text:
\section{Introduction}
   \defcitealias{Kuntzer2016a}{KC16}

Future space-based observatories such as \textit{Euclid}\xspace\footnote{\url{http://www.euclid-ec.org/}} \citep{Euclid} and {\it WFIRST} \citep{WFIRST} require exquisite point-spread function (PSF) stability and measurement to achieve their scientific goals, in particular in view of the weak-lensing cosmological probe \citep{Cropper2013, Massey2013,Schneider2006}. Several instrumental effects have been identified as nuisances in the PSF determination process, such as the colour of the object and colour gradients across the objects \citep[][]{Voigt2012, Semboloni2013}, the brighter-fatter effect \citep{Antilogus2014}, or detector distortions that are due to temperature gradients.

Other astrophysical or atmospheric nuisances also contribute to the degradation of the signal, such as satellite trails, cosmic rays, stray light \citep[e.g.][]{Desai2016}, or PSF distortions by unresolved binary stars \citep[][KC16 hereafter]{Kuntzer2016a}.
We showed in \citetalias{Kuntzer2016a} that unresolved binaries can significantly alter the shape of the \textit{Euclid}\xspace PSF, even if many (thousands of) stellar images are used together to reduce this effect.

In this paper, we propose a technique to identify unresolved binary stars that may hamper a proper PSF measurement. We base our method on the systematic nature of the deformation induced by binary stars. While the deformation of individual stellar images is of the order of the instrumental noise, correlations between different exposures of the same object can lead to the detection of a binary star, or at least to flagging an object as non-reliable for subsequent PSF determination.

A similar approach was used by \citet{Hoekstra2005} to detect eclipsing binaries in the Optical Gravitational Lensing Experiment (OGLE).
Detecting unresolved binaries from the ground using this approach has already proven successful \citep{Terziev2013,Deacon2017}.
We build on the technique pioneered by \citet{Hoekstra2005} but apply more sophisticated algorithms, including artificial neural networks (ANN). We also study the applicability of our techniques to the specific case of \textit{Euclid}\xspace, with the goal of identifying unknown and unresolved binaries that affect the quality of the PSF reconstruction, as described in \citetalias{Kuntzer2016a}.

The observing strategy of \textit{Euclid}\xspace is to take four dithered exposures \citep{Cropper2016} of each field for the weak-lensing surveys.
The deep fields, located near the galactic poles, will be repeatedly observed over the course of the mission, at different orientations.
These mutiple images of the same stars can be used, when combined, to reconstruct an estimation of the PSF to account for its effect.
They can also be individually used to check for systematic biases in the shape estimates of the light profiles. Flagged objects can be removed from the stellar catalogues used in view of the PSF reconstruction.

In this proof-of-concept paper, we choose to work at the catalogue level, and not to measure \textit{Euclid}\xspace-simulated images of single or multiple images.
Our choice to work at the catalogue level is deliberate. Our goal is to show that our approach can be an efficient method to flag unresolved binaries when a catalogue of stars with shape measurements is available.
We do not aim at building a complete PSF shape measurement and reconstruction pipeline.

We show on simulated catalogues of non-dithered observations that we can robustly detect unresolved binary stars in a sample of point sources. This article is organised as follows: we detail the algorithms and associated performance metrics in Sect.~\ref{sec:algos}.
Section~\ref{sec:binaries} presents a summary of the statistical knowledge of the binary systems, and Sect.~\ref{sec:mock_obs} discusses the mock observations of stars with the VIS instrument.
We then illustrate the performance of the algorithm with image simulations in Sect.~\ref{sec:results}. Finally, Sect.~\ref{sec:summary} summarises our findings.

\section{Definitions, scheme, and training data}
\label{sec:algos}

 We propose different algorithms that learn the difference between systematic and stochastic distortions to predict whether an object is a single or a binary star. The examples used for training (in the machine-learning sense) could be drawn from real data of sources whose binary nature is known or from simulated data.

\subsection{Effect of binaries on the PSF shape parameters}
A more in-depth discussion of the effect of binaries on the shape of the PSF is provided in \citetalias{Kuntzer2016a}.
In this paragraph, we summarise the main results of \citetalias{Kuntzer2016a}
very briefly.
Throughout, we use the term contrast for the difference in magnitude $\Delta m$ between the host (or main) and companion star $\Delta m = m_\text{companion} - m_\text{host}$.
The presence of a companion can change the measured PSF ellipticity by an order of 1\% when the angular separation between host and companion is $r=0.\mkern-4mu^{\prime\prime}01$ (i.e. a tenth of the \textit{Euclid}\xspace pixel).
Similarly large effects are seen on the size measurement.
The distortion of the PSF induced by a companion star depends on the angular separation and on the contrast in magnitude \citetalias{Kuntzer2016a}.
The expected distortions of the PSF caused by binary objects are systematic in nature.
On average, the deformations due to binary stars tend to make the PSF rounder and larger.
On a case-by-case basis, however, the deformation of the PSF may not make it rounder, but could increase the ellipticity.

The deformation of the PSF measured on the image of a binary system depends on the relative positions of the host and the companion, and on the magnitude contrast.
For the wide-field survey of the \textit{Euclid}\xspace mission, the exposures are taken in a sequence, thus the physical properties  describing a binary system will not change.
In the deep exposures, taken in a much longer time interval, the properties of the binary system may change.

\subsection{Definitions}

For simplicity, we assume that a given star is imaged in several exposures at similar spatial positions on the detector. The observables used to classify an object are the shape parameters of the PSF. These observables, which are impervious to the measurement method, are the complex ellipticities ($e_1$ and $e_2$) and the PSF ``size'' $R^2$.
These can be defined using quadratic moments as follows.

Let $I(\pmb{\theta})$ be the brightness distribution of an an image. $\bar{\pmb{\theta}}$ would be the position of the centroid.
The tensor of second-order moments $q_{jk}$ computed along the first and second axis of an image is given by
\begin{equation}
  q_{jk}=\frac{\int \left(\theta_j-\bar{\theta}_j\right)\left(\theta_k-\bar{\theta}_k\right) \mathrm{I}(\pmb{\theta}) \mathrm{d}^2\theta}{\int \mathrm{I}(\pmb{\theta}) \mathrm{d}^2\theta},\quad j,k\in\{1, 2\}.
\end{equation}
In a noise-free image, the complex ellipticity components of a light profile are obtained from \citet{Kaiser1995}:
\begin{equation}
  [e_1, e_2] =\left[\frac{q_{11} - q_{22}}{q_{11} + q_{22}}, \frac{2q_{12}}{q_{11} + q_{22}}\right].
\end{equation}
The size of the PSF is defined as
\begin{equation}
  R^2 = q_{11} + q_{22}.
\end{equation}
The standard deviations of the variations in the PSF for the \textit{Euclid}\xspace survey must be smaller than $\sigma(e_i) \leq 2\times 10^{-4}$ for the ellipticity components and $\sigma(R^2)/\langle R^2\rangle \leq 1\times10^{-3}$ for the size according to science requirement 4.2.1.4 \citep{EuclidSciReq,Paulin-Henriksson2008,Euclid}. Each \textit{Euclid}\xspace image will contain from 2\,000 to 3\,000 stars \citep{Cropper2013,Euclid}, depending on Galactic latitude. If the PSF reconstruction scheme is optimal, the knowledge on each individual star is $\sqrt{2\,000}\approx50$ worse than the requirements, that is, 1\% for the ellipticities and 5\% for the size on each individual star.

\subsection{Simulated data}
\label{sec:simdata}
We stress that we work at the catalogue level and not at the pixel level. We assume that the PSF can be measured to within the \textit{Euclid}\xspace expected performances (as stated in the previous paragraph) for each individual star.
From a catalogue, we can flag suspicious observations of what has been considered as a single star.
The precision and accuracy we take on the shape parameters are given by the \textit{Euclid}\xspace requirements.
The preparation of a PSF pipeline that meets the \textit{Euclid}\xspace requirement is beyond the scope of this paper.
For this reason, we chose to work with a catalogue that fulfils the claimed requirements.
In this way, our results do not depend on any given shape measurement method either.

For the observations of single stars, we assume that the observed PSF parameters are noisy around their fiducial value (we discuss this assumption in Sect.~\ref{sec:assumptions}).
In the case of a binary system, however, the observables $e_1, e_2$, and $R^2$ are systematically modified by a value that depends on the angular separation and the contrast of the system.
We assume throughout that the PSF parameters $e_1,e_2,\text{a}\text{nd
}R^2$ are measured to the above precision, that is, 1\% (elipticity)
and 5\% (size) per star observation.

Catalogues of simulated single and binary system observations are produced by the following procedure and described in more details in Sect.~\ref{sec:mock_obs}:

\begin{enumerate}

\item Draw a stellar population from the Besan\c{c}on Galaxy Model \citep[BGM,][]{Robin2003} for a \textit{Euclid}\xspace-like sky.

\item Using empirical knowledge on the distribution of binary stars \citep{Duchene2013}, compute the confidence that any given star is a binary system and compute its characteristics (semi-major axis of the orbit and mass ratio).

\item Draw a realisation of the binary population with selections in angular separation and contrast.

\item Estimate the observed PSF parameters for each object in the catalogue and for each exposure.

