text:
\subsection{Related work}
\label{sec_relatedWork}
The literature related to our approach can be classified into two categories:
\emph{(i)} uncertainty visualization and \emph{(ii)} ensemble visualization. In
the first case, the data variability is explicitly encoded by an estimator of
the probability density function (PDF) of a point-wise random variable (usually
with strong assumptions on its structure)\brian{, while i}{. I}n the second, the variability
arises from a series of global empirical observations.

\noindent
\textbf{(i) Uncertainty visualization:} The analysis and visualization of
uncertainty in data \cite{GUM} is commonly recognized as an important
yet long standing challenge in the visualization community \cite{Pang97,
johnson03,
MacEachren05}, as documented in several surveys \cite{potter12,
bonneau:hal-01060465}. Given an estimator of the PDF of a random variable
modeling the point-wise variability in data values (for instance, a Gaussian
distribution), several techniques have been proposed to represent the
distribution of the uncertainty in the data, by either considering the entropy
of the random variables \cite{potter13}, correlation in uncertainties
\cite{DBLP:journals/cgf/PfaffelmoserW12} or variability in the data
gradient \cite{DBLP:journals/tvcg/PfaffelmoserMW13}.\BrianSays{prev sentence too long} To understand the
positional uncertainty related to \brian{geometrical}{geometric} constructions generated out of
the raw uncertain data, specialized methods have been designed\brian{, f}{. F}or \brian{instance}{example,} \brian{to
estimate}{approaches have estimated} the positional uncertainty of level sets under
various interpolation schemes and PDF models \cite{pfaffelmoser11, poetkow11,
DBLP:journals/cgf/PothkowWH11,
SchlegelKS12,
poetkow13, poethkow13b, AthawaleE13,
AthawaleSE16}. More related to our work, several approaches have been
investigated to estimate the positional variability of critical points under a
Gaussian PDF model \cite{otto10, OttoGT11, petz12, LiebmannS16} or
interval-based pointwise representations of the uncertainty \cite{bhatia12,
szymczak13, tierny_ev14}. However, a common aspect of the above techniques is
that they explicitly rely on strong assumptions regarding the probability
density function (PDF) modeling the random variables, which are often assumed
to be Gaussian or uniform (which is implicitly the case for interval-based
representations).
Such strong assumptions are
 limiting in practice when
considering ensemble data, where PDF reconstructed from the empirical
observations can follow an arbitrary, unknown model. Moreover, most of the
parametric PDF models studied in these techniques do not consider multi-modal
distributions, which is \brian{however necessary}{a necessity} when several trends clearly occur in
the ensemble.

\noindent
\textbf{(ii) Ensemble visualization:} A different category of techniques has
been specifically investigated to visualize variability in ensemble data. In
this setting, a series of global empirical observations (i.e. the members of
the ensemble) are taken
as an input for the actual computation of
\brian{geometrical}{geometric}
constructions, such as level sets or streamlines\brian{, and t}{. T}he
\brian{spatial}{}
variability of \brian{these}{the} constructions
is then analyzed. For instance,
spaghetti plots \cite{diggle02}
have been used intensively to visualize level set variability in weather
ensemble data \cite{DBLP:conf/icdm/PotterWBWDPJ09,
DBLP:journals/tvcg/SanyalZDMAM10}. More advanced representations
\cite{pfaffelmoser13} have also been proposed, for instance with the notion of
contour
boxplot \cite{WhitakerMK13}\brian{ and}{,} their generalization to arbitrary
curves
\cite{DBLP:journals/tvcg/MirzargarWK14}, and also
\brian{applied}{their application} to weather forecast
data \cite{DBLP:journals/tvcg/QuinanM16}.
\brian{}{Additionally,} Hummel et
al. derive a complete framework for the visualization of the variability in
particle advection in ensemble data \cite{DBLP:journals/tvcg/HummelOGJ13}. More
related to our work, \brian{in particular}{particularly} in \brian{its}{their} ambition to analyze trend
variability, \brian{specifically-designed}{specialized} clustering techniques have been investigated
 to cluster isocontours \cite{FerstlKRW16} and streamlines
\cite{DBLP:journals/tvcg/OeltzeLKJTP14, FerstlBW16}\brian{ to identify and analyze
trends in the distribution of these geometric constructions within ensemble
data.}{.} However, \brian{to  our knowledge, }{}this overall strategy has never
been studied for topological \brian{constructions}{features} such as critical points\brian{, which we do
in this paper.}{.}
Our approach aims at identifying trends in critical point layouts within
the ensemble.
\julien{Therefore}{Thus}, a necessary building block of our framework is a
method
to evaluate the similarity between critical point distributions. More
generally, the similarity estimation between topological data representations is
a long-studied problem. Several
\julien{heuristical approaches}{heuristics} have been proposed
to quickly assess structural similarity \cite{hilaga:sig:2001,
SaikiaSW14_branch_decomposition_comparison, thomas14}. At the theoretical
level, several metrics have been carefully studied to evaluate the similarity
between persistence diagrams \cite{cohen-steiner05, interleaving_distance},
merge trees \cite{BeketayevYMWH14} or Reeb graphs \cite{bauer14}. The
computation of barycenters of such constructions (which is relevant to
clustering) has also been studied \cite{TurnerMMH14}. However, while
extensively studied from a theoretical perspective,
the evaluation of these metrics involve\julien{}{s} computationally
\julien{intensive}{expensive}
combinatorial
optimization methods \cite{munkres}\brian{, which}{. This} makes
them challenging to compute, \brian{if not even }{and potentially }impracticable
for real-life cases \cite{botnan18}.
\julien{}{Alternatively, kernel based methods
\cite{ReininghausHBK15,
CarriereCO17} have been
specifically developed for measuring distances between persistence diagrams in
machine learning tasks.}
\julien{More importantly,}{However,}
these approaches \julien{}{(metric or kernel based)} focus on the intrinsic
structural similarity between
topological constructions without considering the \brian{geometrical}{spatial} realization of
these structures in the original\brian{geometrical}{} domain.

In contrast,
our approach
based on persistence maps leverages the soundness of
topological persistence
\cite{edelsbrunner02} while specifically focusing on the \brian{geometrical}{spatial} layout of
the critical points.

\subsection{Contributions}
This paper makes the following new contributions:
\begin{enumerate}[leftmargin=1em]
  \item{
  \vspace{-1ex}
  \emph{An informative representation of critical point layouts}
(\autoref{sec_persistenceMap}): Inspired by distance field transforms, we
introduce the \emph{Persistence Map}, a measure of the
\brian{geometrical}{spatial} density \brian{in}{of}
critical points
which leverages the robustness to noise of topological
persistence to better emphasize salient features.
\brian{C}{In addition, c}omputations \brian{}{to construct
this map }are shown to be trivially parallelizable.}
Experiments demonstrat\brian{e}{ing} the relevance of this representation for comparing
critical point distributions with standard density distance measures \brian{.}{
are provided.}
  \item{
  \vspace{-1ex}
  \emph{A statistical space for critical point layouts}
(\autoref{sec_clustering}):
  We present a  framework which leverages spectral embedding to
  represent persistence maps as points in a low-dimensional Euclidean space,
  where distances between points represent dissimilarities in critical point
layout and where
statistical \brian{tasks}{analysis} can be easily carried out.
The first two dimensions of this space can be used to generate planar views of
the ensemble
to visualize the distribution of critical point layouts
at a global level. We additionally show how
persistence maps can be clustered in this
space to reveal the major trends in critical point layouts and how
relevant automatic suggestions for the number of clusters can be estimated.
  }
  \item{
  \vspace{-1ex}
  \emph{Confidence regions for clusters of critical point
layouts}
(\autoref{sec_confidenceRegions}):
  Based on the clustering of persistence maps, we show how the notion of
mandatory critical point \cite{tierny_ev14} can be leveraged to visualize in
the \brian{geometrical}{spatial} domain the possible outcomes in terms of critical point
layouts, in particular\brian{,}{} with the visualization of
confidence regions along with
their respective probability of appearance. The prediction  accuracy of these
regions is quantitatively evaluated and compared to a baseline
strategy using an off-the-shelf clustering approach.
  }
  \item{
  \vspace{-1ex}
  \emph{Implementation:} We provide a lightweight C++
implementation of our approach that can be used for reproduction purposes.}
\end{enumerate}

\section{Preliminaries}
This section
\julien{
briefly describes our formal setting. We refer the reader to
reference books for introductions to Morse theory \cite{milnor63} and
computational topology \cite{edelsbrunner09}.}{presents
theoretical background on topological data analysis (TDA). It contains
definitions adapted from Tierny et al.~\cite{ttk17}. Reference introduction
books to Morse theory and computational topology have been published by
Milnor~\cite{milnor63} and Edelsbrunner and Harer~\cite{edelsbrunner09}.}

\subsection{Background}
\label{sec:background}
\julien{
We assume our data is\textbf{Data representation:}}{ The input data is given
as an ensemble of $n$ piecewise (PL) linear scalar fields on a
PL $d$-manifold $\mathcal{M}$ (with $d = 2$ or $3$)
$f^{(x)} : \mathcal{M} \rightarrow \mathbb{R}$, \julien{}{with} $x \in [0, ~n-1]$.}
Each individual scalar field $f^{(x)}$
is an \emph{ensemble member}. For brevity,
we will only use the $^{(x)}$ notation in cases where several members
are considered and thus need disambiguation.
\julien{A member $f$ has value at the set of vertices $\mathcal{M}^0$ of $\mathcal{M}$
and
assumed to be linearly interpolated on the simplices of higher dimension.}{Each
member $f$ is valued at the vertices $\mathcal{M}^0$ of $\mathcal{M}$ and is linearly
interpolated with barycentric coordinates on the remaining simplices of
$\mathcal{M}$.}
\julien{We assume and enforce that $f$ is}{In practice, $f$ is enforced to be
injective
\julien{for the vertices of}{on} $\mathcal{M}^0$
\julien{via a variation of}{with}
simulation of simplicity \cite{edelsbrunner90}.
  The set of simplices
  \julien{that have}{having}
  a given simplex $\sigma$ as a face form the \emph{star}
\julien{}{of $\sigma$, } $\stt{\sigma}$.  The set of faces of the
simplices \julien{in}{of} $\stt{\sigma}$
that do not intersect $\sigma$ form the \emph{link} \julien{}{of $\sigma$,}
$\lk{\sigma}$.}
\julien{The topology of our domain can be described by its \emph{Betti
numbers}, where $\beta_i$ are the ranks of its homology
groups~\cite{edelsbrunner09}.}{}

