lectures/Komp_obr/07-iproc_3.tex

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\title[ëÏÍÐØÀÔÅÒÎÁÑ ÏÂÒÁÂÏÔËÁ. ìÅËÃÉÑ 7.3.]{ëÏÍÐØÀÔÅÒÎÁÑ ÏÂÒÁÂÏÔËÁ ÒÅÚÕÌØÔÁÔÏ× ÉÚÍÅÒÅÎÉÊ}
\subtitle{ìÅËÃÉÑ 7.3. ïÂÒÁÂÏÔËÁ ÁÓÔÒÏÎÏÍÉÞÅÓËÉÈ ÉÚÏÂÒÁÖÅÎÉÊ}
\date{5 ÁÐÒÅÌÑ 2021 ÇÏÄÁ}

\begin{document}
% ôÉÔÕÌ
\begin{frame}
\maketitle
\end{frame}
% óÏÄÅÒÖÁÎÉÅ
\begin{frame}
\tableofcontents
\end{frame}

\section{óÉÇÎÁÌ--ÛÕÍ}
\begin{blueframe}{}
\only<1>{
\begin{block}{SNR}
$$\SNR = \frac{N}{\sqrt{N}}= \sqrt{N},\qquad N=N_{star}+N_{sky}\quad\Arr$$
$$\SNR\approx\frac{N_{star}}{\sqrt{N_{star}+2N_{sky}}},\qquad N=t_{exp}\cdot R\quad\Arr$$
$$\SNR\approx\frac{R_{star}\sqrt{t_{exp}}}{\sqrt{R_{star}+2R_{sky}}}\quad\Arr\quad
	\SNR\propto\sqrt{t_{exp}}$$
$$R=R_0\cdot S_{mirror}\propto D_{mirror}^2\quad\Arr\quad \SNR\propto D_{mirror}$$
$$N_{meas}\text{ ËÏÒÏÔËÉÈ ÜËÓÐÏÚÉÃÉÊ ×ÍÅÓÔÏ
ÏÄÎÏÊ:}\quad\sigma_{mean}=\frac{\sigma_{individ}}{\sqrt{N_{meas}}}\propto\frac{\sqrt{S}}{N_{meas}}$$
$$\SNR_{mean}=\frac{S/N_{meas}}{\sigma_{mean}}\propto\sqrt{S}=\SNR_{long}\quad\text{ÔÏÌØËÏ ÅÓÌÉ }
\sigma\approx\sigma_{phot}!!!$$
\end{block}
}
\only<2>{
    \begin{block}{ëÏÒÒÅËÃÉÑ ÁÐÅÒÔÕÒÙ} % CCDPhotometryBook.pdf
        ðÏÞÅÍÕ ÉÚÏÂÒÁÖÅÎÉÅ ÑÒËÏÊ Ú×ÅÚÄÙ ÛÉÒÅ: ÎÅÓÍÏÔÒÑ ÎÁ ÓÏ×ÅÒÛÅÎÎÏ ÏÄÉÎÁËÏ×ÕÀ PSF Õ ÏÂÅÉÈ Ú×ÅÚÄ, ÐÒÉ ÓÅÞÅÎÉÉ
        ÏÄÉÎÁËÏ×ÙÍ ÐÏÒÏÇÏÍ ÑÒËÁÑ Ú×ÅÚÄÁ ×ÓÅÇÄÁ <<ÂÏÌØÛÅ>>. õ×ÅÌÉÞÅÎÉÅ ÁÐÅÒÔÕÒÙ \Arr Õ×ÅÌÉÞÅÎÉÅ ÛÕÍÏ×, ÎÅÏÂÈÏÄÉÍÏ
        ÉÓÐÏÌØÚÏ×ÁÔØ ËÁË ÍÏÖÎÏ ÍÅÎØÛÕÀ ÁÐÅÒÔÕÒÕ.
        $$\Delta_N^{bright} = m(N\cdot \FWHM) - m(1\cdot\FWHM)\quad\Arr\quad
        m^{faint} = m(1\cdot\FWHM) + \Delta_N^{bright},$$
        $m(x)$~-- Ú×ÅÚÄÎÁÑ ×ÅÌÉÞÉÎÁ ÎÁ ÁÐÅÒÔÕÒÅ~$x$.
    \end{block}\vspace*{-1em}
    \img[0.6]{fwhm}
}
\end{blueframe}

