text:
\section{Introduction}

Topological Data Analysis (TDA) has successfully demonstrated the ability to address the needs of data scientists when studying various data types: from scalar fields \cite{Heine2016}, to high dimensional point clouds \cite{Singh2007}. One advantage of TDA is that it provides compact descriptors describing essential features of the data.

Since its versatile applications for data science, a lot of TDA software libraries are available nowadays, implemented in different programming languages, such as R, Python, C++, etc. These libraries include Javaplex \cite{hong_javaplex_2014}, RedHom \cite{hong_capdredhom_2014}, Gudhi \cite{hong_gudhi_2014}, Scikit-tda \cite{scikittda2019}, PHAT \cite{mcgeoch_distributed_2014}, DIPHA \cite{bauer_phat_2017}, Dionysus \cite{morozov2007dionysus}, Perseus \cite{mischaikow_morse_2013}, Ripser \cite{marfil_persistent_2019}, HomCloud \cite{HomCloud}, and Diamorse \cite{delgado2015diamorse}. All these libraries dedicate particular attention to persistent homology, one of the most common tools in TDA.

{\em Persistent homology} \cite{Edelsbrunner2002,zomorodian_computing_2005} provides a multiscale description of the homology of a shape by means of a filtration. Let $\Sigma$ be a simplicial complex, a {\em filtration} is a sequence $\{\Sigma^i\, |\, 0\leq i \leq r\}$ of subsets of $\Sigma$ such that $\emptyset=\Sigma^0 \subseteq \Sigma^1 \subseteq \dots \subseteq \Sigma^r=\Sigma$. Intuitively, given an index $i$, homology captures the cycles of $\Sigma^i$  \cite{munkres_elements_2018}. Given two indices $i$ and $j$, with $i < j$, persistent homology tracks the cycles that appear, or disappear, from $\Sigma^i$ to $\Sigma^j$. Each cycle is characterized by a pair of indices $(i,j)$,  where $i$ is called {\em birth}, $j$ is called {\em death}, and $j-i$ is called {\em persistence}. Each pair is also referred to as a {\em persistence pair}, and the collection of all pairs defines the {\em persistence diagram} \cite{cohen-steiner_stability_2007}.\\

To simplify the analysis of persistence information, most libraries visualize the persistence diagram as a scatter plot representing each persistence pair as a point with the coordinate value $(i,j)$. An alternative representation is a barcode that represents each feature as a line \cite{ghrist_barcodes_2007}. Two libraries offer alternative ways to visualize persistence. Eireen \cite{henselman_matroid_2016} is a toolbox implemented in Julia \cite{bezanson_julia_2017} for the analysis of Vietoris-Rips filtrations \cite{quinn_prospects_1996}. On top of the persistence diagram, it offers the explicit computation and visualization of the associated persistence cycles.

TTK \cite{tierny_topology_2018} focuses on topological data analysis for scientific visualization and defines a unified framework for visualizing different topological structures such as Reeb graphs \cite{reeb1946points}, contour trees \cite{van1997contour}, and Morse-Smale complexes \cite{milnor_morse_1963}. It also provides two ways to visualize persistent homology other than the persistence diagram: The embedded persistence pairs which represent each dot of the persistence diagram as a pair of points in the filtration domain, and the persistence curve which is a line chart plotting the number of pairs at the vary of the persistence values. Recently, a new plugin for TTK has been developed which allows the visualization of persistence cycles \cite{Iuricich2021}, an explicit representation of the boundary of each hole, or void originated by the filtration.\\

Rather than visualizing information related to a single descriptor, this manuscript considers the problem of analyzing multiple descriptors at once. While many metrics have been studied to measure the similarities and differences among persistence-based descriptors \cite{Yan2021}, results are often presented in a "black-box" manner. That is, measures are used to quantify what descriptors are the most similar/distinct without explaining why. In an effort to simplify the interpretation of TDA descriptors for non-experts, this paper proposes TopoEmbedding, an interactive web tool for the analysis and explanation of persistence features similarities.

\section{Overview}
\label{sec:motivation}
The goal of TopoEmbedding is to simplify the analysis of topological descriptors by means of interactive visualizations. In Section \ref{sec:analysis} we describe the TDA pipeline we used to compute the persistence-based descriptors rendered by TopoEmbedding. In Section \ref{sec:visualization} we describe the visual interface used to improve explainability and interactive analysis.

\subsection{Analysis pipeline}
\label{sec:analysis}
To demonstrate the interface functionality, we implemented a representative TDA pipeline analyzing the famous MNIST data set, a collection of 1000 handwritten digits (100 images per digit). The analysis pipeline consists of three different stages.

In the first stage, we compute persistent homology for each handwritten image. If we consider each input image as a piecewise linear scalar field $f: \Sigma \rightarrow \mathbb{R}$ defined on a simplicial complex $\Sigma$, a filtration of $\Sigma$ is naturally defined by the sequence of sublevel sets according to $f$, where the {\em sublevel set} of value $i$ is defined as $\Sigma^i = \{ \Sigma \in \Sigma | f(\Sigma) \leq i \}$. Then, given the filtration $f$ defined on each input image we can obtain the corresponding persistence diagram using the standard algorithm \cite{Edelsbrunner2002}. In this analysis, we focus on points of the persistence diagram representing 1-cycles only and we discard the remaining ones.\\

During the second stage, we generate a persistence image \cite{adams2017persistence} for each persistence diagram computed in the first stage. Intuitively, a {\em persistence image} is a regular grid discretizing the density of persistence pairs in each portion of the persistence diagram. Each persistence pair $(i, j)$ in a persistence diagram is first transformed in a birth-death coordinates representation with $(x, y) = (i, j-i)$. Then, a persistence image is obtained by fitting a Gaussian kernel distribution upon each persistence pair $(x, y)$ defined as,
$$ g(x, y) = \frac{1}{2\pi \sigma^2}e^{-[(x-u_x)^2 + (y-u_y)^2]/2\sigma^2} $$
with mean $u = (u_x, u_y)$ and $\sigma = 0.01$. Each distribution is sampled on a $n=10\times 10$ grid and weighted by the function
\begin{equation*}
    w_b(y) =
    \begin{cases}
        0, & \text{if } y \leq 0 \\
        \frac{y}{b}, & \text{if } 0 < y < b \\
        1, & \text{if } y \geq b,
    \end{cases}
\end{equation*}
where $b$ is the persistence value of the most persistent feature.\\

The idea of the third stage is to interpret the input dataset as a higher dimensional point cloud. After computing the persistence image for each input dataset, we compute a matrix of pairwise distances for all of them with the 2-norm Minkowski distance, which is defined as $D(X, Y) = \sqrt{\sum^{n}_{i=1}|p_i - q_i|^2}$, where $X$ and $Y$ denote two persistence images, $p_i$ and $q_i$ are the $i$-th pixels from persistence image $X$ and $Y$, respectively.

