#  The Koopman representation for self-similar groupoid actions

Valentin Deaconu  Valentin Deaconu
Department of Mathematics (0084)
University of Nevada
Reno NV 89557-0084
USA<EMAIL_ADDRESS>
###### Abstract.

We introduce the $C^{*}$-algebra $C^{*}(\kappa)$ generated by the Koopman
representation $\kappa$ of an étale groupoid $G$ acting on a measure space
$(X,\mu)$. We prove that for a level transitive self-similar action $(G,E)$
with $E$ finite and $|uE^{1}|$ constant, there is an invariant measure $\nu$
on $X=E^{\infty}$ and that $C^{*}(\kappa)$ is residually finite-dimensional
with a normalized self-similar trace. We also discus $p$-fold similarities of
Hilbert spaces in connection to representations of the graph algebra
$C^{*}(E)$ and self-similar representations of $G$ in connection to the Cuntz-
Pimsner algebra $C^{*}(G,E)$.

###### Key words and phrases:

Koopman representation; self-similar groupoid action; residually finite
dimensional; trace; Cuntz-Pimsner algebra.

###### 1991 Mathematics Subject Classification:

Primary 46L05.

##  1\. introduction

The concept of a group action on a space was generalized to a groupoid action
and it has applications to dynamical systems, representation theory and
operator algebras. If groups can roughly be described as the set of symmetries
of certain objects, then groupoids can be thought as the set of symmetries of
fibered objects.

Self-similar group actions are defined by using subgroups of the automorphism
group of a rooted tree which is viewed as the path space of a graph with one
vertex and $n\geq 2$ edges. Given a finite directed graph $E$ with no sources,
the corresponding path space is fibered over $E^{0}$ and it gives rise to a
family of trees (or forest) $T_{E}$. It is natural to consider self-similar
actions of subgroupoids $G$ of PIso$(T_{E})$, the discrete groupoid of partial
isomorphsims of $T_{E}$. The most interesting case is when $|uE^{1}|$ is
constant for $u\in E^{0}$, since then all trees in $T_{E}$ are isomorphic.

We begin by recalling the definition of a groupoid action on a space $X$ and
of the Koopman representation $\kappa$ associated to an invariant measure
$\mu$ on $X$. Self-similar groupoid actions $(G,E)$ were introduced in [5] and
in the presence of an invariant measure on $X=E^{\infty}$, we discuss the
$C^{*}$-algebra $C^{*}(\kappa)$ associated to the induced Koopman
representation. We prove that $C^{*}(\kappa)$ is residually finite-dimensional
(RFD) and it has a self-similar trace $\tau_{0}$.

We then define the wreath recursion for $(G,E)$ and the matrix recursion for
$C_{c}(G)$. We recall the concept of $p$-fold similarities of Hilbert spaces
and relate it to representations of the graph algebra $C^{*}(E)$. We also
define self-similar representations of $G$ and relate these to representations
of the Cuntz-Pimsner algebra $C^{*}(G,E)$. We illustrate with several
examples.

##  2\. Groupoid actions and the Koopman representation

A groupoid $G$ is a small category with inverses. We will use $d$ and $t$ for
the domain and target maps $d,t:G\to G^{(0)}$ to distinguish them from the
source and range maps $s,r:E^{1}\to E^{0}$ on directed graphs. An étale
groupoid is a topological groupoid where the target map $t$ (and necessarily
the domain map $d$) is a local homeomorphism. The unit space $G^{(0)}$ of an
étale groupoid is always an open subset of $G$. The set of composable pairs is
denoted $G^{(2)}$ and the set of $g\in G$ with $d(g)=u,t(g)=v$ is denoted
$G_{u}^{v}$. Two units $x,y\in G^{(0)}$ belong to the same $G$-orbit if there
exists $g\in G$ such that $d(g)=x$ and $t(g)=y$. When every $G$-orbit is dense
in $G^{(0)}$, then the groupoid $G$ is called minimal.

The isotropy group of a unit $x\in G^{(0)}$ is the group

$G_{x}^{x}:=\\{g\in G\;|\;d(g)=t(g)=x\\},$

and the isotropy bundle is

$G^{\prime}:=\\{g\in G\;|\;d(g)=t(g)\\}=\bigcup_{x\in G^{(0)}}G_{x}^{x}.$

A groupoid $G$ is said to be principal if all isotropy groups are trivial, or
equivalently, $G^{\prime}=G^{(0)}$.

######  Definition 2.1.

Let $G$ be an étale groupoid. A bisection is an open subset $U\subseteq G$
such that $d$ and $t$ are both injective when restricted to $U$.

We now recall the definition of a groupoid action on a space given in [8]:

######  Definition 2.2.

A topological groupoid $G$ is said to act (on the left) on a locally compact
space $X$, if there are given a continuous open surjection
$\omega:X\rightarrow G^{(0)}$, called the anchor or moment map, and a
continuous map

$G\ast X\rightarrow X,\quad\text{write}\quad(g,x)\mapsto g\cdot x,$

where

$G\ast X=\\{(g,x)\in G\times X\mid d(g)=\omega(x)\\},$

that satisfy

i) $\omega(g\cdot x)=t(g)$ for all $(g,x)\in G\ast X,$

ii) $(g_{2},x)\in G\ast X,\,\,(g_{1},g_{2})\in G^{(2)}$ implies
$(g_{1}g_{2},x),(g_{1},g_{2}\cdot x)\in G*X$ and

$g_{1}\cdot(g_{2}\cdot x)=(g_{1}g_{2})\cdot x,$

iii) $\omega(x)\cdot x=x$ for all $x\in X$.

We should mention that recently, some authors don’t necessarily assume that
$\omega:X\to G^{(0)}$ is open or surjective (see [12] for example).

The action of $G$ on $X$ is called transitive if given $x,y\in X$, there is
$g\in G$ with $g\cdot x=y$ and is free if $g\cdot x=x$ for some $x$ implies
$g=\omega(x)$.

The set of fixed points is defined as

$X^{G}=\\{x\in X:g\cdot x=x\;\text{for all}\;g\in
G_{\omega(x)}^{\omega(x)}\\}.$

If $G$ has trivial isotropy, then $X^{G}=X$.

For $x\in X$, its stabilizer group is

$G(x)=\\{g\in G:g\cdot x=x\\},$

which is a subgroup of $G_{u}^{u}$ where $u=\omega(x)$. The orbit of $x\in X$
is

$G\cdot x=\\{g\cdot x:g\in G,\;d(g)=\omega(x)\\}.$

The set of orbits is denoted by $G\backslash X$ and has the quotient topology.
The action of $G$ on $X$ is called minimal if every orbit $G\cdot x$ is dense
in $X$.

######  Example 2.3.

A groupoid $G$ with open domain and target maps acts on its unit space
$G^{(0)}$ by $g\cdot d(g)=t(g)$. In this case, $\omega=id$. The groupoid is
called transitive if this action is transitive. Notice that $g\cdot u=u$ for
all $g\in G_{u}^{u}$, in particular $(G^{(0)})^{G}=G^{(0)}$. A transitive
groupoid with discrete unit space is of the form $G^{(0)}\times K\times
G^{(0)}$ where $K$ is the isotropy group.

######  Remark 2.4.

The fibered product $G\ast X=\\{(g,x)\in G\times X\mid d(g)=\omega(x)\\}$ has
a natural structure of groupoid, called the semi-direct product or action
groupoid and is denoted by $G\ltimes X$, where

$(G\ltimes X)^{(2)}=\\{((g_{1},x_{1}),(g_{2},x_{2}))\mid\,\,x_{1}=g_{2}\cdot
x_{2}\\},$

with operations

$(g_{1},g_{2}\cdot
x_{2})(g_{2},x_{2})=(g_{1}g_{2},x_{2}),\;\;(g,x)^{-1}=(g^{-1},g\cdot x).$

The domain and target maps of $G\ltimes X$ are

$d(g,x)=(d(g),x)=(\omega(x),x),\quad t(g,x)=(t(g),g\cdot x)=(\omega(g\cdot
x),g\cdot x),$

and the unit space $(G\ltimes X)^{(0)}$ may be identified with $X$ via the map

$i:X\to G\ltimes X,\;\;i(x)=(\omega(x),x).$

The projection map

$\pi:G\ltimes X\to G,\;\;\pi(g,x)=g$

is a covering of groupoids.

Assume that the topological groupoid $G$ acts on the space $X$ via
$\omega:X\to G^{(0)}$. A Borel measure $\mu$ on $X$ is $G$-invariant if
$\mu(g^{-1}\cdot A)=\mu(A)$ for any Borel set $A\subseteq\omega^{-1}(t(g))$.

######  Definition 2.5.

For a $G$-invariant measure $\mu$ on $X$, the Hilbert space $L^{2}(X,\mu)$ is
fibered over $G^{(0)}$ with fibers $L^{2}(\omega^{-1}(u),\mu)$ for $u\in
G^{(0)}$ and we define the Koopman representation $\kappa$ of $G$ on
$L^{2}(X,\mu)$ by

$\kappa_{g}:L^{2}(\omega^{-1}(d(g)),\mu)\to
L^{2}(\omega^{-1}(t(g)),\mu),\;\;\kappa_{g}(f)(x)=f(g^{-1}\cdot x).$

It can be verified that each $\kappa_{g}$ is a partial isometry in
$\mathcal{L}(L^{2}(X,\mu))$ and if $G$ is étale (our main case of interest),
the representation $\kappa:G\to\mathcal{L}(L^{2}(X,\mu))$ can be extended to
the algebra $C_{c}(G)$ of compactly supported continuous functions. We denote
by $C^{*}(\kappa)$ the closure of $\kappa(C_{c}(G))$ in the operator norm of
$\mathcal{L}(L^{2}(X,\mu))$.

For $G$ a discrete groupoid acting on $X$, we can also define a family of
representations $\\{\lambda_{x}\\}_{x\in X}$ on the orbit Hilbert spaces
$\ell^{2}(G\cdot x)$ such that

$\lambda_{x}(g)\xi(y)=\xi(g^{-1}\cdot y)$

for $\xi\in\ell^{2}(G\cdot x)$. Note that $y=h\cdot x$ where $d(h)=\omega(x)$
and $g^{-1}\cdot y=(g^{-1}h)\cdot x\in G\cdot x$ since
$d(g^{-1})=\omega(h\cdot x)=t(h)$.

If $\mu$ is a measure on $X$, by taking the integral
$\displaystyle\mathcal{H}=\int_{X}\ell^{2}(G\cdot x)d\mu(x)$, we get a
representation $\lambda:G\to\mathcal{L}(\mathcal{H})$ which can be extended to
a representation of $C_{c}(G)$. The relationship between the norm closure
$C^{*}(\lambda)$ of $\lambda(C_{c}(G))$ in $\mathcal{L}(\mathcal{H})$, the
above $C^{*}$-algebra $C^{*}(\kappa)$ obtained from the Koopman representation
and the reduced $C^{*}$-algebra $C^{*}_{r}(G)$ remains to be explored. Recall
that for an étale groupoid, $C^{*}_{r}(G)$ is defined as the completion of
$C_{c}(G)$ in the norm $\|a\|=\sup_{u\in G^{(0)}}\|\pi_{\lambda}^{u}(a)\|$
where

$\pi_{\lambda}^{u}(a)\xi(g)=\sum_{r(h)=r(g)}a(h)\xi(h^{-1}g),$

for $a\in C_{c}(G)$ and $\xi\in\ell^{2}(s^{-1}(u))$.

