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ГРНТИ 29.35 Радиофизика. Физические основы электроники
ГРНТИ 29.31 Оптика
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ОКСО 03.06.01 Физика и астрономия
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BISAC SCI004000 Astronomy
BISAC SCI005000 Physics / Astrophysics
The paper presents a new tool for constructing Doppler tomograms: Tomo-V (https://tomo-v.inasan.ru), developed on the basis of the Algebraic Reconstruction Technique (ART). Previously, the ART method was not widely used in tomography since its direct implementation had high computational complexity. The author developed a fast version of this algorithm, which made it possible to implement it as a web application that runs at an acceptable speed in a browser on a personal computer. Using this method, it is possible to obtain sharp tomographic images from blurred profiles; in addition, the method has shown good results in restoring images from noisy data, from a small number of profiles, and from profiles contaminated with absorption lines and the emission of an expanding envelope. Tomo-V also includes tools for analysing the obtained tomograms, allowing one to display accretion disks and Roche lobes on the tomogram as well as the back projection of the tomographic image onto the flow elements in spatial coordinates.
binaries: spectroscopic
\documentclass{vak2024}
\usepackage[utf8]{inputenc}
\usepackage{url}
\usepackage{hyperref}
\usepackage{aas_macros}
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\begin{document}
\title{Tomo-V -- a new tool for Doppler tomography}
\titlerunning{Tomo-V Doppler tomography}\author{Pavel~Kaygorodov\inst{1}}
\authorrunning{Pavel~Kaygorodov} \institute{Institute of Astronomy of the Russian Academy of Sciences, 119017, Pyatnitskaya str., 48, Moscow, Russia}
\abstract{
The paper presents a new tool for constructing Doppler tomograms -- Tomo-V (\url{https://tomo-v.inasan.ru}), developed on the basis of the Algebraic Reconstruction Technique (ART). Previously, the ART method was not widely used in tomography, since its direct implementation had high computational complexity. The author developed a fast version of this algorithm, which made it possible to implement it within the framework of a web application that runs at an acceptable speed in a browser on a personal computer. Using this method, it is possible to obtain sharp tomographic images from blurred profiles; in addition, the method has shown good results in restoring images from noisy data, from a small number of profiles, as well as from profiles contaminated with absorption lines and emission of the expanding envelope. Tomo-V also includes tools for analysing the obtained tomograms, allowing to display accretion disks and Roche lobes on the tomogram, as well as the back projection of the tomographic image onto flow elements in spatial coordinates.
\keywords{\textcolor{red}{(stars:) binaries: spectroscopic}}
\doi{10.26119/VAK2024-ZZZZ}
}
\maketitle
\section{Introduction}
To construct Doppler tomograms, methods are used similar to those used to construct tomographic images in other fields -- medicine, defectoscopy, etc. The input data for constructing Doppler tomograms are the set of spectral line profiles obtained from observations of binary stars. The peculiarity of these data is their heterogeneity (the state of the object can change during the observation process), noise, fairly low resolution, and a relatively small number of profiles that can be obtained for one object. To obtain tomograms from such data, it is necessary to use special methods, including regularization methods that allow you to restore lost information from inconsistent input data. The most commonly used method is the maximum entropy method (MEM)~\citep{1988MNRAS.235..269M}, which allows you to obtain images with the least possible amount of detail, which is important for interpretation. Also, to construct tomograms, the radio astronomical approach (CLEAN method)~\citep{1997ASPC..125..202A} is used, which is well suited for identifying the boundaries of elements. These methods use a property of tomograms called the ``central theorem'', which allows you to search for a solution in Fourier space, filling in the areas missing from the original data, in accordance with a certain algorithm. In contrast, the Algebraic Reconstruction Techinque (ART)~\citep{GORDON1970471} uses minimisation methods to find a solution directly (for Doppler tomography) in velocity space, without using the Fourier transform. Until now, this method has not been used to obtain Doppler tomograms, probably due to its high computational complexity -- it requires minimisation in a space with a dimension equal to the number of pixels in the desired image. For example, for an image with a resolution of $50\times 50$ pixels, it is necessary to minimise (for example, gradient descent) in a space with a dimension of 2500. When creating Tomo-V, through certain optimizations \textcolor{red}{even without using a GPU -- we can pre-calculate some data, which can be re-used during all of the steps}. It is possible to achieve good performance when using the ART method. This made it possible to create a program open for general use (\url{https://tomo-v.inasan.ru}).
\section{The ART method}
With the ART method we are trying to find a two-dimensional brightness distribution in the velocity space such that the projections of this distribution, taken for phases corresponding to the phases of the observed profiles, has best conformance with the observed profiles. The standard deviation $\chi^2$ is used as a criteria. In Tomo-V, each pixel in the image contributes to the luminance as a point source, convolved with a Gaussian profile having a defined full width at half maximum (FWHM). Due to the overlap of the Gaussian profiles of points, nonlinearity appears, which does not allow the solution to degenerate into a set of superimposed bands.