Then, we use the matrix of pairwise distances to obtain a lower-dimensional embedding of the point cloud, where each point corresponds to an input digit image (associated with a persistence diagram and also a persistence image).

\textcolor{red}{The computation and visualization in the entire pipeline are mainly based on persistent images instead of persistent diagrams for two reasons. First, we use both supervised and unsupervised dimensionality reduction methods for the dataset clustering, and these methods (e.g., tSNE \cite{van2008visualizing}) require vectorized input such as the persistent image. The clustering allows users to explore the similarity between clusters of datasets. Second, a persistent image generalizes persistent pairs into a uniform representation of pixels encoded with the weights of contributing cycles. This allows the filtering of less important cycles and the filtering of less different pixels for the comparison of two datasets. The filtering is helpful to explore the similarity between two datasets.}

A number of supervised and unsupervised dimensionality reduction techniques are easily available to the scope. In our analysis we use three methods simultaneously, namely, Isomap\cite{balasubramanian_isomap_2002}, Multi-dimensional Scaling (MDS)\cite{webb_multidimensional_1995}, and t-distributed Stochastic Neighbor Embedding (t-SNE)\cite{van2008visualizing}. Figure \ref{fig:embedding} shows an example of the embedding obtained with Isomap.

\subsection{Visualization and user interactions}
\label{sec:visualization}

The overall visualization of the lower-dimensional embedding is what can be obtained with standard data science libraries. From the results shown in Figure \ref{fig:embedding}, we can notice that Isomap is well preserving the information about the input digit similarities. We recall that the analysis pipeline focuses on 1-cycles (see Section \ref{sec:analysis}). Thus, it is not surprising that the lower-dimensional embedding highlights three classes of similarity among input digits corresponding to three branches: digits with one distinct 1-cycle (e.g., 6,9,0), digits with two distinct 1-cycles (e.g., 8), and digits with no 1-cycles (e.g., 1,2,3,4,5,7).

\begin{figure}[htp]
   \centering
   \includegraphics[width=0.98\linewidth]{images/Figure-1.png}
   \caption{Three zoomed views of the lower dimensional embedding created by Isomap. Each view is color-coded to indicate where it was sampled from. We can notice that each branch of the Isomap embedding isolates digits with zero highly persistent 1-cycles (blue), one 1-cycle (green), or two 1-cycles (red).}
   \label{fig:embedding}
\end{figure}

It is fairly easy for a user who is experienced in TDA and persistent homology to interpret the representation. However, what about an inexperienced one? To address this need, TopoEmbedding allows any user to visually explore the lower-dimensional embedding and its corresponding persistence images along with the 1-cycles. When the user selects a point in the scatter plot, TopoEmbedding will display the corresponding persistence image (see Figure \ref{fig:persistence}a left) in a diverging blue-red color map. Red pixels indicate high-density regions that contain more persistence pairs than the blue pixels. The visualization of the persistence image is also paired with the input digit image, and the persistence cycles \cite{Iuricich2021} (see Figure \ref{fig:persistence}a right), an explicit visualization of 1-cycles originated and destroyed by the filtration. We recall that persistent homology associates an importance value (i.e., persistence) to each feature according to its lifespan in the filtration. Using this information, we are able to provide a filtering tool and allow users to remove cycles according to their importance (see Figure \ref{fig:persistence}b).\\

\begin{figure}[htp]
   \centering
   \includegraphics[width=0.98\linewidth]{images/Figure-2.png}
   \caption{(a) Example of a persistence image and the corresponding persistence cycles as rendered in TopoEmbedding. The persistence image is visualized as a grid of resolution $n=10\times10$ and with pixel values color-coded according to a diverging (blue-red) color map. The 1-cycles (right) are displayed upon the digit as collections of lines indicating the boundary of a hole originated by the filtration. (b) Cycles can be filtered according to their persistence. This allows to update the visualization focusing on the most important features captured by persistent homology. On the left we show unfiltered cycles as computed by persistent homology. On the right we show only the most persistence cycles.}
   \label{fig:persistence}
\end{figure}

By selecting two digits, the user can compare their corresponding descriptors and topological features. The two persistence images are juxtaposed, and a line chart (see Figure \ref{fig:pixel_threshold}a) shows the pixel-wise difference between two persistence images with values sorted in ascending order.
To highlight differences between the two persistence images, the user can threshold their pixels setting the minimum and maximum pixel-wise difference. As a result, only pixels within such range will be visualized at full opacity (see Figure \ref{fig:pixel_threshold}b).\\

\begin{figure}[htp]
   \centering
   \includegraphics[width=0.95\linewidth]{images/Figure-3.png}
   \caption{(a) Line chart showing the pixel-wise difference between the two persistence images selected. The green background in the line chart indicates the pixel-wise difference selected as threshold. (b) After reducing the thresholding area focusing on pixels with the highest pixel-wise difference, the green area in the chart is resized as well as all pixels in the persistence images excluded by the same threshold are displayed with higher transparency colors.}
   \label{fig:pixel_threshold}
\end{figure}

The interface components can be used to explore similarities and differences between a pair of persistence images. At the same time, differences in the corresponding topological features (i.e., persistence cycles) visualized in the input data domain space explain the causes of such differences in an interpretable manner.

As an example, Figure \ref{fig:similarity} illustrates a possible use case specific to the MNIST dataset. The digit 8 highlighted in blue in Figure \ref{fig:similarity} rightly belongs to the branch of the Isomap embedding containing most 8 digits (i.e., those containing two cycles).

