# The Euler Characteristic:
A General Topological Descriptor for Complex Data

Alexander Smith¶ and Victor M. Zavala¶
¶Department of Chemical and Biological Engineering
University of Wisconsin-Madison, 1415 Engineering Dr, Madison, WI 53706, USA
Corresponding Author<EMAIL_ADDRESS>
###### Abstract

Datasets are mathematical objects (e.g., point clouds, matrices, graphs,
images, fields/functions) that have shape. This shape encodes important
knowledge about the system under study. Topology is an area of mathematics
that provides diverse tools to characterize the shape of data objects. In this
work, we study a specific tool known as the Euler characteristic (EC). The EC
is a general, low-dimensional, and interpretable descriptor of topological
spaces defined by data objects. We revise the mathematical foundations of the
EC and highlight its connections with statistics, linear algebra, field
theory, and graph theory. We discuss advantages offered by the use of the EC
in the characterization of complex datasets; to do so, we illustrate its use
in different applications of interest in chemical engineering such as process
monitoring, flow cytometry, and microscopy. We show that the EC provides a
descriptor that effectively reduces complex datasets and that this reduction
facilitates tasks such as visualization, regression, classification, and
clustering.

##  1 Introduction

Datasets are mathematical objects (e.g., point clouds, matrices, graphs,
images, field/functions) that have shape [45]. Characterizing the shape
(geometrical features) of these objects reduces the dimensionality and
complexity of the data while minimizing information loss, but is not always
straightforward [34]. Popular tools from statistics, linear algebra, and
signal processing (e.g., moments, correlation functions, singular value
decomposition, convolutions, Fourier analysis) do not directly characterize
geometrical features of data objects; instead, such tools are used to
characterize other types of features (e.g., variance and frequency content).

Topology is a branch of mathematics that provides powerful tools to
characterize the shape of data objects. One such tool is the so-called Euler
characteristic (EC); the EC, originally used for the characterization of
polyhedra [14], is now broadly used in scientific areas such as random fields
[2, 3, 4], cosmology [41, 38, 19], material science [31, 6, 9, 42],
thermodynamics [32, 20, 16], and neuro-science [21, 23]. To the best of our
knowledge, the EC has seen limited applications in engineering and of these
applications most are focused on the characterisation of the permeability of
porous media and modelling geometric states of fluids within porous media [42,
30]. However, a fact that is often overlooked in this literature is that the
EC provides a general descriptor of different types of topological spaces
(this enables the characterization of a much wider range of data objects).
This generality arises from the fact that (i) one can use transformations to
map a data object into another type of object and (ii) the EC has fundamental
connections with statistics, field theory, linear algebra, and graph theory.

In a nutshell, the EC is a descriptor that characterizes geometrical features
of a topological space defined by a data object. This characterization is
accomplished by performing a decomposition of the space into a set of
independent topological bases. This decomposition is similar in spirit to an
eigen-decomposition of a matrix; here, the matrix object is decomposed into a
set of independent basis vectors. The EC is a scalar integer quantity that is
defined as the alternating sum of the rank of the topological bases. The EC is
often combined with a transformation technique known as filtration to
characterize the geometry of different objects such as matrices, images,
fields/functions, and weighted graphs. This characterization is summarized in
the form of what is called an EC curve which provides a direct approach to
quantify the topology of an object.

Topological descriptors such as the EC offer advantages over statistical
descriptors [31]. For instance, statistical descriptors such as Moran’s I,
which measures spatial structure via spatial autocorrelation, or correlation
matrices do not directly capture the global structure of the data (thus
limiting the ability to characterize geometrical features) [39, 22]. High-
order statistical descriptors such as 2-point correlation functions, which
have been employed in characterizing the structure of heterogeneous materials,
are also limited at capturing spatial and morphological features of the data
(especially if the data object is irregular) [18, 29]. However, there exists
well-developed theory that connects the EC to the geometry of random fields
[3, 4]. Such work establishes that the EC encodes information of simple
statistical descriptors such as means and variances and of more complex
descriptors (e.g., space-time covariances) [2]. These connections between
topology and statistics are powerful and provide a mechanism to understand the
emergence of topological features from physical behavior (e.g., diffusion
phenomena). The EC also connects with concepts from linear algebra and graph
theory; for example, a matrix or an image can be represented as a weighted
graph and the geometry of such graph can be quantified using the EC (a graph
is a 2D polyhedron). Establishing these connections is important because data
objects encountered in practice are often complex and require the combined use
of different characterization techniques. For instance, in brain analysis, one
often characterizes a multivariate time series (a collection of time series
obtained from different locations in the brain) by constructing a correlation
matrix. The topology of this matrix is then reduced to an EC curve through a
filtration. The EC curve provides a low-dimensional descriptor that
characterizes the spatio-temporal structure of the brain. It is also important
to emphasize that the EC curve is a much simpler topological descriptor than
the so-called persistence diagrams used in most of the topological data
analysis (TDA) literature [45]. Persistent diagrams contain more topological
information but are more difficult to analyze and interpret.

The aim of this paper is to present an applied perspective on the EC. We
briefly discuss the mathematics of the EC and discuss how to use filtration
operations to characterize diverse data objects. This discussion will
establish connections with field theory, graph theory, and linear algebra. We
then bring our focus to applying these concepts to tackle diverse problems
arising in science and engineering; in particular, we discuss how the EC can
be used in process monitoring by analyzing correlation structures. We also
apply the EC in the analysis of both 2D spatial and 3D spatio-temporal fields;
these data objects are derived from reaction-diffusion partial differential
equations (PDEs), micrographs of liquid crystals, and flow cytometry. In these
examples, we show how to use the EC as a data pre-processing (reduction) step
that can facilitate machine learning tasks. We also provide scripts and
datasets to help the interested reader apply these tools.