\brian{}{
  For a vertex $v$, let us define $\lkminus{v}$
to be the \emph{lower link} \julien{}{of $v$} ($\lkminus{v} = \{ \sigma \in
\lk{v} ~ | ~
\forall u \in \sigma : f(u) < f(v)\}$) and $\lkplus{v}$ as the
\emph{upper link} \julien{}{of $v$} ($\lkplus{v} = \{ \sigma \in \lk{v} ~ | ~
\forall u \in \sigma : f(u) > f(v)\}$).
  When both $\lkminus{v}$ and
$\lkplus{v}$ are simply connected, $v$ is called a \emph{regular vertex}.
\julien{Otherwise}{If not}, $v$ is a \emph{critical
point} of $f$ \cite{banchoff70}.
\julien{}{Such points correspond to configurations where the \emph{sub-level
sets} $\sub{i}$ of $f$ (subset of $\mathcal{M}$ valued below the isovalue $i$)}
change their topology
when continuously varying the isovalue $i$.
\julien{Given a $\mathcal{M}$ of dimension $d$, }{
Critical points often correspond to features of interest in applications.
They are usually classified with the notion of
\emph{index} $\mathcal{I}$}, which is equal to $0$, $1$, $(d-1)$ and $d$
respectively for:
minima (empty lower link),
\julien{}{$1$-saddles (2 connected components of lower link),}
$(d-1)$-saddles (2 connected components of upper link), and
maxima (empty upper link) respectively.
}

\begin{figure}
  \includegraphics[height=2.15cm]{colorMap_blueGreen.jpg}
  \hfill
 \includegraphics[height=2.15cm]{persistenceDiagram.jpg}
 \vcaption{
 Critical points (spheres, dark blue: minima, dark green: maxima, other:
saddles) and persistence diagrams of a clean (a) and noisy (b) 2D scalar
field.
From left to right: original 2D
data, 3D terrain
representation, persistence diagram. The diagrams clearly exhibit in both cases
two large pairs, corresponding to the main hills. In the noisy diagram
(b), small bars near the diagonal correspond to noisy features in the
data.}
  \label{fig_persistenceDiagram}
\end{figure}

\label{sec_persistence}
\brian{}{
The \julien{}{population of} critical points of $f$ can be
\julien{illustrated}{visually encoded}
\julien{by a topological abstraction
called the}{with the notion of} \emph{persistence
diagram}~\cite{edelsbrunner02} (\autoref{fig_persistenceDiagram}). This diagram
encodes critical points as pairs \julien{such that}{}$(c,
c')$ \julien{with}{such that} $f(c) < f(c')$
and $\mathcal{I}(c) = \mathcal{I}(c') - 1$.
These pairs follow the Elder rule~\cite{edelsbrunner09},
\julien{such}{which intuitively implies}
that if two \julien{}{topological} features of $\sub{i}$ meet at a critical
point $c'$ of $f$,
the \emph{youngest} feature
(created at the highest function value)
 \emph{dies}, favoring the \emph{oldest} (created at the lowest function
value).
In a persistence diagram $\persistenceDiagram{f}$,
each pair  $(c, c')$ is represented as a point in 2D at coordinates
$\big(f(c), f(c')\big)$, which are
the  \emph{birth} and \emph{death} of the pair respectively.
The  \emph{persistence} of the pair
is given by its height in the diagram, $P(c, c') = |f(c') - f(c)|$. It
describes the lifespan \julien{}{in the range} of \julien{a}{the corresponding}
topological feature.}
\julien{
In the rest of the paper, when discussing persistence diagrams,
we will only
consider critical point pairs of index $(0, 1)$ and $\big((d - 1), d\big)$.}{In
the following, only the critical point pairs involving local extrema,  $(0, 1)$
and $\big((d - 1), d\big)$, will be considered.}
\julien{The
impact of this simplification is discussed in \autoref{sec_results}.}{The
consequence of this simplifying assumption are described in
\autoref{sec_results}.}
\julien{}{Moreover, for genericity purposes,
all persistence evaluations will be normalized with regard to the largest
persistence found in the data ($P(c, c') \in [0, 1]$).}
\julien{}{In practice,
the pairs of the diagram located in the vicinity of the diagonal denote
low-amplitude noise
while prominent features will be associated with persistent pairs, located
far away from the diagonal (\autoref{fig_persistenceDiagram}).}
The
persistence diagram has
been extensively studied from a theoretical perspective and its stability
to perturbations in the input data has been demonstrated
\cite{cohen-steiner05}. This
stability result greatly motivated the use of persistence in applications,
ranging from machine learning \cite{chazal13} to visualization, where it has
been shown to significantly help users distinguish salient features from noise.

\begin{figure*}
  \includegraphics[height=3.95cm]{colorMap_blueGreen.jpg}
  \hfill
  \includegraphics[height=3.95cm]{pipeline.jpg}
 \vcaption{
 Pipeline overview.
 (a) Given an ensemble of scalar fields, our approach
computes the persistence diagram of each member (critical points are scaled by
persistence). (b) To ease the comparison of critical point layouts, each member
is transformed into an alternative representation, the persistence map, which
denotes the local density in salient features. (c) The distance matrix
between the persistence maps is constructed and used to embed each member
in a low dimensional \julien{}{feature} space, whose
first two components can be used to generate planar overviews of the distinct
critical point patterns found in the ensemble (d). In this \julien{}{feature}
space, clusters of
members are automatically estimated. (e) The persistence atlas is
finally composed by considering the mandatory critical points of each cluster
independently (one color per cluster), revealing the main configurations of
critical point layouts in the ensemble in terms of numbers and positions.
\julien{The likelihood of appearance of critical points}{The positional
variability of critical points}
within each
mandatory critical point is \julien{shown}{indicated} with a color map.
The
bar plot (right side) \julien{indicates}{shows} the statistics of appearance of
each
cluster in the ensemble.
}
 \label{fig_overview}
\end{figure*}

\subsection{Overview}
Our approach is composed of three main steps (\autoref{fig_overview}). It takes
as \brian{an }{}input $n$ PL scalar fields defined on the same PL manifold
$\mathcal{M}$.

First (\autoref{sec_persistenceMap}), the persistence map of each ensemble
member is computed. The purpose of this representation is to evaluate the
spatial distribution of the critical points in each member, while
\brian{}{at the same time }balancing the
contribution of each critical point by its
persistence\brian{, in order}{} to
emphasize salient features and reduce the contribution of noise.
\BrianSays{don't like balace in prev sentence}

Second (\autoref{sec_clustering}), we leverage spectral embedding to represent
each member as a point in a low-dimensional Euclidean space\brian{, where d}{. D}istances in
this feature space denote dissimilarities between persistence maps.
This space is conducive to further statistical analysis of the members which
are clustered based on their persistence maps.
The first two dimensions of this space are used to generate planar views
which
enable the direct visualization of
the main trends in the ensemble in terms of critical point layouts.

Third (\autoref{sec_confidenceRegions}), confidence regions in the geometrical
domain are computed for each cluster by leveraging the notion of mandatory
critical point\brian{}{s} \cite{tierny_ev14}. Finally, the confidence regions of all
clusters are composed together into the final persistence
atlas\brian{, which}{. This} enables
the visualization of the regions of occurrence of the most prominent critical
points along with estimations of their probability of appearance.

\section{Persistence maps}
\label{sec_persistenceMap}
In this section, we introduce the notion of persistence map, a representation
of the critical point distribution in each member.

\subsection{Motivation}
The main target of
persistence maps is to facilitate the comparison of two members $f^{(x)}$
and $f^{(y)}$ in terms of the layout of their critical points.
As discussed in \autoref{sec_relatedWork}, existing topological metrics (e.g.
the Bottleneck distance \cite{cohen-steiner05}) do not take into account the
spatial embedding of the critical points in $\mathcal{M}$ and are therefore not
suited for our purpose.
Let $C^{(x)}$ and $C^{(y)}$ be the set of critical points of $f^{(x)}$ and
$f^{(y)}$ respectively, which can be interpreted as point clouds in $\mathcal{M}$.
The problem of comparing the layouts of critical points of $f^{(x)}$
and $f^{(y)}$ then reduces to that of comparing two point clouds, a problem for
which no universal solution exists. For instance, the Haussdorff
distance, \brian{which }{}can be seen as a \emph{worst-case} \brian{metric,
}{metric that} only measures the distance
between the two most distant points of the two sets\brian{, which}{.  This} is too limiting for
our setting since the similarity of the rest of the point cloud is
not assessed.
Moreover, due to the presence of noise in the data, it is highly likely in
practice that a significant number of the critical  points of $f^{(x)}$ and
$f^{(y)}$ are noise artifacts\brian{, even in the case of smooth simulation data}{}.
Such \brian{noise }{}artifacts must be taken into account in the similarity estimation
in order to reduce \brian{the}{their} importance\brian{ of noise}{} and highlight salient features. This
last observation is the main motivation behind persistence maps.