Close to such digit, we found digit 6 highlighted in red. This digit should not belong to this branch since it should be characterized by one 1-cycle. If we compare the most different pixels in the corresponding persistence images, it is still difficult to interpret why one digit is being classified as close to the other. However, if we look at the explicit representation of 1-cycles (Figure \ref{fig:similarity}top), we notice that both filtrations create two high persistent 1-cycles, This explains why digit 6 is being classified similarly to digit 8.

The same reason can be applied to a digit 8 (highlighted in green), which ends far away in the embedding from the first digit 8 (highlighted in blue). Once again, the corresponding persistence image is hard to interpret for humans. However, if we combine the persistence image with an explicit representation of the persistence cycles, we can realize that no highly persistent cycles are being captured for this digit. This explains why this digit is classified as having no cycles and, for this reason, is being correctly grouped with similar digits (e.g., 1,2,3, etc.)

\begin{figure}[htp]
   \centering
   \includegraphics[width=0.95\linewidth]{images/Figure-4.png}
   \caption{While the digit 8 highlighted in blue belongs to the correct branch where most digits 8 are, a few digits in the MNIST dataset are being classified as belonging to the wrong branch of the Isomap embedding. One of these is digit 6 highlighted in red. While the corresponding persistence image does not clarify why the digit is classified similarly to an 8, the explicit visualization of the 1-cycles shows that the underlying filtration created two highly persistent 1-cycles. The other way around, the digit 8 highlighted in green ends up far away from its corresponding branch. Once again, the visualization of the 1-cycles shows that there are not highly persistent 1-cycles in this filtration, explaining why the digit ends up in said position.}
   \label{fig:similarity}
\end{figure}

\section{Conclusions}
\label{sec:conclusions}

We have presented TopoEmbedding, a web-based TDA tool to assist in the visualization and analysis of persistent homology. The source code of our analysis pipeline and the implementation of TopoEmbedding can be found at \url{https://github.com/DaVisLab/TopoEmbedding}.
TopoEmbedding represents a proof-of-concept highlighting the importance of interactive visualizations for TDA. In this work, we used a 2D dataset (e.g., MINIST) to demonstrate the functionality of TopoEmbedding. However, we are interested in applying similar strategies for the analysis of 3D datasets where the direct rendering of volumetric images is unfeasible. In such a context, the visualization of 1-cycles and 2-cycles (surfaces indicating the boundary of voids) could become crucial for interpreting the results of the TDA pipeline.

\bibliographystyle{siamplain}

\section{Introduction.}
Topological Data Analysis (TDA) has successfully demonstrated the ability to address the needs of data scientists when studying various data types: from scalar fields \cite{Heine2016}, to high dimensional point clouds \cite{Singh2007}. One advantage of TDA is that it provides compact descriptors describing essential features of the data.
Many TDA software libraries are available nowadays, implemented in different programming languages, such as R, Python, and C++. These include Javaplex \cite{hong_javaplex_2014}, RedHom \cite{hong_capdredhom_2014}, Gudhi \cite{hong_gudhi_2014}, Scikit-tda \cite{scikittda2019}, PHAT \cite{mcgeoch_distributed_2014}, DIPHA \cite{bauer_phat_2017}, Dionysus \cite{morozov2007dionysus}, Perseus \cite{mischaikow_morse_2013}, Ripser \cite{marfil_persistent_2019}, HomCloud \cite{HomCloud}, and Diamorse \cite{delgado2015diamorse}. All these libraries dedicate particular attention to persistent homology, one of the most common tools in TDA.

{\em Persistent homology} \cite{Edelsbrunner2002,zomorodian_computing_2005} provides a multiscale description of the homology of a shape by means of a filtration. Let $\Sigma$ be a simplicial complex, a {\em filtration} is a sequence $\{\Sigma^i\, |\, 0\leq i \leq r\}$ of subsets of $\Sigma$ such that $\emptyset=\Sigma^0 \subseteq \Sigma^1 \subseteq \dots \subseteq \Sigma^r=\Sigma$. Intuitively, given an index $i$, homology captures the cycles of $\Sigma^i$  \cite{munkres_elements_2018}. Given two indices $i$ and $j$, with $i < j$, persistent homology tracks the cycles that appear, or disappear, from $\Sigma^i$ to $\Sigma^j$. Each cycle is characterized by a pair of indices $(i,j)$,  where $i$ is called {\em birth}, $j$ is called {\em death}, and $j-i$ is called {\em persistence}. Each pair is also referred to as a {\em persistence pair}, and the collection of all pairs defines the {\em persistence diagram} \cite{cohen-steiner_stability_2007}.

To simplify the analysis of persistence information, most libraries visualize the persistence diagram as a scatter plot representing each persistence pair as a point with the coordinate value $(i,j)$. An alternative representation is a barcode that represents each feature as a line \cite{ghrist_barcodes_2007}. Two libraries offer alternative ways to visualize persistence. Eireen \cite{henselman_matroid_2016} is a toolbox implemented in Julia \cite{bezanson_julia_2017} for the analysis of Vietoris-Rips filtrations \cite{quinn_prospects_1996}. On top of the persistence diagram, it offers the explicit computation and visualization of the associated persistence cycles. The Topological ToolKit (TTK) \cite{tierny_topology_2018} focuses on topological data analysis for scientific visualization and defines a unified framework for visualizing different topological structures such as Reeb graphs \cite{reeb1946points}, contour trees \cite{van1997contour}, and Morse-Smale complexes \cite{milnor_morse_1963}. It also provides two ways to visualize persistent homology other than the persistence diagram: the embedded persistence pairs, which represent each dot of the persistence diagram as a pair of points in the filtration domain, and the persistence curve, which is a line chart plotting the number of pairs at the vary of the persistence values. Recently, a new plugin for TTK has been developed which allows the visualization of persistence cycles \cite{Iuricich2021}, an explicit representation of the boundary of each hole, or void originated by the filtration.\\

Rather than visualizing information related to a single descriptor, this manuscript considers the problem of analyzing multiple descriptors at once. While many metrics have been studied to measure the similarities and differences among persistence-based descriptors \cite{Yan2021}, results are often presented in a "black-box" manner. That is, measures are used to quantify what descriptors are the most similar without explaining why. In an effort to simplify the interpretation of TDA descriptors for non-experts, this paper proposes TopoEmbedding, an interactive web tool for the analysis and explanation of persistence features similarities.

