that the mean of all vertices is the origin. We then
compute the radius of the simplicial complex at $t=0$ (i.e. the largest norm
of all vertices after re-centring) and divide all vertices in the sequence of
simplicial complexes by that value. We translate and scale the data in this
way to simplify it, given its limited size. As a result, the initial size of
organoids or any movement throughout time is not considered by any downstream
analysis, including DETECT.

###  3.2. Spatio-temporal synthetic data

We next apply DETECT to a synthetic data set to verify our findings on the
experimental data. We use data generated by a model first presented in Yan et
al. [40]. Yan et al. construct a mechanistic model describing the growth of
cancerous colon organoids containing stem, progenitor and terminally
differentiated cells in 3D. These cells differ in terms of their rates of
mitosis, differentiation and cell death, the rates at which these processes
occur are controlled by model parameters.

We fix all but two of the parameters presented in [40]. We vary
$\lambda_{L}=0.5,\,1,\,2$, a death-rate parameter, and
$\lambda^{\mathrm{SC}}_{m}=\lambda^{\mathrm{CP}}_{m}=0.35,\,0.71,\,1.42$, cell
mitosis parameters for stem cells and committed progenitor cells, respectively
(all parameters are dimensionless). All other parameters are left at the
default values given in [40]. As we observe all possible combinations of death
and proliferation parameters, we get time-course data for nine different
computationally modelled organoids. We visualise examples of the simulated
organoid development in this data set in Figure 4.

Figure 4. Spatio-temporal shape changes in 3D of synthetic organoids. Top
column: Low (LO) mitosis rate. Middle column: Medium (MD) mitosis rate. Bottom
column: High (HI) mitosis rate. In all organoids visualised, the lysis rates
are at the lowest values given in the data set.

Unlike the experimental data, the number of (3D) boundary points varies
proportionally with the size of the simulated organoid. To ensure
computational tractability, we restrict ourselves to 300 boundary points
sampled uniformly at random at each time point. Different 300 point samples do
not lead to any notable perturbations in the downstream analyses. To
triangulate the boundary surface, we first re-centre the organoid such that
the mean of all boundary points is the origin. We then perform a stereographic
projection into the xy-plane and perform a Delaunay triangulation and identify
those points bordering an infinite area 2-cell. After projecting the finite
components of the triangulations back onto the sphere, we add a further point,
which is the mean of all points bordering an infinite area cell in the
previous step, and insert 1 and 2-cells to fill the north-pole area of our
organoid. We credit [11] with this pre-processing procedure. Unlike in the
experimental data, re-scaling is not needed as all simulated organoids are
identical at $t=1$.

We perform this pre-processing step to ensure that the resulting simplicial
complex has a geometric realisation homeomorphic to a sphere (or a union
thereof, if an organoid disconnects). This pre-processing is a necessary step
to ensure that faithful topological features (e.g. the correct number of
components or holes) are present before computing Euler Characteristics.
Random rotations of the data ahead of pre-processing lead to negligible
differences in ECT and thus suggest this pre-processing does not introduce
artificial geometric features.

##  4\. Results

We analyse the shape of organoid boundaries of experimental (2D) and synthetic
(3D) data by first building a simplicial complex representation. We then
compute the SECT of each organoid at each time point to obtain DETECT. When
computing DETECT, we use $a=6$ for the 2D organoids, $a=15$ for the 3D
organoids and $P$ to be 100 evenly spaces points in $[-a,a]$. We then compute
an $m$-dimensional feature embedding ($m=100$ for the experimental data and
$m=30$ for the smaller synthetic data set; both data sets use a Gaussian
kernel with $\lambda=|N|\times|P|$).

###  4.1. Regressing SECT to classical shape statistics

We first compute the SECT of static images of experimental organoids and
demonstrate that the SECT includes information conveyed by classical shape
statistics. The classic shape statistics, diameter, the mean and max centroid
distances, the equivalent diameter, the major and minor axis lengths and the
area of the convex hull, quantify geometric properties of a 2D shape. These
statistics are widely used and invariant under translation and $O(2)$ actions.
As each of these statistics is calculated for a static shape, i.e. for an
organoid boundary at a fixed time frame, we compare these statistics to the
SECT at fixed time frames.