\section{äÅËÏÎ×ÏÌÀÃÉÑ}
\begin{frame}{äÅËÏÎ×ÏÌÀÃÉÑ}
\only<1>{
\begin{block}{}
$$I(x,y) = P(x,y)*O(x,y)+N(x,y),\quad\text{$P$~-- PSF}\quad\text{ÉÌÉ}$$
$$\FT{I}=\FT{O}\cdot\FT{P}+\FT{N}\quad\Arr\quad
\FT{O}=\frac{\FT{I} - \FT{N}}{\FT{P}}$$
$$\text{îÁÉÍÅÎØÛÉÅ Ë×ÁÄÒÁÔÙ:}\quad
\FT{O}=\frac{\FT{P}^*\FT{I}}{|\FT{P}|^2}$$
$$\text{òÅÇÕÌÑÒÉÚÁÃÉÑ ôÉÈÏÎÏ×Á, $\min(J_T)$ ($H$~-- HPF):}\quad
\quad J_T=||I-P*O|| - \lambda||H*O||,$$
$$\FT{O}=\frac{\FT{P}^*\FT{I}}{|\FT{P}|^2+\lambda|\FT{H}|^2}$$
\end{block}
}\only<2>{
\begin{block}{òÅÇÕÌÑÒÉÚÁÃÉÑ ÐÏ âÁÊÅÓÕ}
$$p(O|I)=\frac{p(I|O)\cdot p(O)}{p(I)}$$
$$\text{Maximum likelihood:}\quad \mathrm{ML}(O)=\max_O p(I|O)$$
$$\text{Maximum-a-posteriori solution:}\quad
\mathrm{MAP}(O)=\max_O p(I|O)\cdot p(O)$$
\end{block}
\begin{block}{}
\begin{itemize}
\item éÔÅÒÁÃÉÏÎÎÁÑ ÒÅÇÕÌÑÒÉÚÁÃÉÑ
\item ÷ÅÊ×ÌÅÔ-ÒÅÇÕÌÑÒÉÚÁÃÉÑ
\item \dots
\end{itemize}
\end{block}
}
\end{frame}

\begin{frame}{æÕÎËÃÉÑ ÒÁÓÓÅÑÎÉÑ ÔÏÞËÉ}
\only<1>{\img[0.6]{moffat}}
\only<2>{\begin{block}{}
\begin{itemize}
\item çÁÕÓÓ: $f(x) = f_0\exp\Bigl(\dfrac{-(x-x_0)^2}{2\sigma^2}\Bigr)$, $\FWHM\approx2.355\sigma$
\item íÏÆÆÁÔ: $f(x) = f_0\Bigl(1+\dfrac{(x-x_0)^2}{\alpha^2}\Bigr)^{-\beta}$,
$\FWHM\approx2\alpha\sqrt{2^{1/\beta}-1}$
\item æÒÉÄ: $\FT{f} \propto \exp\Bigl[-(bu)^{5/3}\Bigr]$,
$\FWHM\approx 2.921 b$ (íÏÆÆÁÔ Ó $\beta=4.765$, ÔÉÐÉÞÎÙÅ ÖÅ $\beta=2.5\cdots4.5$).
\end{itemize}
\end{block}
}
\end{frame}

\section{ïÂÎÁÒÕÖÅÎÉÅ}
\begin{frame}{ïÂÎÁÒÕÖÅÎÉÅ}
\begin{block}{ðÒÏÓÔÅÊÛÉÊ ÁÌÇÏÒÉÔÍ}
\begin{enumerate}
\item ÷ÙÞÉÓÌÅÎÉÅ É ×ÙÞÉÔÁÎÉÅ ÆÏÎÁ
\item ó×ÅÒÔËÁ Ó ÍÁÓËÏÊ É ÂÉÎÁÒÉÚÁÃÉÑ
\item ïÂÎÁÒÕÖÅÎÉÅ Ó×ÑÚÎÙÈ ÏÂÌÁÓÔÅÊ
\item õÔÏÞÎÅÎÉÅ ÆÏÎÁ, goto 1
\item ëÌÁÓÓÉÆÉËÁÃÉÑ, ÆÏÔÏÍÅÔÒÉÑ É Ô.Ð.
\end{enumerate}
\end{block}
\end{frame}