\end{enumerate}

\subsection{Principle of the binary detection algorithms}
\label{sec:principle}
\begin{figure}[!h]
 \begin{center}
  \includegraphics[width=0.9\linewidth]{correlation}
 \end{center}
 \caption{\label{fig:correlation} Sketch of the measurement principles. The four observations are compared to the fiducial value of the shape parameter. A binary star will affect the shape of the PSF systematically across the four exposures (bottom), whereas the measurement errors are uncorrelated (top). }
\end{figure}
Our detection algorithms of unresolved binaries follows this procedure and is illustrated in Fig.~\ref{fig:correlation}.
\begin{enumerate}
\item Compute the relative error, $\delta^{(i)}$, based on the PSF parameters $p^{(i)} =e_1^{(i)}, e_2^{(i)}$ or $(R^{2}){^{(i)}}$ for a star $i$ with respect to a fiducial value $p^{(i)}_0$. The relative error is
\begin{equation}\label{eq:relerr}
\delta^{(i)} = \frac{p^{(i)}-p^{(i)}_0}{p^{(i)}_0}.
\end{equation}
For each star $i$, four such relative errors are measured for
each parameter because there are four observations.
We denote access to the relative error in  exposure $t$ as $\delta^{(i)}(t)$.
The fiducial parameter fields are either known {\it \textup{a priori}} or are directly inferred from the measurement PSF parameters using an iterative field estimation and outlier removal algorithm.

When the PSF field is unknown, the PSF at a given point is reconstructed using the parameters of neighbouring single stars. The interpolated value can be interpreted as the fiducial value at that point. When the measured relative errors are too different from the fiducial value, the object is considered as an outlier and is
removed from the list of single stars.
This process of interpolating a fiducial value based on neighbouring single stars and marking it single or outlier is repeated several times. This algorithm is detailed in Sect.~\ref{sec:uknPSF}.
\item Use the resulting relative errors $\delta^{(i)}$ as the features for the algorithms to infer the presence of binaries.

\end{enumerate}

\section{Algorithms to identify unresolved binary stars}
\label{sec:binfind}

We propose three different classifiers that yield a binary output (binary star or single star) or, similarly, a number between 0 and 1 that translates the confidence of detection. The three classifiers tested here are (i) a simple auto-correlation of the input features, and two supervised-learning methods, namely (ii) random forests (RF) and (iii) ANNs. All three methods are evaluated on two binary system selection criteria: their (i) angular separation, and (ii) contrast.

\subsection{Algorithm 1: Auto-correlation function}
\label{sec:method_acf}

 \begin{figure}[!h]
 \begin{center}
  \includegraphics[width=0.9\linewidth]{acf_sketch}
 \end{center}
 \caption{\label{fig:acf_sketch} Illustration of the result of the ACF for a relative error $\pmb{\delta}$ of a given shape parameter.
 We show the ACF for a single star (green) and a binary system (red).
 For simplicity, the noise level in the shape measurement is set to unrealistically low values.
 The shape parameters of the single stars are uncorrelated between exposures, thus the ACF coefficients are all zero, yielding the scalar quantity $\Sigma=\Sigma_{\tau=0}^3 \mathrm{ACF}(\tau)=0$. The shape parameters are correlated for a binary system, and a strong auto-correlation between exposures is observed, with a high scalar quantity $\Sigma=10$. }
\end{figure}

For each object $i$ in the field, we consider a list of four relative errors $\delta^{(i)}$ for each of the shape parameters. These lists can be understood as time series of the relative errors.
As described in Sect.~\ref{sec:principle} and illustrated in Fig.~\ref{fig:correlation}, when the object is a single star, the errors made on the measured parameters are stochastic.
However, when the object is a binary system, the errors are correlated because the relative position of the main and companion star do not change between the first and last exposure.
A measure of the correlation of the signal with itself can distinguish between no correlations (low auto-correlation score) or correlations (high auto-correlation score), which acts as a proxy to the binary or single star nature of the object.

An estimate of the auto-correlation function (ACF) coefficient for a lag of $\tau$ for a vector of relative errors $\pmb{\delta}$ observed at $t=\{1,2,3,4\}$ is given by
\begin{equation} \label{eq:acf}
 \mathrm{ACF}(\tau) =\frac{1}{ (n-1)\sigma^2}\sum_{t=1}^{n=4} [\delta{(t)} - \bar{\delta}][\delta{(t+\tau)} - \bar{\delta}], \,
\end{equation}
where $\sigma^2$ is the sample variance, and $\bar{\delta}$ is
its mean value as computed from $n$ observations of $\delta$. $n=4$ exposures in our work. For simplicity, we dropped the star index $i$ in the above equation. This method is data driven: it is based on the assumption that the relative errors are more strongly correlated for binary systems than for single systems. No physics is included in the model.
We sketch the result of the ACF method for a binary system and a single star in Fig.~\ref{fig:acf_sketch}.

We sum the four estimated ACF coefficients for the four observations to create a scalar quantity $\Sigma=\sum_{\tau=0}^3\mathrm{ACF}(\tau)$.
We choose a value for a separating threshold that classifies highly auto-correlated observables as binary stars and, conversely, links low auto-correlation of the observables to single stars.
Each PSF parameter yields an estimate of the auto-correlation. We refer to each of these classifications as $e_1,e_2$, and $R^2$ channels.
The outputs of the three detection channels can be combined to improve the classifiers' performance.
A three-channel classifier is obtained by taking a weighted average of the output of each individual channel.
The weights are empirically determined by evaluating the individual channel's performance.
The prediction of the three-channel classifier is again compared to a separating threshold.
The ACF classifier is run against a training dataset to determine the separating threshold, which is set such that the false-positive rate is 10\%.

\begin{figure*}
 \begin{center}
  \includegraphics[width=0.45\linewidth]{dist_v_contrast_semilog}
  \includegraphics[width=0.45\linewidth]{dist_v_angsep_semilog}
 \end{center}
 \caption{\label{fig:stellar_pop} Numerical model of the binary star density as a function of distance to Earth and contrast (left) and angular separation (right). The grey scales and the histograms are linear and relative to the maximum bin. We select stars in the magnitude range $18\leq I(AB)\leq24.5$. Both plots exhibit bimodal distributions with low-mass main stars located at shorter distances. More massive stars are not located in our direct neighbourhood, as suggested by the distribution of angular separation with distance. Companions of very low mass stars (down to the brown dwarf limit) are rare with the applied selection cuts. }
\end{figure*}

\subsection{Algorithm 2: Random forest}

When the expected distortions of the shape parameters are known, we can train an algorithm to recognise patterns in the data.
The random forest (RF) algorithm is a large ensemble of simple but weak decision trees.
Decision trees are a class of non-parametric supervised-learning classifier or regression methods that hierarchically approach the data.
The training of a decision tree consists of learning a set of decision rules based on the features and the ground truth \citep{Breiman2001}.
Decision trees taken individually tend to overfit the data. Averaging over a large number of trees -- a forest -- reduces the classification error.

In the same way as for the ACF classifier, the features adopted for the RF are the relative errors made on the PSF parameters as given by Eq.~(\ref{eq:relerr}). The ground truth is the nature of the object (binary or single star).
The depth of the individual trees is not limited, and a forest of 50 trees is constructed.

We used the RF implementation of the {\tt Python} package {\tt scikit-learn} with the Gini impurity criterion \citep{scikit-learn}.

\subsection{Algorithm 3: Artificial neural networks}

Feed-forward artificial neural networks (ANNs) belong to a class of supervised algorithms that can be used for regression or classification tasks. We applied ANNs to classify binary and single stars. For an introduction to ANNs, we refer to \citet{Bishop1995}. The ANNs are made of an input layer, an arbitrary number of hidden layers, and an output layer. The hidden and output layers consist of a number of neurons that take as input a vector $\pmb{x}$ and return a scalar $h(\pmb{x},\pmb{w}, b)$ through
\begin{equation} \label{eq:ANN}
 h(\pmb{x},\pmb{w}, b) = h\left(\sum_{i=1}^N w_i x_i + b\right),
\end{equation}
where $\pmb{w}$ and $b$ are the weights and the bias, respectively. The monotonic and continuous function $h$ is the so-called activation function.
In the following, we use the hyperbolic tangent as activation function.
The ability of an ANN to find patterns shared by the the input data (the features) and the labels of each image (the ground truth and its prediction) depends on the architecture of the network, in that case, the number of neurons and layers.
The architecture must be optimised for specific applications.
We found that for our application, three layers of 15 neurons each performs well.
We trained the network on a standard mean-square-error cost function.
The size of the training and test datasets is given in Sect.~\ref{sec:res_ann}.

The choice of the ANN implementation, like the choice of the RF implementation, is arbitrary from the outset. A comparison
of the details of each implementation is beyond the scope of this paper. We used the {\tt Python} bindings for the Fast Artificial Neural Network Library ({\tt FANN}) implementation and its optimiser described in \citet{Nissen2003}.

The ACF and the RF methods were applied to the simulated data described in Sect.~\ref{sec:results}
The performance of the ANNs is analysed in Section~\ref{sec:res_ann} on worst-case experiments only, namely on an unknown field in
both stellar population and PSF field parameters.