##  2 Introduction to the Euler Characteristic

Around 1735, Euler discovered a relationship between the number of vertices,
edges, and faces of a convex polyhedron (which is now known as the EC). The
study and generalization of this formula, specifically by Cauchy and
L’Huilier, is at the origin of topology. The EC is a scalar integer value that
summarizes the shape of a topological space (an object). A topological space
is a set with a structure defined by continuity and connectedness which also
represents the ideas of limits or closeness based on relationships between the
sets of the space rather than a specific distance or metric [35] . Topological
spaces are a central unifying notion that appears in virtually every branch of
modern mathematics (e.g., capture graphs and manifolds). The EC is a
topological invariant quantity (e.g., does not change with deformations such
rotation, streching, bending). For instance, the topology of a graph is fully
defined by its node-edge connectivity (the location of the nodes and edges is
irrelevant); as such, graphs that appear to be visually distinct might have
the same underlying topology (and thus have the same EC value). In this work,
we will introduce the EC from the perspective of data objects that can be
represented as graphs (a 2D polyhedron) and manifolds (e.g., images and
fields/functions). Graphs and manifolds are special types of topological
spaces but, as we will see, these spaces are sufficient to represent a vast
number of data objects encountered in chemical engineering applications such
as the analysis of industrial chemical process sensor data (graph) and soft
material experimental images (manifold) .

(a) Example object (2D shape)

(b) EC of example object ($\chi$)

Figure 1: Illustration on how topology can be quantified via the Euler
characteristic. (a) A 2D shape that has two connected components and three
holes. (b) The EC is an alternating sum of the number of connected components
and holes and thus $\chi=-1$.

We begin with a general definition of the EC (which we denote as
$\chi\in\mathbb{Z}$) for a 2D manifold (see Figure 1). The EC of the manifold
is an alternating sum of the number of $0$-dimensional topological bases
(known as connected components) and the number of $1$-dimensional topological
bases (known as holes):

$\chi=\\#\ \text{Connected Components}-\\#\ \text{Holes}=\beta_{0}-\beta_{1}$
(2.1)

There are only two sets of topological bases (connected components and holes)
that describe a 2D object. The rank of the first basis is known as the $0$-th
Betti number $\beta_{0}\in\mathbb{Z}_{+}$, while the rank of the second basis
is known as the $1$-st Betti number $\beta_{1}\in\mathbb{Z}_{+}$; here,
$\mathbb{Z}_{+}$ denotes the set of all nonnegative integers. The relationship
between the number of topological bases and dimensions holds for
$n$-dimensional shapes; as such, the EC of an ($n+1$)-dimensional shape is
given by:

$\chi=\sum_{i=0}^{n}(-1)^{i}\beta_{i}$ (2.2)

To simplify our notation, we will use the generalized topological
representation of the _Betti numbers_ $\beta_{n}\in\mathbb{Z}_{+}$; here, the
$n$-th Betti number $\beta_{n}$ is the number of unique $n$-dimensional
topological bases for a given shape [2].

###  2.1 EC for Graphs

A graph object $G(V,E)$ is a 2D topological space (a 2D polyhedron). The EC of
this object is given by:

$\chi=|V|-|E|=\beta_{0}-\beta_{1},$ (2.3)

where $|V|\in\mathbb{Z}_{+}$ is the number of graph vertices (nodes) and
$|E|\in\mathbb{Z}_{+}$ is the number of graph edges. We define
$v(e),v^{\prime}(e)\in V$ as the support nodes of edge $e\in E$. One can show
that $|V|-|E|$ equals the number of connected components of the graph
($\beta_{0}$) minus the number of holes (cycles) of the graph ($\beta_{1}$)
[1].

One can use the EC to characterize _edge-weighted graphs_ $G(V,E,w_{E})$,
where each edge $e\in E$ has an associated scalar weight
$w_{E}(e)\in\mathbb{R}$. Similarly, one can use the EC to characterize _node-
weighted graphs_ $G(V,E,w_{V})$, where each node $v\in V$ has an associated
scalar weight $w_{V}(v)\in\mathbb{R}$. This characterization is done via a
process known as a filtration which leads to the creation of a topological
descriptor known as an EC curve. The ability to deal with weighted graphs
enables analysis of data objects that are represented as discrete fields, such
as matrices and images. For example, a correlation matrix (a square and
symmetric matrix) can be represented as an edge-weighted graph in which the
nodes are the random variables, the edges are the connections between
variables, and the weights are the degrees of correlation between pairs of
random variables (Figure 2). A grayscale image can be represented as a node-
weighted graph in which the nodes are pixel locations, the weights are the
intensity of the pixels, and the edges connect adjacent pixels to form a grid
(Figure 3). A 2D discrete field (e.g., obtained from a discretized PDE) can
also be represented as a node-weighted graph; here, the nodes are locations in
the discretization mesh and the weights are values of a variable of interest
at such locations (e.g., temperature). It is important to emphasize that
graphs are topological spaces that do not live in a Euclidean space; as such,
the location of the nodes and edges is irrelevant (topology is fully dictated
by the node-edge connectivity). Thus, the EC and associated EC curve are
focused on the global topology of the graph during a filtration, and not on
specific connectivity information (e.g. the number of edges connected to node
$x$) which it does not track.

To characterize the topology of an edge-weighted graph $G(V,E,w_{E})$, we
perform a filtration of the graph by eliminating edges that have weights less
than or equal to a certain threshold $w_{E}(e)\leq\ell$ (with
$\ell\in\mathbb{R}$). Filtration gives a graph that is sparser than the
original graph and that has an associated EC value. We can repeat the
filtering process by progressively increasing the threshold value $\ell$ and
with this obtain new graphs and associated EC values. One stops the process
once the threshold reaches the largest weight in the original graph
$G(V,E,w_{E})$; this gives the original graph itself and and its associated EC
value. To formalize the filtration process, we define the following
_filtration function_ (see Figure 2).

######  Definition 2.1.

Graph Edge Filtration Function ($f_{E}$): For an undirected edge-weighted
graph $G:=G(V,E,w_{E})\in\mathbb{G}$ with scalar edge weight values
$\\{w_{E}(e)\in\mathbb{R}:e\in E\\}$ the filtration function
$f_{E}:\mathbb{G}\rightarrow\mathbb{R}$ is defined such that
$f_{E}(G)=\max_{e\in E}w_{E}(e)$. The pre-image $f_{E}^{-1}(\ell)$, with
$\ell\in\mathbb{R}$, is given by the graph
$G_{\ell}:=G(V,E_{\ell},w_{E_{\ell}})$ where $E_{\ell}=\\{e\in E\ :\
w_{E}(e)\leq\ell\\}$.