There are other interesting $C^{*}$-algebras associated to a groupoid action,
like $C^{*}(G\ltimes X)\cong C_{0}(X)\rtimes G$. In the case of a self-similar
groupoid action $(G,E)$, the Cuntz-Pimsner algebra $C^{*}(G,E)$ is constructed
from a $C^{*}(G)$-correspondence or from an étale groupoid $\mathcal{G}(G,E)$,
see [2] for details.

##  3\. Self-similar actions

Let $E=(E^{0},E^{1},r,s)$ be a finite directed graph with no sources. For
$k\geq 2$, define the set of paths of length $k$ in $E$ as

$E^{k}=\\{e_{1}e_{2}\cdots e_{k}:e_{i}\in E^{1},\;r(e_{i+1})=s(e_{i})\\}.$

The maps $r,s$ are naturally extended to $E^{k}$ by taking

$r(e_{1}e_{2}\cdots e_{k})=r(e_{1}),\;\;s(e_{1}e_{2}\cdots e_{k})=s(e_{k}).$

We denote by $E^{*}:=\bigcup_{k\geq 0}E^{k}$ the space of finite paths
(including vertices) and by $E^{\infty}$ the infinite path space of $E$ with
the usual topology given by the cylinder sets $Z(\alpha)=\\{\alpha\xi:\xi\in
E^{\infty}\\}$ for $\alpha\in E^{*}$. Both spaces $E^{*}$ and $E^{\infty}$ are
fibered over $E^{0}$ using the range map. We know that $E^{\infty}$ is a
Cantor space precisely when $E$ satisfies condition (L), i.e. every cycle has
an entrance.

We can visualize the set $E^{*}$ as indexing the vertices of a union of rooted
trees or forest $T_{E}$ given by $T_{E}^{0}=E^{*}$ and with edges

$T_{E}^{1}=\\{(\alpha,\alpha e):\alpha\in E^{*},e\in
E^{1}\;\text{and}\;s(\alpha)=r(e)\\}.$

######  Example 3.1.

For the graph

$\scriptstyle u$$\scriptstyle v$$\scriptstyle w$$\scriptstyle
e_{2}$$\scriptstyle e_{3}$$\scriptstyle e_{4}$$\scriptstyle
e_{5}$$\scriptstyle e_{6}$$\scriptstyle e_{1}$

the forest $T_{E}$ looks like

$\scriptstyle u$$\scriptstyle v$$\scriptstyle w$$\scriptstyle
e_{1}$$\scriptstyle e_{3}$$\scriptstyle e_{1}e_{1}$$\vdots$$\scriptstyle
e_{1}e_{3}$$\vdots$$\scriptstyle e_{3}e_{2}$$\vdots$$\scriptstyle
e_{3}e_{6}$$\vdots$$\scriptstyle e_{2}$$\scriptstyle e_{6}$$\scriptstyle
e_{2}e_{1}$$\vdots$$\scriptstyle e_{2}e_{3}$$\vdots$$\scriptstyle
e_{6}e_{4}$$\vdots$$\scriptstyle e_{6}e_{5}$$\vdots$$\scriptstyle
e_{4}$$\scriptstyle e_{5}$$\scriptstyle e_{4}e_{2}$$\vdots$$\scriptstyle
e_{4}e_{6}$$\vdots$$\scriptstyle e_{5}e_{2}$$\vdots$$\scriptstyle
e_{5}e_{6}$$\vdots$

Note that the level $n$ of $T_{E}$ has $|E^{n}|$ vertices.

Recall that a partial isomorphism of the forest $T_{E}$ corresponding to a
given directed graph $E$ consists of a pair $(v,w)\in E^{0}\times E^{0}$ and a
bijection $g:vE^{*}\to wE^{*}$ such that

  * •

$g|_{vE^{k}}:vE^{k}\to wE^{k}$ is bijective for all $k\geq 1$.

  * •

$g(\xi e)\in g(\xi)E^{1}$ for $\xi\in vE^{*}$ and $e\in E^{1}$ with
$r(e)=s(\xi)$.

Here $vE^{k}$ denotes the set of paths $\xi\in E^{k}$ with $r(\xi)=v$ and
similarly for $vE^{*}$. The set of partial isomorphisms of $T_{E}$ with the
usual operations forms a discrete groupoid PIso$(T_{E})$ with unit space
$E^{0}$. The identity morphisms are $id_{v}:vE^{*}\to vE^{*}$, the inverse of
$g:vE^{*}\to wE^{*}$ is $g^{-1}:wE^{*}\to vE^{*}$, and the multiplication is
composition. We often identify $v\in E^{0}$ with $id_{v}\in$ PIso$(T_{E})$.

In Example 3.1, the sum of the entries in each row of the graph adjacency
matrix

$A_{E}=\left(\begin{array}[]{ccc}1&1&0\\\ 1&0&1\\\ 0&2&0\end{array}\right)$

is the same, so the set of edges ending at each vertex has the same
cardinality and the corresponding trees are isomorphic. In general it could
happen that for some vertices $v,w$ there is no bijection $g:vE^{*}\to wE^{*}$
as above. In particular, PIso$(T_{E})$ could be a group bundle, see Example
3.2. If $E$ has a single vertex, then PIso$(T_{E})=$Aut$(T_{E})$ is a discrete
group.

######  Example 3.2.

Let $E$ be the graph

$\scriptstyle u$$\scriptstyle v$$\scriptstyle e_{3}$$\scriptstyle
e_{1}$$\scriptstyle e_{2}$$\scriptstyle e_{4}$$\scriptstyle e_{5}$

with $E^{0}=\\{u,v\\},E^{1}=\\{e_{1},e_{2},e_{3},e_{4},e_{5}\\}$ and forest
$T_{E}$

$\scriptstyle u$$\scriptstyle v$$\scriptstyle e_{1}$$\scriptstyle
e_{2}$$\scriptstyle e_{1}e_{1}$$\vdots$$\scriptstyle
e_{1}e_{2}$$\vdots$$\scriptstyle e_{2}e_{2}$$\vdots$$\scriptstyle
e_{2}e_{1}$$\vdots$$\scriptstyle e_{3}$$\scriptstyle e_{4}$$\scriptstyle
e_{5}$$\scriptstyle e_{3}e_{1}$$\vdots$$\scriptstyle
e_{3}e_{2}$$\vdots$$\scriptstyle e_{4}e_{3}$$\vdots$$\scriptstyle
e_{4}e_{4}$$\vdots$$\scriptstyle e_{4}e_{5}$$\vdots$$\scriptstyle
e_{5}e_{3}$$\vdots$$\scriptstyle e_{5}e_{4}$$\vdots$$\scriptstyle
e_{5}e_{5}$$\vdots$

Obviously, the trees $uE^{*}$ and $vE^{*}$ are not isomorphic.

We are interested in self-similar actions of groupoids on the path space of
directed graphs $E$ as introduced and studied in [5].

######  Definition 3.3.

Let $E$ be a finite directed graph with no sources, and let $G$ be a discrete
groupoid with unit space $G^{(0)}=E^{0}$. A self-similar action $(G,E)$ of $G$
on the path space of $E$ is given by a faithful homomorphism $G\to$
PIso$(T_{E})$ such that for every $g\in G$ and $e\in d(g)E^{1}$ there exists a
unique $h\in G$ denoted also by $g|_{e}$ and called the restriction of $g$ to
$e$ such that

$g\cdot(e\xi)=(g\cdot e)(h\cdot\xi)\;\;\text{for all}\;\;\xi\in s(e)E^{*}.$

######  Remark 3.4.

We have

$d(g|_{e})=s(e),\;t(g|_{e})=s(g\cdot e)=g|_{e}\cdot s(e),\;r(g\cdot e)=g\cdot
r(e).$

This can be visualized as

$e$$g\cdot e$$r(e)=d(g)$$g\cdot r(e)$$g$$s(e)$$s(g\cdot e)$$g|_{e}$

In particular, in general $s(g\cdot e)\neq g\cdot s(e)$, i.e. the source map
is not $G$-equivariant as it is assumed for group actions in [4]. It is shown
in Appendix A of [5] that a self-similar group action $(H,E)$ as in [4]
determines a self-similar groupoid action $(H\times E^{0},E)$ as in Definition
3.3, where $H\times E^{0}$ is the action groupoid of the group $H$. But not
any self-similar groupoid action comes from a self-similar group action.

It is possible that $g|_{e}=g$ for all $e\in d(g)E^{1}$, in which case

$g\cdot(e_{1}e_{2}\cdots e_{n})=(g\cdot e_{1})\cdots(g\cdot e_{n}).$

######  Remark 3.5.

A self-similar groupoid action $(G,E)$ determines an action of $G$ on the
graph $T_{E}$, in the sense that $G$ acts on both the vertex space $T_{E}^{0}$
and the edge space $T_{E}^{1}$ which are fibered over $G^{(0)}=E^{0}$ and
intertwines the range and the source maps of $T_{E}$, see Definition 4.1 in
[3]. A self-similar groupoid action extends to an action of $G$ on
$E^{\infty}$ such that $g\cdot\xi=\eta$ if for all $n$ we have
$g\cdot(\xi_{1}\cdots\xi_{n})=\eta_{1}\cdots\eta_{n}$.

The faithfulness condition ensures that for each $g\in G$ and $\xi\in E^{*}$
with $d(g)=r(\xi)$, there is a unique element $g|_{\xi}\in G$ satisfying

$g\cdot(\xi\eta)=(g\cdot\xi)(g|_{\xi}\cdot\eta)\;\text{ for all}\;\eta\in
s(\xi)E^{*}.$

By Proposition 3.6 of [5], self-similar groupoid actions have the following
properties:

(1) $g|_{\xi\eta}=(g|_{\xi})|_{\eta}$

(2) $id_{r(\xi)}|_{\xi}=id_{s(\xi)}$

(3) if $(h,g)\in G^{(2)}$, then $(h|_{g\cdot\xi},g|_{\xi})\in G^{(2)}$ and
$(hg)|_{\xi}=(h|_{g\cdot\xi})(g|_{\xi})$

(4) $g^{-1}|_{\xi}=(g|_{g^{-1}\cdot\xi})^{-1}$

for $g,h\in G,\xi\in d(g)E^{*}$, and $\eta\in s(\xi)E^{*}$.

######  Definition 3.6.

A self-similar action $(G,E)$ is said to be level transitive if the induced
action on $E^{*}$ is transitive on each $E^{n}$. It follows that the action is
level transitive iff it is minimal on the infinite path space $E^{\infty}$.

######  Example 3.7.

Let $E$ be the graph from Example 3.1

$\scriptstyle u$$\scriptstyle v$$\scriptstyle w$$\scriptstyle
e_{2}$$\scriptstyle e_{3}$$\scriptstyle e_{4}$$\scriptstyle
e_{5}$$\scriptstyle e_{6}$$\scriptstyle e_{1}$

with $E^{0}=\\{u,v,w\\},E^{1}=\\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\\}$.
Consider the groupoid $G$ with unit space $G^{(0)}=\\{u,v,w\\}$ and generators
$a,b,c$ such that $d(a)=u,\;t(a)=v=d(b)=t(c),\;t(b)=w=d(c)$ as in the picture

$\scriptstyle u$$\scriptstyle v$$\scriptstyle w$$\scriptstyle a$$\scriptstyle
b$$\scriptstyle c$

Define the self-similar action $(G,E)$ given by

$a\cdot e_{1}=e_{2},\;\;a|_{e_{1}}=u,\;\;a\cdot e_{3}=e_{6},\;\;a|_{e_{3}}=b,$
$b\cdot e_{2}=e_{5},\;\;b|_{e_{2}}=a,\;\;b\cdot e_{6}=e_{4},\;\;b|_{e_{6}}=c,$
$c\cdot e_{4}=e_{2},\;\;c|_{e_{4}}=a^{-1},\;\;c\cdot
e_{5}=e_{6},\;\;c|_{e_{5}}=b.$

Note that the action of $G$ is level transitive. It can be shown (see [2])
that $G$ is a transitive groupoid with isotropy $\mathbb{Z}$ and therefore
$C^{*}(G)\cong C_{r}^{*}(G)\cong M_{3}\otimes C(\mathbb{T})$.