\subsection{Metrics for the detection performance}
\label{metrics}

We used three metrics to quantify the effectiveness of detecting binary stars:

\begin{itemize}
\item The \emph{\textup{receiver operating characteristic}} (ROC) is a graphical representation of the performance when the decision threshold is varied. The abscissa represents the false-positive rate (FPR, or fall-out),
\begin{equation}
 \mathrm{FPR} = \frac{\mathrm{false\ positives}}{\mathrm{false\ positives + true\ negatives}},
\end{equation}
while the ordinate encodes the true-positive rate (TPR, also called sensitivity and recall),
\begin{equation}
 \mathrm{TPR} = \frac{\mathrm{true\ positives}}{\mathrm{true\ positives + false\ negatives}}.
\end{equation}
An algorithm that would randomly classify a star as a binary lives on the diagonal FPR$=$TPR.
An ideal classifier would be represented by a curve passing through the coordinate $[$FPR$=$0, TPR$=$1$]$. The ROC allows for a quick and reliable comparison between different classifiers, marginalising over the detection threshold. The choice of the separating threshold can be based on the ROC by trading off the TPR for the FPR \citep{Kleinbaum2010}.
\item The \emph{area under the curve} (AUC) is the integral of the ROC over the FPR. It summarises the ROC as a single scalar. A random algorithm would score $\text{AUC}=0.5,$ while an ideal method would reach $\text{AUC}=1$. For this metric to be high, the true positive rate and the precision (the ratio of the true positives to the sum of true and false positives) must both
be high.
\item The $F_1$\emph{ score} is a scalar metric widely used for binary classification, defined as
\begin{equation}
 F_1 = \frac{2\mathrm{TP}}{2\mathrm{TP}+\mathrm{FN}+\mathrm{FP}},
\end{equation}
 where $\mathrm{TP}$, $\mathrm{FN}$, and $\mathrm{FP}$ are the numbers of true-positive, false-negative, and false-positive classifications, respectively \citep[e.g.][]{Herlocker2004}.
This metric is not only sensitive to the number of correctly detected binaries (TP), but also to the number of binaries classified as single stars (FN) and inversely (FP). A perfect classification would yield $F_1 = 1$, while a completely random decision-making algorithm would have an $F_1$ score that tends to zero.
\end{itemize}

\section{Binary star population in \textit{Euclid}\xspace}
\label{sec:binaries}

\begin{figure}
 \begin{center}
  \includegraphics[width=1.\linewidth]{dang_summary}
 \end{center}
\caption{\label{fig:dang_star} 95-percentile relative errors of the PSF shape parameters $e_1$ and the size. The relative errors in $e_2$ are about twice smaller. The values of the relative errors are given in the colour bar for the ellipticity (top number) and for the size (bottom number). The magnitude selection is $18 \leq I(AB) \leq 24.5$. The distributions of relative errors contain all binaries that have at most an angular separation on the $x$-axis and at minimum a contrast on the $y$-axis.}
\end{figure}

In order to test our binary star finders, we designed \textit{Euclid}\xspace-like simulated images, building on a stellar population with realistic binary fractions. The latter as well as the orbital parameter distributions and luminosity contrasts have been studied in volume-limited searches.
Early searches were restricted to low numbers of stars, hence leading to incomplete samples \citep[e.g.][]{Abt1976,Marcy1989,Duquennoy1991}. More recent searches harvested more data, but the total sample of stars with a detailed characterisation is still limited to a few hundred
\citep{Raghavan2010,Milone2012,Cummings2014,Nardiello2015,Riddle2015,Rodriguez2015}.

\citet{Duchene2013} compiled the available data and proposed distributions of multiple systems in terms of multiple star fraction, mass ratio to the main star, and angular separation.
We used these parameters to a create stellar population based on BGM realisations and with realistic binary fractions.
Figure~\ref{fig:stellar_pop} shows the contrast and the angular separation of binary stars as a function of distance to Earth for stars located in the direction of the anti-Galactic centre, with Galactic latitude in the range $+15^\circ<b<+75^\circ$.
We selected stars when their apparent magnitude was in the range of the expected magnitude for the \textit{Euclid}\xspace wide-field survey (i.e. $18 \lesssim I(AB) \lesssim 24.5$).
The bimodal nature of the distributions in Fig.~\ref{fig:stellar_pop} is explained by the mass of the main star. Massive stars in the
selected magnitude range are rare in the neighbourhood of the Sun, and inversely, low-mass stars are not selected by our cut in magnitude at large distances.
Figure~\ref{fig:stellar_pop} also presents histograms of the stars as a function of contrast and angular separation.
They demonstrate that most binary systems that will be seen by \textit{Euclid}\xspace have a low contrast, meaning that both stars are similar in terms of magnitude and have a very small angular separation, with about 75\% of the binaries separated by less than $3.5$~milliarcseconds (mas) and 50\% below $1.7$~mas.
Most companions have a magnitude similar to the main star of the system, but a wide diversity of contrast is expected.
Systems that contain brown dwarf companions are also predicted.

Figure~\ref{fig:dang_star} illustrates the magnitude of the relative errors due to binaries at small separations and low contrast.
We show the 95-percentile value of the expected relative errors in the shape of PSFs over a given binary population.
The intuition that larger relative errors occur at larger angular separation and low contrast is confirmed.
Quantitatively, the relative errors follow the same trends, but the effect is strongest on the $e_1$ component, followed by $e_2$, whose relative errors are about half as important.
The relative errors induced on the size are an order of magnitude weaker than on $e_2$. Since the size is also predicted to be measured five times less accurately than the ellipticities (see Sect.~\ref{sec:simdata}), the latter are more sensitive indicators of the presence of binaries than the former.

The change in PSF shape due to the presence of binary stars is not yet budgeted in \textit{Euclid}\xspace, and there is no requirement available so far on the performance of any binary rejection method.
However, Fig.~\ref{fig:dang_star} indicates a detection limit of $\sim3$~mas in separation almost independently of contrast.
This ensures that the distortions due to binaries are of the same order of magnitude as the measurement error per star for the ellipticity, that is, $\sim1\%$.

\section{Mock catalogues of PSF shape parameters}
\label{sec:mock_obs}

In this section, we explain how we create the catalogues of PSF parameters for any given single and binary system at any spatial position in the four central CCDs of the \textit{Euclid}\xspace VIS detector.

For the shape parameters contained in the catalogues to be in a realistic range, we measured super-sampled noise-free \textit{Euclid}\xspace-simulated PSFs.
We simulated 600 \textit{Euclid}\xspace PSFs at randomly selected positions on the detector using the ray-tracing tool {\tt Zemax} \citep{NgoleMboula2015}. The images of these PSFs are super-sampled by a factor of 12 (i.e. the pixel is one-twelfth of $0.\mkern-4mu^{\prime\prime}1$).
The shapes of the simulated single and binary stars were measured using the adaptive-moment scheme proposed by \citet{Bernstein2002} and based on re-Gaussianisation \citep{Hirata2003} as implemented in {\tt Galsim} \citep{Rowe2015}.
To obtain a realistic stellar population, we used the BGM, with the \textit{Euclid}\xspace cuts on magnitude ($18 \lesssim I(AB) \lesssim 24.5$) combined with the information on the estimates of the fraction of multiple stars and orbital parameter distributions derived from \citet{Duchene2013}.
We restricted the stellar population to astrophysical binaries. Coincidental binaries of field stars are negligible given the star density and field of view (of the order of 0.001\%).
We derived the rate of coincidental binaries from our catalogues of stars, estimating a high density of 5\,000 stars in the \textit{Euclid}\xspace field of view assuming a uniform distribution of the stellar positions.
As a simplification, images of binary and single stars were produced using a flat spectral energy distribution (SED).
In real data, the stellar SEDs will be provided by the \emph{Gaia} catalogue \citep{Bruijne2012,GaiaCollaboration2016}, ground-based multi-band data, or methods such as the VIS single-image spectral classifier as proposed in \citet{Kuntzer2016b}.
The measurements provide a set of fiducial shape parameters and sample the distortions due to binary stars.
Mock catalogues can be drawn from these fiducial values and distortions. These catalogues contain the shape parameters ($e_1, e_2$ and the size) for each object in the observed field.

To reflect the \textit{Euclid}\xspace observing strategy, four realisations of mock catalogues were prepared of the same field.
The noisy shape parameters were assumed to be known, as we described in Sect.~\ref{sec:algos}, to 1\% for the ellipticities and 5\% for the size.
The shape parameters were computed by interpolating their fiducial values from a set of the 600 \textit{Euclid}\xspace PSFs.
The distortions due to a given binary system were also interpolated.
The distortions were interpolated at their spatial position on the image, but also in terms of position of the binary with respect to its host and contrast.
The mock catalogues of the stars contained four noisy values for each shape parameter, corresponding to the four dithered observations.

After computing the interpolation, the PSF shape parameters for any system (single or binary in all its variety) can be derived and at any position.
Selection cuts were imposed on the contrast and the angular separation of the binary system during the preparation of the catalogues.
Regardless of the value of the criteria, the fraction of binaries was artificially maintained to 30\% to avoid a very low fraction of positive samples in the data, which would bias the metrics.

In the following numerical experiments, we only work on the catalogues.
\section{Numerical experiments}
\label{sec:results}

The binary classifier algorithms were applied to two different experiments: (i) the PSF field is known to the \textit{Euclid}\xspace requirements, a best-case experiment, and (ii) the PSF field is not known beforehand, the worst-case experiment. For each experiment, a training and test dataset were prepared according to Sect.~\ref{sec:mock_obs}.
The training set was necessary in the case of the ACF method to determine the separating threshold. For RF it was used to build decision trees.