The pre-image of the filtration function is used to create a set of nested
graphs; this is done by defining a sequence of increasing thresholds
$\ell_{1}<\ell_{2}<...<\ell_{m}$ with associated graphs:

$G_{\ell_{1}}\subseteq G_{\ell_{2}}\subseteq...\subseteq G_{\ell_{m}}\subseteq
G$ (2.4)

Here, we note that $G_{\ell_{m}}=G$ if $\ell_{m}=\max_{e\in E}w_{E}(e)$ (the
last graph in the filtration is the original graph). We also re-emphasize that
the density of the graph (its number of edges) increases with the threshold
value; specifically, $G_{\ell_{1}}$ is the graph with lowest density (highest
sparsity) and $G_{\ell_{m}}$ is the graph of highest density (lowest
sparsity).

We can define a similar filtration for node-weighted graphs; here, nodes are
filtered/eliminated based on their weight values. The _filtration function_
for this object (see Figure 3) can be defined as follows.

######  Definition 2.2.

Graph Node Filtration Function ($f_{V}$): For an undirected node-weighted
graph $G:=G(V,E,w_{V})\in\mathbb{G}$ with scalar node weight values
$\\{w_{V}(v)\in\mathbb{R}:v\in V\\}$ the filtration function
$f_{V}:\mathbb{G}\rightarrow\mathbb{R}$ is defined such that
$f_{V}(G)=\max_{v\in V}w_{V}(v)$. The pre-image $f_{V}^{-1}(\ell)$ is given by
the graph $G_{\ell}:=G(V_{\ell},E_{\ell},w_{V_{\ell}})$ where
$V_{\ell}=\\{v\in V\ :\ w_{V}(v)\leq\ell\\}$ and $E_{\ell}=\\{e\in
E:v(e),v^{\prime}(e)\in V_{\ell}\\}$.

As in the edge-weighted case, the pre-image of the node filtration function is
used to create a set of nested graphs:

$G_{\ell_{1}}\subseteq G_{\ell_{2}}\subseteq...\subseteq G_{\ell_{m}}\subseteq
G$ (2.5)

Here, we highlight that node filtration has the effect of eliminating both
nodes and edges; in other words, if a node is eliminated, we also eliminate
its supported edges. We again note that $G_{\ell_{m}}=G$ if
$\ell_{m}=\max_{v\in V}w_{V}(v)$ (the last graph in the filtration is the
original graph).

For each graph $G_{\ell}$ in the node or edge filtration process, we compute
and record its EC value $\chi_{\ell}$ . This information is used to construct
the so-called EC curve, which contains the pairs $(\ell,\chi_{\ell})$. The EC
curve thus provides a topological descriptor for edge- or node-weighted
graphs. Furthermore, it is important to note that the edge and/or node weights
are a critical component of this analysis as this is what allows for the
filtration to be performed. Without this information only a single EC value
can be computed which is ineffective at distinguishing graphs with different
topologies if they have the same ratio of nodes and edges.

(a) Edge weighted graph filtration.

(b) EC Curve for the graph filtration.

Figure 2: (a) Filtration of a correlation matrix, which is represented as an
edge-weighted graph. The random variables are treated as nodes, and the
correlation between the variables are treated as edge weights $w_{E}(e)$. As
the level set of the filtration function $f_{E}$ increases in value, edges
that are below the given threshold are added into the resulting graph
$f^{-1}_{E}$, revealing the topology of the correlation matrix over multiple
correlation thresholds. (b) Resulting EC curve for the filtration of the edge-
weighted graph in (a).

(a) Filtration of node-weighted graph (image).

(b) EC Curve for the graph filtration.

Figure 3: (a) Filtration of a simple grayscale image, which is represented as
a node-weighted graph, with the graph node filtration function $f_{V}$. (b)
Resulting EC curve for the filtration of the grayscale image in (a).

##  3 EC for Manifolds

The EC can also be used for characterizing the geometry of topological spaces
known as manifolds [48, 8, 24]. Specifically, we consider an $n$-dimensional
manifold $M$ that is defined by a chart $(X,f)$, where
$X\subseteq\mathbb{R}^{n}$ is an open subset of $M$ (a Euclidean space) and
the field/function $f:X\to\mathbb{R}$ is a homeomorphism (a one-to-one, onto,
and continuous mapping with an inverse mapping that is also continuous). In
simple terms, the manifolds of interest involve an $n$-dimensional continuous
domain $X$ and field $f:X\to\mathbb{R}$. We will say that field $f$ is
$n$-dimensional if $X\subseteq\mathbb{R}^{n}$ and we say that this field is
embedded in an $(n+1)$-dimensional manifold (because the dimension of the
chart $(X,f)$ is $n+1$). For instance, in Figure 4, we show a 1-dimensional
(1D) field/function $f:X\rightarrow\mathbb{R}$ with domain
$X\subseteq\mathbb{R}$; this field is embedded in a 2D manifold $M$ defined by
the chart $(X,f)$. In our 1D field example, the horizontal coordinate of the
chart is $X$ and the vertical coordinate is $f$. A 1D field arise, for
instance, as the solution of an ordinary differential equation (e.g., the
domain is defined by a time coordinate). Another example of a 1D field is the
probability density function of a univariate random variable. Similarly, a 2D
field $f:X\rightarrow\mathbb{R}$ with $X\subseteq\mathbb{R}^{2}$ is embedded
in a 3D manifold $M$ (given by a 3D chart). Formalizing these definitions is
important in characterizing the topology of fields. Higher dimensional fields
arise, for instance, as solutions of PDEs (e.g., the domain is defined by
space-time coordinates) or can be probability density functions of mutivariate
random variables. These high-dimensional fields often have complex topology
and are notoriously difficult to analyze/characterize.