A reason for the difficulty in estimating the similarity between the
point clouds  $C^{(x)}$ and $C^{(y)}$ is that
there exists no canonical parameterization of these sets allowing for a
straightforward comparison with established distance measures, as
can be done for streamlines for instance \cite{FerstlBW16}.
 $C^{(x)}$ and $C^{(y)}$  may not even be of the same size.
This observation
motivates the transformation of $C^{(x)}$ and $C^{(y)}$ into an alternate
representation that would yield a natural parameterization directly usable with
standard distance measures.

\begin{figure}
  \includegraphics[height=3.85cm]{colorMap_blueGreen.jpg}
  \hfill
  \includegraphics[height=3.85cm]{persistenceMap.jpg}
 \vcaption{
 Persistence maps (bottom, maxima only) for four 2D scalar fields (top, inset:
terrain view with critical points): $f^{(0)}$ (a), $f^{(1)}$ (b), $f^{(2)}$
(c), $f^{(3)}$ (d).
  When using constant \julien{functions}{values} for $\alpha(c)$ and
$\sigma(c)$, $\phi$
estimates the local density in critical points (b, right inset with maxima in
red). In this example, the $L_2$ distance separating $f^{(0)}$ from $f^{(1)}$,
$f^{(2)}$ and $f^{(3)}$ is
$200.79$,
$41.03$ and
$14.18$ respectively. In
contrast, the same distance from $\phi^{(0)}$ to $\phi^{(1)}$,
$\phi^{(2)}$ and $\phi^{(3)}$ is
$11.09$,
$2.84$ and
$49.01$ respectively.
This different ordering indicates that the $L_2$ distance between persistence
maps is less sensitive to noise and global shifts in data values and better
discriminates changes in salient features.
 }
 \label{fig_persistenceMap}
\end{figure}

\newcommand{\vspace{-1ex}}{\vspace{-1ex}}

\subsection{Formulation}
\label{sec_persistenceMap_eq}
Breckner and M\"oller \cite{DBLP:journals/cgf/BrucknerM10} faced a similar
problem in the context of isosurface comparison and introduced a signed
distance field transform, measuring the distance between each vertex of
$\mathcal{M}$ and the considered isosurface.
Then, the similarity between two isosurfaces can be evaluated based on the
standard $L_2$ distance between their distance transforms.
The same idea has been later used by
Ferstl et al. \cite{FerstlKRW16} in a context that\brian{highly}{} resembles
our
setting
(isocontour clustering for level set variability analysis and visualization).
We build upon this strategy \brian{for}{to construct} persistence maps. In particular, one could
derive \brian{such}{} a distance transform for a critical point set $C$, by
considering for each vertex $v \in \mathcal{M}$, the distance to the closest
critical point of $C$.
However, such a distance transform would be highly \brian{sensible}{sensitive} to the presence of
noise in the data since all the critical points of $f$ would be
considered for its computation.
Therefore, it
is necessary to develop a
transformation where the contribution of each critical point could be weighted
by an importance measure, such as topological persistence
\cite{edelsbrunner02}. While such a weighting strategy is difficult to elaborate
for distance fields, it is much easier to derive for sums of gaussian radial
basis functions. In particular, let $\phi : \mathcal{M} \rightarrow \mathbb{R}^+$
be the following scalar function, where $\alpha(c)$ and $\sigma(c)$ are scalars
controlling the
amplitude and spatial spread of the contribution of \julien{}{the critical
point} $c$:
\begin{eqnarray}
\vspace{-1ex}
  \label{eq_persistenceMap}
 \phi(v) = \sum_{c \in C} \alpha(c) e^{-{{||v -
c||_2^2}\over{2 \sigma(c)^2}}}
\vspace{-1ex}
\end{eqnarray}
If constant values are considered for both $\alpha(c)$ and $\sigma(c)$, $\phi$
is a measure of the local critical point density
(\autoref{fig_persistenceMap}, inset).
To limit the importance of
noisy critical
points in this density estimation and to highlight  salient
features\julien{instead}{}, we
use\julien{topological}{} persistence as an importance measure in the
expressions of
$\alpha(c)$ and $\sigma(c)$  as follows, where $P(c)$ stands for the persistence
of the critical point pair containing $c$ in $\persistenceDiagram{f}$:
\begin{eqnarray}
\vspace{-1ex}
 \alpha(c) =  P(c), \quad \sigma(c) = \gamma P(c)
\vspace{-1ex}
 \label{eq_persistence}
\end{eqnarray}
$\gamma$
controls the focus
that is
given
to salient features in terms of their spread in the \brian{geometrical}{spatial}
domain.
\brian{(d}{D}istances are normalized with regard to the bounding box
diagonal\brian{)}{}. We have found that
$\gamma = 0.1$
is a good value in practice.
\julien{Note
that
t}This representation resembles the
notion of persistence images \cite{persistenceImages}, which
focuses on range rather than domain density.

\subsection{Distances}
\label{sec_distanceMatrix}
Since they are both defined on the same \brian{geometrical}{spatial} domain $\mathcal{M}$, the
persistence maps $\phi^{(x)}$ and $\phi^{(y)}$ of two critical points sets
$C^{(x)}$ and $C^{(y)}$ benefit from a common parameterization and their
distance can be estimated with standard distance measures, such as the $L_2$
norm:
\begin{eqnarray}
\vspace{-1ex}
 ||\phi^{(x)} - \phi^{(y)}||_2 = \sqrt{\sum_{v \in \mathcal{M}} \big(\phi^{(x)}(v)
- \phi^{(y)}(v)\big)^2}
\vspace{-1ex}
\label{eq_distance}
\end{eqnarray}
By design, this metric is robust to noise, since the contribution of critical
points to the persistence maps is weighted by their
persistence \julien{.}{ (\equref{eq_persistence}). Hence, small persistence
pairs
(typically corresponding to low amplitude noise,
\autoref{fig_persistenceDiagram}(b))
will have a
negligible contribution in practice to the persistence maps
(\autoref{fig_persistenceMap}(b),
further discussion in
\autoref{sec_limitations}).
} This is
important since small scale additive noise often occur in practice even for
\brian{}{assumed} smooth simulation data.
This metric is also robust by design to global
variations in data values which do not change the critical point spatial
layout, since the actual data values are not taken into account in the
persistence map. In contrast,
the standard $L_2$ distance $||f^{(x)} - f^{(y)}||_2$
would tend to miss the possible preservation
of salient features in the presence of global shifts in data values,
as\brian{it}{} can
be the case with seasonal effects in climate data.
Finally, the distance
$||\phi^{(x)} - \phi^{(y)}||_2$
is specifically designed to penalize changes in the layout of salient
critical points.
The above properties are illustrated in \autoref{fig_persistenceMap}, which
shows persistence maps on a toy example,
$f^{(0)}$, along with three variants: $f^{(1)}$ with additive noise, $f^{(2)}$
which contains a global shift in data values (slope), and $f^{(3)}$ which
contains an additional salient feature. For this data, we have:
$||f^{(0)} - f^{(3)}||_2 < ||f^{(0)} - f^{(2)}||_2 < ||f^{(0)} - f^{(1)}||_2$.
In other words, with the $L_2$ distance between the actual data values,
the noise affected dataset ($f^{(1)}$) is the most distant to the original \brian{one}{}
($f^{(0)}$), while the dataset with a drastic change in critical point layout
($f^{(3)}$) is the closest. In contrast, the $L_2$ distance between the
corresponding persistence maps results in a different ordering:
$||\phi^{(0)} - \phi^{(2)}||_2 < ||\phi^{(0)} - \phi^{(1)}||_2 < ||\phi^{(0)} -
\phi^{(3)}||_2$. In other words, with the persistence map metric, the closest
data set from
the original \brian{one}{} ($f^{(0)}$) is the one which better preserves the critical
point layout ($f^{(2)}$), while the most distant \brian{one}{ }is the one which
changes it the most ($f^{(3)}$).
This indicates that the metric $||\phi^{(x)} - \phi^{(y)}||_2$
is indeed more robust to noise and global shift in data values than
$||f^{(x)} - f^{(y)}||_2$ and that it better describes variations in
the layout of salient critical points.
\julien{}{Our distance (Eq.~\ref{eq_distance}) resembles the
\emph{kernel distance} defined for generic point cloud data
\cite{PhillipsWZ15}. In contrast, persistence maps focus on the critical points
of a scalar field (instead of generic point clouds). This allows to
additionally consider in the density estimation
the persistence of each critical point
 as an importance measure
(Eq.~\ref{eq_persistence}),
to highlight salient
features and reduce the effect of noise.
}

\section{Space of persistence maps}
\label{sec_clustering}

As described above, the $L_2$ distance between persistence maps is a good
candidate to compare the spatial layout of critical points between two members.
Based on this metric, a distance matrix $\Phi$ is computed
for the entire ensemble, with $\Phi_{xy} = ||\phi^{(x)} - \phi^{(y)}||_2$,
and then normalized.
In this section, we exploit this distance matrix to visualize and
identify the main trends in critical point layouts within the
ensemble.

\subsection{Low dimensional embedding}
\label{sec_eigenMap}
To directly visualize the global trends in critical point layouts, we first
consider a low dimensional embedding of the ensemble into a space of
persistence maps, noted $\mathcal{P}$,
where each map $\phi$ is represented by a point and where distances between
points denote distances between persistence maps. For this, we employ
established
methods
for non-linear dimensionality
reduction \cite{wickelmaier2003introduction, borg05}.
In particular, we focus on the spectral approach by Belkin et al.
\cite{belkin2003laplacian} based on Laplacian eigenmaps, which has been shown
to
better preserve locality than standard methods such as principal component
analysis \cite{surveyPCA} or Isomap \cite{tenenbaum00}. This property is
particularly beneficial if clustering is subsequently considered, which is the
case in our framework (\autoref{sec_subClustering}). For completeness, we
briefly sketch the main steps of the Laplacian eigenmap approach and we refer
the reader to
\cite{belkin2003laplacian} for further details.