\section{Overview.}
\label{sec:motivation}
The goal of TopoEmbedding is to simplify the analysis of topological descriptors by means of interactive visualizations. In Section \ref{sec:analysis} we describe the TDA pipeline we used to compute the persistence-based descriptors rendered by TopoEmbedding. In Section \ref{sec:visualization} we describe the visual interface used to improve explainability and interactive analysis.

\subsection{Analysis pipeline.}
\label{sec:analysis}
To demonstrate the interface functionality, we implemented a representative TDA pipeline analyzing the famous MNIST data set, a collection of 1000 handwritten digits (100 images per digit). The analysis pipeline consists of three different stages.

In the first stage, we compute persistent homology for each handwritten image. If we consider each input image as a piecewise linear scalar field $f: \Sigma \rightarrow \mathbb{R}$ defined on a simplicial complex $\Sigma$, a filtration of $\Sigma$ is naturally defined by the sequence of sublevel sets according to $f$, where the {\em sublevel set} of value $i$ is defined as $\Sigma^i = \{ \sigma \in \Sigma | f(\sigma) \leq i \}$. Then, given the filtration $f$ defined on each input image we can obtain the corresponding persistence diagram using the standard algorithm \cite{Edelsbrunner2002}. In this analysis, we focus on points of the persistence diagram representing 1-cycles only and we discard the remaining ones.\\

During the second stage, we generate a persistence image \cite{adams2017persistence} for each persistence diagram computed in the first stage. Intuitively, a {\em persistence image} is a regular grid discretizing the density of persistence pairs in each portion of the persistence diagram. Each persistence pair $(i, j)$ in a persistence diagram is first transformed into a birth-death coordinates representation with $(x, y) = (i, j-i)$. Then, a persistence image is obtained by fitting a Gaussian kernel distribution upon each persistence pair $(x, y)$ defined as:
$$ g(x, y) = \frac{1}{2\pi \sigma^2}e^{-[(x-u_x)^2 + (y-u_y)^2]/2\sigma^2} $$
with mean $u = (u_x, u_y)$ and $\sigma = 0.01$. Each distribution is sampled on a $n=10\times 10$ grid and weighted by the function
\begin{equation*}\label{e1.1}
    w_b(y) =
    \begin{cases}
        0, & \text{if } y \leq 0 \\
        \frac{y}{b}, & \text{if } 0 < y < b \\
        1, & \text{if } y \geq b,
    \end{cases}
\end{equation*}
where $b$ is the persistence value of the most persistent feature.\\

The idea of the third stage is to interpret the input dataset as a higher dimensional point cloud. After computing the persistence image for each input dataset, we compute a matrix of pairwise distances for all of them with the 2-norm Minkowski distance, which is defined as $D(X, Y) = \sqrt{\sum^{n}_{i=1}|p_i - q_i|^2}$, where $X$ and $Y$ denote two persistence images, $p_i$ and $q_i$ are the $i$-th pixels from persistence image $X$ and $Y$, respectively.

Notice we use persistent images because of their robustness and scalability \cite{adams2017persistence}. However, the metrics $D$ could be computed directly on the persistent diagram using bottleneck distance \cite{cohen-steiner_stability_2007}, Wasserstein distance \cite{cohen-steiner_lipschitz_2010}, or similar metrics \cite{yan_scalar_2021}.
 We use the matrix of pairwise distances to obtain a lower-dimensional embedding of the point cloud, where each point corresponds to an input digit image (associated with a persistence diagram and also a persistence image).

A number of supervised and unsupervised dimensionality reduction techniques are easily available to the scope. In our analysis we use three methods simultaneously, namely, Isomap\cite{balasubramanian_isomap_2002}, Multi-dimensional Scaling (MDS)\cite{webb_multidimensional_1995}, and t-distributed Stochastic Neighbor Embedding (t-SNE)\cite{van2008visualizing}. Figure \ref{fig:embedding} shows an example of the embedding obtained with Isomap.

The whole process is computationally efficient and scalable. All our experiments were run on a Desktop computer mounting a Core i7-8700 3.20 GHz processor and 32 GB of RAM. In the first stage we used a dedicated TTK plugin for computing persistent homology and persistence cycles on each image \cite{Iuricich2021}. This step requires less than 50MB of RAM and less than a second per image. In the second and third stages, the peak memory usage for computing the persistent images and the lower dimension embedding (including the distance matrix) is less than 100MB, and the computation time is around 120 seconds. All computations are performed offline and the results saved in dedicated files used during the interactive phase.

\subsection{Visualization and user interactions.}
\label{sec:visualization}

The lower-dimensional embedding obtained with standard data science libraries \cite{balasubramanian_isomap_2002,webb_multidimensional_1995,van2008visualizing} is directly visualized as a 2D scatterplot.
Figure \ref{fig:embedding} shows the results obtained with Isomap \cite{balasubramanian_isomap_2002}. We can notice that Isomap is preserving the information about the input digit similarities. We recall that the analysis pipeline focuses on 1-cycles (see Section \ref{sec:analysis}). Thus, it is not surprising that the lower-dimensional embedding highlights three classes of similarity among the input digits: digits with one distinct 1-cycle (e.g., 6,9,0), digits with two distinct 1-cycles (e.g., 8), and digits with no 1-cycles (e.g., 1,2,3,4,5,7).

\begin{figure}[htp]
   \centering
   \includegraphics[width=0.8\linewidth]{images/Figure-1.png}
   \caption{Three zoomed views of the lower dimensional embedding created by Isomap. Each view is color-coded to indicate where it was sampled from. We can notice that each branch of the Isomap embedding isolates digits with zero highly persistent 1-cycles (blue), two 1-cycles (red), or one 1-cycle (green).}
   \label{fig:embedding}
\end{figure}