The chart $(X,f)$ is also often referred to as the graph of field $f$. For
$n=2$ (2D), the chart $(X,f)$ is analogous to the concept of a node-weighted
graph $G(V,E,w_{V})$ arising in graph theory, in the sense that the domain $X$
captures spatial locations (the nodes $V$) and the function $f$ captures
values at the spatial locations (the node weights $w_{V}$). This analogy
becomes clearer when the domain $X$ is a 2D box and $G(V,E,w_{V})$ has a mesh
topology; here, the mesh can be seen as a discrete approximation of the
continuous domain $X$. Mesh topologies arise in images, matrices, and
discretized PDEs. We will define the chart associated with a given manifold
using the notation $G(X,f)$ and we will refer to this as a field graph. This
definition introduces some abuse of notation (analogous to $G(V,E,w_{V})$) but
we do this in order to emphasize connections between field graphs and node-
weighted graphs. This will facilitate the explanation of the concept of
filtration in a field context. We also emphasize that the concept of a field
graph generalizes to arbitrarily high dimensions (while a graph is inherently
a 2D object). We also emphasize that a graph does not live in a Euclidean
space (while $X$ does). The fact that $X$ is a Euclidean space indicates that
there is a notion of order.

A filtration can be applied to continuous fields in the same way that it is
applied to node-weighted graphs (discrete case); however, this now requires a
filtration function that is defined over continuous domains. We can define a
filtration function via superlevel sets of the field [37].

######  Definition 3.1.

Superlevel Set: Given a manifold $M$ with field $f:X\rightarrow\mathbb{R}$ and
domain $X\subseteq\mathbb{R}^{n}$, the superlevel set $X_{\ell}$ at a
threshold $\ell\in\mathbb{R}$ is defined as:

$X_{\ell}=\\{x\in X:f(x)\geq\ell\\}.$ (3.6)

The super level set is has an associated field graph
$G_{\ell}:=G(X_{\ell},f_{\ell})$ with $f_{\ell}$ defined over $X_{\ell}$.

The field graph $G_{\ell}=(X_{\ell},f_{\ell})$ contains all points of the
manifold $M$ that have a function value greater than or equal to $\ell$ (it is
a filtration of the manifold). Similar to the node-weighted graph case, the
filtration creates a nested set of field graphs which are obtained by defining
a sequence of decreasing filtration values $\ell_{1}>\ell_{2}>...>\ell_{m}$
with associated field graphs:

$G_{\ell_{1}}\subseteq G_{\ell_{2}}\subseteq...\subseteq G_{\ell_{m}}\subseteq
G$ (3.7)

Here, the field graphs are sparser with larger threshold values; we also have
that $G_{\ell_{m}}=G(X,f)$ (the original graph) if $\ell_{m}=\min_{x\in
X}\;f(x)$. For each superlevel set we obtain the field graph $G_{\ell}$ and we
compute and record its EC value $\chi_{\ell}$ (e.g., we determine the number
of connected components and number of holes). This information is used to
construct an EC curve, which contains the pairs $(\ell,\chi_{\ell})$.

It is important to highlight that the EC curve provides a topological
descriptor for a general $n$-dimensional field. The types of topological bases
change with dimension; for instance, for a 1D field (e.g., a temporal field)
we only have connected components ($\chi=\beta_{0}$) while for a 2D field
(e.g., a space-time field) we have connected components and holes
($\chi=\beta_{0}-\beta_{1}$). It is also often convenient to track the
evolution of the individual Betti numbers through the filtration; in other
words, we keep track of the pairs $(\ell,\beta_{\ell})$.

In Figure 4, we present multiple superlevel sets of a 1D field and the
corresponding EC curve. Since this is a 1D field, the EC of a given superlevel
set only involves the number of connected components. Filtration captures
topology by revealing a local maximum (connected component is formed) or a
local minimum (two components are connected); thus, the EC compactly encodes
information about the critical points of the field and their relations with
respect to the function shape, which are the topologically interesting
characteristics of a continuous function or field. The critical points of
fields are key because they define their topological features.

(a) Superlevel set filtration

(b) EC curve

Figure 4: (a) Superlevel set filtration of a 1D field (embedded in a 2D
manifold). The horizontal lines represent the thresholds of the filtration and
these cut through the 2D field graph. The topology of the graphs $G_{\ell}$
captures the critical points of the field. When a local maximum is passed, a
new connected component is formed ($\beta_{0}$ increases). When a local
minimum is passed, two separate components are joined into one component
($\beta_{0}$ decreases). (b) Resulting EC curve for the filtration of the
function in (a). The EC curve captures the location of critical points in the
function and their relationships via the shape of the function.

Superlevel set filtration is generalizable and extends to higher dimensional
fields. Such fields appear in scientific areas such as geophysics [33, 15,
10], climatology [10, 28], astrophysics [53], and medical imaging [36, 7, 52,
54]. Fields can be used to represent many important data objects such as
images (2D fields embedded in a 3D manifold), volumes (3D fields embedded in a
4D manifold), and spatio-temporal fields obtained from PDEs (4D fields
embedded in a 5D manifold). Theoretical connections between integral geometry,
statistics, and topology have shown that the EC provides a general descriptor
to characterize the behavior of these complex data objects, such as
identifying higher order statistical characteristics of these data objects
(e.g. statistics of derivatives of the data) or capturing global properties of
the data, such as the magnitude and frequency of spatial or temporal
flucturations, without the need for assumptions such as isotropy or
stationarity [5, 2].

(a) Superlevel set filtration

(b) EC curve

Figure 5: (a) Superlevel set filtration of a 2D field (embedded in a 3D
manifold). The plane represents the threshold of the filtration and this cuts
through the 3D field graph. As the filtration threshold is passed from top to
bottom, the EC of the resulting field graph $G_{\ell}$ is computed. (b) The EC
curve constructed from the filtration of the field. The 2D fields capture the
evolution of the topology of the varying superlevel sets during the filtration
(note emergence of connected components and holes).  Figure 6: Connected
components ($\beta_{0}$) and holes ($\beta_{1}$) that make up the EC curve for
the 2D field in Figure 5. The structure of the EC curve is a direct reflection
of the changing topology of the superlevel sets during the filtration. The
local maxima of the field represent connected components ($\beta_{0}$) which
are eventually joined via saddle points to form holes ($\beta_{1}$) that are
filled in once local minima of the field are passed by the filtration.