######  Example 3.8.

Let $E$ be the graph

$\scriptstyle u$$\scriptstyle v$$\scriptstyle e_{3}$$\scriptstyle
e_{1}$$\scriptstyle e_{2}$$\scriptstyle e_{4}$

with $E^{0}=\\{u,v\\},E^{1}=\\{e_{1},e_{2},e_{3},e_{4}\\}$ and forest $T_{E}$

$\scriptstyle u$$\scriptstyle v$$\scriptstyle e_{1}$$\scriptstyle
e_{2}$$\scriptstyle e_{1}e_{1}$$\vdots$$\scriptstyle
e_{1}e_{2}$$\vdots$$\scriptstyle e_{2}e_{2}$$\vdots$$\scriptstyle
e_{2}e_{1}$$\vdots$$\scriptstyle e_{3}$$\scriptstyle e_{4}$$\scriptstyle
e_{3}e_{1}$$\vdots$$\scriptstyle e_{3}e_{2}$$\vdots$$\scriptstyle
e_{4}e_{3}$$\vdots$$\scriptstyle e_{4}e_{4}$$\vdots$

Consider the groupoid $G=\langle a,b,c\rangle$ with
$d(a)=d(b)=d(c)=u=t(a)=t(b)$ and $t(c)=v$ as in the picture

$\scriptstyle u$$\scriptstyle v$$\scriptstyle c$$\scriptstyle a$$\scriptstyle
b$

Define the self-similar action $(G,E)$ by

$a\cdot e_{1}=e_{2},\;a|_{e_{1}}=b,\;a\cdot e_{2}=e_{1},\;a|_{e_{2}}=a,$
$b\cdot e_{1}=e_{1},\;b|_{e_{1}}=b,\;b\cdot e_{2}=e_{2},\;b|_{e_{2}}=a,$
$c\cdot e_{1}=e_{3},\;c|_{e_{1}}=a,\;c\cdot e_{2}=e_{4},\;c|_{e_{2}}=c.$

Then the action is level transitive and $G_{u}^{u}=\langle a,b\rangle$ is
isomorphic to the lamplighter group
$L=(\mathbb{Z}_{2})^{\mathbb{Z}}\rtimes\mathbb{Z}$, where the action is
translation. It can be shown that the groupoid $G$ is transitive and hence
$C^{*}(G)\cong M_{2}\otimes C^{*}(L)$.

######  Example 3.9.

Let $E$ be the graph from Example 3.2 with $E^{0}=\\{u,v\\}$ and
$E^{1}=\\{e_{1},e_{2},e_{3},e_{4},e_{5}\\}$. The groupoid $G=\langle
a,b\rangle$ where $d(a)=t(a)=u,d(b)=t(b)=v$ acts on $E^{*}$ by

$a\cdot e_{1}=e_{2},\;a|_{e_{1}}=u,\;a\cdot e_{2}=e_{1},\;a|_{e_{2}}=a,$
$b\cdot e_{3}=e_{3},\;b|_{e_{3}}=a,\;b\cdot
e_{4}=e_{5},\;b|_{e_{4}}=v,\;b\cdot e_{5}=e_{4},\;b|_{e_{5}}=b.$

This action is not level transitive. In fact, $G$ is a group bundle with
$G_{u}^{u}\cong G_{v}^{v}\cong\mathbb{Z}$, so $C^{*}(G)\cong
C(\mathbb{T})\oplus C(\mathbb{T})$.

##  4\. The $C^{*}$-algebra of the Koopman representation

In this section, we study the $C^{*}$-algebra $C^{*}(\kappa)$ associated to
the Koopman representation of a level transitive self-similar action $(G,E)$
for an invariant probability measure $\nu$ on $E^{\infty}$. We assume that
$|uE^{1}|=p\geq 2$ is constant for all $u\in E^{0}$, such that all trees
$uE^{*}$ are isomorphic. In particular, $E^{\infty}$ is the union of Cantor
sets $uE^{\infty}$ for $u\in E^{0}$ and it has a uniform probability measure
$\nu$. The measure $\nu$ is obtained as the direct product of uniform measures
on each $uE^{1}$ such that $\displaystyle\nu(uE^{\infty})=\frac{1}{|E^{0}|}$.
Since the action is level transitive, the measure $\nu$ is $G$-invariant.

Recall that a trace on a $C^{*}$-algebra $A$ is a linear functional
$\tau:A\to\mathbb{C}$ such that $\tau(x^{*}x)\geq 0$ and $\tau(xy)=\tau(yx)$
for all $x,y\in A$. A trace is faithful if $\tau(x^{*}x)=0$ implies $x=0$ and
is normalized if $A$ is unital and $\tau(1)=1$.

Given an étale groupoid $G$ with compact unit space, a regular Borel measure
$\mu$ on $G^{(0)}$ is said to be $G$-invariant if for any bisection $U$ we
have $\mu(t(U))=\mu(d(U))$. Such a regular, $G$-invariant Borel probability
measure $\mu$ on $G^{(0)}$ determines a trace on $C_{r}^{*}(G)$ and on
$C^{*}(G)$ via the formula

$\tau(f)=\int_{G^{(0)}}f(u)d\mu(u)$

for $f\in C_{c}(G)$.

For a discrete groupoid $G$, there is a representation $\iota$ of $G$ on
$\ell^{2}(G^{(0)})$ in which $\iota_{g}$ is the matrix unit $e_{t(g),d(g)}$.
This is the analogue of the trivial representation of a group. The
corresponding homomorphism $\pi:C^{*}(G)\to\mathcal{L}(\ell^{2}(G^{(0)}))$
takes elements of $C_{c}(G)$ to finite rank operators. When $G^{(0)}$ is
finite, $\pi$ takes values in $M_{|G^{(0)}|}(\mathbb{C})$ and composing with
the normalized trace, we obtain a tracial state on $C^{*}(G)$. If $G$ is not
transitive, then $\pi$ is not onto, and the normalized traces on simple
summands will give other tracial states on $C^{*}(G)$.

Traces on $C^{*}(G)$ can also be obtained from traces on the isotropy groups,
see section 7 in [5].

######  Definition 4.1.

A $C^{*}$-algebra $A$ is called residually finite-dimensional (RFD) if there
exist $*$-homomorphisms $\pi_{n}:A\to\mathbb{M}_{k(n)}(\mathbb{C})$ into
matrix algebras which determine a faithful map $\displaystyle A\to\prod_{n\geq
1}\mathbb{M}_{k(n)}(\mathbb{C})$. In other words, $A$ is RFD if it has a
separating family of finite dimensional representations.

Given an embedding $\displaystyle A\to\prod_{k\geq
1}\mathbb{M}_{k}(\mathbb{C})$ of an RFD $C^{*}$-algebra $A$, if $\tau_{k}$ is
the standard normalized trace on $\mathbb{M}_{k}(\mathbb{C})$, then the trace

$\tau:\prod_{k\geq
1}\mathbb{M}_{k}(\mathbb{C})\to\mathbb{C},\;\;\tau(a_{1},a_{2},...)=\sum_{k=1}^{\infty}2^{-n}\tau_{k}(a_{k})$

is faithful and its restriction to $A$ yields a faithful trace. The
coefficients $2^{-n}$ can be changed to possibly obtain other traces on $A$.

######  Theorem 4.2.

Let $(G,E)$ be a level transitive self-similar action such that
$|uE^{1}|=p\geq 2$ is constant for all $u\in E^{0}$, and let $\nu$ be the
uniform probability measure on $X=E^{\infty}$. Then the $C^{*}$-algebra
$C^{*}(\kappa)$ of the Koopman representation of $G$ on $L^{2}(X,\nu)$ is RFD
and it has a normalized trace $\tau_{0}$.

###### Proof.

The Hilbert space $\mathcal{H}=L^{2}(E^{\infty},\nu)$ is generated by the
characteristic functions $\chi_{Z(\alpha)}$ of the cylinder sets
$Z(\alpha)=\\{\alpha\xi:\xi\in E^{\infty}\\}$ for $\alpha\in E^{*}$. Recall
that the Koopman representation of $G$ is given by

$\kappa_{g}:L^{2}(d(g)E^{\infty},\nu)\to
L^{2}(t(g)E^{\infty},\nu),\;\kappa_{g}(f)(x)=f(g^{-1}\cdot x)$

and it follows that $\kappa_{g}(\chi_{Z(\alpha)})=\chi_{Z(g\cdot\alpha)}$.

For $n\geq 0$ let $\mathcal{H}_{n}$ be the span of all $\chi_{Z(\alpha)}$ with
$|\alpha|=n$. Then $\mathcal{H}_{n}$ is finite dimensional with dimension
$|E^{0}|p^{n}$. We have an embedding
$j_{n}:\mathcal{H}_{n}\to\mathcal{H}_{n+1}$ given by

$\chi_{Z(\alpha_{1}\cdots\alpha_{n})}\mapsto\sum_{r(e)=s(\alpha_{n})}\chi_{Z(\alpha_{1}\cdots\alpha_{n}e)}.$

If $\mathcal{H}_{n}^{\prime}$ denotes the orthogonal complement of
$\mathcal{H}_{n-1}$ in $\mathcal{H}_{n}$ for $n\geq 1$, then
$\dim\mathcal{H}_{n}^{\prime}=|E^{0}|p^{n-1}(p-1)=q_{n}$ and

$\mathcal{H}=L^{2}(E^{\infty},\nu)=\mathcal{H}_{0}\oplus\bigoplus_{n\geq
1}\mathcal{H}_{n}^{\prime}$

with $\dim\mathcal{H}_{0}=|E^{0}|=q_{0}$. Since each of the partitions
$\mathcal{P}_{n}$ of $E^{\infty}$ into cylinder sets $Z(\alpha)$ with
$|\alpha|=n$ is $G$-invariant and the spaces $\mathcal{H}_{n}$ and
$\mathcal{H}_{n}^{\prime}$ are $G$-invariant, it follows that the Koopman
representation $\kappa:G\to\mathcal{L}(L^{2}(X,\nu))$ is a direct sum of
finite dimensional representations $\pi_{n}^{\prime}$ and $C^{*}(\kappa)$
embedds into $\displaystyle\prod_{n\geq 0}\mathbb{M}_{q_{n}}$, so it is
residually finite-dimensional. Note that $\pi_{n}=\kappa|_{\mathcal{H}_{n}}$
is isomorphic to the representation $\sigma_{n}$ on $\ell^{2}(E^{n})$ induced
by the action of the groupoid on the $n$-th level $E^{n}$ of the forest
$T_{E}$.