In the worst-case experiment, the training set was used to optimise the weights and biases of the ANNs and the threshold was determined on an additional validation set.
The results reported in the following were measured on the test set that was similar to the training set, but was not shown to the algorithm during the training.
The selection criteria (contrast and angular separation of the binaries) in both experiments evolved on a grid. The angular separations ranged between 1 and 15 mas.
The angular separation is a lower bound criterion, while the contrast is an upper bound criterion.

\subsection{Known PSF field experiment}

\begin{figure}
 \begin{center}
  \includegraphics[width=1.\linewidth]{knPSF_roc_sep5_con10}
 \end{center}
\caption{\label{fig:knPSF_roc} ROC curve for a $5$ mas minimum separation and a contrast of up to one magnitude. The curves for the single channel ($e_1$, $e_2$, and $R^2$, denoted $N(e_1)$, $N(e_2)$, and $N(R^2)$) are shown along with the combined three-channel classifier (denoted ACF) and the RF method. All curves are colour-coded with the $F_1$ score computed at each threshold. The dashed line shows the performance of a random-guess algorithm. The numerical value in the legend is the AUC of the method.}
\end{figure}
\begin{figure}
 \begin{center}
  \includegraphics[width=0.99\linewidth]{knPSF_all_f1auc_alpha_contr}
 \end{center}
\caption{\label{fig:knPSF_perf} Performance metrics $F_1$ (left) and AUC (right) for different combinations of selection criteria in the known PSF experiment, using the ACF method (top panel) and RF (lower panel) to detect the binaries. The parameters of the PSF shapes are known to 1\% for the ellipticity and 5\% for the size.}
\end{figure}

\begin{figure}
 \begin{center}
  \includegraphics[width=1.\linewidth]{knPSF_featimp_sep8_con5}
 \end{center}
\caption{\label{fig:knPSF_featimp} Feature importance for an RF classifier with the selection criteria set at an $8$ mas minimum separation and a contrast of up to half a magnitude. Features 0 to 3 encode $e_1$ relative errors for each exposure, features 4 to 7 are the same for the $e_2$ component, and features 8 to 11 are the same for the size. The error bar shows the 1$\sigma$ deviations of the importance of the feature on the learned dataset.}
\end{figure}

In this experiment, we assumed that the PSF shape parameters are known to the \textit{Euclid}\xspace requirements at any spatial position, namely $\sigma(e_i) \leq 2\times 10^{-4}$ for the ellipticity components and $\sigma(R^2)/\langle R^2\rangle \leq 1\times10^{-3}$ for the size \citep{EuclidSciReq,Paulin-Henriksson2008,Cropper2013,Massey2013}. The number of objects in the datasets depends slightly on selection criteria, but amounts on average to a few thousand objects.

For every pair of criteria, the threshold selected on the training set produced the targeted 10\% FPR in the testing phase. An ROC curve was calculated for each of the individual channels (i.e. classifying on each of the shape parameters individually) and the three-channel classifier (i.e. taking into account all shape parameters to classify).
The ROC curves are displayed in Fig.~\ref{fig:knPSF_roc}.
The resulting AUC, an indicator of the performance of each of the channel, was used as weight for the three-channel ACF.
The weights were left fixed for all criteria pairs.

The most useful information to predict the nature of the object is inferred from the $e_1$ relative error features (Eq.~(\ref{eq:relerr})). The second most useful information is the second ellipticity component. While its overall performance is poorer than the $e_1$ channel, the $e_2$ channel has a better TPR at low FPR. This better performance at low FPR implies that a combination of the two ellipticity channels leads to a better overall classification. The ACF size channel classifier is a very poor predictor of binary systems.
This remains true for most choices of selection criteria, short of very large angular separations.

As expected, better performances in terms of both AUC and $F_1$
were obtained when the angular separation was increased. This is illustrated in Fig.~\ref{fig:knPSF_perf}. At small angular separations, the ACF classifier is barely better than a random classifier, with $F_1\sim 0.2-0.3$. At large angular separations however, the performance of the classifier is greatly improved. In general, the performance of the classifier mostly
depends on the angular separation and very little on the maximum luminosity contrast.
The separating threshold chosen was chosen on the training dataset such that the FPR was 10\%. The value of this separating threshold was the same for any choice of selection criteria.
The separating threshold was thus not influenced by the selection criteria, which simplified the application of the ACF technique.

We then tested the RF method using the same training and test datasets as for the ACF. The RF method consistently obtained better scores than the ACF, as is shown in Fig.~\ref{fig:knPSF_perf}. The ROC curves for the RF method (Fig.~\ref{fig:knPSF_roc}) also indicate a better AUC and a higher $F_1$ score for the same FPR.
The separating threshold is determined during the RF training to reach the goal FPR of 10\%. The relative importances of the features in the RF algorithm show, according to the ACF method, that the $e_1$ relative errors (see Fig.~\ref{fig:knPSF_featimp}) encompass the most decisive information.
The $e_2$ features come in second, while the size information is third and negligible in the decision-making process.
The RF method is only weakly dependent on the contrast, like the ACF. When the PSF field is known (to the the \textit{Euclid}\xspace requirement), the RF method in particular can be used to find binary systems even at small angular separations.

\subsection{Unknown PSF parameter experiment} \label{sec:uknPSF}

We now turn to an experiment where we assumed no prior know\-ledge of the PSF field. To follow the procedure laid out in Sect.~\ref{sec:algos}, the PSF shape parameters must first be determined at the position of the objects of interest and the field of PSF shape parameters estimated.
To this end, we adopted a "leave-one-out" scheme, described in algorithm~\ref{algo:leaveoneout}. The PSF shape parameters at the position of the objects are interpolated from the nearest ten and assumed single neighbours and taken as fiducial parameters.
The shape parameters of the objects are then compared to the fiducial parameters. The resulting relative errors are then used as the vector of features for the classifiers.
The object is then classified into as a single or binary star.
At the first iteration of algorithm~\ref{algo:leaveoneout}, we assumed that all stars in the field are single.
The above procedure was repeated to iteratively construct a good estimate of the PSF field, and to determine where the binaries are in the field of view.

\begin{algorithm}
 \begin{algorithmic}[1]
 \Procedure{UknPSF}{star\_list, $p$}
 \State  single\_stars $\gets$ star\_list
 \For {iteration $<$ max\_iterations}
 \State  iter\_single\_list $\gets$ []
 \For {{\bf each} star {\bf in} star\_list}
        \State usable\_stars $\gets$ single\_stars $-$ star
        \State K $\gets$ Get10ClosestStars(usable\_stars, star)
        \State\# K is the list of the 10 nearest stars
        \State $w \gets$ InverseDistanceSquare(K, star)
        \State $P^{(\text{K})}\gets$MedianOverExposures($p^{(\text{K})}$)
        \State\# $P^{(\text{K})}$ is an array of K stars $\times$ 3 shape parameters, averaged over exposures
        \State $p^{(\text{star})}_{0} \gets $ WeightedAverage($P^{(\text{K})}$, $w$)
        \State\# $p^{(\text{star})}_{0} $ is a vector of the 3 shape parameters      \State $\delta^{(\text{star})} \gets (p^{(\text{star})}-p^{(\text{star})}_{0})/ p^{(\text{star})}_{0}$
        \State\# $\delta^{(\text{star})}$ is an array of 4 exposures $\times$ 3 parameters
        \If {Classification($\delta^{(\text{star})}) ==$ Single }
        \State append(star, iter\_single\_list)
        \EndIf
 \EndFor
 \State single\_stars $\gets$  iter\_single\_list
 \EndFor
\State binary\_list $\gets$ star\_list -  single\_stars
 \State \textbf{return} binary\_list
 \EndProcedure
 \end{algorithmic}
 \caption{``Leave-one-out'' reconstruction scheme used in the case of an unknown PSF field for the ACF and RF methods in the training phase. $p$ denotes the PSF parameters for all stars and the four exposures, and $\delta$ is the relative errors.}
 \label{algo:leaveoneout}
\end{algorithm}

\begin{figure}
 \begin{center}
  \includegraphics[width=0.99\linewidth]{uknPSF_roc_sep10_con10} \end{center}
\caption{\label{fig:uknPSF_roc} ROC curves for the unknown PSF experiment. The angular selection criterion is 10 mas and the
contrast is of up to one magnitude.
The curves for the single channel ($e_1$, $e_2$, and $R^2$, denoted $N(e_1)$, $N(e_2)$, and $N(R^2)$) are shown along with the combined three-channel classifier (denoted ACF), the RF, and the ANN methods. All curves are colour-coded with the $F_1$ score computed at each threshold. The dashed line shows the performance of a random-guess algorithm. The numerical value in the legend is the AUC of the method.}
\end{figure}

The VIS instrument will be comprised of 6 x 6 CCDs, each with 4k $\times$ 4k pixels, with a field of view of $0.787\times 0.709$ deg$^2$ \citep{Racca2016}. About 2\,000-3\,000 objects will be used to reconstruct the PSF field in any pointing \citep{Cropper2013,Euclid}.
In the present work, we only have PSF estimates for the four central detectors.
The number of stars seen in this reduced field is 280, and we still use a binary fraction of 30\%.