The filtration of a high-dimensional field is analogous to that used in the 1D
case. The difference is the number and nature of the topological features
captured in higher dimensions. For instance, for a 2D field (embedded in a 3D
manifold), we capture both connected components and holes. Specifically, the
threshold is a 2D plane that cuts through the 3D graph. When the plane passes
through a local maximum we have that connected components are formed, when it
passes a saddle point components are joined to form holes, and when a local
minimum is passed holes are filled. This reveals that filtration captures
incidence of different types of critical points in the field (its topological
features) and this information is summarized in the EC curve. This process is
illustrated in Figure 5; here, we see that the EC curve contains a single
minimum and a single maximum. The reason for this structure is revealed when
we visualize the individual topological bases for our 2D field (see Figure 6).
Here, we see two distinct areas of the filtration, the first is dominated by
connected components ($\beta_{0}$) which causes the maximum and the second is
dominated by holes ($\beta_{1}$) which causes the minimum. These topologically
distinct components of the filtration represent the presence of local maxima
(high $\beta_{0}$ values) and saddle points/local minima (high $\beta_{1}$
values) in the 2D field. Betti numbers reveal differences in the topology of a
data object at varying thresholds and have been used in analysis that is
complementary to the EC itself [38]. Specifically, analyzing individual Betti
numbers can provide additional insight into the appearance/disappearance of
specific topological features throughout the filtration process. This approach
is at the core of the so-called persistence diagrams (which summarize the
appearance/disappearance of topological features). Persistent diagrams, while
more informative, are difficult to interpret and analyze primarily because the
diagram is constructed from an unordered set of intervals which are not
amenable to statistical computations (e.g. means and variances) [49, 17]. For
this reason, it is difficult to analyze persistence diagrams directly or to
use them as a data preprocessing step for further analysis in statistical and
machine learning methods without further transformation.

##  4 Applications

In this section, we illustrate how to use the EC to characterize diverse
datasets arising in applications of interest to chemical and biological
engineers . All scripts and data needed to reproduce the results can be found
here https://github.com/zavalab/ML/tree/master/ECpaper.

###  4.1 Brain and Process Monitoring

One is often interested in characterizing the topology of graphs (such as
those arising in brain networks or process networks) as a way to identify
abnormal behavior. As an example, we might want to relate the topological
structure of the functional connections of brains at different stages of
development (adult vs. child) or to identify diseases. Here, we illustrate how
to do this using a real dataset taken from the work of Richardson and co-
workers available in OpenNeuro (ds000228) [40]. We then show how to use this
same approach to identify faults in chemical processes.

In the brain dataset, adults and children watch a short film and the activity
in different regions of the brain are measured with functional magnetic
resonance imaging (fMRI). Here, the signal in each region $i=1,...,n$ is a
univariate random variable $Y_{i}$ and we denote the collection of signals as
the multivariate random vector $Y=(Y_{1},...,Y_{n})$. We denote the
observation of the signals at time $t=1,...,m$ as $Y(t)\in\mathbb{R}^{n}$.
This dataset is thus a multivariate time series; the series is used to
construct a functional network, which is given by the sample inverse
covariance matrix Cov$[Y]^{-1}\in\mathbb{R}^{n\times n}$ (also known as
precision matrix). This procedure is summarized in Figure 7. The precision
matrix is represented as an edge-weighted graph $G(V,E,w_{E})$; here, vertices
represent the different regions of the brain and edge weights $w(e)\in[0,1]$
represent the absolute value of the partial correlation between different
regions. From Figure 7, we can see that these functional networks have a
complex structure, making them difficult to characterize.

(a) Multivariate time series

(b) Precision matrix

(c) Functional network

Figure 7: (a) Brain signals measured during an fMRI study of a developed brain
while watching a film. The signals measure brain activity in different regions
of the brain. (b) The precision matrix constructed from the brain signals. (c)
Functional network representation of the precision matrix. The width of the
edges represent the strength of the partial correlation between the different
regions of the brain.

We perform a filtration in order to characterize the precision matrix of
different brains. This filtration gives an EC curve for each brain that is
used to understand topological differences between developed and
underdeveloped brains (see Figure 8). Here, we can see that there is a
perceptible difference between the average EC curves for different brain
types. This illustrates the effectiveness of the EC curve in identifying
structural differences in complex functional networks [11, 12]. Intuitively, a
developed brain has a more widespread correlation structure, and thus the EC
curve decays more slowly.

(a) Brain signal time series.

(b) EC curve from precision matrix filtration.

Figure 8: (a) A representation of the filtration process showing the addition
of new edges as the level set of correlation is increased. The addition of the
edges reduces the number of connected components ($\beta_{0}$) and increases
the number of cycles ($\beta_{1}$). (b) Comparison of the average EC curves
for the brain functional networks of individuals with developed and
underdeveloped brains while watching a film. The difference between the two
curves demonstrates that the EC, along with the associated filtration, can be
an effective tool in differentiating complex networks .

Interestingly, the brain monitoring problem is analogous (from a mathematical
viewpoint) to the problem of chemical process monitoring. This is because,
fundamentally, any multivariate time series can be represented as a precision
matrix. In brain monitoring methods, such as fMRI, the goal is to identify
differences in brain activity that may be a result of genetic defects,
disease, or different stimuli. In chemical process monitoring, we seek to
identify the presence of faults, disturbances, or problems in equipment
operation. Brain monitoring typically uses observations on electrical signals,
blood flow, or oxygen levels; in a chemical processes we use observations on
temperatures, pressures, and flows (see Figure 9). To highlight this analogy,
we provide a fault detection case study for the Tennessee Eastman process
dataset [13]. This dataset contains a simulation of a chemical process that is
monitored using 52 variables (Figure 9). The goal is to use time series for
such variables to identify faulty behavior. The variables monitored by the
different sensors of a chemical process (in different locations) are
$Y_{i},\,i=1,2,...,n$ and the observations are $Y(t)\in\mathbb{R}^{n}$ for
$t=1,...,m$. As in the brain example, this multivariate time series is used to
construct the precision matrix Cov$[{Y}]^{-1}\in\mathbb{R}^{n\times n}$. We
again represent this matrix as an edge-weighted graph $G(V,E,w_{E})$ and use
filtration to determine its EC curve. We compute EC curves for precision
matrices obtained from different multivariate time series (containing
different types of faults).