First, an adjacency graph is constructed, where the $x^{th}$ node represents
the $x^{th}$ ensemble member and where arcs are introduced between the node
$x$ and its $n_n$ nearest neighbors
(according to the distance matrix $\Phi$).
In practice, we set $n_n$ to a default recommended value ($5$).
 Next, a weight matrix $W$ is constructed such that
$
W_{xy} = 1
$
if $x$ and $y$ are connected in the adjacency graph and $0$ otherwise.
A diagonal matrix $D$ is also established such that $D_{xx} = \sum_{y}
W_{xy}$. Then, the Laplacian, $L$, of the adjacency graph is considered as $L = D
- W$, which is a symmetric, positive semidefinite $n \times n$ matrix
\cite{belkin2003laplacian}. Finally,
the low-dimensonal space $\mathcal{P}$ is constructed by
projecting each ensemble member along the $n_d$ first eigenvectors
$\psi \in \mathbb{R}^n \times \mathbb{R}^n$ of $L$, which are solutions of the
generalized eigenvector problem: $L \psi = \lambda D \psi$ (where $\lambda \in
\mathbb{R}^n$ stands for the $n$ eigenvalues of $L$). In practice, the first
eigenvector $\psi_0$ is discarded, as suggested by Belkin et al.
\cite{belkin2003laplacian}. Thus, the $x^{th}$ ensemble member is then
embedded at position
$\psi^{(x)} = \big(\psi_1(x), \dots, \psi_{n_d}(x)\big)$.
Since the first eigenvectors of $L$ are usually considered to be the most
informative \cite{vonLuxburg2007}, for visualization purpose, we typically
represent planar layouts of the space of persistence maps $\mathcal{P}$ by only
considering the first two components of this vector ($\psi_1(x), \psi_2(x)$).

\subsection{Persistence map clusters}
\label{sec_subClustering}

\autoref{fig_overview}(d) shows a typical 2D layout of the first two dimensions
of
 the space of persistence maps, $\mathcal{P}$,
for a
toy ensemble dataset. As shown in this example, clear \brian{patterns,}{patterns
that}
correspond\brian{ing}{} to distinct trends in critical point layout emerge from this
visualization. To quantitatively analyze these patterns, we next employ
 clustering algorithms.
In
particular, we employ the popular $k$-means algorithm \cite{historicKmeans},
which has been shown to be well suited for a combined usage with spectral
emdedding (\autoref{sec_eigenMap}), yiedling the notion of \emph{spectral
clustering} \cite{DBLP:conf/cvpr/ShiM97}. This algorithm is based on the
classical Lloyd relaxation scheme \cite{lloyd57} which,
given an initial assignment of $k$ cluster centroids chosen among the data
points,
assigns each data point to the cluster of its closest
centroid. Next, for each cluster, a new centroid is selected as the point being
the closest to the new cluster barycenter and the procedure is iterated until
convergence. Note that for the above clustering procedure, the spectral
clustering literature recommends to only use the $k$ first components of
$\psi^{(x)}$ \cite{vonLuxburg2007}, although we found in practice that with our
implementation, the most stable results were obtained for $n_d = k - 1$.

The number $k$ of clusters to be considered is particularly important as it
directly corresponds to the number of trends which can be visualized in the
ensemble. While we offer users the possibility to explicitly specify
pre-defined values of $k$, we also provide an automatic estimation procedure.
Several statistical measures have been studied for the automatic estimation of
$k$, such as the Bayesian Information Criterion \cite{pelleg00}. In the
specific case of spectral clustering however, it has been shown that the
eigenvalues of the Laplacian matrix (\autoref{sec_eigenMap})
already exhibit important hints regarding cluster numbers and
that they are particularly
useful to identify
proper values for $k$.
In particular, the first eigenvalue $\lambda_k$ resulting in a significant
\emph{eigengap} $\delta_k = |\lambda_k - \lambda_{k+1}|$ is usually considered
as a good value for $k$ (see von Luxburg \cite{vonLuxburg2007}
for formal
arguments based on perturbation theory). Thus, in practice, we provide
as an
initial guess for $k$, the position of the first local maximum of eigengap
$\delta_k$.
\autoref{fig_eigenGaps} plots the evolution of the eigengaps for the example of
\autoref{fig_overview}. As shown in this figure, the appropriate number of
clusters for this specific dataset indeed corresponds to the first local
maximum of
eigengap ($k = 3$). Note that several other local maxima of eigengaps occur for
higher eigenvalues. We also offer users the possibility to interactively
explore them individually.

\begin{figure}
  \centering
  \includegraphics[width=0.9\linewidth]{eigengaps.jpg}
  \vspace{-1.5ex}
 \vcaption{Eigengaps computed for the example shown in \autoref{fig_overview}.
Our approach uses the position of the first local maximum of eigengap as an
initial value for the number of clusters $k$. Our framework also offers the
possibility to explore the other maxima, as well as arbitrary $k$ values.}
\vspace{-1ex}
  \label{fig_eigenGaps}
\end{figure}

\section{Confidence regions for persistence map clusters}
\label{sec_confidenceRegions}
The major trends in critical point layout in the ensemble
can be identified by clustering the persistence maps (\autoref{sec_clustering}).
In this section, we describe how to visualize\brian{, back on the geometrical domain,}{}
the spatial variability of critical points within each of the identified
clusters.

\subsection{Per cluster variability analysis}
\label{sec_mandatoryCriticalPoints}
The clustering procedure described in the previous section
identifies
disjoints subsets of ensemble members which share a common
pattern in critical point layout. Let $f^{(X)} = \{f^{(x_0)}, f^{(x_1)}, \dots
f^{(x_m)}\}$ be such a subset ($m < n$). To understand the variability
of critical points within this subset, one needs  first \emph{(i)} to
identify a common topological structure among all of the members of $f^{(X)}$
and second \emph{(ii)} to analyze its spatial variability.
As discussed in \autoref{sec_relatedWork}, several approaches have been
proposed to study the positional uncertainty of critical points. Among those,
we focus on the approach based on mandatory critical points
\cite{tierny_ev14}
since it
is
based on point-wise intervals
and is\brian{}{,} therefore\brian{}{,} well suited for the analysis of
ensemble data, where no specific assumption can be made about the structure of
the point-wise random variables locally modeling the data variability. For
completeness, we briefly sketch the main steps of this method and refer the
reader to \cite{tierny_ev14} for further details.

First, pointwise scalar value bounds are extracted as two scalar fields $f^- :
\mathcal{M} \rightarrow \mathbb{R}$ and $f^+ : \mathcal{M} \rightarrow \mathbb{R}$,
such that $f^-(v) = \min_{f^{(x)} \in f^{(X)}} f^{(x)}(v)$ and $f^+(v) =
\max_{f^{(x)} \in f^{(X)}} f^{(x)}(v)$.
Given an isovalue $i$, let $C^-(i)$ be a connected component of sub-level set
of $f^-$ (blue region in \autoref{fig_candidateRegions}). By construction, for
each vertex
$v$ in $C^-(i)$, there exists at least one member $f^{(x)}
\in f^{(X)}$ for which $f^{(x)}(v) \leq i$. Then, there exists a
member $f^{(x)}$
for which
a connected component of sub-level set $C^{(x)}$  passes through $v$ at isovalue
$i$ (gray regions in \autoref{fig_candidateRegions}). Then, $C^-(i)$ is called a
\emph{candidate region} for the appearance of a
local
minimum (responsible for the creation of the component $C^{(x)}$ in $f^{(x)}$).

\begin{figure}
\vspace{-2ex}
  \centering
\includegraphics[width=.9\linewidth]{mandatoryCriticalPoint.jpg}
\includegraphics[width=0.975\linewidth]{colorMap_blackWhite_h.jpg}
 \vcaption{
Sub-level sets of $f^-$ (blue) and $f^+$ (green) at three different isovalues
$i < f^+(m_1) < f^+(m_2)$ with $m_1$ and $m_2$ being minima of $f^+$. The
sub-level set components $C^{(x)}$ and $C^{(y)}$ of two ensemble members
$f^{(x)}$ (left) and $f^{(y)}$ (right) are shown in gray.}
 \label{fig_candidateRegions}
\end{figure}

Let $m_1$ be a minimum of
$f^+$. Since $f^-$ and $f^+$ are nested, $m_1$ must be located inside a
connected component of sub-level set of $f^-$ at isovalue $f^+(m_1)$.
Let $C^-\big(f^+(m_1)\big)$ be that region and let us first consider that $m_1$
is the only minimum of $f^+$ in it.
At
isovalue $f^+(m_1)$, by construction, \emph{all} the members $f^{(x)} \in
f^{(X)}$ are such that $f^{(x)}(m_1) \leq f^+(m_1)$. This means, that for
\emph{all} the members $f^{(x)}$ of the subset $f^{(X)}$, there exists a
connected component of sub-level set
$C^{(x)}$
passing through $m_1$ (gray components containing $m_1$ in
\autoref{fig_candidateRegions}(c) and \autoref{fig_candidateRegions}(d)). In
particular, this connected component was created at an earlier isovalue, at
one of the vertices of the corresponding candidate region,
$C^-\big(f^+(m_1)\big)$. Overall, this means that $C^-\big(f^+(m_1)\big)$ must
contain at least one minimum (responsible for the
initial creation of the component $C^{(x)}$) for all the members
of $f^{(X)}$. Thus, the region $C^-\big(f^+(m_1)\big)$ is
called  a \emph{mandatory minimum}: a minimal connected component $C^-$
of $\mathcal{M}$, associated with a minimal interval $I^- = [min_{v \in C^-}
f^-(v), min_{v \in C^-}f^+(v)]$, such that any $f^{(x)}$ contains at least one
minimum $m_1$ in $C^-$ with $f^{(x)}(m_1) \in I^-$.