To compare the aforementioned shape statistics with the SECT, we apply a
standard linear regression model. The SECT of each organoid at each time is
represented by a Nystroem feature embedding. We project the feature embedding
to 50 dimensions using PCA [31]. The PCA vectors give the independent
variables, while the classic shape statistics listed above are viewed as the
dependent variables. We pass the square-root values of the convex hull area to
the regression model, as it has a squared relationship with the remaining
metrics in the (default) symmetric cases. An illustration of the main notions
is given in Figure 5. We perform a 50-fold cross-validation for each metric
and report mean coefficients of determination and standard deviations of the
coefficient of determination in Table 1.

|  $R^{2}$-scores | std
---|---|---
equivalent diameter | 0.880 | 0.073
max. centroid distance | 0.916 | 0.041
mean centroid distance | 0.894 | 0.067
major axis length | 0.880 | 0.052
minor axis length | 0.630 | 0.246
perimeter | 0.972 | 0.024
$\sqrt{\text{convex area}}$ | 0.868 | 0.105
Table 1. Mean coefficients of determinations ($R^{2}$-scores) and their
standard deviations (std) in a 50-fold cross-validation of a Linear regression
in which the Nystroem features of the SECT give the independent variables and
the 8 variables above give the dependent variables. Figure 5. Left: The red
point gives the centroid (point which minimises mean squared distance to
boundary) of the organoid and the green line gives the distance to a boundary
point. The maximal and mean lengths of such green lines give the max. and mean
centroid distance. Centre: The red ellipse gives the ellipse with the best fit
to the boundary. The purple line gives the major axis and the green line the
minor axis. Right: the area of the convex hull (convex area) is visualised in
opaque red.

We find that the SECT regresses multiple classical shape statistics with high
accuracy. We note that the diameter of an organoid should be proportional to
both the major axis length and the maximum centroid distance.

The SECT has high predictive accuracy of equivalent diameters and perimeter
and, as a result, can also detect symmetry breaking. The lower accuracy of the
minor axis length and convex area suggests that the SECT is more limited in
its ability to capture the (mean) size of indentations, compared to detecting
size and elongation. We remark that this limitation may be related to
segmentation accuracy, and therefore, we consider organoid shapes with known
segmentation (i.e. synthetic data) in Section 4.3.

In addition to the above regression analysis, we can decompose the covariance
matrix of the aforementioned classical shape statistics by its singular
values. We remark that we standardise the data in each feature before
computing the covariances. We observe that the first four principal components
of the classical shape statistics explain over 90% of the variance in this
data set (see Figure 6). We then perform a canonical correlation analysis
(CCA) [38] between these four principal components and the PCA-transformed
SECT data. We find that the first four pairs of canonical variables have a
perfect correlation score of 1.0 and conclude that the SECT can explain over
90% of the variance in the classical shape statistics on the given data set.

Figure 6. The eigenvalues ($y$-axis) of the covariance matrix of the classical
shape descriptors in decreasing order (indices of eigenvalues on $x$-axis).
The data was standardised in each component before the covariance matrix was
computed. The first four eigenvalues (to the left of the dashed line) account
for 90.5% of the variance in the data.

###  4.2. Classification of Organoids

We focus on classifying p53-knock-out organoids into treated and untreated
groups. Before training a classifier, we seek to exclude some organoids where
the segmentation does not accurately trace the organoid boundary in the video.
In these cases, the segmented boundary is significantly larger than the true
organoid boundary. Therefore, we exclude organoids whose radius grows by a
factor of more than two. For p53-knock-out organoids, this procedure excludes
two out of 98 organoids. Of the remaining 96 organoids, 55 are untreated.

We use random forest classification [4] to classify p53-knock-out organoids
into untreated and VC-treated experimental groups. Random forest
classification then trains an ensemble of decision trees trained on random
subsets of the Nystroem-transformed DETECT data. The trees classify data
points by majority vote. We use the scikit-learn [32] implementation of random
forest classification and optimise the hyper-parameters of the maximum tree
depth, the minimum number of samples allowed to define a split and the minimum
number of samples per tree leaf by cross-validation grid search (GridSearchCV
in scikit-learn). Based on this optimisation, maximum tree depth is five,
minimum number of samples to define a split is five and minimum number of
samples per tree leaf is three. The 5-fold cross-validation for these
parameters gives a mean classification accuracy of 68.8% with a standard
deviation of 2.7%. We remark that setting the number of Nystroem features to
$m=500$ increases the mean accuracy further to 70.0% but also increases the
standard deviation to 7.5%. The higher standard deviation suggests that
setting $m=500$ could result in overfitting.

The accuracy of classification results based on DETECT exceeds those based on
all classical statistics (e.g. area and perimeter) which give a mean
classification accuracy of 60.5% and a standard deviation of 4.8% when we use
the pipeline and cross-validation method described above. Classification based
on DETECT also exceeds the baseline accuracy associated with guessing, which
is 57.4% as there are slightly more untreated than treated organoids in our
data set. Further, we have shown that combining DETECT with machine learning
can distinguish organoids treated with valproic acid and GSK3 inhibitor based
on quantification of their shape dynamics as well as regress out classical
shape statistics.