If we write $x\in C_{c}(G)$ as $x=(x_{0},x_{1},...)$ where
$x_{n}\in\mathbb{M}_{q_{n}}$ for $n\geq 0$ and

$[x]_{n}=\left[\begin{array}[]{ccccc}x_{0}&0&\cdots&0\\\ 0&x_{1}&\cdots&0\\\
\vdots&\vdots&\cdots&\vdots\\\ 0&0&\cdots&x_{n}\end{array}\right],$

then we define

$\tau_{0}(x)=\lim_{n\to\infty}\frac{1}{|E^{0}|p^{n}}\text{Tr}[x]_{n},$

where Tr is the sum of the diagonal entries in a matrix. Obviously,
$\tau_{0}(1)=1$.

To see that the limit exists, for $g\in G$ we denote by $[g]_{n}$ the matrix
of order $|E^{0}|p^{n}$ corresponding to the operator $\pi_{n}(g)$, in other
words a partial isometry matrix with entries $0$ or $1$. If we represent
$[g]_{n}$ as a block matrix of order $|E^{0}|p^{n-1}$ with diagonal blocks
$g_{ii}$ of order $p$, since $[g]_{n}$ is obtained from the matrix $[g]_{n-1}$
by replacing the zero matrix elements with zero matrices of order $p$ and by
replacing the entries equal to $1$ with the corresponding partial isometry
matrix of order $p$ which has normalized trace $\leq 1$, we get

$\frac{1}{|E^{0}|p^{n}}\text{Tr}[g]_{n}=\frac{1}{|E^{0}|p^{n-1}}\sum_{i=1}^{|E^{0}|p^{n-1}}\frac{1}{p}\text{Tr}(g_{ii})\leq\frac{1}{|E^{0}|p^{n-1}}\text{Tr}[g]_{n-1},$

so the limit defining $\tau_{0}$ exists for $g\in G$ and for any $x\in
C_{c}(G)$ as well.

The property $\tau_{0}(xy)=\tau_{0}(yx)$ follows from the fact that
Tr$(AB)=$Tr$(BA)$. For $x\in C_{c}(G)$ self-adjoint we have
$\|x\|_{\kappa}=\sup_{n\geq 0}\|x_{n}\|$ for $x=(x_{0},x_{1},...)$, where on
the right hand side we have the operator norm of $\mathbb{M}_{q_{n}}$. We
obtain

$\tau_{0}(x)=\lim_{n\to\infty}\frac{1}{|E^{0}|p^{n}}\text{Tr}[x]_{n}=$
$=\lim_{n\to\infty}\left[\frac{1}{p^{n}}\frac{\text{Tr}(x_{0})}{q_{0}}+\frac{p-1}{p^{n}}\frac{\text{Tr}(x_{1})}{q_{1}}+\frac{p-1}{p^{n-1}}\frac{\text{Tr}(x_{2})}{q_{2}}+\cdots+\frac{p-1}{p}\frac{\text{Tr}(x_{n})}{q_{n}}\right]\leq$
$\leq\lim_{n\to\infty}\left[\left(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots\right)\sup_{1\leq
i\leq n}\frac{\text{Tr}(x_{i})}{q_{i}}\right]\leq 2\sup_{n}\|x_{n}\|,$

since Tr$(B)\leq m\|B\|$ for a self-adjoint $m\times m$ matrix $B$. We
conclude that the linear functional $\tau_{0}$ can be extended to the whole
algebra $C^{*}(\kappa)$, the norm closure of $C_{c}(G)$ in
$\mathcal{L}(\mathcal{H})$, since any element in $C_{c}(G)$ is a linear
combination of two self-adjoint elements.

∎

######  Example 4.3.

Let’s determine the trace $\tau_{0}$ on $C^{*}(\kappa)$, the $C^{*}$-algebra
of the Koopman representation on $L^{2}(E^{\infty},\nu)$ for the self-similar
groupoid action $(G,E)$ in Example 3.7. Recall that $|E^{0}|=3$ and $p=2$, so
$\dim\mathcal{H}_{n}=3\cdot 2^{n}$. We get

$\pi_{0}(a)=\left[\begin{array}[]{ccc}0&0&0\\\ 1&0&0\\\
0&0&0\end{array}\right],\;\pi_{0}(b)=\left[\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\
0&1&0\end{array}\right],\;\pi_{0}(c)=\left[\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\ 0&0&0\end{array}\right].$

The $6\times 6$ matrix $\pi_{1}(a)$ is obtained by replacing the $21$ entry
$1$ of $\pi_{0}(a)$ by the $2\times 2$ matrix
$\displaystyle\left[\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right]$ and the
$0$ entries by $2\times 2$ zero matrices. The $6\times 6$ matrix $\pi_{1}(b)$
replaces the $32$ entry $1$ of $\pi_{0}(b)$ by the $2\times 2$ matrix
$\displaystyle\left[\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right]$ and the
$0$ entries by $2\times 2$ zero matrices. The $6\times 6$ matrix $\pi_{1}(c)$
replaces the $23$ entry $1$ of $\pi_{0}(b)$ by the $2\times 2$ matrix
$\displaystyle\left[\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right]$ and the
$0$ entries by $2\times 2$ zero matrices. Iterating this process, it is clear
that since the diagonal entries of the $3\cdot 2^{n}\times 3\cdot 2^{n}$
matrices $\pi_{n}(a),\pi_{n}(b)$ and $\pi_{n}(c)$ are zero, we get

$\tau_{0}(a)=\tau_{0}(b)=\tau_{0}(c)=0.$

Note that $\displaystyle\tau_{0}(u)=\tau_{0}(v)=\tau_{0}(w)=\frac{1}{3}$.

##  5\. Self-similar representations

Consider a self-similar groupoid action $(G,E)$ such that $|uE^{1}|=p\geq 2$.
Every $g\in G$ determines a bijection $\sigma_{g}:d(g)E^{1}\to t(g)E^{1}$
given by the action $e\mapsto g\cdot e$ for $e\in d(g)E^{1}$, and an element
in $G^{d(g)E^{1}}$ equal to the function $\varphi_{g}:e\mapsto g|_{e}$ from
$d(g)E^{1}$ to $G$, identified with a $p$-tuple of elements in $G$. Let
$\displaystyle\mathfrak{S}(E^{1})$ be the set of bijections $uE^{1}\to vE^{1}$
for $u,v\in E^{0}$, which has a groupoid structure with unit space $E^{0}$.
Note that $\mathfrak{S}(E^{1})$ acts on $G^{p}$. The associated wreath
recursion for $(G,E)$ is the groupoid homomorphism

$\phi:G\to\mathfrak{S}(E^{1})\ltimes
G^{p},\;\phi(g)=(\sigma_{g},\varphi_{g}).$

Indeed, we have

$(\sigma_{g},\varphi_{g})(\sigma_{h},\varphi_{h})=(\sigma_{gh},\varphi_{gh})$

for $(g,h)\in G^{(2)}$. Recall that $(gh)|_{e}=(g|_{h\cdot e})(h|_{e})$.

######  Definition 5.1.

A $p$-fold similarity of an infinite dimensional Hilbert space $\mathcal{H}$
is an isomorphism
$\psi:\mathcal{H}\to\mathcal{H}^{p}=\underbrace{\mathcal{H}\oplus\cdots\oplus\mathcal{H}}_{p}$.

######  Example 5.2.

For a level transitive self-similar groupoid action $(G,E)$ such that
$|uE^{1}|=p\geq 2$, let $\nu$ be the invariant probability measure on
$E^{\infty}$. The Hilbert spaces $L^{2}(uE^{\infty},\nu)$ are decomposed into
direct sums

$L^{2}(uE^{\infty},\nu)=\bigoplus_{e\in uE^{1}}L^{2}(eE^{\infty},\nu)$

and the spaces $L^{2}(eE^{\infty},\nu)$ are naturaly isomorphic to
$L^{2}(s(e)E^{\infty},\nu)$ via

$V_{e}:L^{2}(eE^{\infty},\nu)\to
L^{2}(s(e)E^{\infty},\nu),\;V_{e}(f)(\xi)=\frac{1}{\sqrt{p}}f(e\xi).$

We can view $V_{e}$ as partial isometries of $L^{2}(E^{\infty},\nu)$ and then
$\displaystyle\sum_{e\in E^{1}}V_{e}$ is a $p$-fold similarity of the Hilbert
space $L^{2}(E^{\infty},\nu)$.

######  Remark 5.3.

Recall that the graph $C^{*}$-algebra $C^{*}(E)$ is generated by partial
isometries $S_{e}$ for $e\in E^{1}$ and projections $P_{u}$ for $u\in E^{0}$
such that

$S_{e}^{*}S_{e}=P_{s(e)},\;\sum_{r(e)=u}S_{e}S_{e}^{*}=P_{u}.$

Given a representation $\pi:C^{*}(E)\to\mathcal{L}(\mathcal{H})$ on a Hilbert
space $\mathcal{H}$, since $\displaystyle\sum_{u\in E^{0}}P_{u}=I$, we get a
decomposition $\displaystyle\mathcal{H}=\bigoplus_{u\in E^{0}}\mathcal{H}_{u}$
with $\mathcal{H}_{u}=\pi(P_{u})\mathcal{H}$ and since $|uE^{1}|=p$ is
constant, we get a $p$-fold similarity $\mathcal{H}_{u}\to\mathcal{H}_{u}^{p}$
given by
$\xi\mapsto(\pi(S^{*}_{e_{1}})(\xi),\pi(S^{*}_{e_{2}})(\xi),...,\pi(S^{*}_{e_{p}})(\xi))$
for $uE^{1}=\\{e_{1},e_{2},...,e_{p}\\}$. Conversely, a decomposition
$\displaystyle\mathcal{H}=\bigoplus_{u\in E^{0}}\mathcal{H}_{u}$ and a
$p$-fold similarity $\psi:\mathcal{H}_{u}\to\mathcal{H}_{u}^{p}$ for each
$u\in E^{0}$ determines a representation of $C^{*}(E)$ by
$\pi(S_{e_{j}})(\xi)=\psi^{-1}(0,...,\xi,...,0)$ with $\xi$ in position $j$
for $e_{j}\in uE^{1}$.

######  Example 5.4.

The representation $\pi$ of $C^{*}(E)$ associated with the $p$-fold similarity
of $\displaystyle L^{2}(E^{\infty},\nu)=\bigoplus_{u\in
E^{0}}L^{2}(uE^{\infty},\nu)$ is generated by the partial isometries
$\pi(S_{e})=T_{e}$ given by

$T_{e}(f)(\xi)=\begin{cases}\sqrt{p}f(\xi^{\prime})&\text{if}\;\xi=e\xi^{\prime}\\\
0\;&\text{otherwise}\end{cases}$

for $f\in L^{2}(E^{\infty},\nu)$.

######  Definition 5.5.

Let $(G,E)$ be a self-similar groupoid action. If $\mathcal{H}$ is a Hilbert
space bundle over $E^{0}$ such that each fiber $\mathcal{H}_{u}$ for $u\in
E^{0}$ has a $p$-fold similarity and $\pi:C^{*}(E)\to\mathcal{L}(\mathcal{H})$
is the associated representation of the graph algebra with generators $S_{e}$,
then a representation $\rho$ of $G$ on $\mathcal{H}$ is called self-similar if
$\rho(g)\pi(S_{e})=\pi(S_{g\cdot e})\rho(g|_{e})$.

######  Example 5.6.

Let $E$ be a finite graph with $|uE^{1}|=p\geq 2$ constant for each $u\in
E^{0}$. If the groupoid $G$ acts on $\displaystyle E^{*}=\bigcup_{u\in
E^{0}}uE^{*}$ and the induced action on $E^{\infty}$ preserves the measure
$\nu$, then we get a representation $\rho$ of $G$ on $L^{2}(E^{\infty},\nu)$
which is self-similar.