\autoref{fig_candidateRegions} illustrates this process where candidate regions
(blue) may contain several connected components $C^{(x)}$ (gray) of
sub-level set of ensemble members ($m_1$ is shown in green).
Note that if the candidate region contains a second minimum $m_2$ such that
$f^+(m_1) < f^+(m_2)$, this implies that the sub-level set of all members pass
through $m_2$ as well. However, they may do so with the component $C^{(x)}$
which already contains $m_1$ (\autoref{fig_candidateRegions}(e)). Thus, the
existence of such a second minimum $m_2$ does not necessarily imply the
existence of an additional minimum in $f^{(x)}$, as it is the case in
\autoref{fig_candidateRegions}(e) (as opposed to
\autoref{fig_candidateRegions}(f)). As discussed in \autoref{sec_results}, this
observation may have important practical implications,
as it may prevent the detection of a mandatory critical point in case of
high pointwise value variability $|f^+(v) - f^-(v)|$.

Other types of
mandatory critical points are extracted similarly, as
described in \cite{tierny_ev14}. Eventually, each cluster $f^{(X)}$ is
associated with a collection of mandatory critical points, which describe the
spatial variability of the common topological structure found among its
members.

\subsection{Global visualization}
The mandatory critical points can be visualized for each cluster independently,
by displaying each critical component with a colored region.
Additionally, \julien{a
likelihood of appearance of critical points can be computed and visualized
within each region.}{the positional variability of critical points within each
region can be estimated and visualized as follows.}
Given a histogram
representation of the data values taken by a vertex $v$ in $f^{(X)}$, we
estimate this \julien{likelihood}{variability} as the probability of $v$ to
admit a scalar value
within the critical interval of each mandatory critical point. Finally, we
estimate the overall probability of appearance of a mandatory critical point as
the proportion between the size of $f^{(X)}$ and the total number\brian{}{,} $n$\brian{}{,} of
members in the ensemble. As shown in \autoref{fig_overview} (right), this
probability can be visualized in the form of a barplot.
\brian{Overall t}{T}he Persistence Atlas is then \brian{composed of }{created
from }a collection of confidence maps
(\brian{which can be }{}composed together) \brian{which provides,}{that provide} for each major trend found in
the ensemble, confidence regions for the \brian{appareance}{appearance} of critical points\brian{,}{} along
with their probability of appearance, as well as\julien{an estimation of}{,}
their \julien{}{individual critical point} spatial
variability given by the above\julien{likelihood}{}
estimation \julien{\brian{,}{.}
\brian{as shown in}{See }
\autoref{fig_overview}.}{(\autoref{fig_overview}).}

\section{Results}
\label{sec_results}

This section presents experimental results obtained on a desktop computer with
a Xeon CPU (2.6 GHz, 2x6 cores), with 64 GB of RAM. For the computation
of the persistence diagrams, we used the
Topology ToolKit (TTK) \cite{ttk17}.
For the spectral embedding and clustering, we adapted  classes
from the \emph{scikit-learn} package \cite{scikitlearn}. The other components
of our
approach have been implemented as TTK modules.

\begin{figure}
\centering
  \vspace{-2ex}
\includegraphics[height=5cm]{colorMap_blackWhite.jpg}
\hfill
\includegraphics[height=5cm]{vortexStreet_criticalPoints.jpg}
\hfill
\includegraphics[height=5cm]{colorMap_blueGreen.jpg}
 \vcaption{Five representative members of the ensemble of
\autoref{fig_teaser}. Critical points (minima and
maxima) are shown with spheres scaled by their persistence (left).
Persistence maps are shown on the right.}
  \label{fig_vortexStreet_pm}
\end{figure}

\begin{figure}
  \vspace{-2ex}
 \includegraphics[width=\linewidth]{isabella_criticalPoints.jpg}

 \includegraphics[width=\linewidth]{colorMap_blueGreen_h.jpg}

 \vcaption{Three representative members for the \emph{Isabel}
volumetric ensemble (left). Local maxima (scaled by
persistence) of velocity magnitude capture the eyewall of the hurricane,
high wind speed peripheral regions as well as the hurricane's tail (second and
third
row). The persistence maps (local maxima only, right) capture
these
subtle features by construction, while taking less into account noisy critical
points (smallest spheres).
}
\label{fig_isabella_cp}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[height=2.3cm]{colorMap_blueGreen.jpg}
  \hfill
  \includegraphics[height=2.3cm]{distanceMatrix.jpg}
 \vcaption{Two representative members of the \emph{Starting vortex} ensemble
((a) and (b)) along with their critical points scaled by persistence. Given the
small spatial extent of these features, the $L_2$ norm between the actual data
values fails at capturing similarities between members belonging to the
bottom configuration (b), as denoted by the corresponding distance matrix (c),
where distances are important (green) in the upper-right corner. In contrast,
the
distance matrix (d) computed from persistence maps ((e) and (f)) exhibits much
smaller distances (blue) between these members, facilitating their gathering in
the
low dimensional space $\mathcal{P}$.}
 \label{fig_distanceMatrix}
\end{figure}

\begin{table}
  \centering
  \vcaption{Running time (in seconds, with 12 cores) for the different steps of
  our approach: persistence maps ($P.M.$, \autoref{sec_persistenceMap_eq}),
  distance matrix ($D.M.$, \autoref{sec_distanceMatrix}),
  low-dimensional embedding ($E.$, \autoref{sec_eigenMap}),
  clustering ($C.$, \autoref{sec_subClustering}) and mandatory critical points
($M.C.P.$, \autoref{sec_confidenceRegions}).}
  \label{tab_runningTimes}
  \scalebox{0.65}{
    \centering
    \begin{tabular}{l|rr|r|rrrr|r}
    \toprule
    Dataset &$n$ & $|\mathcal{M}^0|$ &
    P.M. &
D.M. & E. & C. & M.C.P. & \textbf{Total}\\
    \midrule
    Gaussians (\autoref{fig_overview}) & 100 & 262,144
      & 57.28
      & 1.03
      & 0.67
      & 0.08
      & 2.53
      & \textbf{61.59} \\
    Vortex street (\autoref{fig_teaser}) & 45 & 30,000
      & 2.28
      & 0.02
      & 0.67
      & 0.09
      & 0.22
      & \textbf{3.28} \\
    Starting vortex (\autoref{fig_startingVortex}) & 12 & 1,500,000
      & 61.44
      & 0.09
      & 0.65
      & 0.07
      & 9.08
      & \textbf{71.33} \\
    Isabel (\autoref{fig_isabella}) & 12 & 3,125,000
      & 168.70
      & 0.18
      & 0.63
      & 0.07
      & 41.84
      & \textbf{211.68} \\
    Sea Surface Height (\autoref{fig_sea}) & 48 & 1,036,800
      & 290.25
      & 0.99
      & 0.65
      & 0.08
      & 8.38
      & \textbf{300.35} \\
    \bottomrule
    \end{tabular}
  }
\end{table}

\begin{figure*}
  \vspace{-2ex}
\centering
  \includegraphics[height=2.75cm]{colorMap_blueGreen.jpg}
  \hspace{2ex}
 \includegraphics[height=2.75cm]{startingVortex.jpg}
 \vcaption{Mandatory critical point for the entire \emph{Starting vortex}
ensemble \julien{}{(left)} and the corresponding
\julien{likelihood of critical point appearance}{critical point positional
variability estimation} (\julien{}{color map}, left).
Given the trend variability of this ensemble, this global extraction identifies
only one, very large, mandatory maximum (colored region) describing both
regimes, although these two vortices never occur simultaneously in the data.
The persistence atlas automatically identifies the two trends present in the
data, as shown in the planar view (center), resulting in much more accurate
predictions for the appearance of the distinctly identified vortices (green
and blue regions, right).}
\label{fig_startingVortex}
\end{figure*}

\begin{figure*}
  \centering
  \includegraphics[height=3.95cm]{colorMap_blueGreen.jpg}
  \hspace{4ex}
 \includegraphics[height=3.95cm]{isabella_atlas.jpg}
 \vcaption{Mandatory critical point for the\julien{entire}{} \emph{Isabel}
ensemble,
viewed from above and below (a). Given the trend variability of this ensemble,
this\julien{global}{}
extraction identifies only one, very large, mandatory maximum
\julien{(colored region)}{(region colored by critical point positional
variability)} merging the three
distinct states of the hurricane
(formation, drift and landfall). In contrast, the persistence atlas manages to
isolate these three states and provides\julien{much}{} more accurate
confidence regions
for the position of the hurricane eyewall (colored regions).}
\label{fig_isabella}
\vspace{-10pt}
\end{figure*}