\begin{blueframe}{}
\img{objdet}
\end{blueframe}

\begin{blueframe}{éÚÏÆÏÔÙ}
\only<1>{\begin{block}{íÅÔÏÄ ÛÁÇÁÀÝÉÈ Ë×ÁÄÒÁÔÏ×}
âÉÎÁÒÉÚÕÅÍ ÉÚÏÂÒÁÖÅÎÉÅ ÐÏ ÚÁÄÁÎÎÏÍÕ ÐÏÒÏÇÕ. ðÏ ÓÏÓÅÄÑÍ ËÁÖÄÏÇÏ ÐÉËÓÅÌÑ ×ÙÞÉÓÌÑÅÍ ÂÉÔÏ×ÕÀ ÍÁÓËÕ
($0\div15$). ïÔ ÔÏÞËÉ $1\div14$ ÓÔÒÏÉÍ ÉÚÏÌÉÎÉÀ, ÓÏÏÔ×ÅÔÓÔ×ÅÎÎÏ ÍÅÎÑÑ ÚÎÁÞÅÎÉÑ × ÐÉËÓÅÌÑÈ ÍÁÓËÉ. ëÁÖÄÙÊ ÕÚÅÌ
ÉÚÏÌÉÎÉÉ~--- ÌÉÎÅÊÎÁÑ ÉÌÉ ÄÒÕÇÁÑ ÉÎÔÅÒÐÏÌÑÃÉÑ ÉÎÔÅÎÓÉ×ÎÏÓÔÉ × ÐÉËÓÅÌÑÈ ÏÒÉÇÉÎÁÌÁ.
\end{block}
\img[0.5]{isophotes}
}
\only<2>{\img{Marching_squares_algorithm}}
\end{blueframe}

\begin{frame}{WCS-ÐÒÉ×ÑÚËÁ}
\only<1>{
\img[0.6]{WCS_triangles}
\centerline{A.~P\'al, G.\'A.~Bakos. PASP {\bf 118}: 1474--1483, 2006. }}
\only<2>{
\img[0.65]{WCS_quad}
\centerline{\url{astrometry.net}}}
\only<3>{\begin{block}{}
\begin{itemize}
\item ðÏÌÏÖÅÎÉÅ ÎÅÓËÏÌØËÉÈ Ú×ÅÚÄ ÈÁÒÁËÔÅÒÉÚÕÅÔÓÑ ÐÁÒÁÍÅÔÒÏÍ, ÉÎ×ÁÒÉÁÎÔÎÙÍ Ë ÚÅÒËÁÌÉÒÏ×ÁÎÉÀ, ÍÁÓÛÔÁÂÉÒÏ×ÁÎÉÀ,
×ÒÁÝÅÎÉÀ É ÐÅÒÅÎÏÓÕ. õÓÔÏÊÞÉ×ÙÍ Ë ÛÕÍÕ.
\item ë×ÁÄÒÁÔÕ ABCD ÓÏÏÔ×ÅÔÓÔ×ÕÅÔ ÞÅÔÙÒÅÈÍÅÒÎÙÊ ËÏÄ × ÏÔÎÏÓÉÔÅÌØÎÙÈ ËÏÏÒÄÉÎÁÔÁÈ ÔÏÞÅË C É D.
\item ðÒÏÂÌÅÍÁ ×ÙÒÏÖÄÅÎÉÑ: ÐÒÉ ÓÍÅÎÅ ÐÏÒÑÄËÁ A, B ÉÌÉ C, D ËÏÄ <<ÏÔÒÁÖÁÅÔÓÑ>>.
\item îÁ ÎÅÂÅ ÓÔÒÏÉÔÓÑ ÓÅÔËÁ Ó ÍÁÓÛÔÁÂÉÒÕÅÍÙÍ ÛÁÇÏÍ, ÐÏ ËÁÔÁÌÏÖÎÙÍ ÄÁÎÎÙÍ × ÅÅ ÑÞÅÊËÁÈ ÏÐÒÅÄÅÌÑÀÔÓÑ Ë×ÁÄÒÁÔÙ
Ó ÎÉÓÐÁÄÁÀÝÅÊ ÑÒËÏÓÔØÀ Ú×ÅÚÄ.
\item ðÏÌÕÞÅÎÎÙÊ ÎÁÂÏÒ ËÏÄÏ× ÐÏÚ×ÏÌÑÅÔ ÉÄÅÎÔÉÆÉÃÉÒÏ×ÁÔØ ÕÞÁÓÔËÉ ÎÅÂÁ ×ÐÌÏÔØ ÄÏ ÓÁÍÙÈ ÍÅÌËÉÈ ÍÁÓÛÔÁÂÏ× (ÎÕÖÎÙ
ÈÏÔÑ ÂÙ ÞÅÔÙÒÅ Ú×ÅÚÄÙ × ËÁÄÒÅ).
\item þÅÍ ÂÏÌØÛÅ Ú×ÅÚÄ ÎÁ ËÁÄÒÅ, ÔÅÍ ÎÁÄÅÖÎÅÊ ÂÕÄÅÔ ÉÄÅÎÔÉÆÉËÁÃÉÑ.
\end{itemize}
\end{block}
}
\end{frame}