######  Proposition 5.7.

Let $(G,E)$ be a self-similar action as above, and let
$\pi:C^{*}(E)\to\mathcal{L}(\mathcal{H})$ be a representation associated to
the similarity of a Hilbert bundle $\mathcal{H}$. A representation $\rho$ of
$G$ on $\mathcal{H}$ is self-similar if and only if $\rho$ and $\pi$ generate
a representation of the Cuntz-Pimsner algebra
$C^{*}(G,E)=C^{*}(\mathcal{G}(G,E))$. Hence self-similar representations of
$G$ are precisely restrictions onto $G$ of representations of $C^{*}(G,E)$.

###### Proof.

Given a self-similar groupoid action $(G,E)$, recall that the $C^{*}$-algebra
$C^{*}(G,E)$ is defined as the Cuntz-Pimsner algebra of the
$C^{*}$-correspondence

$\mathcal{M}=\mathcal{M}(G,E)=\mathcal{X}(E)\otimes_{C(E^{0})}C^{*}(G)$

over $C^{*}(G)$. Here $\mathcal{X}(E)=C(E^{1})$ is the $C^{*}$-correspondence
over $C(E^{0})$ associated to the graph $E$ and $C(E^{0})=C(G^{(0)})\subseteq
C^{*}(G)$. The right action of $C^{*}(G)$ on $\mathcal{M}$ is the usual one
and the left action is determined by the representation

$W:G\to\mathcal{L}(\mathcal{M}),\;\;W_{g}(i_{e}\otimes
a)=\begin{cases}i_{g\cdot e}\otimes i_{g|_{e}}a\;\;\text{if}\;d(g)=r(e)\\\
0\;\;\text{otherwise,}\end{cases}$

where $i_{e}\in C(E^{1})$ and $i_{g}\in C_{c}(G)$ are point masses for $e\in
E^{1},g\in G$ and $a\in C^{*}(G)$. The inner product of $\mathcal{M}$ is given
by

$\langle\xi\otimes a,\eta\otimes
b\rangle=\langle\langle\eta,\xi\rangle\;a,b\rangle=a^{*}\langle\xi,\eta\rangle\;b$

for $\xi,\eta\in C(E^{1})$ and $a,b\in C^{*}(G)$.

The Cuntz-Pimsner algebra $C^{*}(G,E)$ is generated by $U_{g},P_{v}$ and
$S_{e}$, the images of $g\in G,v\in E^{0}=G^{(0)}$ and of $e\in E^{1}$ which
satisfy

(1) $g\mapsto U_{g}$ is a representation by partial isometries of $G$ with
$U_{v}=P_{v}$ for $v\in E^{0}$;

(2) $S_{e}$ are partial isometries with $S_{e}^{*}S_{e}=P_{s(e)}$ and
$\displaystyle\sum_{r(e)=v}S_{e}S_{e}^{*}=P_{v}$;

(3) $U_{g}S_{e}=\begin{cases}S_{g\cdot e}U_{g|_{e}}\;\mbox{if}\;d(g)=r(e)\\\
0,\;\mbox{otherwise;}\end{cases}$

(4) $U_{g}P_{v}=\begin{cases}P_{g\cdot v}U_{g}\;\mbox{if}\;d(g)=v\\\
0,\;\mbox{otherwise.}\;\end{cases}$

A representation $\pi:C^{*}(E)\to\mathcal{L}(\mathcal{H})$ and a
representation $\rho$ of $G$ on $\mathcal{H}$ will extend to a representation
of the Cuntz-Pimsner algebra $C^{*}(G,E)=C^{*}(\mathcal{G}(G,E))$ if and only
if $\rho(g)\pi(S_{e})=\pi(S_{g\cdot e})\rho(g|_{e})$.

∎

######  Remark 5.8.

Let $(G,E)$ be a self-similar action with $|uE^{1}|=p$. If
$\rho:G\to\mathcal{L}(\mathcal{H})$ is a self-similar representation of $G$ on
a Hilbert bundle, then with respect to the decomposition
$\psi:\mathcal{H}\to\mathcal{H}^{p}$, each $\rho(g)$ has a $p\times p$ matrix
$\rho(g)=(A_{yx})$ for $x\in d(g)E^{1}$ and $y\in t(g)E^{1}$, where

$A_{yx}=\begin{cases}\rho(g|_{x})&\text{if}\;g\cdot x=y\\\
0&\text{otherwise.}\end{cases}$

We have a homomorphism $\phi:C_{c}(G)\to\mathbb{M}_{p}(C_{c}(G))$ of the
algebra $C_{c}(G)$ called matrix recursion, which is the linear extension of
$\phi(g)=(A_{yx})$ for $x\in d(g)E^{1}$ and $y\in t(g)E^{1}$, where

$A_{yx}=\begin{cases}g|_{x}&\text{if}\;g\cdot x=y\\\
0&\text{otherwise.}\end{cases}$

######  Example 5.9.

For the self-similar action $(G,E)$ of the groupoid $G=\langle a,b,c\rangle$
in Example 3.7 we have

$\sigma_{a}:\\{e_{1},e_{3}\\}\to\\{e_{2},e_{6}\\},\;\sigma_{b}:\\{e_{2},e_{6}\\}\to\\{e_{4},e_{5}\\},\;\sigma_{c}:\\{e_{4},e_{5}\\}\to\\{e_{2},e_{6}\\},$
$\varphi_{a}=(u,b),\;\varphi_{b}=(a,c),\;\varphi_{c}=(a^{-1},b)$

and $\phi:C_{c}(G)\to M_{2}(C_{c}(G))$ is given by

$\phi(a)=\left[\begin{array}[]{cc}u&0\\\
0&b\end{array}\right],\;\phi(b)=\left[\begin{array}[]{cc}0&a\\\
c&0\end{array}\right],\;\phi(c)=\left[\begin{array}[]{cc}a^{-1}&0\\\
0&b\end{array}\right].$

## References

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  * [3] V. Deaconu, Groupoid actions on $C^{*}$-correspondences, New York J. of Math. 24 (2018), 1020–1038.
  * [4] R. Exel, E. Pardo, Self-Similar graphs: a unified treatment of Katsura and Nekrashevych algebras, Adv. Math. 306(2017), 1046–1129.
  * [5] M. Laca, I. Raeburn, J. Ramagge, M. F. Whittaker, Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs, Adv. Math. 331 (2018), 268–325.
  * [6] R. I. Grigorchuk Some problems of the dynamics of group actions on rooted trees. (Russian) Tr. Mat. Inst. Steklova 273 (2011), Sovremennye Problemy Matematiki, 7–191 translation in Proc. Steklov Inst. Math. 273 (2011), no. 1, 64–175
  * [7] R. I. Grigorchuk and V. Nekrashevych, Self-similar groups, operator algebras and Schur complement, J. of Modern Dyn. vol. 1, no. 3 (2007), 323–370.
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  * [9] V. Nekrashevych, Cuntz-Pimsner algebras of group actions, J. Operator Theory, 52 (2004), 223–249.
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  * [12] Williams, Dana P. A tool kit for groupoid $C^{*}$-algebras. Mathematical Surveys and Monographs, 241. American Mathematical Society, Providence, RI, 2019. xv+398 pp.


# Shape Matters: Detecting Vertebral Fractures Using Differentiable Point-
Based Shape Decoding

Hellena Hempe
Institute of Medical Informatics
University of Luebeck
Luebeck, Germany
<EMAIL_ADDRESS>
& Alexander Bigalke
Institute of Medical Informatics
University of Luebeck
Luebeck, Germany
<EMAIL_ADDRESS>
& Mattias P. Heinrich
Institute of Medical Informatics
University of Luebeck
Luebeck, Germany
<EMAIL_ADDRESS>
Alexander Bigalke is also affiliated with Dräger, Drägerwerk AG & Co. KGaA

###### Abstract

Degenerative spinal pathologies are highly prevalent among the elderly
population. Timely diagnosis of osteoporotic fractures and other degenerative
deformities facilitates proactive measures to mitigate the risk of severe back
pain and disability. Objective: In this study, we specifically explore the use
of shape auto-encoders for vertebrae, taking advantage of advancements in
automated multi-label segmentation and the availability of large datasets for
unsupervised learning. Methods: Our shape auto-encoders are trained on a large
set of vertebrae surface patches, leveraging the vast amount of available data
for vertebra segmentation. This addresses the label scarcity problem faced
when learning shape information of vertebrae from image intensities. Based on
the learned shape features we train an MLP to detect vertebral body fractures.
Results: Using segmentation masks that were automatically generated using the
TotalSegmentator, our proposed method achieves an AUC of 0.901 on the VerSe19
testset. This outperforms image-based and surface-based end-to-end trained
models. Additionally, our results demonstrate that pre-training the models in
an unsupervised manner enhances geometric methods like PointNet and DGCNN.
Conclusion: Our findings emphasise the advantages of explicitly learning shape
features for diagnosing osteoporotic vertebrae fractures. This approach
improves the reliability of classification results and reduces the need for
annotated labels. Significance: This study provides novel insights into the
effectiveness of various encoder-decoder models for shape analysis of
vertebrae and proposes a new decoder architecture: the point-based shape
decoder. The source code will be made available on GitHub:
https://github.com/multimodallearning/shape_matters

_Keywords_ Auto-encoder $\cdot$ Computer-aided diagnosis $\cdot$ Deep Learning
$\cdot$ Shape analysis $\cdot$ Spine $\cdot$ Vertebral body fractures

##  1 Introduction

Degenerative pathology of the spine such as osteoporotic fractures of
vertebral bodies, spinal canal stenosis, spondylolisthesis and other
degenerative deformations of vertebrae, are a common healthcare risk in the
elderly population [1]. Early detection of degenerative diseases may enable
preventative measures to reduce the risk of chronic severe back pain and
disability [2]. In recent years, deep learning has emerged as a powerful tool
for many medical applications. Advances in deep learning in computer-aided
diagnosis have paved the way for more timely interventions and improved
patient outcomes. In this study, we specifically focus on the intersection of
deep learning and the diagnosis of vertebral body fractures.

So far, most deep learning-based approaches for classification of vertebral
body fractures are trained on intensity images in an end-to-end manner. Even
after the VerSe dataset/benchmark for segmentation and localisation of
vertebrae was introduced in ’19 and ’20 [3, 4, 5], most introduced methods
rely on the localisation of vertebrae instead of leveraging the available
segmentation masks [6, 7, 8]. The amount of available ground truth
segmentation masks of vertebrae was further increased by the TotalSegmentator
dataset [9, 10]. To date, only few recently proposed methods [11, 12] employ
the information provided by segmentation masks in diagnostic downstream tasks.

We identified three relevant challenges that are not sufficiently addressed by
existing methods and have so far prevented a wider clinical adoption of
vertebral CAD. a) Image-based classifiers are prone to deterioration under
domain shifts, i.e. they are limited in their adaptability for variations of
the image intensity, scanner acquisition settings and other shifts in the data
distribution. Furthermore, 3D CNNs trained in a fully-supervised setting tend
to overfit and require a larger amount of labelled data of sufficient quality.
b) Surface-based geometric learning models [13, 14] have so far been less
expressive than 3D CNNs and may not achieve the required accuracy on limited
labelled data. c) Shape encoder-decoder architectures [11, 12] may help to
overcome the label scarcity by shifting a larger proportion of the training to
an unsupervised setting for which even healthy spines can be employed, but
they may still fail to learn representative latent space representations, due
to an over-pronounced weight of the decoder (that is later discarded for the
classification task).