\subsection{Experiments}
Figures \ref{fig_teaser} and \ref{fig_vortexStreet_pm} to \ref{fig_sea}
report various experiments on simulated and acquired 2D and 3D ensemble
datasets. \autoref{fig_teaser} presents our entire approach on an
ensemble of 45 von K\'arm\'an vortex streets, where the considered scalar
data is the orthogonal component of the curl taken at a fixed time-step, for
five different fluids of distinct viscosity (9 runs per fluid, each run with
varying Reynolds numbers).
\julien{}{For such scalar fields, local extrema
are typically considered as
reliable estimations of the center of the vortices.}
\julien{The critical points}{Extrema}
of a few representative
members
(\autoref{fig_teaser}(a)) exhibit clearly distinct layout patterns, in terms of
both the position and number of vortices, revealing high \emph{spatial} and
\emph{trend} variabilities within the ensemble. The mandatory critical points
estimated for the entire ensemble are particularly conservative given these
variabilities: only one region is extracted for each side of the street (one
for minima, one for maxima). The persistence atlas manages to automatically
identify five clusters in the ensemble, corresponding to distinct critical
point layouts (one per viscosity regime). The mandatory critical points
extracted from these clusters provide more accurate and useful predictions for
the appearance of vortices (colored regions in (d) to (h), one color per
cluster). In particular, the persistence atlas reveals that the number of
vortices increases with the Reynolds numbers (from left to right: 6, 10, 12, 14
and 15 vortices) while the spatial variability of each vortex tends to decrease
for increasing Reynolds numbers (smaller mandatory critical points).
\autoref{fig_vortexStreet_pm} illustrates persistence maps for five
representative members of the ensemble and shows how salient features are
captured by this representation.
\autoref{fig_isabella_cp} shows persistence
maps on a volumetric ensemble composed of groups of key timesteps
(formation, drift and landfall) in the simulation of the Isabel hurricane
\cite{scivisIsabel}.
\julien{}{For such datasets, the eyewall of the hurricane is typically
characterized by high wind velocities (green regions,
\autoref{fig_isabella_cp}, left) and contains salient maxima.}
In particular, this figure shows that subtle features of
the hurricane (eyewall, high wind speed peripheral regions and hurricane's
tail) are well captured by local maxima of the wind velocity magnitude and by
the corresponding persistence maps. As discussed in
\autoref{sec_distanceMatrix}, the $L_2$ norm between persistence maps is more
suited to our purpose than the $L_2$ norm between the actual data values, since
it is more robust to noise and global shifts in data values, while better
discriminating changes in salient features (\autoref{fig_persistenceMap}).
\autoref{fig_distanceMatrix} further exemplifies this observation on the
\emph{Starting vortex} ensemble, which includes 12 runs of a 2D
simulation of the formation of a vortex behind a wing, for two distinct wing
configurations. The considered scalar field is the curl orthogonal
component \julien{}{and salient extrema are expected at the center of
vortices}.
Given the
small spatial extent of the features behind the wing, the $L_2$ norm between
the actual data
values fails at capturing similarities between members belonging to the
second configuration, as denoted by the corresponding distance matrix,
where distances are important in the upper-right corner.
In particular, two members belonging to the same wing configuration are
reported by this distance as the two furthest members (darkest green entry).
In contrast, the
distance matrix computed from persistence maps exhibits much
smaller (resp. higher) distances between the members belonging to a common
(resp. distinct) wing configuration.

\autoref{fig_startingVortex} shows the persistence atlas for the \emph{Starting
vortex} ensemble. Given the trend variability of this dataset, the mandatory
critical points computed from the entire ensemble exhibit
only one, very large, mandatory maximum (colored region) describing the
appareance of vortices for both wing configurations,
although these two vortices never occur simultaneously in the data.
The persistence atlas automatically identifies the two trends present in the
data, as shown in the planar view (center), resulting in much more accurate
predictions for the appearance of the distinctly identified vortices (green
and blue region, right).
\autoref{fig_isabella} shows the persistence atlas for
the \emph{Isabel} ensemble. Similarly to the previous example, mandatory
critical points computed from the entire ensemble identify only one, very
large, mandatory maximum, which merges the three distinct states of the
hurricane. In contrast, the persistence atlas manages to isolate these three
states and provides much more accurate confidence regions for the position of
the hurricane eyewall. Note that this example is the only dataset for which the
initial automatic suggestion for the number of clusters $k$
provided by the eigengap heuristic needed
adjustment. All the other results have been generated with the automatic
suggestion.
\autoref{fig_sea} shows the persistence atlas for the \emph{Sea surface height}
ensemble, which is composed of 48 observations taken in January, April, July
and October 2012
(\href{https://ecco.jpl.nasa.gov/products/all/}{https://ecco.jpl.nasa.gov/products/all/}).
\julien{}{For such datasets, salient extrema in the height are expected at the
center of eddies.}
The mandatory critical points globally extracted on the entire ensemble identify
only few features (\autoref{fig_sea}(a)), due to the high pointwise data
variability (\autoref{sec_mandatoryCriticalPoints}). The clustering
automatically performed by our approach based on the persistence maps correctly
identifies four clusters, corresponding to the four seasons: winter (c), spring
(d), summer (e) and fall (f). This seasonal decomposition drastically
reduces pointwise data variability and enables mandatory critical points to
identify many more structures, corresponding to clockwise and counterwise
vortices (minima and maxima) and revealing complex structures in the Gulf
stream area (insets). Note that, for this example, due to the high number of
critical points and their respective proximity, the parameter $\gamma$,
controlling the spread of salient features in the persistence map, has been set
to $0.01$ instead of the default value ($0.1$).

\subsection{Time performance}
\label{sec_performance}
 \autoref{tab_runningTimes} presents the running times we obtained
\brian{with}{for} the datasets presented in this paper.
The most time consuming \brian{part}{portion} of our approach is the computation of the
persistence maps, which typically needs to be run for each ensemble member
\brian{in}{as} a
pre-process. Since the number of pairs in the diagram is typically proportional
to the number $|\mathcal{M}^0|$ of vertices in the domain, this part
requires $O(n\times|\mathcal{M}^0|^2)$ steps overall.
In practice, to accelerate this computation, we
ignore all pairs with a persistence less than $1\%$ of the total function
range.
The distance matrix computation
takes $O(n^2 \times|\mathcal{M}^0| )$ steps, but
since $n$ is
typically much smaller than $|\mathcal{M}^0|$, the computation time for this step
is small in practice. Both the spectral
embedding (\autoref{sec_eigenMap}) and clustering (\autoref{sec_subClustering})
employ iterative solvers but these computations are typically the fastest steps
of the pipeline. \julien{Finally, t}{T}he computation of the mandatory critical
points for
each cluster admits quadratic complexity  $O(|\mathcal{M}^0|^2)$.

Most of these steps can be trivially parallelized.
The persistence diagram is computed in parallel \cite{gueunet_ldav17} and the
persistence map can be evaluated independently for each vertex.
Each entry of the distance matrix $\Phi$
(\autoref{sec_distanceMatrix}) can be computed independently.
Finally, mandatory critical points are computed in parallel for
each cluster. As reported in \autoref{tab_runningTimes},
once the persistence maps have been computed in a pre-process, the rest
of the framework is sufficiently fast to allow interactiv\julien{e
exploration}{ity}.

\subsection{Comparison}
In this section, we compare our approach to
\julien{
a \emph{baseline} approach,
which consists in simply clustering persistent critical points in the
geometrical domain, by using a vanilla implementation of spectral clustering,
combined with our eigengap heuristic for the automatic suggestion
of the number of clusters (\autoref{sec_subClustering}).}{alternative critical
point clustering strategies. First, we consider a baseline approach,
which consists in simply clustering persistent critical points in the
spatial domain, by using a vanilla implementation of spectral clustering,
combined with our eigengap heuristic for the automatic suggestion
of the number of clusters (\autoref{sec_subClustering}).}
Once such clusters
have been computed, this baseline strategy evaluates confidence regions for the
appearance of critical points by considering the convex hull of each cluster
in the \julien{geometrical}{spatial} domain.
As shown in \autoref{fig_comparison}(a), this simple strategy provides
unsatisfactory results for the von K\'arm\'an vortex street ensemble
(\autoref{fig_teaser}) since features which never occur simultaneously in the
ensemble are clustered based on their proximity. In particular, the extracted
clusters mix the two types of vortices (right and left) and group them based
on their distance from the obstacle (bottom).

\julien{To compare this
strategy to the persistence atlas,}{To further evaluate our approach,} we
consider the ensemble from
\autoref{fig_overview}, which we split in half\brian{,}{} into a training and \brian{a }{}test
ensemble. The training ensemble is analyzed with \julien{}{\emph{(i)}} the
baseline \julien{strategy}{approach}
(\autoref{fig_comparison}(b))\julien{}{, \julien{}{\emph{(ii)}} a strategy
based on the kernel method
by Reininghaus et al. \cite{ReininghausHBK15}
(\autoref{fig_comparison}(c), where the distance matrix considered for
clustering has been generated
with the authors' implementation of the kernel method
\cite{implemKernel} run with default parameters)} and
\julien{}{\emph{(iii)}}
the persistence atlas
(\autoref{fig_comparison}(\julien{d}{e})). To quantitatively evaluate the
prediction
performance of \julien{the two}{these} approaches, we consider the persistent
critical points of
the test ensemble having a persistence higher than 20\% of the function range.
Next, the test critical points are
assigned to the confidence region in which they land in the domain
(spheres of matching colors in \autoref{fig_comparison}). As shown in
\autoref{fig_comparison}(b),
the baseline \julien{strategy}{approach}
\julien{yields an incorrect}{overestimates the} number of clusters.
\julien{Moreover}{In particular},
it
fails at clustering together features which always occur
simultaneously
(dark green and light blue clusters in \autoref{fig_comparison}(b)).
\julien{}{Kernel based methods for persistence diagrams \cite{ReininghausHBK15,
implemKernel} do not take the spatial embedding of
critical points into account (\autoref{fig_comparison}(c)) and
cluster members with the same persistence profile, irrespective of
the features' location. This leads to an
underestimated number of clusters: the blue and white clusters of
\autoref{fig_overview}, which both include a single very persistent maximum,
are erroneously merged although the corresponding features never occur
simultaneously in the ensemble.
Moreover, convex hulls obtained from this clustering overestimate the size of
the confidence regions in the presence of multiple salient features per
cluster.}
Even
when the correct clustering is explicitly provided
(\autoref{fig_comparison}\julien{(c)}{(d)}),
confidence regions based on convex hulls
miss
21\% of the persistent critical points of the test ensemble. In contrast, the
persistence atlas \julien{}{(\autoref{fig_comparison}(e))}
provides a correct prediction for 100\% of the critical points
of the test ensemble, which  illustrates the quantitative
performance of the persistence atlas regarding critical point prediction.

\begin{figure}
  \vspace{-2ex}
  \centering
 \includegraphics[width=0.9\linewidth]{seaSurfaceHeight_atlas.jpg}