We determined the threshold for the ACF and inferred the decision rules for the RF on the training set in a similar way as in the known PSF field experiment.
To test the performance of the methods, we created 20 different fields, thus different spatial positions and binary populations, each containing 280 stars. The reported metrics were averaged over the 20 fields. The relative error estimator during the testing phase was modified from Eq.~\ref{eq:relerr} to
\begin{equation}\label{eq:relerr2}
\delta^{(i)} = \frac{p^{(i)}-p^{(i)}_0}{p^{(i)}_0 + \epsilon},\end{equation}
where $\epsilon$ is a calibration coefficient accounting for the PSF field reconstruction errors.
For RF, $\epsilon_\text{RF}$ varies from 0.03 to 0.07 as a function of the selection criteria, while $\epsilon_\text{ACF}$ is in the range $[0.01,0.035]$.
If this modification is not enforced, the FPR increases several
times.

The ACF method performs well in the case of an unknown PSF field.
Even if its overall performance is reduced, it remains close to the known PSF field experiment.
Figure~\ref{fig:uknPSF_roc} shows the ROC curve in the unknown PSF case. The ACF principle is simpler in the sense that it does not need to train on the features themselves.
It only has to find a threshold between single and binary stars based on the degree of correlation of the features.
The RF method relies on the individual relative error estimates to infer the binary nature.
The interpolation of PSF parameters and determination of a binary star loop is not encoded in the training of the RF, which reduces the performance.
For both methods, the FPR is significantly reduced between the first iteration (in which no binaries are assumed in the field) and the second iteration to settle to about 10\% as required. After this, the FPR stays roughly constant, while the $F_1$ score tends to increase slightly.
The ACF seems to outperform the RF technique, both in terms of AUC (reaching a maximum of about 0.9 --  Fig.~\ref{fig:uknPSF_perf}) and $F_1$ score, especially at high angular separations.
The performance of the classification again mostly depends on the minimum angular separation and not on maximum contrast.
In Figures~\ref{fig:knPSF_perf} and \ref{fig:uknPSF_perf}, the selection is made based on maximum contrast.
If the selection were made on minimum contrast, there would be a stronger effect.
However, we are interested in removing the most damaging objects from the PSF samples, and systems with a high contrast will not be the most damaging during the PSF reconstruction (see Fig.~\ref{fig:dang_star}).
The inclusion of the hyperparameter $\epsilon$ does succeed in producing good detection, but it is determined manually, which can hamper the image reduction pipeline.
For this reason, we present in the next paragraph the ANN approach, which does not require the inclusion of the hyperparameter $\epsilon$.

\begin{figure}
 \begin{center}
  \includegraphics[width=0.99\linewidth]{uknPSF_all_f1auc_alpha_contr}
 \end{center}
\caption{\label{fig:uknPSF_perf} Performance metrics $F_1$ (left) and AUC (right) for different combinations of selection criteria in the unknown PSF experiment, using the ACF method (top panel), RF (mid panel), and ANN (bottom panel).}
\end{figure}

\subsection{Applying ANNs to unknown PSF field experiments} \label{sec:res_ann}

The reconstruction scheme, based on the calculation of the relative errors and correlations between exposures, is subject to poor performance if there is a significant dither between exposures. The PSF can indeed spatially vary to such a degree that there is only little correlation in the PSF parameters. A more flexible approach to capture this effect is to teach the method about the deviations from the fiducial value and how it evolves with the position of the object on the chip. This latter scheme by construction handles any dithering and has the additional advantage of not having to tune an additional parameter such as $\epsilon$ in the relative error as in Eq.~\ref{eq:relerr2} to calibrate out the errors that are due to the PSF field reconstruction.

The method was trained on parameters observed on 25 unknown fields, each containing 280 stars, with the fiducial parameters computed by the ``leave-one-out'' reconstruction scheme. We give as features the deviations from the estimated parameters and the estimated parameters. The procedure of identifying the binary star is similar to the ACF or RF procedures: first, a run on all objects, with no prior on which are binaries. Then a second pass is made, with an interpolation ignoring the objects that were blacklisted as binaries in the first run. This algorithm is almost identical to algorithm~\ref{algo:leaveoneout}, with the difference that the features used to classify are the interpolated fiducial parameters and the measured parameters. The second pass reduces the FPR by a few percentage points and slightly increases the true-positive rate. Two passes are sufficient to reach the best performance, while ACF and RF typically require three to five passes. In the current implementation, there are 15 input features (three deviations times four exposures plus the three interpolated fiducial values of the parameters) with a hidden three 15-neuron layers and 1 output neuron.
The architecture used in this work is thus: 15 inputs, 15 hidden neurons, 15 hidden neurons, 15 hidden neurons, and 1 output.
We validated the training on a set of five fields to select the separating threshold value and tested 15 fields of 280 stars.
Overall, the ANN method performs better than either ACF and RF, especially at high minimum angular separation, reaching an AUC and $F_1$ score of almost one, as shown in Fig.~\ref{fig:uknPSF_roc}. The $F_1$ score, as shown in Fig.~\ref{fig:uknPSF_perf}, is higher for most of the selection criteria. The main drawback of this method is the complexity of the training scheme.

\subsection{Performance dependence on the assumptions} \label{sec:assumptions}

In the following, we discuss different effects or assumptions that can significantly affect the performances of the algorithms.

\begin{itemize}
\item The measurements of the shape parameters are assumed to be accurate.
The features should reflect a systematic bias between the exposures. If there is a systematic additive bias in the shape measurement, the algorithms could be misled into classifying single stars as binaries.
For the ACF classifier, the performance depends on the value of the additive bias. A slow decrease in metrics value is observed until an additive bias levels of the order of $10^{-3}$.
Past this threshold, the ACF classifier is essentially random.
For the machine-learning classifiers, if the additive bias is included in the training, no significant degradation of the performance is noted. If the bias is not included, the same threshold in bias level is observed with the same consequences.
\item The errors on the shape measurements are assumed to be Gaussian. We tested the shape measurement algorithm based on adaptive moments on noisy PSF images at the \textit{Euclid}\xspace resolution.
Although the errors are larger than the \textit{Euclid}\xspace requirements, their distribution is Gaussian.
\item If the precision on the measurement of the size is improved from 5\% to $0.5-1\%$, the size channel carries much more weight.
If the precision of measurement on the ellipticities is improved by a factor of two from the baseline of 1\%, the performance increases by a few percent. A degradation of the same magnitude is observed if the precision is worsened by a factor of two.
 \item The number of stars per field in the \textit{Euclid}\xspace survey will vary depending on Galactic latitude. The estimates range from 1,800 objects for high Galactic latitude to twice this number at low Galactic latitudes \citep{Cropper2013, Euclid}.
For the ACF, both performance metrics are stable for a number of stars per field higher than 1\,000.
For the RF, the AUC metrics increase by 5-10\% between fields where 1\,000 stars are present and 3\,000 stars per field.
The AUC for the ANN similarly improves in the same conditions.
\item We finally discuss the completeness of the detection when the algorithms are trained with a full binary population, that is, with no selection of the binaries for training.
As expected from Fig.~\ref{fig:uknPSF_perf}, the AUC value is 0.5 and the overall $F_1$ score is poor. The completeness of the detection of binaries at large angular separation is also significantly reduced compared to methods trained on binaries selected for their large angular separation. The value of the completeness is roughly divided by two between a classifier trained to find unresolved binaries in an unknown PSF field at 10 mas minimum and a classifier trained to detect all binaries.
\end{itemize}

\section{Conclusion} \label{sec:summary}

Unresolved binary stars can create significant biases on the PSF determination in space-based weak-lensing surveys like \textit{Euclid}\xspace. As binary stars are ubiquitous in the sky, their observation cannot be avoided. The catalogues of single and binary stars provided by \emph{Gaia} will be useful to flag a number of undesired objects, but the binary stars identified by \emph{Gaia} are not expected to match the depth of \textit{Euclid}\xspace or match its footprint on the sky \citep{Eyer2015}.

We here presented an approach to detect unresolved binaries using catalogues of shape parameters of PSFs observed multiple times.
Repeated measurements of the same objects are provided by the dithering plan and the deep field observations.

We used relative errors of the complex ellipticity and size to their fiducial value as input features for our classification algorithms. We proposed three methods. The first (ACF) is based on the auto-correlation of the relative errors of the parameters. The two others are supervised machine-learning algorithms: random forest (RF), and artificial neural networks (ANN). The methods were tested using two numerical experiments: a best case, in which the PSF parameters are known to the \textit{Euclid}\xspace requirements, and a worst case, in which the fiducial PSF parameters are unknown.
Based on an analysis of the distortions caused by binary stars in a realistic \textit{Euclid}\xspace setting, we suggested a detection limit for binaries separated by at least $\sim 3$~mas with a contrast lower than 1.5 magnitude to remove binaries whose distortions on the PSF ellipticity is of the order of the measurement error.

Supervised machine-learning approaches perform well in the best-case scenario, and the detection limit of 3 mas is reached. The tests on the ACF method indicate that about 50\% of the binary stars above the detection limit are correctly flagged. The influence of the contrast is weak in the detection performance, which depends mostly on the angular separation. In the worst-case experiment, the performances are degraded because the fiducial PSF shape parameters must first be determined. The RF method is limited in the worst-case scenario. ANNs performs better than the other two methods because of the increased ability of learning the relation between the deviations from the fiducial values and the binary nature of the object. The ANN approach is able to detect about 50\% of the binaries at a angular separation of 5 mas minimum. We stress that in this worse-case scenario, no prior knowledge of the PSF was used at all.