As a remedy, we make the following suggestions: a) Instead of training image-
based classifiers we propose to leverage information from a preceding
segmentation model and directly operate on shape information of vertebrae
(surface contour) for the classification task. This allows trained deep
learning models to be independent of shifts in image intensities and moves the
demand for labelled data away from the classification task. b) Leveraging the
vast amount of available segmentation masks, unsupervised learning can help
geometric learning models overcome problems related to limited labelled data.
c) To ensure a more representative latent space representation, we introduce a
novel decoder architecture that ensures that most learned parameters are
located in the encoder model and thus relevant for the classification task.

Contributions:  1) We believe that the effectiveness of recently proposed AE
models is limited due to the sub-optimal design of encoder-decoder components.
Therefore, we perform an in-depth analysis of the effectiveness of various AE
architectures for the diagnosis of vertebral body fractures. 2) For this
purpose, we develop a modular encoder-decoder framework which allows arbitrary
combinations of point- and image-based encoder and decoder architectures. 3)
To address the problem of over-pronounced weights in the decoder, we designed
a novel point-based shape decoder model that employs a differentiable sampling
step to bridge the gap between point- and voxel-based solutions. 4) By
performing extensive experiments to analyse various combinations of encoder
and decoder architectures we verified that the detection of vertebral
fractures, which by definition is a geometric problem, is better solved using
shape-based methods compared to image-intensity-based. 5) The results of our
experiments furthermore demonstrate the particular advantages of employing our
novel point-based shape decoder compared to other models.

Outline: In the subsequent section 2 we provide a comprehensive review of
related literature, emphasising the existing disparity between image- and
point-based analyses. Following this, our comprehensive shape-based
classification framework 3.1, incorporating the proposed point-based shape
decoder 3.2, is meticulously detailed in the methods section 3. The
experiments and results section 4 elaborates on our setup, encompassing data
particulars and implementation details 4.1. Afterwards, the results of our
experiments are presented in 4.2 and 4.3, followed by a thorough discussion of
the outcomes 5 and a concise summary of our findings 6.

##  2 Related Works

Radiologists commonly assess osteoporotic fractures using the Genant score
[15]. The Genant Score is a semiquantitative measurement by which a fracture
is defined by the height reduction of the vertebral body (in lateral X-Ray)
and categorised into 0-healthy, 1-mildly fractured, 2-moderately fractured and
3-severely fractured. In 3D-CT scans this is a bit more complicated as an
optimal slice for measuring the height difference needs to be determined.
However, automated detection of vertebral fractures is often performed on CT
scans especially on quantitative CT scans for measurements of bone mineral
density which is directly linked to osteoporosis [16].

When categorising recently proposed methods for computer-aided detection of
vertebral compression fractures, we can observe that most recently proposed
methods operate directly on the CT intensity data e.g. [6, 7, 17]. The method
proposed by Nicolaes et al. [6] suggests a two-step approach which detects
vertebral fractures in CT scans using a voxel-based CNN to perform voxel-level
classification on patches containing vertebrae. Yilmaz et al. [7] present a
pipeline in which the vertebrae are firstly detected using a hierarchical
neural network and secondly apply a CNN on the identified patches that are
extracted from the original CT scan. The Grading Loss as proposed by Husseini
et al. [17], addresses the problem of severe data imbalance and minor
differences in appearance between healthy and fractured vertebrae in CT scans
by exploring a representation learning-based approach that was inspired by
Genant grades. These methods do not exploit the shape information provided by
the segmentation mask. Consequently, the shape of the vertebrae and the
associated fracture status need to be learned implicitly, which leads to a
complication of the learning process.

As a remedy, two related previous works aim to explicitly use the shape of
vertebrae. Firstly, the Spine-VAE [11] employs the masked image data of
vertebrae as input to a conditioned VAE to capture the large variations of
healthy and fractured vertebra shapes. The conditioning is performed by
concatenating information about the label of the corresponding to the vertebra
(T1-L5 split into 5 groups) as one-hot-encoded vector. After training the VAE
until the reconstruction loss converges, the encoder model is further in
conjunction with a Multi-Layer Perceptron to classify vertebral body
fractures. Secondly, the purely geometric approach proposed by Sekuboyina et
al. [12] employs a surface point-based probabilistic AE to learn shape
representations of vertebrae. The task of detecting vertebral fractures is
then treated as out-of-distribution detection task by computing the
reconstruction error based on a model that was only trained on healthy
vertebrae subjects. Both methods rely on accurate segmentation masks of
vertebrae during test-time and do not involve unsupervised pre-training on a
larger separate dataset to learn shape features.

In many computer vision applications, geometric deep learning methods play a
crucial role in extracting meaningful features from 3D data by capturing
spatial relationships for shape analysis. Notably, two state-of-the-art
techniques in this domain are PointNet [13] and DGCNN (Dynamic Graph
Convolutional Neural Network)[18]. PointNet stands out for its ability to
process unstructured point clouds directly, showcasing robust performance in
tasks like shape recognition and segmentation. DGCNN, on the other hand,
utilises dynamic graph structures to capture both local and global geometric
relationships within point cloud data, enhancing the model’s capacity to
discern intricate patterns. These geometric deep learning approaches have
significantly advanced the accuracy and efficiency of medical image analysis,
particularly in scenarios where a nuanced understanding of spatial
relationships and complex structures within volumetric data is essential.

Figure 1: To determine the most suitable architecture for our task we employ
combinations of several encoder-decoder architectures including traditional
convolutional methods and geometric methods. The AEs are trained to
reconstruct either a point-cloud representation or a volumetric surface
representation of vertebrae, which are derived from the previously computed
segmentation mask. As Shape encoder (A) we employ a convolutional method, as
well as a point-based and a graph-based method to predict the embedding
$\bm{z}$. As Shape decoder (B) we employ a convolutional method as well as a
point-based method and propose a novel point-based shape decoder. The Shape
classifier (C) is then trained separately on the embedding $\bm{z}$ for each
encoder-decoder combination using the same MLP-model. Note that only the
weights of the MLP are trained in a supervised manner, whereas the weights of
the encoder are fixed.

Geometric learning approaches are already being applied in medical imaging
tasks, e.g. assessment for spinal curvature in X-Ray [19] and detection of
pedicles for spinal surgery in CT scans [20]. However, detecting vertebral
fractures in 3D CT scans allows for automated opportunistic screening in CT
scans that were taken for other purposes [21]. Furthermore, the required
landmarks for the measurement of vertebral height in 2D lie on the surface of
the vertebral body in 3D, which is why we believe that geometric learning
approaches are beneficial for learning the important shape features from the
contour of vertebrae.

##  3 Methods

To investigate the potential of shape features for detecting vertebral
fractures, we experiment with several encoder-decoder architectures and deploy
their latent space vector for classification. In particular, we compare
combinations of encoder and decoder trained to reconstruct shape features of
vertebrae based on the surface of their respective segmentation mask. The
training of our AE architecture comparison pipeline is split into two stages.
In the first stage, we train our AE models in an unsupervised manner on a
large-scale dataset without classification labels or a specific high
occurrence of spine diseases to generate meaningful shape features in the
latent space. During the second stage, we employ the generated shape features
of the encoder (freezing the encoder layers) and train an MLP for detection of
fractures based on these shape features on a smaller labelled dataset.

The problem setup can be defined as follows: Based on multi-label segmentation
masks of vertebrae performed on a CT Scan, extract patches
$\mathcal{S}_{in}\in\mathbb{R}^{D\times H\times W}$ of the segmentation mask
contour that is computed using an edge-filter. For geometric learning methods,
we extract a point cloud representation
$\mathcal{P}_{in}\in\mathbb{R}^{3\times N}$ of $N$ 3D points by employing a
threshold on the grid. Firstly, train an AE-model comprised of a shape encoder
$f$ and a shape decoder $g$ on an unsupervised reconstruction task by
minimising the smoothL1-Loss of the Chamfer-Loss between the input ${S}_{in}$
or ${P}_{in}$ and the prediction ${S}_{out}$ or ${P}_{out}$. Secondly, using
the latent space vector $\bm{z}\in\mathbb{R}^{M}$ computed by the shape
encoder $f$ as input, train an MLP as classifier $h$ to predict the fracture
status (healthy or fractured) by minimising the Cross-Entropy loss function
between the target class ${c}_{target}$ and the predicted class ${c}_{pred}$.
As encoders we employ 3 approaches, a simple convolutional-encoder
${f}_{conv}$, a point-encoder ${f}_{point}$ based on the PointNet architecture
[13] and a Graph encoder ${f}_{graph}$ based on the DGCNN [18]. As decoders,
we employ 3 approaches, a convolutional decoder ${g}_{conv}$, our a point-
decoder ${g}_{point}$ and a novel point-based shape decoder ${g}_{pbShape}$
The main idea of the general framework for the employed image- or point-based
vertebrae auto-encoder (AE) framework is depicted in Fig. 1 Section A. Our
newly proposed decoder architecture is visualised in Fig. 2 Section B.

###  3.1 Architectural building blocks

$\color[rgb]{0.1328125,0.54296875,0.1328125}{\mdblksquare}$ Convolutional-
encoder: Given a 3D surface patch $\mathcal{S}_{in}\in\mathbb{R}^{D\times
H\times W}$ of depth $\bm{D}$, height $\bm{H}$ and width $\bm{W}$, extracted
from an automatic multi-label 3D CT segmentation [9, 10] that is separated
into binary individual vertebrae contours. We aim to encode the vertebral
shape into a low-dimensional embedding $\bm{z}\in\mathbb{R}^{M}$ with
$\bm{M=2048}$ using a fully-convolutional encoder network
$f_{conv}(\mathcal{S}_{in})$ parameterised by trainable weights
$\bm{w}_{f_{conv}}$.

$\color[rgb]{0.0,0.0,0.5}{\mdblksquare}$ Point-encoder:  Based on the 3D
surface patches $\mathcal{S}_{in}$, we extract a 3D point cloud
$\mathcal{P}_{in}\in\mathbb{R}^{3\times N}$ where $\bm{N}$ corresponds to the
number of key-points in the point cloud that are sampled from voxel-
representation by applying a threshold on the values in the voxel grid. Using
a PointNet [13] model $f_{point}(\mathcal{P}_{in})$ with weights
$\bm{w}_{f_{point}}$, we generate a low-dimensional embedding
$\bm{z}\in\mathbb{R}^{M}$ with $\bm{M=2048}$.

$\color[rgb]{1.0,0.0,0.0}{\mdblksquare}$ Graph-encoder:  As for the point-
encoder, the graph-encoder utilises an extracted 3D surface point cloud
$\mathcal{P}_{in}\in\mathbb{R}^{3\times N}$ for each individual vertebra. We
employ a DGCNN [18] $f_{graph,k}(\mathcal{P}_{in})$ with parameter $\bm{k}$
for the k-Nearest Neighbour (kNN) graphs and weights $\bm{w}_{f_{graph}}$ to
compute an embedding $\bm{z}\in\mathbb{R}^{M}$ with $\bm{M=2048}$.

$\color[rgb]{0.1845703125,0.3095703125,0.3095703125}{\mdblksquare}$
Convolutional-decoder:  After generating the embedding $\bm{z}$ a
convolutional decoder $g_{conv}(\bm{z})$ with weights $\bm{w}_{g_{conv}}$ is
used to map $\bm{z}$ back into $\mathcal{S_{out}}\in\mathbb{R}^{D\times
H\times W}$. During training $\mathcal{S}_{out}$ is used to minimise a smooth
L1-Loss function $\mathcal{L}_{smoothL1}$ to reconstruct $\mathcal{S}_{in}$.
The utilised convolutional decoder model is based on PixelShuffle layers.