 \includegraphics[width=\linewidth]{colorMap_blueGreen_h.jpg}
 \vspace{-3ex}
 \vcaption{Persistence atlas for the \emph{Sea surface height} ensemble.
 (a)  Mandatory critical points for the entire ensemble miss many features and
over-estimate the spatial variability of the extracted structures. (b)
The planar layout of the space of persistence maps shows the automatically
evaluated clustering of the ensemble (one color per cluster), which correspond
to the four seasons.
(c), (d), (e) and (f) The mandatory critical points for each of the
 identified clusters (respectively: winter, spring, summer and
fall) provide a more precise critical point variability estimation, revealing
hundreds of vortices (blue: minima, green: maxima).\vspace{-1.75ex}}
 \label{fig_sea}
\end{figure}
\subsection{Limitations}
\label{sec_limitations}
\julien{}{Our entire pipeline assumes that the input data is given as a
collection of piecewise linear scalar fields (\autoref{sec:background}). In
many applications~\cite{ScheuermannTH99}, this may be too restrictive
(motivating taylored interpolants for uncertainty modeling
\cite{SchlegelKS12}). However, generalizing the TDA arsenal to a larger set of
interpolants is a vast research topic (see \cite{NuchaBHN17} for an example)
which goes beyond the scope of this paper.}
Our approach focuses on $(0, 1)$ and $\big((d-1), d\big))$ persistence pairs,
which correspond to pairs only involving minima and maxima\brian{, and
p}{. P}ersistence
maps (\equref{eq_persistenceMap}) are \brian{}{therefore} only computed based
on
the location of \brian{}{either} the minima or maxima
(Figs.~\ref{fig_overview},
\ref{fig_startingVortex}
and \ref{fig_isabella_cp})\brian{}{,} or
both (Figs.~\ref{fig_vortexStreet_pm} and \ref{fig_sea}).
\BrianSays{doublecheck the change} Hence, saddle points
are not taken into account by our framework in its current form. However,
we \brian{}{have}
found that in practice the correspondence between saddle points and features of
interest was less clear in \brian{}{our} applications. Also, when the data
exhibits salient large
flat plateaus, persistent critical points can appear in arbitrary locations
inside these plateaus\brian{, }{. This can} potentially impair\brian{ing}{} the
stability of the persistence
map. However, we did not observe this behavior in practice on our datasets as
large plateaus, when they occurred, were not collocated with salient features.
We found in practice that using constant weights ($W_{xy} = 1$) for the
evaluation of the Laplacian of the adjacency graph of the persistence maps
(\autoref{sec_clustering}) resulted in more stable and accurate clusterings
than the second weighting strategy (based on heat kernels) described by Belkin
et al. \cite{belkin2003laplacian}. However, constant weights result in the
limitation that several members can be projected to the exact same point in the
low dimensional space when they exhibit a very similar neighborhood pattern in
the adjacency graph. \brian{Then}{In this case,} the number of visible points in
our planar layouts
may be smaller than the actual number of members. However, this non-uniqueness
in the embedding only occurs for persistence maps which are very close \brian{from}{to}
each other, hence it does not impact negatively the clustering \brian{}{or
analysis}.
\julien{Finally, a}{A}lthough the automatic suggestion for the number of
clusters $k$
provided satisfactory results for all but one example (where it needed to be
changed from $2$ to $3$, \autoref{fig_isabella}), an exhaustive interactive
exploration may be needed when there is no clear trend in the ensemble.
\julien{}{Finally, the persistence atlas currently displays simultaneously
mandatory critical points for all clusters. This may result in cluttered
visualizations due to overlapping. Although we provide users with the
possibility of refining this visualization to a selected subset of clusters,
improved strategies for the overall visualization of the atlas could be
researched in the future.}

\begin{figure}
  \vspace{-2ex}
  \includegraphics[width=\linewidth]{baseline.jpg}
  \vspace{-2ex}
  \vcaption{
  \julien{Comparison to a baseline strategy clustering critical points in the
geometrical domain.}{Comparison to alternative critical point clustering
strategies.}
\julien{(a) Features which never occur simultaneously in the ensemble but which
are
close from each other can be clustered together erroneously (dataset from
\autoref{fig_teaser}).}{(a) Clustering the critical points directly in the
spatial domain can cluster erroneously features which never occur simultaneously
in the ensemble but which are close from each other in the spatial domain
(dataset from
\autoref{fig_teaser}).}
(b) On the contrary, features which always occur simultaneously
(\autoref{fig_overview}) may not be clustered together if they are too distant
from each other, leading to
an \julien{incorrect}{overestimated} number of clusters (black spheres).
\julien{}{(c) Kernel based methods for persistence diagrams
\cite{ReininghausHBK15, implemKernel} do not take spatial embedding into account
and cluster together members with the same persistence profile
irrespective of the features' location, leading to an underestimated number
of
clusters.}
\julien{(c)}{(d)}
Even when the clustering is explicitly \julien{provided}{given}, confidence
regions based on
convex hulls miss 21\% of the critical points of the test ensemble (red
spheres). \julien{(d)}{(e)} The persistence atlas estimated from the training
ensemble provides a correct prediction for 100\%
of the critical points of the test ensemble (colored spheres).
\vspace{-1.5ex}}
  \label{fig_comparison}
\end{figure}

\section{Conclusion}
In this paper, we presented the \emph{Persistence Atlas}, an approach for the
visual analysis of the spatial variability of features of interest represented
by critical points in ensemble data. By analyzing the structure of the ensemble
in terms of patterns of critical point layouts, our method addresses
\emph{trend} variability, by identifying clusters of ensemble members which
share a common geometrical configuration of critical points. By computing
mandatory critical points for each cluster, our approach addresses
\emph{spatial} variability, by showing minimal regions where at least
one critical point is guaranteed to occur for each member of the cluster, hence
conveying the
 possible
extent of features
for each trend.
Our approach is based on the new notion of \emph{Persistence Map}, which
describes the local density in critical points and leverages topological
persistence to emphasize salient features,
and which has been shown to be well suited for the
purpose of comparing
geometrical layouts of critical points. We showed how to leverage spectral
embedding methods to provide low-dimensional views representing the main trends
found in the ensemble. We also showed how to leverage spectral clustering to
automatically identify revelant clusters of ensemble members and how to provide
relevant automatic guesses based on eigengaps for the number of clusters.
In practice, our approach has been shown to provide more accurate descriptions
of the variability of critical points than global methods, such as the original
mandatory critical points \cite{tierny_ev14}, which
either miss features or
considerably over-estimate spatial variability
in the presence of trend variability.
\julien{Moreover, in contrast to these methods,
our technique manages to reveal,
thanks to its clustering capabilities,
subtle dependences in the occurrence of features of interest, by separating in
different clusters features which never occur simultaneously in the ensemble
members.}{} We quantitatively evaluated the prediction accuracy of our method
and
showed that it compared favorably to a baseline strategy based on an
off-the-shelf clustering approach.
\julien{From a performance point of view,
the computation of persistence maps can be time consuming. However, this step
can be run as a pre-process and the rest of our framework can be run at
interactive rates, allowing interactive exploration.}{}
Our work extends recent advances in the visual analysis of spatial variability
in ensembles of geometrical objects, such as level sets \cite{FerstlKRW16}
or streamlines \cite{FerstlBW16}, to topological structures. In particular, we
focused in this paper on features of interest represented by critical points.
However, many more topological constructions
could benefit from a similar variability analysis based on such tailored
clustering strategies. For instance, the separatrices of the Morse-Smale
complex \cite{gyulassy_vis08, robins11} have been shown to excel at
representing filament structures in various applications, such as chemistry
\cite{chemistry_vis14, harshChemistry} or astrophysics \cite{sousbie11,
shivashankar2016felix}, and studying their trend and spatial variabilities
would be of tremedous help for the understanding of non-deterministic models in
these applications.
By first focusing on critical points, we believe we made a
first step in this direction, which will be helpful and inspirational for
future generalizations to other topological constructions.

\acknowledgments{
\vspace{-.5ex}
\footnotesize{
This work is partially supported by the BPI grant ``AVIDO'' (PIA
FSN2, reference P112017-2661376/DOS0021427),
NSF CRII 1657020, and NSF/NIH QuBBD 1664848.
We would like to
thank the reviewers for their thoughtful remarks and
suggestions. Julien Tierny
would like to
dedicate this paper to his son Otis.}
}