It is fairly easy for a user who is experienced in TDA and persistent homology to interpret the representation. However, what about an inexperienced one? To address this need, TopoEmbedding allows any user to visually explore the lower-dimensional embedding and its corresponding persistence images along with the 1-cycles. When the user selects a point in the scatter plot, TopoEmbedding will display the corresponding persistence image (see Figure \ref{fig:persistence}a left) in a diverging blue-red color map. Red pixels indicate high-density regions that contain more persistence pairs than the blue pixels. The visualization of the persistence image is also paired with the input digit image, and the persistence cycles \cite{Iuricich2021} (see Figure \ref{fig:persistence}a right), an explicit visualization of 1-cycles originated and destroyed by the filtration. We recall that persistent homology associates an importance value (i.e., persistence) to each feature according to its lifespan in the filtration. Using this information, we are able to provide a filtering tool allowing users to remove cycles according to their importance (see Figure \ref{fig:persistence}b).
\begin{figure}[htb]
   \centering
   \includegraphics[width=0.8\linewidth]{images/Figure-2.png}
   \caption{(a) Example of a persistence image and the corresponding persistence cycles as rendered in TopoEmbedding. The persistence image is visualized as a grid of resolution $n=10\times10$ and with pixel values color-coded according to a diverging (blue-red) color map. The 1-cycles (right) are displayed upon the digit as collections of blue lines indicating the boundary of a hole originated by the filtration. (b) Cycles can be filtered according to their persistence. This allows to update the visualization focusing on the most important features captured by persistent homology. On the left we show unfiltered cycles as computed by persistent homology. On the right we show only the most persistent cycles.}
   \label{fig:persistence}
\end{figure}\\

By selecting two digits, the user can compare their corresponding descriptors and topological features. The two persistence images are juxtaposed, and a line chart (see Figure \ref{fig:pixel_threshold}a) shows the pixel-wise difference between two persistence images with values sorted in ascending order.
To highlight differences between the two persistence images, the user can threshold their pixels setting the minimum and maximum pixel-wise difference. As a result, only pixels within such range will be visualized at full opacity (see Figure \ref{fig:pixel_threshold}b).
\begin{figure}[htb]
   \centering
   \includegraphics[width=0.8\linewidth]{images/Figure-3.png}
   \caption{(a) Line chart showing the pixel-wise difference between the two persistence images selected. The green background in the line chart indicates the pixel-wise difference selected as threshold. (b) After reducing the thresholding area focusing on pixels with the highest pixel-wise difference, the green area in the chart is resized as well as all pixels in the persistence images excluded by the same threshold are displayed with higher transparency colors.}
   \label{fig:pixel_threshold}
\end{figure}\\

The interface components can be used to explore similarities and differences between a pair of persistence images. At the same time, differences in the corresponding topological features (i.e., persistence cycles) visualized in the input data domain space explain the causes of such differences in an interpretable manner.

As an example, Figure \ref{fig:similarity} illustrates a possible use case specific to the MNIST dataset. The digit 8 highlighted in blue in Figure \ref{fig:similarity} correctly belongs to the branch of the Isomap embedding containing most 8 digits (i.e., those containing two cycles).
\begin{figure}[htp]
   \centering
   \includegraphics[width=0.8\linewidth]{images/Figure-4.png}
   \caption{While the digit 8 highlighted in blue belongs to the correct branch where most digits 8 are, a few digits in the MNIST dataset are being classified as belonging to the wrong branch of the Isomap embedding. One of these is digit 6 highlighted in red. While the corresponding persistence image does not clarify why the digit is classified similarly to an 8, the explicit visualization of the 1-cycles shows that the underlying filtration created two highly persistent 1-cycles. The other way around, the digit 8 highlighted in green ends up far away from its corresponding branch. Once again, the visualization of the 1-cycles shows that there are not highly persistent 1-cycles in this filtration, explaining why the digit ends up in said position.}
   \label{fig:similarity}
\end{figure}\\

Close to such digit, we found digit 6 highlighted in red. This digit should not belong to this branch since it should be characterized by one 1-cycle. If we compare the most different pixels in the corresponding persistence images, it is still difficult to interpret why one digit is being classified as close to the other. However, if we look at the explicit representation of 1-cycles (Figure \ref{fig:similarity}), we notice that both filtrations create two high persistent 1-cycles. This explains why digit 6 is being classified similarly to digit 8.

The same reason can be applied to a digit 8 (highlighted in green), which ends far away in the embedding from the first digit 8 (highlighted in blue). Once again, the corresponding persistence image is hard to interpret for humans. However, if we combine the persistence image with an explicit representation of the persistence cycles, we can realize that no highly persistent cycles are being captured for this digit. This explains why this digit is classified as having no cycles and, for this reason, is being correctly grouped with similar digits (e.g., 1,2,3, etc.)

\section{Conclusions}
\label{sec:conclusions}

We have presented TopoEmbedding, a web-based TDA tool to assist in the visualization and analysis of persistent homology. The source code of our analysis pipeline and the implementation of TopoEmbedding can be found at \url{https://github.com/DaVisLab/TopoEmbedding}.
TopoEmbedding represents a proof-of-concept highlighting the importance of interactive visualizations for TDA. In this work, we used a 2D dataset (e.g., MINIST) to demonstrate the functionality of TopoEmbedding. However, we are interested in applying similar strategies for the analysis of 3D datasets where the direct rendering of volumetric images is unfeasible. In such a context, the visualization of 1-cycles and 2-cycles (surfaces indicating the boundary of voids) could become crucial for interpreting the results of the TDA pipeline.

\bibliographystyle{siamplain}

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The endowed Fund for Educational Excellence (FFEE) was created in 1990 by a group of business, school, and private community leaders for the benefit of Linn County K-12 students. Since that time FFEE has annually made grant awards to support teachers in creative, innovative, and effective classroom projects.
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New York State Department of Transportation (DOT) was originally the New York State Highway Department in 1909, which became the Department of Public Works in 1927. The office was located in the Wills Block. Frank L. Bisbee was the state engineer and superintendent. He continued in this office until the fall of 1949, when he retired and was replaced by P. Raymond Keyes. The plans for the new office and storage buildings on upper Main Street were completed in 1947 with the awarding of contracts in 1951. The office was completed in 1953. The low bidder on the construction work was Adirondack Construction Company of Glens Falls with a bid of $184,786. The total of all contracts was $238,065. In 1967 the NYS Department of Public Works (DPW) became the Department of Transportation (DOT).