The methods proposed here are given as a proof-of-concept. Dithering between exposures is left for future implementation. The treatment of dithering can be taught to a machine-learning approach without any major change in the method, as shown by our tests with the ANN.
Priors based on the apparent spectral class of an object can be added to overcome the difficulty of the dependence of the PSF shape on the spectra of the observed point source.
The incorporation of PSF parameters know\-ledge, even if not at the \textit{Euclid}\xspace requirements, can significantly increase the effectiveness of the proposed methods.

The code corresponding to the algorithms used in this work and all scripts to reproduce the results are publicly available from a GitHub repository accessible via \url{http://lastro.epfl.ch/software}.

\begin{acknowledgements}
The authors would like to thank J\'er\^ome Amiaux, Patrick Hudelot, Koryo Okumura, and Samuel Ronayette for providing the simulated \textit{Euclid}\xspace\ PSFs.
We would like to thank Henk Hoekstra and Joana Frontera-Pons for reading a draft of this paper and Vivien Bonvin for useful suggestions.
We thank the anonymous referee for their time and valuable comments.
This research has made use of NASA's Astrophysics Data System.
We are grateful to the authors of the following {\tt Python} packages: {\tt Astropy}, a community-developed core Python package for Astronomy \citep{astropy}, {\tt Matplotlib} \citep{matplotlib}, {\tt Scipy} \citep{scipy} and {\tt Numpy} \citep{numpy}.
This work is supported by the Swiss National Science Foundation (SNSF).
\end{acknowledgements}

\bibliographystyle{aa}

redpajama_set_name: RedPajamaArXiv


Physical (body), Mental (thoughts), Emotional (feelings) and Spiritual (connection to Authentic Self, personal meaning and purpose).
Primarily, in my work I use Positive Psychology, Solution Focused Therapy, Spiritual Psychology, Person Centered Therapy, Holistic Counseling, Energy Psychology and EMDR.
I've written more about each of these areas below for those interested in more detail.
Positive psychology focuses on helping people to thrive. Using positive psychology, I use effective interventions to build on client strengths, and help clients reduce anxiety and depression, and become happier. Martin Seligman is the founder of positive psychology, a field of study that examines healthy states, such as happiness, building on strengths and optimism.
Solution Focused Therapy is designed to empower clients to recognize their own resources to resolve their issues. Clients determine the goals for treatment, and report weekly on improvements, ripple effects from the changes, rate their progress, receive feedback from the therapist and figure out their next steps toward resolution. Concrete and goal oriented, this fits well with my active coaching style.
Spiritual psychology returns to the true definition of psychology… 'study of the soul'. It recognizes spiritual reality and begins with the idea that we are Souls having a human experience. For many, this is a radical new idea! I assist my clients in recognizing the Beauty, Wisdom, and Compassion within them and inherent in every human being. Spiritual psychology isn't referring to religion. It speaks to the authentic essence of who you are — the True Self. I love working with clients who resonate with this, empowering them to find more meaning and create more balanced lives, practice self-nurturing, and connect with their intuition.
Created by Carl Rogers, the person-centered approach views the client as their own best authority on their own experience, and fully capable of fulfilling their own potential for growth. Rogers strongly believed that therapists should be warm, genuine, authentic and understanding as they journey with the client. I do my best.
Often emotional symptoms like anxiety and depression result from physical problems, and sometimes they occur as side effects from health problems. I'm not a physician, but part of my role is educating clients. Natural health strategies such as clean diet, pure water, restful sleep and walking in sunshine help improve mood. My orientation is toward seeking the cause of disturbance.
EFT or 'Tapping' is a technique I recommend to optimize emotional health. Emotional health is essential to your physical health, and you won't achieve ideal health with emotional barriers in your way. EFT is self-administered, and easy to learn. It's a form of psychological acupressure, based on the same energy meridians used in traditional acupuncture to treat physical and emotional ailments for over five thousand years, but without needles. Instead, simple tapping with the fingertips is used. This combination of tapping the energy meridians and voicing positive affirmation works to clear the "short-circuit" – the emotional block – from your body's bioenergy system, restoring balance. EFT has a very high rate of success.


text:
Dashtab (, also Romanized as Dashtāb) is a village in Bakharz Rural District, in the Central District of Bakharz County, Razavi Khorasan Province, Iran. At the 2006 census, its population was 183, in 44 families.

References

Populated places in Bakharz County

redpajama_set_name: RedPajamaWikipedia


text:
import json
from jinja2 import Template

HTML_TEMPLATE_PATH = 'stackexchange_analyzer/templates/result.html'

def export_to_json(result, export_path):
    with open(export_path, 'w') as export_file:
        export_file.write(json.dumps(result))

def export_to_html(result, export_path):
    with open(HTML_TEMPLATE_PATH) as template_file:
        template = Template(template_file.read())
    rendered_html = template.render(questions=result)
    with open(export_path, 'w') as export_file:
        export_file.write(rendered_html)

redpajama_set_name: RedPajamaGithub


1/ @coincheckjp hack update: NEM is creating an automated tagging system that will be ready in 24-48 hours. This automated system will follow the money and tag any account that receives tainted money. NEM has already shown exchanges how to check if an account has been tagged.
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5/5 My interview with Jeff McDonald @thejabo38 NEM VP of NEM Foundation coming soon. Stay tuned, NEM community.
1.) NEM will not fork.
2.) XEM (as well as any coin) is tracked on the blockchain. We can see where the hacked funds are and flag the accounts. What we cannot do is alter/disable the funds.
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On Saturday, April 28th 2017, The Burlington Township Police Department will be allowing individuals to dispose of medications anonymously. No questions or requests for identification will be made. Individuals (No Businesses) will be able to deposit medication freely and independently.
Individuals may dispose of the medication in its original container and will place the medication inside the provided drug disposal box. If the original container is submitted, the individual is encouraged to remove the prescription label from the container.


text:
Who Is The Best Author Of All Time
July 13, 2021 by adminbagus Leave a Comment
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Charles Dickens 168. 100 rows Authors such as Miguel de Cervantes Jane Austen Alexandre Dumas Charles Dickens. Ray Bradbury 3 William Peter Blatty 3 Now these are all classic well-respected horror authors who wrote groundbreaking novels or who took the genre in new and unexpected directions.
In addition to being one of the bestselling authors of all time Sheldon also worked as a TV producer who created the iconic show I Dream of Jeannie It is estimated that Sheldons books have sold around 300 million copies. 1 WILLIAM SHAKESPEARE William Shakespeare is an English playwright poet and actor born in April 1564. Grown-ups are quirky creatures full of quirks and secrets The 19th century author Roald Dahl is well known for his incredible childrens stories.
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Filed Under: wallpaper Tagged With: author, best, time

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THURSDAY, Feb. 4, 2016 (HealthDay News) -- Giving steroids to pregnant women at risk for late preterm delivery may reduce the risk of severe respiratory problems in their babies, a new study finds.
The study included more than 2,800 pregnant women with a high risk of late preterm delivery (34 to 36 weeks of pregnancy) who were randomly selected to receive two injections over 24 hours of either the steroid betamethasone or a placebo.
The steroid is commonly used to prevent complications in babies born before 34 weeks of pregnancy. A full-term pregnancy is considered to be 40 weeks.
Compared to babies born to mothers who received the placebo, babies born to mothers who received the steroid were much less likely to have severe respiratory complications shortly after birth, to require a long stay in the neonatal intensive care unit, or to need respiratory treatments.
The findings were published Feb. 4 in the New England Journal of Medicine.
"Our study demonstrates that administering a medication that is commonly used to prevent complications in babies born before 34 weeks of gestation can also reduce the risk of many serious complications in babies delivered just a few weeks before term," lead investigator Dr. Cynthia Gyamfi-Bannerman, an associate professor of women's health at Columbia University Medical Center in New York City, said in a Columbia news release.
"This will transform the way we care for mothers at risk for late preterm delivery," added Gyamfi-Bannerman, who is also an obstetrician and maternal-fetal medicine specialist at New York-Presbyterian Hospital, in New York City.
Each year, about 8 percent (more than 300,000) of babies born in the United States are late preterm births. About 12 percent of those babies have respiratory or other serious complications and require long stays in the neonatal intensive care unit, the researchers said.
"While survival among late preterm infants is comparable to that of babies born at term, the rate of respiratory problems and other serious complications in this group is not comparable and remains unacceptably high," Gyamfi-Bannerman said.
"Expanding the use of a well-studied, safe medication to improve lung development before birth offers a means of preventing many of these complications," she said.


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Non-sparking aluminum models break down to fit into spill kits as small as 5-gallon pails. Assemble in just seconds.
Model #0404 features a polyethylene scoop and excellent chemical resistance.
Dimensions: Assembled: 29" x 10" x 11" Collapsed: 15" x 10" x 3"


Browser-based ShopZone Has Desktop Upgrade - InternetNews.
Breakthrough Software announced the availability of ShopZone Web Edition, a browser-based e-commerce solution that includes a desktop upgrade option.
Aimed at ISPs, telcos, and portals that want to host online malls, it is designed for users to import their online stores in the desktop version for greater functionality and advanced customization--then publish them via the same ISP or host.
"Time is money, so we made ShopZone Web Edition as easy as possible for a host to setup and configure, and for a merchant to get started building their professional e-commerce site," said Trevor Stout, Breakthrough Software's co-founder and CEO.
"And because ShopZone Web Edition is the only browser-based e-commerce solution with a desktop upgrade path for advanced development, customers using our product save time in the long run because they won't have to start over from scratch."
ShopZone Web Edition provides a store setup wizard to guide users through the online store creation process. Advanced features include customizable site style sheets, advanced page editing, discussion groups, online forms, site statistics, and product specials. There are full site administration capabilities for the merchant, plus facilities for online payments, taxes, and shipping rates.
The vendor said it can provide ShopZone Web Edition immediately to ISPs and other large hosting services.