\clearpage

\bibliographystyle{abbrv-doi}

redpajama_set_name: RedPajamaArXiv


text:
You are here: Main » Tottenham's Kane has 'significant' ankle ligament damage
Tottenham's Kane has 'significant' ankle ligament damage
by Perry Erickson	- at April 13, 2019
Injured in the first quarter-final match of the Champions League against Manchester City, the Tottenham striker Harry Kane will miss three months, MSN reported.
The club gave no timetable for his return.
Tottenham have offered an update on the status of Harry Kane and Dele Alli following their respective injuries against Manchester City in the Champions League quarter-final first leg, with Kane suffering "significant" ligament damage in his left ankle.
But the club indicated he could play again this term as he "will continue to be assessed by our medical staff over the coming week".
We are going to do everything to recover him as soon as possible but there is only five weeks of the season to recover.
He added: 'He was so upset after the game, it was such a hard moment for him of course. e feel so disappointed to accept the situation but now it's much better.
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Kane has scored 22 goals in 37 appearances for England, who face the Netherlands in the semi-finals of the Nations League on 6 June.
Spurs said of Alli: "The England midfielder will now be assessed to determine his availability for this weekend's [Premier League] fixture against Huddersfield Town".
The second leg of their Champions League tie against City is on Wednesday at the Etihad Stadium. He damaged ligaments in the same ankle in January, which led to him missing almost two months and seven games.
Kane's absence could have huge ramifications for Spurs' season.
However on Friday Pochettino insisted: 'It's a stat that means nothing, Harry is one of the best players in the world and of course we're going to miss him.
Pochettino undoubtedly possesses other capable goalscorers in his squad but it is clear that not being able to call on Kane during the run-in equates to a huge blow, with Fernando Llorente the Argentine manager's only other option at centre forward.
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redpajama_set_name: RedPajamaCommonCrawl


text:
Tim Ellis (DJ Dynamic) Opens Bernie Sanders Rally
You may have heard of hip hop artist Killer Mike coming out in support of Bernie Sanders alongside a slough of other celebrity endorsements. You may have seen artists like Zedd speak out against candidates like Donald Trump, saying that he'd "Move Back To Germany" if he was elected president. Its becoming common news to see those who align with a progressive stance support him in his endeavors to become president. The Bernie Sanders campaign has notably picked up endorsements from the likes of Diplo, Steve Aoki, Porter Robinson and more. Not until today did a DJ actually help open up a Bernie Sanders Rally.
Today, Tim Ellis, who is known as DJ Dynamic to his friends and fans alike, opened the floor for a Bernie Sanders Rally in Rochester, New York, hosted at Monroe Community College. He's not a house hold name but is known from Rochester to Toronto and anywhere else happy hardcore has been popular in North America. Tim is a DJ who was spent a good portion of his life in the electronic music community and has seen both the positive and negative sides of the spectrum. He runs Nocturnal Commisions and has MC/DJ'd since the early 2000's, so he has seen this community at its best and its worst. This is what he said to supporters of Bernie Sanders this morning in Rochester.
I'm not Bernie, I'm sorry. My name is Tim I'm here, one of the field organizers for Bernie Sanders and I'm very happy to be back home in Rochester. The way that I got here, got involved with this campaign was a little different. I actually have been a Rave DJ for many years and you might say "How do you get here from there, right?" The thing is that when you go out to these raves you see a really warm and caring community and you also see a lot of people who are left behind. People who go to these parties and have nowhere to go afterwards, they go out to these parties because they have nowhere else to go. You might see them there every weekend and then during the week see them out in the street. And it doesn't have to be like that. And I knew that, and I felt that. And it was very exciting and heartwarming for me to find someone stand up and say that we could do better, and that's Bernie Sanders.
The message is clear, let's take care of each other. Check out Tim's opening speech from earlier today.
Want to see the full speech from the Bernie Sanders Rally in Rochester today? Click HERE!
Follow Tim Ellis aka DJ Dynamic on Social Media:
Website | Twitter | SoundCloud


Education is essential. This means that mother and father have a duty to be energetic in their kid's education. The restricted programs are permitting students to arrange for explicit jobs as opposed to giving them a normal education on a topic. One classroom trainer and twenty-5 to thirty young minds to fill with data can be a very troublesome feat to accomplish.
In the past twenty years or so, dad and mom did not see education to be essential for their youngster as they believed that their little one solely needs data on certain points. Two crucial concerns, yet many dad and mom fail to acknowledge the importance of being concerned in their child's education.
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Our staff right this moment are used to watching CNN with speaking video, a crawler message along the bottom, and the weather forecast on the facet-all while having four on-line chats with their mates. If these students are already so aggressive at school simply think about how aggressive would they be after they transfer on to higher education and after that, work.… Read More..
Education has modified significantly within the final twenty years. The growing understanding is that globalization is being mirrored in an educational agenda that permits for various, and numerous, enhancements upon the education system that allows the educators themselves to develop on their teaching, and current students with real world situations that require them to "suppose outdoors the field", or exterior the realm of their specific field, if you will.
Without query, the majority of what a younger little one learns will take place in class, however there may be additionally way more that youngsters will study, and should be taught, outdoors the classroom. Expertise additionally allows college students to cheat extra simply. Some individuals really feel the federal government is doing so due to stress from "larger powers" to extend the educational well-being of students without receiving any opposition to the modifications.
A distance learning course is a web-based based mostly course that has helped people who may already be working or those who need to remain at house obtain a degree. The one simple thing that parents ought to do to assist in the formal education of their baby is to take an curiosity in all aspects of their child's school actions: academics, further-curricular actions and relationships.
Because of its lower costs, online education also advantages decrease-income students. Having the new worker demonstrate abilities for a manager shows you two issues: how good the coach was, and that the worker can do the capabilities of the job.
Which means dad and mom have a accountability to be energetic in their child's education. The restricted programs are permitting college students to arrange for explicit jobs versus giving them a common education on a topic. One classroom trainer and twenty-5 to thirty younger minds to fill with knowledge could be a very tough feat to perform.… Read More..
This which means of Education would give rise to the Goal, as disseminating formally, non-formally, culturally, nationally, scientifically and ritually-expertise, literacy, data, norm and values, as pedagogies of the institutions giving rise to the goal.… Read More..


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Spidertracks Customer Spotlight: Zatzworks, Alaska
If you've ever watched a nature show about Alaska, chances are you've seen some of Daniel Zatz's work in action. A veteran of the film business, Daniel's been living in Alaska for the better part of three decades and has established himself and his company ZatzWorks as the go-tos for aerial imagery of this stunning part of North America.
With four Emmy awards and more than 4,000 hours of filming time under his belt, it's no surprise that Daniel's built a client roster including giants such as the BBC, Discovery, and National Geographic. But flying in wilderness like Alaska poses risks — remote regions where communication gets hairy, weather changes forcing re-routes — so to keep tabs on ZatzWorks's R66 helicopter and to increase his overall safety, Daniel uses Spidertracks. Here's what he has to say about our product:
Tell us a bit about you and your organisation. What does Zatzworks do? How long have you been in operation?
ZatzWorks is an aerial cinematography business that mainly operates in Alaska and shoots television and film production. Our R66 helicopter is in the air around 300 to 400 hours per year in some of the most remote regions of North America. These areas don't have radar, cell, or reliable marine-band radio links to the world.
How did you become aware of Spidertracks, and when did you begin using us? What was involved in your decision to become a Spidertracks customer?
I learned about Spidertracks from another Alaskan pilot who flew a float plane in a commercial operation. To anyone who understands the system, it's instantly apparent that Spidertracks is far superior to traditional flight plans — especially for operators who can easily be diverted because of weather, light, or wildlife travel. The cost is trivial compared with the operation of a safe flying business.
What Spidertracks features and functionalities are the most important for your organisation, and why?
Our primary use for Spidertracks is as a reliable safety device, but it goes far beyond that. I regularly send an 'I'm OK' text to my family while working in the bush, which is immensely helpful to them. When working over Arctic and sub-Arctic waters and other remote areas to film wildlife like whales, otters or sea lions, I check in more regularly.We also use the Spidertracks website to review our tracks, especially when doing non-wildlife filming like power line inspection work.
What would you say are the biggest strengths of the Spidertracks system?
There are two aspects of the system that we love.
The first is reliability. We've learned over the decade that it's rock-solid and that if anything happens to us while in the bush, we'll be found quickly.
We also love having real people on the other end of a phone line who can support our team members and family in case of a false alarm — like in the times when we forget to turn off the Watch system. Having good, knowledgeable Spidertracks support is amazing and a great reflection on the quality of the company.
What key issues for your business has Spidertracks helped with?
Spidertracks makes our operation safer. When we're flying in the Arctic and a herd of caribou leads us in an unexpected direction or a flock of migrating geese leads us beyond the reach of our originally intended destination, we always know that the folks back home are keeping up with our new route and destination. Spidertracks allows us to respond to changes in the natural world without adding stress.
Schedule a demo to understand what Spidertracks could do for your aviation business.
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How to Live Together
In 1977, Roland Barthes taught a course of lectures at the Collège de France dedicated to the theme of human relationships, their various forms and social models. How to Live Together: Novelistic Simulations of Some Everyday Spaces is the first publication of archival notes by Barthes for his eponymous course.
"We possess archives of two types—sound and written archives. From one hand, these are lectures recorded on tape, and from the other hand, the "lecture notes" which Barthes used. <…> Here we offer the written archive to the reader. The sound archive was prepared separately for the edition and is available in MP3 audio-format," Eric Marti writes in the introduction. The book is first of all meant for the professional audience and provides a large number of appendix materials—abstract of the course by Barthes himself, introduction and notes by Claude Coste, bibliography, indexes of names and events, as well as a glossary of Greek terms.
In its structure, the book presents a sequence of notes made during the process of preparation for the lectures, divided into separate mini-chapters and not meant for publishing. Thus we have a sketchy style, abundance of signs, abbreviations, and thesis-like judgments. The semantic units from which the text is built Barthes calls traits, or, according to the traditional translation into Russian, "figures." In comparison with the chapter or other complete form, trait is "only a first step, a sketch of thought, sometimes only of a descriptive character. Every trait, every problem implies incompleteness—to stimulate personal thinking activity of the listeners," Coste writes. These "figures" are not structured as a narration but put in the conditional—alphabetic—order.
Thematically, Barthes continues his contemplation which he began at the École pratique des hautes études, and which later became a book A Lover's Discourse: Fragments. One of the key concepts introduced in the course at the Collège de France becomes the idiorrhythmie—according to Barthes, "a form of living together which does not exclude individual freedom." Analyzing this phenomenon, he refers to the history of oriental monasteries, their rules and social relationships. Barthes also uses the fiction by Thomas Mann, André Gide, Émile Zola, Daniel Defoe, and often refers to Proust to explore these ideas further.
As always for Barthes, social is inseparable from ethical, and in Nietzsche's opposition of Method and Culture he chooses the latter. In other words, both the course and the notes do not make a student's book on the history of human communities, but present a free-style contemplation about the phenomena connected to, in Barthes's terminology, the concept of Living-Together: "I lay out the figures in a lecture hall instead of mixing them at home at my desk. The only difference is that we do not see the whole picture here: the best scenario is that you'll have to paint it yourself."
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