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« Girls, claim your rights!
Cote d'Ivoire anyone? »
Cote d'Ivoire and the thinking trap
March 20, 2011 by Linda Raftree
On a flight home from Kenya last summer, I sat next to a young man from Burundi. His country was just holding elections, and we got to talking about them. I'd heard people calling the elections a sham because the opposition boycotted them, and there was only one candidate (from the ruling party) to even vote for. I asked what he thought about that situation and the current president. He said 'Well, you know, we Africans…. sometimes these guys have their own style of doing things which others may not understand.' We got to talking about politics in countries near Burundi and African big men who don't want to step down. He kept joking, with each subsequent leader: 'Hey… well, you know, I guess he has his own style too.' Seems there is a lot of personal style to be reckoned with.
Consider Laurent Gbagbo in Cote d'Ivoire who's refused to step down as president and whose conflict with internationally recognized president-elect Ouattara is bringing (has brought?) the country to civil war. I'm about as far from a Cote d'Ivoire expert as you can get, never having visited the country and not having a great handle on the history or much knowledge about the current situation. So I went digging for some news articles and opinion pieces so I could understand things better.
Alex Thurston gives a good round-up on Sahel Blog and John James describes the day-to-day in his article Ivory Coast's Descent into Madness (heard through @scarlettlion). This sobering article, West Africa lurches towards war, from March 11, explains how the Cote d'Ivoire conflict could impact the whole region. The International Crisis Group sums up the situation like this:
'The election was part of a peace process that began after the September 2002 rebellion and was endorsed by several accords, the latest the 2007 Ouagadougou Political Agreement that all candidates, including Gbagbo, accepted and that set out compromises on organisation and security for the balloting. Ouattara won the run-off with a margin of more than 350,000 votes over Gbagbo.
The UN certified that result, but Gbagbo used the country's highest court to throw out votes arbitrarily so he could stage a constitutional coup. Since then, he has relied on violence and ultra-nationalist rhetoric to cling to power. Over 300 people have been killed, dozens raped and many more abducted and disappeared by security forces. ECOWAS and the African Union (AU) have recognised Ouattara as president-elect and asked Gbagbo to step down, but he is apparently prepared to resist to the end, even if it means throwing Côte d'Ivoire into anarchy, war and economic disaster with terrible consequences for the entire region.'
Aaron Bady of ZunguZungu goes deeper and analyzes the historical roots of the conflict in his post Our deafening silence on the Coast of Africa: Cote d'Ivoire and our Myth of Continents, which I'd encourage you to read in full.
'This is, in other words, about the political power structure working to limit who gets to politically represent and be represented in this "African" nation, precisely by drawing a line to separating "North Africa" from "West Africa," in politically interested ways, to maintain political control for the 'real' Ivoirians in the south. Without denying that there might be other bad guys in this story, the fact remains — virtually unspoken — that ten years ago, Ouattara's ethnic origin as Burkinabé — his parents were from Burkina Faso — was used by politicians from the predominantly Christian southern part of the country to disqualify him from running for the presidency; a political campaign of "Ivorité" worked to distinguish between "indigenous Ivorians" and "Ivorians of immigrant ancestry," as a thoroughly programmatic effort to dis-enfranchise the (largely Muslim) north by lumping them all together with the (many) immigrants in the north who originated Burkina Faso. The current violence started with a disturbingly analogous move: after Ouattara was declared to have won the presidential election — by the UN, ECOWAS, the AU, and all international observers — the Gbagbo controlled Constitutional Council threw out 660,000 votes from the north and gave the election to Gbagbo.
It's all very complicated. But for now, my point is simply to note that these geographical categories are not only part of the problem in Côte d'Ivoire itself. In this case, they are part of the reason why we find it so difficult to even talk coherently about the problem.'
The organization where I work operates in Liberia, and so I've been getting updates about the situation on the Cote d'Ivoire-Liberia border, where we're providing some support to the large number of refugees streaming into Liberia. These updates describe a deteriorating situation. Ghana, Guinea, Mali and Burkina Faso haven't received as many refugees but are still starting to feel the effects of the conflict heating up. The past week saw a spill-over of violence onto the Liberian side of the border as armed Ivorians entered the Liberian territory and attacked Liberian and Ivorian nationals who were found along the North Eastern Liberia-Cote d'Ivoire border. UN security troops were deployed to put an end to the fighting and gain control. Serious shooting has also been reported along the South East Liberia-Cote d'Ivoire border, accelerating the influx of refugees into the counties of Maryland and Grand Gedeh. One report says that in Cote d'Ivoire, rapes, abductions and killings are being committed by people supporting both sides and influential leaders appear to be deliberately stimulating attacks against political opponents, other ethnic groups, nationals from other West African countries, as well as UN staff and operations.
I was Skyping Wednesday with my boss, who lives in Burkina. She oversees our programs in the West Africa region, including emergency and humanitarian crises. She said there are over 80,000 refugees who've come into Liberia already, 'and it's only going to get worse.'
On top of those who have fled into Liberia, there are at least 300,000 internally displaced. 'The area where the refugees are in Liberia is totally undeveloped,' she said. 'It's 26 villages with no infrastructure which now have to cope with the influx of people. People don't want to go to a camp. They prefer to stay in the villages. There are security worries as there is fighting very close to the border and armed men from Cote d'Ivoire are crossing the border to Liberia and started firing on the Liberian side of the border. They were shut down by UN troops, but it's worrying.'
Since she and I talked the situation has gotten worse still. Foreign Policy Blog reports:
'This is the real thing. In a message on state television last night, Gbagbo — who has refused to step down after losing an internationally certified election in November — asked civilians to get involved in the fight to "neutralize" his opponents. That's about as close as you get to saying pick up your machetes and join me. Or forget machetes; Ivory Coast is still heavily armed from its civil war last decade. Gbagbo certainly hasn't been shy about using military force lately; his forces are thought to have been behind a mortar attack on a market yesterday that left 40 people injured. Earlier this week there were reports of his using helicopters and tanks to mow down suspected opposition supporters.'
My boss also worries there will be a cholera outbreak. The rains started early and the hygiene conditions are bad. Ghana has a cholera outbreak right now, and it's likely that with much people movement, rains and inadequate sanitary conditions, there will be an epidemic. 'And people from Cote d'Ivoire are hungry,' she said. They haven't had access to cash for weeks as all the banks are closed and food stocks are used up. We are feeling this even here in Burkina as food coming from Cote d'Ivoire starts to get short and prices increase…. And many of the refugees are adolescents. We are worried about them being hired into armed forces, so we are looking into options to try to organize the youth to simply give them something to do.'
On March 3, the International Crisis Group issued the steps that they see as necessary to avoid a civil war and conflict that will shake the whole region:
'The requirements to avoid a disastrous new conflict include Gbagbo stepping down; Ouattara offering to negotiate, with civil society help, an agreement for unity, national reconciliation and an interim transitional government with him at its head (but without the irreconcilable former president); the UN peace-keeping mission standing firm to carry out its civilian protection mandate; and the international community unequivocally supporting any decisions of the Economic Community of West African States (ECOWAS), including deployment of a military mission.'
I wonder if these actions will actually be taken, or if the escalating situation will allow for them to be taken, as each passing day seems to throw the country further towards a seemingly impossible-to-resolve situation. It's tempting to think that people are not paying attention to this crisis because of other crises happening right now in Japan, Egypt and Libya. I'm not convinced. Aaron Bady's post, referred to above, spells out other possible reasons for the lack of attention to the situation, including the common narratives about Africa, 'the dark continent', that color our perceptions and reactions. Again, I'd really encourage you to read the whole piece, but here's a bit of it:
'It isn't that "we" don't want to understand; it's that we don't know how to see beyond the initial same-old-story-ness of this story, when we hear it. Which is why, I would suggest, we end up where we started: a sense that, because there is violence, we should pay attention to what is happening, followed by the discovery that there is no news there; just the same "turmoil in Africa" narratives we sort of quietly presume to be going on across the continent all the time, and nothing we can think anything new about.'
I'm someone who rants quite a bit about the negative images we associate so often with Africa and Africans. But looking at the information available about the current situation, I'm having a hard time finding any hope there. I wonder what solutions there really are, where they come from, and if and when these cycles will end. Is it because I haven't visited Cote d'Ivoire myself that it's hard to see beyond the negative narratives? Is it because I don't personally know the day-to-day 'place of friendly people, amazing fresh fruit, long and unspoilt tropical beaches ' that John James describes? In my mind, I keep coming back to the narrative about Africa's 'big men' and their 'personal styles'.
Seems I'm also caught in the thinking trap that Aaron laid out.
Posted in Cote d'Ivoire, disaster and emergencies | Tagged civil war, cote d'ivoire, Crisis, emergency, ivory coast, liberia, solutions, thinking trap | 4 Comments
on March 20, 2011 at 1:22 pm | Reply Cote d'Ivoire and the thinking trap | Global Health Hub
[…] the original post: Cote d'Ivoire and the thinking trap AKPC_IDS += […]
on March 21, 2011 at 10:27 am | Reply John Palmkvistt
From my perspective the UN has unnecessarily turned an electoral crisis into a humanitarian crisis. I monitored the events from Monrovia, Liberia as they unfolded and it was plain to see that the UN, foreign humanitarian community, and sympathetic journalists and bloggers communicated the urgent need to make war against Gbagbo. This in turn would create a humanitarian crisis that would create the potential for grant money and work for the above mentioned groups.
Shortly following the election the UN sent more troops to Cote d'Ivoire, and aid journalists issued alarming reports. The foreign humanitarian community fed the press with distortions and openly demanded of the Liberian government the permission to build a refugee camp. The Liberian government was against the construction of the refugee camp because they did not want to create a magnate for human migration. But since its soverignty is ceded to the UN, they eventually agreed, but with reluctance. Meanwhile, the INGOs in Liberia quietly discussed the urgent need for a humanitarian crisis in Cote d'Ivoire so that their imperiled cash situations could be buoyed by new grant money.
Now all of the foreigners have their wish. The UN is hanging tough and deepening its influence in the region, the press is giving breathtaking accounts of human misery, and the humanitarian community is crowing about the good work they are doing. Meanwhile people's lives are being wrecked in Cote d'Ivoire and Liberia. It is a disturbing equation in which increased misery for Africans equals increased opportunity for foreigners.
on March 23, 2011 at 8:14 am | Reply Kirafrique
I too am worried & know the most wonderful country in Africa with it's hospitable people, long sandy beaches, great food and mixed population; la belle Cote d'Ivoire
Last night I was in touch with a mayor of a town in the east of the country, they have 420+ refugees, they hadn't finished counting and are still getting more arriving daily on foot, by taxi … They're worried about where to house them, how to feed them all and the ensuing illnesses that could happen with this extra burden on a small town.
There are many forgotten places like this. I'm also trying to get 38 orphaned children to safety as the President of the centre has mis-managed the situation & the children aren't able to eat.
The UNOCI has royally messed up the situation, I heard them being described as 'camera touting tourists who just count the dead' – which I have to say having seen them in 'action' 2 months ago in Abidjan and elsewhere is not far from the truth.
The 'big guys' are being left to carry out their own battle whilst the civilians suffer, killed, kidnapped, injured & lost
Who can help if not the UN???
on March 26, 2011 at 11:15 am | Reply Blog For Cote d'Ivoire #IvoryCoast #civ2010 #blog4civ
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redpajama_set_name: RedPajamaCommonCrawl