\begin{blueframe}{ôÒÉÁÎÇÕÌÑÃÉÑ äÅÌÏÎÅ}
\img[0.6]{delaunay}
\end{blueframe}

\begin{blueframe}{äÉÁÇÒÁÍÍÙ ÷ÏÒÏÎÏÇÏ}
\only<1>{\img[0.6]{voronoi}}
\only<2>{\img[0.6]{delvor}}
\end{blueframe}

\begin{frame}{ó×ÏÊÓÔ×Á ÔÒÉÁÎÇÕÌÑÃÉÉ äÅÌÏÎÅ}
\begin{block}{}
\begin{itemize}
\item ôä ×ÚÁÉÍÎÏ ÏÄÎÏÚÎÁÞÎÏ ÓÏÏÔ×ÅÔÓÔ×ÕÅÔ ÄÉÁÇÒÁÍÍÅ ÷ÏÒÏÎÏÇÏ ÄÌÑ ÔÏÇÏ ÖÅ ÍÎÏÖÅÓÔ×Á ÔÏÞÅË.
ëÁË ÓÌÅÄÓÔ×ÉÅ: ÅÓÌÉ ÎÉËÁËÉÅ ÞÅÔÙÒÅ ÔÏÞËÉ ÎÅ ÌÅÖÁÔ ÎÁ ÏÄÎÏÊ ÏËÒÕÖÎÏÓÔÉ, ôä ÅÄÉÎÓÔ×ÅÎÎÁ.
\item ôä ÍÁËÓÉÍÉÚÉÒÕÅÔ ÍÉÎÉÍÁÌØÎÙÊ ÕÇÏÌ ÓÒÅÄÉ ×ÓÅÈ ÕÇÌÏ× ×ÓÅÈ ÐÏÓÔÒÏÅÎÎÙÈ ÔÒÅÕÇÏÌØÎÉËÏ×, ÔÅÍ
ÓÁÍÙÍ ÉÚÂÅÇÁÀÔÓÑ <<ÔÏÎËÉÅ>> ÔÒÅÕÇÏÌØÎÉËÉ.
\item ôä ÍÁËÓÉÍÉÚÉÒÕÅÔ ÓÕÍÍÕ ÒÁÄÉÕÓÏ× ×ÐÉÓÁÎÎÙÈ ÏËÒÕÖÎÏÓÔÅÊ.
\item ôä ÍÉÎÉÍÉÚÉÒÕÅÔ ÍÁËÓÉÍÁÌØÎÙÊ ÒÁÄÉÕÓ ÍÉÎÉÍÁÌØÎÏÇÏ ÏÂßÅÍÌÀÝÅÇÏ ÛÁÒÁ.
\item ôä  ÎÁ ÐÌÏÓËÏÓÔÉ ÏÂÌÁÄÁÅÔ ÍÉÎÉÍÁÌØÎÏÊ ÓÕÍÍÏÊ ÒÁÄÉÕÓÏ× ÏËÒÕÖÎÏÓÔÅÊ, ÏÐÉÓÁÎÎÙÈ ÏËÏÌÏ
ÔÒÅÕÇÏÌØÎÉËÏ×, ÓÒÅÄÉ ×ÓÅÈ ×ÏÚÍÏÖÎÙÈ ÔÒÉÁÎÇÕÌÑÃÉÊ.
\end{itemize}
\end{block}
\end{frame}