Our after school karate classes ended last week. A student of the year was awarded for Tuesday and Wednesday's groups.The club proved to be very popular with pupils and their parents. This club is financed by a grant from the Mercy Order Northern Province. Ronan Collins ,instructor to the club,has invited any interested pupils to join his evening classes for the summer to continue their karate training.


The Protocol Office manages the reservation of the Misteri d'Elx Hall located in the Torrevaíllo Building and the Rectorate and Social Council Building Function Hall.
For online applications send "the technical needs and risk management form" to the Vice-rector for material resources and equipment. vdo.recmater@umh.es.


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Estinzione (The Sixth Extinction) – romanzo di James Rollins del 2014
 La sesta estinzione: una storia innaturale (The Sixth Extinction: An Unnatural History) – saggio di Elizabeth Kolbert del 2014


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James Bond Muscle Car Heads To Auction
Yes, Agent 007 had a taste for American muscle…
Usually, when people think of the cars from the James Bond movies, their mind immediately goes to the Aston Martins. From there they might think of the Lotus Esprit submarine or any number of other luxury vehicles, mostly British but also some German. However, in the movie On Her Majesty's Secret Service, James Bond is along for the ride as Contessa Teresa di Vicenzo engages in a high-speed, snowy chase with skis strapped to the back of a 1969 Mercury Cougar XR7 convertible. Now, that American muscle car is headed to auction and it's the first time that's ever happened, so collectors are understandably excited.
Special ordered in the UK, this Mercury was built on February 6, 1969. It features a red over red scheme, steel wheels, and the Jet Ram V8 engine. There were only 127 of these XR7 convertibles to use the Jet Ram V8, so even without its movie history this vehicle would be quite desirable.
photo credit: Bonhams
For the movie, a ski rack was fitted to the trunk lid and a pair of Kneissl skis were strapped on. The car still has both the rack and skis, instantly signaling to Bond fans this is no ordinary Cougar.
For filming, it's often the case that multiple cars are used. There were actually three of these Mercury Cougar XR7 convertibles. Research suggests this muscle car was used in the famous barn scene where James Bond proposed as the two sought shelter from a blizzard.
The subject of a recent restoration, this former movie car is now looking its best.
The Cougar XR7 is anticipated to hammer at between $130,000 and $200,000 USD. Of course, since this car has never crossed an auction block before, there's a sense that estimated range could be off, maybe by quite a bit. We'll see on December 16 when Bonhams auctions it as part of The Bond Street Sale. Check out the lot for yourself here.


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Q: Setting exact view area (rect) of QGraphicsScene in QGraphicsView - a better fitInView()? I am failing in trying to restore a particular view (visible area) in a subclassed QGraphicsView on a subclassed QGraphicsScene.
The visible rectangle of the scene viewed should be
mapToScene( rect() ).boundingRect()

which is
QRectF(27.8261,26.9565 673.043x507.826)

I obtain these values in the destructor of QGraphicsView, they seem valid.
After restoring the window geometry, I am using said values in the first QGraphicsView::resizeEvent (the first event has an invalid old size QSize(-1, -1)) in the attempt to restore the area shown in view using
fitInView( QRectF(....) , Qt::IgnoreAspectRatio);

This triggers a number of scrollContentsBy events before showing the view, another resize and scroll event after, then the mainwindow showEvent fires, causing more resize scroll and resize events.
I am sure this sequence is neccessary for the GUI layout to build, but I am now confused as to how I could run fitInView once everything is set up.
Using a QTimer::singleShot to run the function well after the GUI is shown, I get approximately the desired result, but the matrix and viewed area differ:
ui->graphicsView->matrix();
ui->graphicsView->mapToScene( ui->graphicsView->rect()).boundingRect();

Before:
"[m11=1 m12=0 m21=0 m22=1 dx=0 dy=0] (x=27 y=27 w=774 h=508)"

Restored:
"[m11=0.954724 m12=0 m21=0 m22=0.954724 dx=0 dy=0] (x=17.8062 y=24.0907 w=810.705 h=532.091)"

So fitInView() does not serve my purpose very well - is there another, reliable way?
I'm using Qt 4.8.1 with MSVC2010

Another approach is to restore transform and window scroll position like so:
settings->setValue("view", mapToScene( rect() ).boundingRect() );
settings->setValue("transform", transform() );
settings->sync();

and restore them
QTransform trans = settings->value("transform", QTransform()).value<QTransform>();
QRectF view = settings->value( "view", QRectF() ).value<QRectF>();
setTransform( trans );
centerOn( view.center() );

But this method, too, has an offset.
Before
"[m11=4.96282 m12=0 m21=0 m22=4.96282 dx=0 dy=0] (x=29.6203 y=29.4188 w=155.96 h=104.981)"

After
"[m11=4.96282 m12=0 m21=0 m22=4.96282 dx=0 dy=0] (x=54.8076 y=53.8001 w=155.96 h=104.981)"

An offset also is present when scrollbars are hidden.
Moving the code to showEvent() does not affect result.

A: What is finally working: in showEvent, restore the transform, restore scroll-bar values:
setTransform( trans );
verticalScrollBar()->setValue( v );
horizontalScrollBar()->setValue( h );

Scroll bar visibility does not affect the view position and size, they are just "overlays".

redpajama_set_name: RedPajamaStackExchange


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in: Articles incorporating text from Wikipedia, Royal Air Force aircraft squadrons, Special Operations Executive
No. 161 Squadron RAF
9 May 1918 - 17 August 1918
15 February 1942 - 2 June 1945
Special Duties
No. 3 Group
RAF Newmarket Feb 1942-March 1942
RAF Graveley March 1942-April 1942
RAF Tempsford April 1942-June 1945
Motto(s)
Liberate
Lockheed Hudson, Westland Lysander, Handley Page Halifax, Douglas Havoc, Shorts Stirling, Armstrong Whitworth Whitley
MA / JR
No. 161 (Special Duties) Squadron was a highly secretive unit of the Royal Air Force, which, together with 138 Squadron, was tasked with missions of the Special Operations Executive during the Second World War. Their primary role was to drop and collect secret agents and equipment into and from Nazi-occupied Europe. The squadron had a secondary role in acting as the King's Flight.
2 Aircraft
A proposal was made on 9 May 1918 to create a Squadron flying the Airco DH.9A in a daylight bombing role. The scheduled formation date was postponed several times before the plans were entirely cancelled. The squadron was reformed at RAF Newmarket on 15 February 1942 when the King's Flight was combined with elements of 138 Squadron. In April, 161 Squadron moved to RAF Tempsford in Bedfordshire where it would remain until disbandment on 2 June 1945.[1]
Several types of aircraft were used by the squadron in the course of their duties.
Westland Lysander February 1942 - November 1944
Armstrong Whitworth Whitley V February 1942 - December 1942
Havoc I February 1942 - December 1943
Handley Page Halifax B.Mk II September 1942 - December 1942
Handley Page Halifax B.Mk V November 1942 to November 1944
Lockheed Hudson III / V October 1943-June 1945
Short Stirling III and IV September 1944 - June 1945
The Lysanders were used for the landing and collection of agents, while the other types were used for parachuting agents and supplies.
↑ http://www.setbook.us/books/955644.html
Wikimedia Commons has media related to No. 161 Squadron RAF.
History of No. 161 Squadron
http://www.rafweb.org/Sqn161-165.htm
[1] Information on the Tempsford Squadrons
[2] Final flight, and recovery of, Hudson FK790.
This aircraft and the remains of the pilot were discovered 53 years, to the day, after it went missing on an operation.
Royal Air Force aircraft squadrons
2 Charles E. Grant


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Q: Free a NULL pointer
Possible Duplicate:
Does free(ptr) where ptr is NULL corrupt memory?
Is it good practice to free a NULL pointer in C?

I have a question concerning freeing a null pointer.
char *p = NULL;
free(p);

Could the free(NULL) cause a crash?
Or does it depend on the compiler?

A: From man page of free
void  free(void *ptr);

The free() function deallocates the memory allocation pointed to by
  ptr. If ptr is a NULL pointer, no operation is performed.

If you want to get confirmation from C manual itself

The free function causes the space pointed to by ptr to be
  deallocated, that is, made available for further allocation. If ptr is
  a null pointer, no action occurs.

See page 313 of this document.