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Aug. 24, 2019 / 10:01 AM	/ Updated Aug. 24, 2019 at 7:05 PM
Trump arrives in France for G7 summit amid protests, meets Macron
Sommer Brokaw
President Donald Trump appears with French President Emmanuel Macron during lunch at the Hotel du Palais in Biarritz, southwest France on Saturday. Photo by Ludovic Marin/EPA-EFE/Pool
President Donald Trump speaks with members of the media as he departs the White House in Washington, D.C., on Friday night en route to the G7 summit in Biarritz, France. Photo by Leigh Vogel/UPI | License Photo
Aug. 24 (UPI) -- President Donald Trump arrived in France for a G7 summit as clashes between protesters and police turned violent Saturday.
Trump's first stop was the Hotel du Palais in Biarritz, France, where he's expected to stay for the duration of the annual summit of the world's seven economic leaders, pool reports said.
The summit's host, French President Emmanuel Macron, said he wants to address raging wildfires in the Amazon rainforest, but the slowing global economy appears to be the most pressing matter.
Macron brought up G7 summit talking points while having an impromptu lunch with Trump on Saturday.
RELATED Environmentalists: Amazon fires in Brazil human-caused
Among the issues, Macron said Libya, Syria, Ukraine, North Korea and the Iran crises will be discussed, along with climate protections, gender equality, digital technology, the economy and trade.
Macron said that on Iran they share common ground to make sure Iran doesn't get access to a nuclear weapon and stability of the region.
"We will discuss, as well, about the economic global situation, how to decrease tensions and fix the situation in terms of trade," Macron said. "Because I think if we manage to fix the situation, seven of us -- we fix a great part of the world."
RELATED Bolsonaro: Macron shows 'colonialist mindset' by urging G7 to act on Amazon fires
Trump spoke briefly after Macron and said they "have a lot in common," and called him a friend.
"We've been friends a long time. And every once in awhile, we go at it just a little bit -- not very much. But we get along very well. We have a very good relationship. Sort of, I think I can say, a special relationship."
Trump and first lady Melania Trump then had a dinner with world leaders Saturday night at the base of a tall white lighthouse atop a rocky cliff overlooking Biarritz, the summit's venue, with formal talks starting Sunday.
RELATED Trump credits Putin, slams Obama in call for Russia to rejoin G7
Demonstrators took to the streets in Urrugne to protest the G7 summit Friday evening. Police said protesters threw objects, including fireworks, which injured four officers. Police responded by firing flash balls and tear gas.
Police arrested more than a dozen of the protesters as they attempted to return to their base camp. They faced charges of covering their faces and failing to disperse.
On Friday, Trump saw the stock market plummet as the the Dow Jones Industrial Average closed down 623.34 points, a 2.4 percent drop, amid economic uncertainty as the U.S.-China trade war escalated with tit-for-tat tariffs, adding to anxiety over the economy ahead of the G7 meeting.
Some G7 leaders believe these tariffs are contributing a to a global economic slowdown.
In his remarks on the South Lawn, before departing for France, Trump said he imposed more tariffs on China because "we want it to stop," the cheating.
Trump also said he was "not happy with Jay Powell," referring to Federal Reserve Chairman Jerome Powell.
He has downplayed the stock market plunge of around 600 points Friday, saying stock markets have risen since his election, "so don't talk to me about 600 points."
An administration official said Trump will boast of the U.S. economy's success at the summit's Sunday morning session.
"I would anticipate President Trump will be speaking quite frankly about the policies he's seen work in his own economy and really wanting to work with other countries in the G7 to figure out how we can jumpstart growth in economies all around to ensure that there are markets and opportunities for all of our workers and people," the administration official said.
"Trump will work with allies to build a future of opportunity and promise for all our nations," the office of the press secretary added in pool reports. "The president wants other nations to share in our success and urges leaders around the world to implement policies like tax reform and deregulation that promote strong economic growth."