\section{Working with Tomo-V}
\begin{figure}[t]
\begin{center}
\begin{tabular}{ccccccccc}
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterAblur.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterAblur.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{dogblur.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{starblur.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA300blured.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA600blured.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{dog300blured.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{dog600blured.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{star300blured.pdf}}\\
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA11result.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA5result.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{dog5result.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{star5result.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA300result.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA600result.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{dog300result.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{dog600result.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{star300result.pdf}}\\
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA11trail.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA5trail.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{dog5trail.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{star5trail.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA300trail.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{letterA600trail.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{dog300trail.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{dog600trail.pdf}} &
\raisebox{-\totalheight}{\includegraphics[width=1.6cm]{star300trail.pdf}}\\
(a) & (b) & (c) & (d) & (e) & (f) & (g) & (h) & (i)
\end{tabular}
\end{center}
\caption{The results of restoration for the symbol ``A'' by 11 profiles (a), the symbol ``A'' by 5 profiles (b), the symbol ``@'' by 5 profiles (c) and the symbol ``*'' by 5 profiles (d) with FWHM=100; image restoration with FWHM=300 km/s (e, g, i -- 500 iterations) and 600 km/s (f, h -- 1000 iterations). The first row -- original images (convolved with a Gaussian with the corresponding FWHM), the second row -- restored images, the third row -- profiles, by which restoration was performed.}\label{fig}
\end{figure}
Tomo-V as a web application is available at~\url{https://tomo-v.inasan.ru}. \textcolor{red}{All of the computations are performed in the user's browser, without any significant CPU load on the server side.} On the page, the user will see a screen divided into two parts - in the left column (initially empty) a list of saved images, an area for loading previously exported data (files with the .tmv extension, see below), and a button for creating a new image. When clicking this button, the user will be offered a choice -- to create a test image (``Sample data\dots'' for mastering the program, as well as for experimenting with the settings, or an image built from real data (``Real data\dots'').
\textcolor{red}{If you select ``Sample data'', a window will open on the right side of the screen, there will be an image of a test symbol and a settings block. Next to this block is a graph for displaying profiles. By comparing the ``real'' (red) profiles with the ``synthetic'' (blue) ones, you can control the quality of the tomogram reconstruction. The right part of the window shows the reconstructed image, a graph with the absorption line (blue) and additional emission (red), as well as elements related to the settings and control of the solution search process. To take into account the weights (for very noisy source data), you can enable the ``Low SNR mode'' option. The ``Noize treshhold'' and ``Low-SNR degree'' parameters can be used to adjust the weights when the ``Low SNR'' mode is enabled. The ``Start reconstruction'' button is used to start the image construction process. In addition, on the block header you can see a button for setting general settings –- the velocity range, FWHM and resolution of the tomogram, as well as the buttons to save/export/delete solver state and a ``close'' button.
The method shows rather good results when restoring images from low number of profiles (see Fig.~\ref{fig}, panels a-d). Even a complex shapes, like symbols ``@'' and ``*'' was reconstructed well from 5 profiles. Also it works well with highly blurred profiles (see Fig.~\ref{fig}, panels e-i). From a highly noisy data (up to SNR=2) it also can restore images with satisfactory quality, but the results cannot be included in this paper due page limitations.}
If you select ``Real data\dots'' when creating a new image, a window for constructing an image based on real data will open. In general, it is similar to the test one, but on the left side there will be a button for selecting files, a table of loaded files, input fields for setting the continuum level (1 by default) and system parameters (component masses and orbital period). The loaded files must be text files and contain two columns -- velocity (in km/s) and intensity. The intensities must be normalised to the continuum level, preferably so that the continuum intensity is taken as one. For each loaded file, it is necessary to specify the phase (in the right part of the table, then click the ``update'' button), also in the right part of the table of loaded files opposite each file there is a check mark, by unchecking which you can remove the corresponding profile from the calculations. Loaded profiles (the part of them that fits into the specified speed range) are immediately shown on the graph, to the right of the file table. You can also load a file by specifying the phases for each profile (``Load phases from file'').
\section*{Funding}
Supported by joint Russian-Bulgarian grant RFBR 20-52-18015 / KP-06-Russia/2-2020
\bibliographystyle{aa}
\bibliography{kaygorodov}
\end{document}
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2. Gordon R., Bender R., Herman G.T., 1970, Journal of Theoretical Biology, 29, p. 471
3. Marsh T.R. and Horne K., 1988, Monthly Notices of the Royal Astronomical Society, 235, p. 269