$\color[rgb]{0.1171875,0.5625,1.0}{\mdblksquare}$ Point-decoder:  The aim of
the point-decoder $g_{point}(\bm{z})$ is to map $\bm{z}$ back to a 3D Point
Cloud representation $\mathcal{P}_{out}\in\mathbb{R}^{3\times N}$ where
$\bm{N}$ corresponds to the number of points in $\mathcal{P}_{out}$. Similar
to the PixelShuffle layers, this is done by subsequently transferring from
network channels $\bm{C}$ into the spatial dimension $\bm{N}$. In the last
step $\bm{C}$ is set to 3 to predict $\mathcal{P^{\prime}}$. During training,
we minimise a Loss function based on the Chamfer distance
$\mathcal{L}_{chamfer}$ between $\mathcal{P}_{in}$ and $\mathcal{P}_{out}$

$\color[rgb]{1.0,0.647,0.0}{\mdblksquare}$ Point-based shape decoder (pbShape-
decoder):  Our point-based shape decoder $g_{pb-shape}(\bm{z})$ can be
described as a combination of point decoder and convolutional decoder. In a
first step, $\bm{z}$ is mapped to an embedding in shape of key-points
$\mathcal{z}_{point}\in\mathbb{R}^{3\times N}$ using a a 3-layer MLP
$g_{point-mlp}(\bm{z})$. In a second step, a differentiable extrapolation
$g_{sampler}(\bm{z})_{point}$ is performed to map back into a volumetric
representation $\mathcal{S}_{out}\mathbb{R}^{D\times H\times W}$. This allows
us to then minimise a smooth L1-Loss function $\mathcal{L}_{smoothL1}$ to
reconstruct $\mathcal{S}_{in}$ from $\mathcal{S}_{out}$. In contrast to the
convolutional-decoder and the point-decoder, our point-based shape-decoder
relies on a considerably smaller amount of trainable parameters. A more
detailed description of our proposed method is provided in the next section.

MLP:  After training of the encoder-decoder models was performed in an
unsupervised manner on a large dataset, a small MLP $h(\bm{z})$ is trained on
a smaller dataset with fracture labels to predict a binary fracture status
from the embedding $\bm{z}$. For this step, the parameters of the previously
trained encoder $\bm{w}_{f}$ are fixed, and only the weights of the MLP
$\bm{w}_{c}$ are optimised using a Cross-Entropy Loss function
$\mathcal{L}_{BCE}$.

Data augmentation:  To improve the generalisability of our models, we
introduce affine augmentation on our input data. For encoder models that take
a volumetric patch as input, the augmentation is performed before the forward
pass during training on each newly drawn batch. For our geometric models, we
also apply the affine augmentation on the volumetric patch after drawing the
batch and sample a random set of key-points from the augmented volumetric
patch using a threshold.

Table 1: Assessment of architectural building blocks employed within our framework including the number of trainable parameters, total size of the model and computational demands in Mult-Add operations.  | #Parameter | Total size (MB) | Mult-Adds
---|---|---|---
Shape encoder: |  |  |
$\color[rgb]{0.1328125,0.54296875,0.1328125}{\mdlgblksquare}$ Conv | 6.9M | 69.36 | 5.56G
$\color[rgb]{0.0,0.0,0.5}{\mdlgblksquare}$ Point | 2.8M | 34.1 | 1.39G
$\color[rgb]{1.0,0.0,0.0}{\mdlgblksquare}$ Graph | 4.5M | 678.92 | 7.96G
Shape decoder: |  |  |
$\color[rgb]{0.1845703125,0.3095703125,0.3095703125}{\mdlgblksquare}$ Conv | 155k | 24.88 | 1.37G
$\color[rgb]{0.1171875,0.5625,1.0}{\mdlgblksquare}$ Point | 62k | 0.59 | 0.9M
$\color[rgb]{1.0,0.647,0.0}{\mdlgblksquare}$ Pb- Shape | 62k | 0.66 | 0.68M
Shape Classifier: |  |  |
MLP | 541k | 2.18 | 0.54M

We provide details about the capacity and computational complexity of our AE
models in Table 1. Our proposed pbShape-decoder requires 600k fewer
computations and has only 62k trainable parameters compared to its
convolutional counterpart (155k). During the AE-model training, the encoders
and decoders are combined, whereas, during training of the MLP classification,
only the MLP parameters are adapted. During the test time of the
classification, the encoder and MLP parameters are employed.

###  3.2 Point-based shape decoder

Normally, the decoder $g(\bm{z})$ would follow an inverse architecture
comprising transposed 3D convolutions or up-sampling layers and introduce a
similar number of additional trainable weights $\bm{w}_{g}$ as the encoder
$\bm{w}_{f}$. This makes the latent space less constrained as information
about the shape can also be ”encoded" in decoder layers. Consequently, we
propose a completely new strategy: As opposed to conventional AEs, we map the
latent space to represent geometric 3D point coordinates using a small MLP
$g_{point-mlp}(\bm{z})$ to compute a point embedding
$\mathcal{z}_{point}\in\mathbb{R}^{3\times N}$, which is then used as input to
a differentiable sampler $g_{sampler}(\bm{z}_{point})$to reconstruct the
original input $\mathcal{S}_{in}$. The trainable weights in $w_{g}$ are
limited to $w_{g_{point-mlp}}$ as $w_{g_{sampler}}\rightarrow\varnothing$
(parameters $y$ are fixed after training). The structure of our proposed
decoder is displayed in Fig. 2.

Figure 2: Point-based Shape decoder:  From the embedding vector $\bm{z}$ a
point-representation of $N$ key-points is computed using an MLP. The layers
each consist of a 1D-Convolution with the channel size denoted by white font
within the blocks, InstanceNorm and ReLU. The number on top of the blocks
denotes the size of the spatial dimension. Afterwards, a differentiable
sampling operation is applied on the key-points to obtain a volumetric
representation. This step requires $N$ additional parameters $\bm{y}$.

We define $g_{sampler}$ as a differentiable geometric operation that
extrapolates a 3D volume from a small number keypoint coordinate and value
pairs. Recall that spatial transformer networks [22] perform differentiable
interpolation of regular 2D/3D feature maps (or images) given a grid of
coordinates that is predicted by a localiser network to obtain sampled values.
Each resampled value on an off-grid location depends on 4 (2D) or 8 (3D)
neighbouring integer locations that are used within bi-/trilinear
interpolation. Here, following [23] we employ the reverse operation in that
spatial coordinates $\bm{x}\in\mathbb{R}^{3}$ and values $\bm{y}\in\mathbb{R}$
are used as input to extrapolate a 3D grid using their respective trilinear
extrapolation coefficients.

The sampling operation $g_{sampler}$ is implemented as a reverse grid sampling
operation in PyTorch, and we compute the derivative for both sampling values
$\bm{y}$ and coordinates $\mathcal{z}_{point}$ to make this step
differentiable. To mitigate noise and ensure a smooth output, we apply a cubic
spline kernel to spatially filter the extrapolation results. Note that the
values $\bm{y}$ attached to each key-point are treated as additional trainable
parameters, which are shared across all training samples and can be learned
restricted within a range of 0 to 1 with a sigmoid. This enables our method to
”paint" a detailed surface, as a zero value placed close to one effectively
erases or overwrites parts of a thicker line.

Furthermore, we hypothesise that only $\bm{N}$ representative (key-point)
coordinates $\bm{x}$ are required to reconstruct a detailed vertebral surface
and that those key-points can be efficiently predicted from the input using
$f$ and $g_{point-mlp}$. Hence, our latent space $\bm{z}_{point}$ now
represents a geometric entity that reflects a compact representation of the
vertebral shape. This is useful for understanding the impact of osteoporotic
fractures on vertebral geometry and can hence lead to improved classification
accuracy. By keeping the number of trainable parameters $w_{g_{point-mlp}}$
small, we ensure that these advantageous properties of $\bm{z}_{point}$ are
mirrored in $\bm{z}$.

##  4 Experiments and Results

To explore the efficacy of utilising unsupervised pre-training of vertebral
shapes for detecting vertebral body fractures, we conducted two experiments.
In the initial experiment, we assessed and compared the classification
outcomes of various AE architectures and end-to-end trained models. We also
scrutinised the adaptability of segmentation masks generated through deep
learning for subsequent diagnostic tasks. The second experiment aimed to
determine the robustness of the AE models by executing a data-hold-out
experiment on the labelled dataset, specifically during the supervised
training phase of the MLP. Subsequently, we outline the prerequisites for our
experiments and delve into the implementation details of both the unsupervised
pre-training stage for our AE models and the supervised training phase for the
MLP. Additionally, we provide implementation details concerning the models
subjected to end-to-end training.

###  4.1 Datasets and implementation details

To conduct our experiments as outlined in Fig.1, we utilise the following two
public datasets:

TotalSegmentator dataset [9, 10]: The dataset contains 1204 CT scans/patients
of different parts of the body and corresponding segmentation masks of 104
volumes of interest including 24 vertebrae (C1-L5). From this dataset, we
extracted patches of nearly 13k vertebrae surface masks, which are employed
for unsupervised training of our AEs. Using our best classifier model we
estimate that 2.9% are fractured.

VerSe19 dataset [3, 4, 24]111(https://osf.io/923ap/, Ethics approval: TUM
Proposal 27/19 S-SR, Licence: CC BY-SA 4.0): The dataset is split into
80/40/40 for training, validation and test set and originally serves as a
benchmark for the vertebrae segmentation and localisation task. Since fracture
labels are also provided for the dataset, we use them to train and test our
vertebral fracture classification. The amounts of extracted vertebrae are
770/385/380 respectively. We have also determined that when excluding mild
fractures (grade 1) [15], approximately 10% of vertebrae of each subset are
fractured.

Unsupervised Pre-Training: The training of our AE models is performed using
the same training regime and hyper-parameters for all encoder-decoder
combinations, to ensure comparability. Each model is trained for 15000
iterations using a batch size of 64. The models are optimised by deploying the
Adam optimiser with an initial learning rate of 0.005 and CosineAnnealing
scheduler with two warm restarts. During training, we augment our data by
applying affine transformations. For geometric approaches, we have fixed the
number of points to 3840, as we have found that the shape of vertebrae
depicted in our given patch size can be accurately represented with this
amount of key-points. For the DGCNN (graph-encoder) model, we have fixed the
hyper-parameter for graph neighbours to $k=20$.

Supervised Training: For our MLP-Classifiers, the training is conducted for
6000 iterations using batch-size 16. We again use the Adam optimiser with an
initial learning rate of 0.0001 and CosineAnnealing learning rate scheduling
with warm restarts. To enhance the diversity of our input data to the
classification model, we augment the input patch to the encoder using affine
transformations and we also keep the parameters of the encoder fixed. For
geometric encoder models, we utilise the augmented input to construct a
randomly sampled point cloud of the vertebral surface. During test time, this
point cloud is sub-sampled by employing farthest point sampling to determine
3840 surface points instead of drawing random points from an intensity
threshold. While we use the ground truth (GT) segmentation masks for training
our classifier models, we employ automatically generated segmentation masks
using the TotalSegmentator model [9, 10] at test time.

End-to-End Training: In addition to our AE architecture comparison, we train
our encoder models from scratch, namely the convolutional encoder, the
PointNet and the DGCNN, in combination with the MLP classifier in an end-to-
end fashion on the Verse19 dataset. Furthermore, the convolutional encoder is
trained on image-patches containing the intensities values as information as
well as on surface patches.