(a) Brain functional network filtration.

(b) Chemical process schematic.

Figure 9: (a) Representation of the brain and the different area signals
obtained during fMRI. The edges represent interactions between the regions of
the brain computed via the precision matrix. (b) A simplified representation
of the Tennessee Eastman process. The process is monitored using temperature
sensors (TS), pressure sensors (PS), flow sensors (FS) and level sensors (LS)
distributed in different regions. While the context of the two systems are
vastly different, the type of data produced (i.e. multivariate time series)
are identical. This suggests that methods used in the analysis of brain
functional networks for the detection of disease can be directly applied to
chemical process systems to detect faults or process issues.

The Tennessee Eastman dataset contains multiple simulations of the chemical
process; here, some simulations contain faults and others do not. For each
simulation, we construct an edge-weighted graph from the precision matrix and
leverage the EC curve to identify whether or not the given process is
experiencing a fault based upon the topological structure induced by the
correlations between the process variables (Figure 10). Figure 11 shows
examples of the precision matrices derived from process simulations that
either contain or do not contain faults, demonstrating that the identification
of faults from the precision matrix is not a trivial task.

(a) Process sensor time series.

(b) Precision matrix.

(c) Functional network.

Figure 10: (a) Process sensor measurements during the operation of the
Tennesee Eastman process system simulation. The measurements represent the
output of the various temperature, pressure, flow, and level indicators during
the simulation. (b) The precision matrix constructed from the process sensor
measurements in (a). This precision matrix is then used to construct an EC
filtration. (c) Simplified graphical representation of the precision matrix
derived from the process sensor measurements. Figure 11: Visualization of
multiple precision matrices derived from chemical process simulations with and
without faults. Distinguishing these matrices is difficult, demonstrating the
inherent complexity in identifying faults in a process with the precision
matrix alone.

Figure 12 demonstrates that there is a quantifiable difference between the
process operating with no faults, and the process operating with faults. We
also note that there is a separation into two groups within the faulty
systems. The two groups of faults deal with with either feed temperature and
reactor faults, or feed composition and condenser faults. The reason for this
separation is of interest and will be explored in future work. This
demonstrates that the EC can be used to detect faults based purely on the
topological structure of the precision matrix. Note that this is a method that
accounts for the space-time relationships of the entire process (all-at-once)
and does not require statistical assumptions on the data (e.g., independence
of observations). The simplicity of (2.3) also ensures that the EC can be
rapidly calculated via the number of edges and vertices of a graph. The
computation of the EC and the associated EC curve requires a simple
thresholding operation (to obtain the number of nodes and edges in the
filtered graph) and a few addition and subtraction operations (to sum the
number of edges and nodes and compute the EC value) at each point in the
filtration, because of this the method scales well with large networks as the
required computations are simple and efficient.

(a) Chemical process network filtration.

(b) EC curves constructed from filtration of network.

Figure 12: (a) Representation of the filtration process showing the addition
of new edges as the level set threshold for correlation is increased. (b)
Comparison of the EC curves for the Tennesee Eastman process under faults and
no faults. There is also a notable separation of the faults into two groups,
one which represents faults primarily associated with feed temperatures or
reactor conditions, and the other associated with feed composition changes or
condenser faults.

###  4.2 Spatio-Temporal Data Analysis

We now use the EC to characterize the spatio-temporal behavior of fields for a
reaction diffusion system. Here, the fields are solutions of a PDE model with
different diffusion ($D\in\mathbb{R}$) and reaction ($R\in\mathbb{R}$)
coefficients. The coefficients can be manipulated to generate fields with
different topological features, which capture different mechanistic behavior
(e.g., reaction-limited or diffusion-limited). The model is described by the
coupled PDEs:

$\displaystyle\frac{\partial u(x,t)}{\partial t}$
$\displaystyle=D\left(\frac{\partial^{2}u(x,t)}{\partial
x_{1}^{2}}+\frac{\partial^{2}u(x,t)}{\partial
x_{2}^{2}}\right)+R(v(x,t)-u(x,t))$ (4.8a) $\displaystyle\frac{\partial
v(x,t)}{\partial t}$ $\displaystyle=D\left(\frac{\partial^{2}v(x,t)}{\partial
x_{1}^{2}}+\frac{\partial^{2}v(x,t)}{\partial
x_{2}^{2}}\right)+R(u(x,t)-v(x,t))$ (4.8b)

Here, $u(x,t):\mathcal{D}_{x}\times\mathcal{D}_{t}\rightarrow\mathbb{R}$ and
$v(x,t):\mathcal{D}_{x}\times\mathcal{D}_{t}\rightarrow\mathbb{R}$ represent
the concentrations for the reactants over space and time. The spatial domain
is continuous and given by
$\mathcal{D}_{x}:=[0,n]\times[0,n]\subset\mathbb{R}^{2}$; the temporal domain
is $\mathcal{D}_{t}:=[0,T]\subset\mathbb{R}$. We thus have that $u(x,t)$ and
$v(x,t)$ are 3D fields (embedded in a 4D manifold). Here, we hypothesize that
the topological features of these fields are expected to change with the
parameter pair $(D,R)$ (the structure changes with governing mechanism).

An example field generated for a given parameter pair ($D,R$) is shown in
Figure 13. Here, we focus on characterizing the topology of $u(x,t)$ for
different values of the parameters. To do so, we generate 30 fields (obtained
using different random initial conditions) for each of the following
combinations: ($D=3,R=0.8$), ($D=3,R=0.4$), and ($D=6,R=0.8$). The goal in the
analysis of this dataset is to cluster the realizations into groups that
reflect the parameters of the models (e.g., to detect changes in the
underlying mechanism). For each simulation, we represent $u(x,t)$ as a spatio-
temporal field (Figure 13). We construct superlevel sets for each spatio-
temporal field and record the EC of the resulting superlevel sets. This
process is similar to the examples shown for the 1D and 2D cases (Figures 4
and 14); however, the filtration performed here can no longer be visualized as
a simple plane that slices the field. This highlights the versatility of using
the EC to characterize datasets over high dimensions.