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Рогатка звичайна (Ceratophrys cornuta) — вид земноводних з роду Рогатка родини Рогаткові. Інші назви «ітанія», «бразильська рогатка», «суринамська рогатка», «амазонська рогатка».

Опис
Загальна довжина досягає 7,2—20 см (без задніх кінцівок). Спостерігається статевий диморфізм: самиця більша за самця. Має щільний ропухоподібній тулуб, надзвичайно велику і широку голову й відповідний їй за розміром рот. Язик з глибоким вирізом ззаду. Над очима є загострені вирости, як й у всіх рогаток, що являють собою витягнуте у високе вістрі верхню повіку. Присутні також високі бородавчасті гребені на окостенілій шкірі голови.

Широка смуга, що проходить від кінця морди через всю спину, у самця помаранчево-червона, а у самки яскраво-зелена. Інші частини тіла зверху забарвлені у червоно-бурі, чорно-бурі і зеленуваті тони, вони розташовуються у вигляді плям та смуг. Черево посередині жовтувато—біле, з боків жовте і всіяне червоно-бурими плямами. У забарвленні спостерігаються значні варіації.

Спосіб життя
Полюбляє тропічні вологі ліси. Актвина вночі. Зустрічається на висоті до 400 м над рівнем моря. Часто заривається у листову підстилку або у ґрунт. Закопавшись вона сидить так, що назовні видається тільки велика голова. У такому положенні ця рогатка чекає на здобич. Живиться безхребетними, мишоподібними гризунами, жабами, ящірками.

Розмноження відбувається під час сезону дощів. Самиця відкладає до 1000 яєць. Пуголовки доволі ненажерливі. Живляться пуголовками інших земноводних.

Розповсюдження
Мешкає у басейні річки Амазонка (Гвіана, Суринам, Гаяна, Бразилія, Венесуела, Колумбія, Еквадор, Перу, Болівія).

Джерела
 Lynch, J.D. (1982). Relationships of the frogs of the genus Ceratophrys (Leptodactylidae) and their bearing on hypotheses of Pleistocene forest refugia in South America and punctuated equilibria. Systematic Zoology, 31, 166–179.

Рогаткові
Тварини, описані 1758

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Sorry but the registration is closed for this year!
Composers are invited to submit their works of electro-acoustic music (EAM) created after 1st January 2015. We place emphasis on the inner meaningfulness of the composition.
Category A is open to compositions of autonomous art EAM or only one independent movement.
Category B is open to compositions of autonomous art EAM, or only one independent movement, for acoustic instrument/voice/ensemble and electronic media (it have to include live performing part). Send a complete version of the composition, please, not only EA part. Incomplete compositions will be excluded.
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Awarded compositions will be presented during the Concert of laureats in December 2018 in Prague.