###  4.3. Distinguishing shape evolution 3D

Finally, we apply our methodology to the synthetic data generated by the model
of Yan, Kostorum and Lowengrub [40]. As described in Section 3, this data set
contains 9 organoids and thus is too small to apply linear regression or
random forest classification. We therefore only report the first two principal
components of the Nystroem-transformed DETECT signatures. These outputs
demonstrate that our methods generalise well to 3D shapes and identify
important structure in the synthetic data, for which we know the ground truth.

(a)

(b)

Figure 7. (A) The first two principal components of the Nystroem-transformed
DETECT for the 3D analysis. We see that these Nystroem features of the DETECT
signatures cluster by mitosis rate, which is the dominant signal in the data.
(B) Example of 2D projections of an organoid with given mitosis rate at the
final time-point.

This analysis, visualised in Figure 7, shows that organoids cluster together
by their mitosis rates. This behaviour is consistent with watching the videos
for these 9 organoids. We see that low-mitosis-rate organoids exhibit little
growth and virtually no symmetry breaking (see Figure 4). Medium-mitosis-rate
organoids show a little more growth than low-proliferation organoids and
notable buckling of their boundary (see Figure 4). Finally, high-mitosis-rate
organoids exhibit a large degree of growth and strong development of
protrusions. In fact, the development of protrusions is so pronounced that
several protrusions disconnect from the main organoid at later time points
(see Figure 4).

The principal components in Figure 7 therefore appear to pick up the major
signal in the synthetic 3D data set. We hypothesise that the first principal
component is proportional to the size of the organoid. Similarly, we
conjecture that the second principal component corresponds to the geometric
complexity of the organoids. In particular, low-mitosis-rate organoids have
the lowest geometric complexity while medium-mitosis-rate exhibit significant
symmetry breaking. The high-mitosis-rate organoids lie in between the two
former groups of organoids in terms of geometric complexity, as their
protrusion development is so pronounced that protrusions disconnect. The
resulting connected tissues are relatively spherical.

##  5\. Discussion

In this work, we have introduced a new technique from the field of topological
data analysis, for detecting temporal shape changes with the Euler
Characteristic Transform (DETECT). We have highlighted its utility by studying
organoid morphology. We first showed that several classical shape descriptors,
including the diameter, the mean and maximum centroid distances, the
equivalent diameter, the major and minor axis lengths and the area of the
convex hull, can be regressed from the Smooth Euler Characteristic Transform
(SECT) with high accuracy. We applied DETECT, with kernel approximation
methods and random forests, to a data set of experimental p53 knock-out mouse
small intestine organoids, and showed that our approach can distinguish VC-
treated organoids from untreated organoids. We remark that this integration
with kernel approximations enables larger data sets to be analysed because
when kernel methods are used the runtime complexity of ECT can be reduced from
being cubic to approximately linear in the number of data points. We
highlighted this methodology generalises to 3D by applying DETECT to an _in
silico_ data set derived from a continuum model of organoid growth [40].
DETECT enables the synthetically generated organoids to be clustered according
to mitosis rate (one of the parameters which were varied in the synthetic
data).

In the future, we aim to extend our findings - both in 2D and 3D - to data
sets of different types of organoids (derived from different organs, with
different genetic backgrounds and/or cultured under different conditions). We
aim also to study information loss between 3D data and their 2D projections.
We will accomplish this by first considering synthetic data generated from 3D
mechanistic models [40], which neglect certain biophysical processes (e.g. the
effects of gravity, the production of extracellular matrix, mechanical stress)
[40]. This will enable us to focus on establishing relationships between the
DETECT signatures and the values of key model parameters (e.g., cell
proliferation and differentiation rates).

We plan to extend this analysis to other types of morphological data that do
not have regularised and smooth boundaries. For example, there are no random
perturbations in the synthetic data studied here. In practice, data sets
analysed by the ECT and its extensions may be noisier than the data sets
analysed in this paper. The ECT (and, by extension, its variants, including
DETECT) are not stable with respect to small perturbations. Theoretical work
studying the stability properties of the ECT and extensions would provide a
significant step forward for their application to noisy morphological data
sets.

## Acknowledgements

We thank Huaming Yan, Anna Konstorum and John Lowengrub for helpful
discussions and for providing us with data generated by their model of
organoid growth. HAH gratefully acknowledges funding from a Royal Society
University Research Fellowship. LM, HMB and HAH are members of the Centre for
Topological Data Analysis, which is funded by the EPSRC grant ‘New Approaches
to Data Science: Application Driven Topological Data Analysis’ EP/R018472/1.
FYZ was funded by the Ludwig Institute for Cancer Research and an EPSRC Life
Sciences Interface Doctoral Training Centre Fellowship, EP/F500394/1. XQ was
funded by Ludwig Cancer Institute, China Scholarship Council, and Oxford
University. For the purpose of Open Access, the authors have applied a CC BY
public copyright licence to any Author Accepted Manuscript (AAM) version
arising from this submission.

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