(a) 2D snapshots

(b) 3D space-time field

(c) 2D snapshots

(d) 3D space-time field

Figure 13: (a,c) Snapshots obtained during evolution of reaction-diffusion
system(4.8a)- (4.8b) with different parameter sets ($D,R$). (b,c) Collection
of snapshots in 3D space-time field. It is difficult to distinguish the
realizations of the models with different parameter values, regardless of
whether they are viewed via individual snapshots or as 3D space-time fields.

The average EC curves for the three different parameter settings are presented
in Figure 14. Here, it is clear that the EC reveals a change in the
topological structure. To perform clustering, we represent the EC curve for
sample $j=1,...,n$ as a vector $\chi_{j}\in\mathbb{R}^{m}$; the entries of
this vector are the EC value of the level set. Each EC vector can be stacked
into a matrix $[\chi_{1}\ \chi_{2}\ \chi_{3}\ ...\
\chi_{n}]^{T}\in\mathbb{R}^{n\times m}$. We obtain a matrix for each of the
three parameter settings. We apply a singular value decomposition to these
matrices and visualize the data projected onto the two leading principal
components. Figure 14 shows the results; note that there is a distinct
clustering of the data into three separate groups. This confirms that the EC
curve captures the topological differences of the fields obtained under
different parameter settings. For comparison, we compute the Fourier transform
of the data to obtain the frequency spectrum and project the spectrum to two
dimensions using a singular value decomposition. The results are shown in
Figure 14; here, we can see that the frequency spectrum of the data does not
contain enough information to separate the different types of fields. This
indicates that the EC contains information that cannot be captured by the
frequency spectrum.

(a) Average ECs

(b) SVD of ECs

(c) SVD of Fourier spectrum

Figure 14: Demonstration of the EC’s ability to capture topological
differences induced by different reaction-diffusion parameters. (a) Average EC
curves for the three different parameter settings. The EC is able to separate
the realizations of the reaction-diffusion system. (b) SVD projection of the
EC curves onto their two leading principal components, revealing a distinct
clustering of the different types of space-time fields. (c) SVD projection of
Fourier spectrum of the space-time fields; here, there is no distinct
separation between the different fields.

###  4.3 Image Analysis

We explore the topological characterization of simulated micrographs for
liquid crystal (LC) systems. These micrographs capture their responses to
different reactive gaseous environments [44, 47]. These LC systems start with
homeotropic alignment of a thin film on a functionalized surface. When the LC
system is exposed to an analyte, the analyte diffuses through the LC film and
disrupts the binding between the LC and the surface. This disruption triggers
a reorientation of the LC film and forms complex optical patterns and textures
simulated via random fields. Figure 15 provides images (micrographs) that
capture the response of an LC system to a couple of different environments.
For this dataset, we want to characterize the topological differences between
the textures of the LC systems when exposed to the different envrionments and
use this information to classify the datasets. This provides a mechanism to
design gas sensors. Such classification tasks have been recently performed
succesfully using convolutional neural networks [44]; these machine learning
models, however, contain an extremely large number of parameters and are
difficult to train.

(a) LC response to environment 1

(b) LC response to environment 2

Figure 15: Comparison of the visual response of an LC system to two different
gaseous environments. We can see that there are perceptible differences
between the responses, but these differences are difficult to quantify due to
the heterogeneity present. The EC is able to effectively summarize the
topological differences and similarities in the images, allowing for an
accurate separation of the responses.

The micrographs of the LC systems at their endpoint are discrete fields
(matrices), which we represent as node-weighted graphs. The EC curves for such
graphs are obtained via filtration. The filtration process seeks to
characterize regions of high and low intensity in the A* channel. We highlight
that one can also think of an image as a discrete approximation of a
continuous 2D field; this emphasizes that the filtration process for a
discrete field is analogous to the filtration process of a node-weighted graph
(we perform filtration over weight values instead of over field values).

The average EC curves for the LC systems exposed to the two gaseous
environments are presented in Figure 16. Here, we also show the projection of
the EC curves onto the first two principal components using SVD. We can see
that there is a strong separation between the two datasets. As a comparison,
we cluster the micrographs by applying SVD directly to the images (as opposed
to the EC curves) and by computing the Fourier transform of the images, which
decomposes an image into a weighted set of spatial frequency functions. We
chose these methods for comparison because they are commonly used in
characterizing image textures [55, 26]. We can see these traditional
techniques do not provide a clear separation (Figure 17). Figure 18 shows the
distribution of the first principal component of the EC values and compares
this with the distribuiton of Moran’s I values which capture average spatial
autocorrelation in the images by computing spatial autocorrelation in
neighborhoods around each pixel in the image and averaging results over the
entire image [25]. It is clear that the ECs provide a sharper separation than
Moran’s I; we conclude that EC is a more effective descriptor. We attribute
the limitations encountered with traditional tools (such as SVD, Fourier, and
Moran’s I) to the spatial heterogeneity of the images (see Figure 15) [29].

(a) Average EC for LC system response.

(b) PCA of EC curves from LC system responses.

Figure 16: (a) The average EC curve of the LC system responses to the two
different gaseous environments. (b) SVD performed on the EC curves. This
highlights that the EC is able to produce a strong, linear separation of the
LC responses.

(a) SVD on images

(b) SVD on Fourier spectrum of images

Figure 17: SVD of the LC systems responses using (a) the raw image data and
(b) the Fourier spectrum of the images. Under these approaches, there is no
obvious separation of the data.

(a) Distribution of 1st PC of EC curves

(b) Distribution of Moran’s I values

Figure 18: (a) Distribution of the 1st principal components for the EC curves.
(b) Distribution of the Moran’s I values. We can see that the ECs provide a
sharper separation than Moran’s I values; as such, ECs are a more informative
descriptor of the data.