###  4.2 Automated fracture detection pipeline

We evaluate the accuracy of our shape-based vertebral fracture detection
through a binary classification that distinguishes between healthy and
fractured vertebrae. Since mild fractures (Genant grade 1)[15] of the
vertebral bodies are oftentimes indistinct and thus harder to annotate
consistently (higher inter-rater variability), vertebrae with label fracture
grade 1 are left out for this analysis.

Table 2: Median AUC on the VerSe19 test set over 10 seeds | GT-masks | TS-masks
---|---|---
Model |  |
End-to-end |  |
Conv-encoder (img) $\color[rgb]{0.1328125,0.54296875,0.1328125}{\mdlgblksquare}$ | 0.756 | -
Conv-encoder (surf) $\color[rgb]{0.1328125,0.54296875,0.1328125}{\mdlgblksquare}$ | 0.883 | 0.847
Point-encoder $\color[rgb]{0.0,0.0,0.5}{\mdlgblksquare}$ | 0.842 | 0.803
Graph-encoder $\color[rgb]{1.0,0.0,0.0}{\mdlgblksquare}$ | 0.615 | 0.634
Conv-encoder - |  |
Conv-decoder $\color[rgb]{0.1328125,0.54296875,0.1328125}{\mdlgblksquare}$-$\color[rgb]{0.1845703125,0.3095703125,0.3095703125}{\mdlgblksquare}$ | 0.890 | 0.814
Point-decoder $\color[rgb]{0.1328125,0.54296875,0.1328125}{\mdlgblksquare}$-$\color[rgb]{0.1171875,0.5625,1.0}{\mdlgblksquare}$ | 0.704 | 0.712
pbShape-decoder $\color[rgb]{0.1328125,0.54296875,0.1328125}{\mdlgblksquare}$-$\color[rgb]{1.0,0.647,0.0}{\mdlgblksquare}$ | 0.924 | 0.914
Point-encoder - |  |
Conv-decoder $\color[rgb]{0.0,0.0,0.5}{\mdlgblksquare}$-$\color[rgb]{0.1845703125,0.3095703125,0.3095703125}{\mdlgblksquare}$ | 0.930 | 0.882
Point-decoder $\color[rgb]{0.0,0.0,0.5}{\mdlgblksquare}$-$\color[rgb]{0.1171875,0.5625,1.0}{\mdlgblksquare}$ | 0.911 | 0.859
pbShape-decoder $\color[rgb]{0.0,0.0,0.5}{\mdlgblksquare}$-$\color[rgb]{1.0,0.647,0.0}{\mdlgblksquare}$ | 0.937 | 0.901
Graph-encoder - |  |
Conv-decoder $\color[rgb]{1.0,0.0,0.0}{\mdlgblksquare}$-$\color[rgb]{0.1845703125,0.3095703125,0.3095703125}{\mdlgblksquare}$ | 0.871 | 0.825
Point-decoder $\color[rgb]{1.0,0.0,0.0}{\mdlgblksquare}$-$\color[rgb]{0.1171875,0.5625,1.0}{\mdlgblksquare}$ | 0.843 | 0.760
pbShape-decoder $\color[rgb]{1.0,0.0,0.0}{\mdlgblksquare}$-$\color[rgb]{1.0,0.647,0.0}{\mdlgblksquare}$ | 0.920 | 0.856

In Table 2 we show the median Area-Under-Curve (AUC) of fracture detection
results for 10 seeds of training the MLP for various encoder-decoder
architectures and additional end-to-end trained models on the test set. We
report the results both for using ground truth segmentation masks for
preprocessing and for automatically generated segmentation masks using the
TotalSegmentator [9][10].

The results illustrate the benefits of pre-training on shape reconstruction
for the detection of vertebral fractures. Furthermore, the results convey that
explicitly training on the shape of vertebrae improves classification
performance compared to training on image intensities directly. The highest
AUC score in the setup utilising GT segmentation masks to generate the model
input is achieved with the Point-encoder and pbShape decoder architecture,
resulting in an AUC of 0.937. When employing TotalSegmentator (TS)
segmentation masks, the convolutional-encoder in conjunction with the pbShape
decoder attains an AUC of 0.914 Notice that the pbShape-decoder consistently
achieves the highest AUC for each encoder model, highlighting the benefits of
the robustness of our proposed decoder.

When examining the outcomes between AE models and end-to-end trained models,
the AE models consistently produce more accurate and robust classification
results. Especially employing the same convolutional-encoder architecture,
outperforms its image-based end-to-end trained counterpart by more than 12.8%
in AUC. Even when using segmentation masks from TotalSegmentator, there’s a 9%
improvement.

Another noteworthy observation arises when considering realistic segmentation
errors by adapting TS-segmentation labels, especially in models trained with
graph-based encoding a more severe drop in classification performance can be
discerned in contrast to other encoder models.

To investigate the performance of our models in greater detail, we present the
receiver-operator curves (ROC-curves) in Fig. 3. The depicted results are
computed using TS-generated segmentation masks for preprocessing. From left to
right the ROC-curves are provided for convolutional-encoder, point-encoder and
graph-encoder models.

When interpreting the plot for the convolutional encoder, identifying the
threshold with the highest possible sensitivity and specificity would be in
favour of the ConvEnc-pbDec model which also reaches the highest overall AUC
in this setup (AUC=0.914).

For ROC-curves obtained for both point-encoder and graph-encoder models, we
observe that by choosing an optimal threshold either the convolutional decoder
model or the pbShape decoder produces the most accurate classification
outcome.

###  4.3 Data-hold-out experiment

In our data-hold-out experiment, we explore the required amount of supervised
training data for achieving robust classification outcomes. AUC values,
calculated across 10 random seeds for each data split (2%, 5%, 10%, 50%, 75%,
and 100%), are presented as both scatter plots and box plots over the 10 seeds
(see Fig. 4). The experiment demonstrates that the robustness is increased by
adding more training data. Notably, employing just 25% of training data
already yields as accurate and robust classification results as models trained
on the complete training dataset. The increased number of outliers in
experiments involving training data below 25% indicates that specific data
splits offer more advantageous conditions for the classification task, while
others present less favourable circumstances.

Figure 3: ROC-Curve and corresponding AUC for encoder-decoder combinations of
the median AUC of 10 seeds. The encoders are grouped by colour and linestyle,
whereas the decoders are grouped by colour and marker. The corresponding Area
Under Curve (AUC) is listed inside the legend.

Figure 4: Results of our data-hold-out experiment as boxplots and scatterplots
of the AUC obtained for 10 random seeds each. The plots are separated by the
employed encoder architecture, and provide the classification results obtained
with the respective decoder. Top-Left:  Convolutional-encoder Top-Right:
Point-encoder models, Bottom-Left:  Graph-encoder models Bottom-Right:  End-
to-End trained models including the traditional CNN trained on image
intensities instead and trained on vertebra surface (denoted as Conv-encoder
img and surf).

##  5 Discussion

In this work, we examined the benefits of directly utilising shape information
in the diagnosis of vertebral body fractures. We introduced a shape encoding
approach that takes a volumetric surface representation of a vertebra as
input, encoding it into a latent vector, which is decoded by transforming the
latent vector into point representations (using an MLP) and then applying a
novel differentiable point sampling method: the Point-based Shape decoder
(pbShape-decoder).

Within our vertebrae AE framework, we analyse the performance of our proposed
decoder and other AE building blocks that are pre-trained on a large-scale
dataset to reconstruct the shape of an input vertebra surface patch and
additionally employ end-to-end trained models for comparison. Whereas the vast
majority of weights in our AE models (the whole encoder part) are trained in
an unsupervised fashion, only a light-weight MLP classifier using the sparse
shape representation from the latent space requires training with fracture
labels. The findings of our end-to-end trained models indicate that, with
supervised training on surface data, good classification results (AUC above
0.9) can already be obtained using a 3D CNN model and our results demonstrate
that using automatically generated segmentation data yields preferable results
in comparison to training on image-intensity data directly. This shows the
advantage of explicitly employing shape information from large-scale multi-
label segmentation models over implicitly learning shape information in
intensity-based CNNs. Furthermore, employing surface information that was
previously already learned, for instance by the nnUNet in the TotalSegmentator
model[9, 10] enhances the robustness of the classification task against domain
shifts. The results affirm that the quality and robustness of the generated
TotalSegmentator [9, 10] masks are adequate for this task. Furthermore, our
findings highlight that the choice of architecture influences the model’s
robustness in the face of inaccuracies in segmentation masks. In a clinical
setting, obtaining ground truth segmentation masks can be challenging. The
fact that our model achieves an AUC of 0.914, holds significant potential for
automated diagnosis of osteoporotic fractures in the spine with limited
availability of labelled data.

Our findings further suggest that, when trained in an end-to-end fashion,
geometric learning approaches such as PointNet [13] and DGCNN [18] do not
perform as well as 3D-convolutional approaches. However, when integrated as an
encoder into an AE architecture and pre-trained on a large-scale dataset, we
demonstrate their ability to achieve more accurate results. Moreover, our
findings indicate that models employing a PointNet as an encoder produce
accurate and robust results with only a small deviation between the different
decoder models (AUC pbShape: .901 conv: -1.9% and point: -4.2%) with fewer
trainable parameters being incorporated. This underscores the effectiveness
and efficacy of the PointNet architecture. Contrarily, the graph-based
encoder, appears to under-perform on this specific task. This could be
attributed to the nature of the task of detecting vertebral fractures which
involves a more global geometric relationship based on height differences,
rendering local graph structures less relevant. Consequently, capturing finer
geometric features may not contribute significantly to this task but might be
of interest when detecting other degenerative deformations of vertebrae.

By performing a data-hold-out experiment, we highlight the effectiveness and
robustness of our models and find that an AUC above 0.9 can be obtained even
when training on 5% of data (51 healthy and 5 fractured vertebrae), but
robustness increases when training on a larger dataset. The results of this
experiment furthermore show the importance and impact of data distributions
and the prevalence of vertebral fractures for this task increases the
difficulty to compare methods between multiple datasets. We thus consider our
point-encoder point-decoder model similar to the approach proposed as pAE by
Sekuboyina et al. [12], who also utilise a point cloud generated from ground
truth segmentation masks as input into their point-based AE architecture.
However, they treat the detection of vertebral body fracture as an anomaly
detection task by training the reconstruction of the point cloud on healthy
samples only achieving an AUC of 0.759 with the best model on their dataset
using accurate segmentation masks. In comparison, our purely point-based AE
model reaches an AUC of 0.895 (0.911 on ground truth segmentation masks),
setting a new state-of-the-art.

##  6 Conclusion

In summary, our study explores AE architectures for effective shape encoding
of vertebrae to diagnose osteoporotic fractures of vertebrae. We introduce a
novel approach, utilising a differentiable geometrical decoder for
unsupervised pre-training on the TotalSegmentator dataset. By combining
convolutional or point-based encoders with our proposed shape decoder, we
achieve a meaningful and robust shape representation in the latent space,
facilitating the detection of osteoporotic fractures. Our approach
demonstrates robustness through comparisons with ground truth segmentation
masks, showcasing commendable classification results even when applying
automatically generated segmentation masks. Moreover, this approach can be
extended to other shape-related tasks, e.g. diagnosing degenerative
deformations, spinal canal stenosis or lymph nodes.

## Acknowledgment

This work was financially supported by the German Federal Ministry of
Education and Research under Grant: 01EC1908D.

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