Anti cancer, rich in fiber. The wild-shaped deeply lobed LEAVES are where the real vitamins and nutrients are. contain more calcium than any other vegetable. Help keep teeth white and gums healthy, helps fight body odor. Juice or blend in smoothies. In mustard family.
Seeds investigated for the antibacterial activity were Brassica campestris (Sarson), Ricinus communis (Arund) and Eruca sativa (Taramira).
Prophylaxis of inflammatory reaction and benign prostate hyperplasia with pollen extracts from Brassica campestris.
40 (ab) Test crop Brassica campestris Ficus subincisa Leaf Radicle 3.
Leaf physiological aspects of nitrogen-use efficiency in Brassica campestris L.
Key words: Episurphus balteatus, Brassica campestris L, Myzus persicae, correlation, regression, Hyderabad.
As temperaturas constantes favorecem a germinacao de algumas especies, como Brassica campestris a 30[degrees]C (DE LA ROSA-IBARRA et al.
The herbicidal activity of title compound was determined, the results showed the title compound displayed excellent herbicidal activity against Brassica campestris.
38] noticed no significant alteration in the concentration of protein due to salinity in Brassica campestris.
For instance, treatment of skin diseases was done by him both with a combination of leaves of Rauwolfia serpentina along with oil of Brassica campestris, as well as roots of Rauwolfia serpentina along with seeds of Piper nigrum and rhizomes of Zingiber officinale.
Brevicoryne brassicae uses 2 related host plant species that occur sympatrically in Chiapas, Mexico (the weed Brassica campestris L.
Brassica campestris lipid transfer protein (Lin et al.
Brassica campestris was stimulated for length but no stimulation was found in its dry matter production while G.


Born, raised, made in Knoxville: Marc Nelson Denim | Oh, the Places We See . . .
Dogwood Art DeTour 2015 took us a lot of places from the open studios of Diana Dee, Melanie Wood, and Shannon Davis to the heart of Knoxville on E. Depot Avenue for the art of manufacturing at Marc Nelson Denim. Oh, sure. We'd seen the ads of a handsome couple sporting whiskey-stained denim, but never knew the story — or even the location — of denim jeans causing a buzz in the fashion world.
Andy explained that denim starts out white and then takes on the colors given to it — like the indigo blue we're all familiar with. Or orange, if you live in Big Orange Country!
Go, Big Orange — and white, and tan, and indigo . .
Marc Nelson Denim jeans can be bought from racks or customized to a specific fit and size. But you need to wear them. A lot. Or that's what Andy told us: They take on your shape and become part of you. That's why Andy — and the instructions on their denimology fact sheet on the website — recommend not washing them — well, for at least for six months. Surprised? We were. But now it kinda makes sense.
Wear 'em for six months so they conform to you!
The newest for Marc Nelson? The color of whiskey. Andy again: That's right. We put whiskey in the wash with the denim. And this is the color we get.
They felt softer to us: whiskey-steeped jeans!
Housed in an industrial district, the Marc Nelson Denim showroom is urban all the way –from displays on rough-hewn wood panels to tables who had a past life in a manufacturing setting.
Love the wheels on this table and concrete floor!
Accessories like Wolverine boots on sale at Marc Nelson Denim.
Animal-skin rug, industrial table: Marc Nelson Denim showroom.
We wouldn't have necessarily thought of Marc Nelson Denim being a favorite stop on an art tour. But with the story, the craftsmanship, the unique quality of the garments — well, why not? This IS art. And, thankfully, it's right here in Knoxville.
For more posts on the Dogwood Arts Festival 2015, click here.
This entry was posted in Retail Shops, Tennessee, Travel and tagged denim, Dogwood Art DeTour 2015, Dogwood Arts Festival 2015, jeans, L. C. Nelson, Marc Hall, Marc Nelson Denim, urban decor. Bookmark the permalink.
Sherri, so sorry you're having problems with other comments coming your way. I don't know how to change something you've clicked, but here's what I'll try. I'll remove your comment from the list. If you want to put another one on, you can. Just don't click "Notify me" when other comments are posted. Thanks for visiting our blog.
small business Vo – IP services will help your company save money.
if you write honest and detailed reviews about them.
Thanks so much for the comment. The folks at Marc Nelson Denim also recommend that you get in the shower wearing your new jeans so the jeans will conform to your body. I'd never heard of that, but it sounds doable. It's funny you said you might not ever wear these. I asked Andy (in the post) if there were any plans to produce jeans for Boomer women. He said they were coming. So, who knows? Every body. Every price range. It could happen!


text:
Q: Stop Process on Network Machine with .Net In my project, I am going to do that :
There is a server in restaurant, and some pos machines which are running on Windows XP called Radiant systems. There will be a .exe on server, and this .exe will stop running some process on Radiant Poses. It is possible with .Net Remoting. But it requires to write an .exe on the Radiant Poses, but I dont want to do that. Think there is no problem on network access on Main Server in restaurant and Radiant Poses in restaurant. Is it possible to do that without using .Net Remoting?

A: One way could be, to use Remoting to call a KillProcess() method that would in turn call the Process.Kill method to kill a process. This article is worth looking at.

A: Do you really need a programming solution for this? Take a look at the command line tools that are available with Windows XP, namely the taskkill command. It allows to kill processes on a local or remote system.

Syntax
taskkill [/s Computer] [/u Domain\User [/p Password]]]
         [/fi FilterName] [/pid ProcessID]|[/im ImageName] [/f][/t]

...
Examples
The following examples show how you can use the taskkill command:
taskkill /pid 1230 /pid 1241 /pid 1253
taskkill /f /fi "USERNAME eq NT AUTHORITY\SYSTEM" /im notepad.exe
taskkill /s srvmain /f /im notepad.exe
taskkill /s srvmain /u maindom\hiropln /p p@ssW23 /fi "IMAGENAME eq note*" /im *
taskkill /s srvmain /u maindom\hiropln /fi "USERNAME ne NT*" /im *
taskkill /f /fi "PID ge 1000" /im *


What do you do when your kids are grown and gone, but you still hope to speak into their lives?
Clay Holderman, a dad from our church came up with this cool idea. He would get a question from one of his adulting kids. Then he would ask the dads in his small group Bible study to record their honest thoughts. Then he would ask a pastor to get a Biblical perspective.
After recording, he would cut and paste and edit the mess into a thoughtful, helpful, 15-minute DadCast, and send it back to his kids.
Or it will be if any of our kids listen to it.
The first question came from his oldest son, Cole. Could he ever become someone his parents couldn't love? What if he became the worst of the worst of humanity?
Listen to the first DadCast on an app below and sign up or share if you like it.


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Total Solar Eclipse July 2010 and World Events
Total solar eclipse will take place on July 11, 2010. This total solar eclipse will be visible in Southern Pacific Ocean regions. The solar eclipse of July 11, 2010, begins at 17:10 hrs UTC (Universal Time) and ends at 21:58 hrs. The eclipse point is 25 degrees Gemini and is aspected by Saturn from Virgo. The eclipse point falls on sensitive points of horoscopes of many nations and hence has the potential to influence world events.
The planetary positions at the moment of total eclipse is shown in the table. Eclipse has significant effect when eclipse point is in the rasi or nakshatra of natal planets or near sensitive points in the horoscope. An Eclipse in the same rashi as another planet will affect the fortune of the person indicated by that planet whereas in the same nakshatra will affect the health of the individual indicated by that planet. The effects of an eclipse can be experienced around the time before or after the eclipse. If the luminary is placed before ascendant lord/sun/lord of the house, then the effects arise prior to the eclipse date.
The 5 Eclipse Astrological Influence Essentials are
1. Eclipse points are energy junctions
2. The eclipse lunar node type, Rasi and Nakshatra are significant astrological influence factors.
3. The eclipse influences the matters related to the eclipse point house from the ascendant.
4. The planets in conjunction with or aspecting the eclipse point are eclipse significators .
5. Eclipse point degrees and Individual horoscope planetary longitudes, Rasi and Navamsa Charts should be analyzed together.
We note here the relation of the eclipse point of the total solar eclipse of July 11th, 2010 to sensitive points of horoscopes of many nations.
1. The eclipse point falls on Ketu in second house of horoscope of Kyrgyzstan, See Kyrgyzstan Unrest Astrology The conjunction suggests more trouble ahead.
2. The eclipse point is in exact opposition to natal Mercury in 4th house in the chart of Indian Republic Day on January 26th 1950.. See Indian Republic Astrology 2010 July Solar Eclipse
3. The eclipse point is in conjunction with natal moon position in the chart of independence of Pakistan. See Total solar Eclipse and Pakistan 2010 Astrology
4. The eclipse point is in opposition to natal moon position in the chart of independence of Morocco.
5. The eclipse point is in opposition to Venus in 6th house in SAMVA USA chart based on the event when Maryland became the 13th and final state to approve the ratification of the Articles of Confederation on February 2, 1781 in Annapolis, Maryland making USA effectively into being as a Perpetual Union.
6. The eclipse point is in conjunction with natal Venus and saturn in 10th house of Indonesia.
Find your future predictions and effects of July eclipse at http://decisioncare.org/astrology/. There are many free astrological tools and horoscopes at http://decisioncare.org/astrology/ which help analyze your own individual horoscope on a daily basis .
It is a set of astrological tools that we believe you will love and learn to use more and more to feel the power of predictive astrology. Please login at our member site http://decisioncare.org/astrology/
Previous Solar Eclipse July Horoscope Astrology
Next Total Solar Eclipse 2010 and World Cup


text:
Q: Is it possible to install the Web Deployment Handler (WMSVC) on Windows 7? I would like to install the Web Deployment Handler service on my development machine (Windows 7 with IIS7) for test purposes, but according to the documentation on MSDN (http://technet.microsoft.com/en-us/library/dd722796(WS.10).aspx) it seems like it can only be installed on Windows Server 2008 with IIS7.
Anyone knows of a way around this?

A: It is not possible to install WMSVC in client SKUs (service wont work), only Server SKUs support it.
However, you should be able to install the Web Deploy Service which will give you almost all functionality except delegated administration, so only administrators can connect to it.