To further demonstrate the usefulness of the EC; we used the EC curve (a
vector) as an input to a support vector machine (SVM) for classification of
the two datasets. We compare this classification approach against approaches
that use SVM with: (i) raw image as input and (ii) Fourier spectrum as input.
Using the EC vector as an input, we were able to classify the two datasets
with $95\pm 6\%$ accuracy, compared to $66\pm 7\%$ and $68\pm 7\%$ accuracy
obtained with raw images and Fourier spectrum. It is particularly remarkable
that, after reducing the images to an EC curve, it is possible to separate the
datasets using a simple (linear) SVM classifier. This highlights how the EC
can be used to pre-process data and how this can facilitate machine learning
tasks. For instance, in this case, it is not necessary to use a more
sophisticated machine learning model (e.g., a convolutional neural net) to
perform image classification.

###  4.4 Point Cloud Analysis

We now shift our focus to the use of the EC to analyze the structure of point
clouds (also known as scatter fields). Here, we consider 2D point clouds that
are realizations of a bivariate random variable. As such, the point clouds
emanate from a 2D joint probability density function (pdf). The analysis of
the shape of univariate pdfs is typically performed by using summarizing
statistics (e.g., moments); analysis in higher dimensions is quite complicated
(no good descriptors exist for multi-dimensional joint pdfs) [51]. This
limitation is particularly relevant when the pdf has a complex shape (e.g., it
is non-Gaussian). Here, we will see that one can characterize the complex
shapes of a multi-variate pdf by using an EC curve. Moreover, the results that
we present highlight that the EC can be used to characterize pdfs in higher
dimensions (even if it is not possible to visualize them).

We use an experimental flow cytometry dataset to illustrate how this can be
done. The dataset was obtained through the FlowRepository (Repository ID: FR-
FCM-ZZC9) [46]. This dataset is obtained in a study of the kinetics of gene
transcription and protein translation within stimulated human blood
mononuclear cells through the quantification of proteins (CD4 and
IFN-$\gamma$) and mRNA (CD4 and IFN-$\gamma$) [50]. In our study, we focus on
the evolution of the concentration of CD4 mRNA and IFN-$\gamma$ mRNA in a
given cell which is measured via a flow cytometer. At each time point, a
number of cells ($\sim 15,000$) are passed through the flow cytometer; each of
these cells provides an observation vector $y\in\mathbb{R}^{2}$ (corresponding
to CD4 mRNA and IFN-$\gamma$ mRNA). These observations are typically
visualized as point clouds in a 2D scatter plot. The evolution of the point
clouds over time is shown in Figure 19. The goal is to characterize how the
shape of the point clouds evolves over time; for instance, it is clear that
the point cloud progressively separates into two distinct domains.

To characterize the shape of the point clouds, we convert them into a
continuous 2D field by applying smoothing. It is important to note that the
point clouds are realizations of a bivariate random variable (CD4 mRNA and
IFN-$\gamma$ mRNA); as such, one can obtain a 2D histogram for them (by
counting the number of points in a bin). The histogram is an empirical
approximation of the joint pdf of CD4 mRNA and IFN-$\gamma$ mRNA. The
continuous 2D field obtained via smoothing is a smooth representation of the
2D histogram. Our approach provides an alternative to traditional heuristic
methods such as gating, which are difficult to tune as they are highly
sensitive to potential noise and outliers in the data [27].

(a) Time = 0 Minutes.

(b) Time = 30 Minutes.

(c) Time = 60 Minutes.

(d) Time = 90 Minutes.

Figure 19: Deformation of a $2$D scatter field over time.

(a) Sample distribution.

(b) Density estimate.

(c) Density surface.

Figure 20: Transforming the point cloud to a field. The raw data (a) is
smoothed via a Gaussian kernel and then the smoothed 2D field (b) is
represented in 3D and processed via a filtration (c). Figure 21: Euclidean
distance between the EC curves as a function of time. There is a continuous
evolution of the distance that characterizes the change in topology.

In order to create the 2D fields, we utilize a Gaussian kernel smoothing [43].
Figure 20 shows the smoothing of a scatter plot. The 2D field is a projection
of a 3D function (it is embedded in a 3D manifold). This is illustrated in
Figure 20; as such, we can characterize the field by using a filtration on the
values of the pdf. Our goal is thus to compute an EC curve for the pdf at
different points in time (to analyze how its shape evolves). To visualize this
deformation, we compute the Euclidean norm of the difference between EC curve
at a time point to that at the initial time. From Figure 21, we can see that
the distance exhibits a strong and continuous dependence with time (indicating
that there is a strong change in the shape of the density function).
Importantly, this also suggests that there exists a continuous mapping between
the EC and time (the topological deformation is continuous with respect to
time). For instance, one could construct a dynamical model that predicts the
evolution of the shape with time. This indicates that the EC provides a useful
descriptor to monitor the evolution of complex shapes. This could help, for
instance, to detect times at which the change in shape is fastest/slowest.

##  5 Conclusions and Future Work

The Euler characteristic (EC) is a powerful tool for the characterization of
complex data objects such as point clouds, graphs, matrices, images, and
functions/fields. The EC summarizes the topological characteristics of such
data objects. We have demonstrated that the EC can be applied to a wide
variety of applications by using creative data transformations (e.g., point
clouds to fields and multivariate time series to correlation matrices). We
have shown that the EC provides an effective descriptor and can be used as a
pre-processing step that simplifies visualization, clustering, and
classification tasks. We believe the inclusion of the EC and other topology
based methods into engineering will forge connections between long standing
fields of science (such as neuroscience, as explored here). As part of future
work, we will explore the use of EC to characterize more complex data objects
(such as random fields, tensors, and simplices) and we will explore new
applications. For instance, it is possible to characterize the topology of
objective functions in optimization problems in order to determine the
complexity in solving such problems. Moreover, one can seek to optimize time-
dependent or space-dependent functions by optimizing their EC. Furthermore,
the EC curve is represented as a vector which makes it difficult to perform
common statistical and hypothesis tests that require a single numerical value.
Thus, our future work will also focus on the development of statistical
methods for the EC curve, such as hypothesis testing, which are needed if the
EC is to be broadly applied in science and engineering.

##  6 Acknowledgments

We acknowledge funding from the U.S. National Science Foundation (NSF) under
BIGDATA grant IIS-1837812.

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