\begin{blueframe}{K-nearest}
\begin{columns}
\column{0.5\textwidth}
\begin{block}{}
ëÌÁÓÓÉÆÉËÁÃÉÑ ÏÂßÅËÔÁ ÐÏ $k$~ÂÌÉÖÁÊÛÉÍ ÓÏÓÅÄÑÍ. ÷ ÓÌÕÞÁÅ ÐÅÒ×ÏÊ ×ÙÂÏÒËÉ~--- ÔÒÅÕÇÏÌØÎÉË, × ÓÌÕÞÁÅ ×ÔÏÒÏÊ~---
Ë×ÁÄÒÁÔ.

$k$ ÍÏÖÅÔ ÂÙÔØ ÄÒÏÂÎÙÍ, ÅÓÌÉ ÐÒÉÍÅÎÑÔØ ×Ú×ÅÛÅÎÎÙÅ ÒÁÓÓÔÏÑÎÉÑ.
\end{block}
\column{0.5\textwidth}
\img{knearest}
\end{columns}
\end{blueframe}

\section{áÎÁÌÉÚ ÒÁÚÒÅÖÅÎÎÙÈ ÄÁÎÎÙÈ}
\begin{frame}{áÎÁÌÉÚ ÒÁÚÒÅÖÅÎÎÙÈ ÄÁÎÎÙÈ}
\only<1>{
\begin{block}{ëÏÒÒÅÌÑÃÉÑ}
$$C(\tau)=\frac{[a(t)-\aver{a}][b(t+\tau)-\aver{b}]}{\sigma_a\sigma_b}$$
$$\text{Unbinned: } U_{ij}=\frac{(a_i-\aver{a})(b_j-\aver{b})}{
\sqrt{(\sigma_a^2-e^2_a)(\sigma_b^2-e^2_b)}},\qquad \Delta t_{ij}=t_j-t_i\qquad$$
$$C(\tau)=\frac1{N_\tau}U_{ij,\tau},\qquad
\tau-\Delta\tau/2\le\Delta t_{ij}\le\tau+\Delta\tau/2$$
îÅ ÎÕÖÎÁ ÉÎÔÅÒÐÏÌÑÃÉÑ!
\end{block}
}\only<2>{
\img[0.8]{scatter_corr}\centerline{ðÕÎËÔÉÒ~--- ËÏÒÒÅÌÑÃÉÑ ÞÅÒÅÚ ÉÎÔÅÒÐÏÌÑÃÉÀ}}
\end{frame}

\begin{frame}{}
\only<1>{
\begin{block}{ðÅÒÉÏÄÏÇÒÁÍÍÁ ìÏÍÂÁ--óËÁÒÇÌÁ (Lomb--Scargle)}
$$P_{x}(\omega )={\frac  {1}{2}}\left({\frac  {\left[\sum _{j}X_{j}\cos \omega (t_{j}-\tau
)\right]^{2}}{\sum _{j}\cos ^{2}\omega (t_{j}-\tau )}}+{\frac  {\left[\sum _{j}X_{j}\sin \omega
(t_{j}-\tau )\right]^{2}}{\sum _{j}\sin ^{2}\omega (t_{j}-\tau )}}\right)$$
$$\tg{2\omega \tau }={\frac  {\sum _{j}\sin 2\omega t_{j}}{\sum _{j}\cos 2\omega t_{j}}}$$
\end{block}
\img{lombscargle}
}\only<2>{
\begin{block}{ðÒÅÏÂÒÁÚÏ×ÁÎÉÅ æÕÒØÅ}
$$ P_m=\Sum_{n=1}^N p_n\e^{-i\frac{2\pi}{N}mn}\quad\Arr\quad
P_m=\Sum_{n=1}^N p_n\e^{-i\frac{2\pi}{T}mt_n},\quad T=t_N-t_1$$
÷ octave: \tt{irsa\_dft(X,Y,freq)}:
$\displaystyle P(\nu)=\Sum_{n=1}^N p_n\e^{-i \nu_n\cdot t_n}$
\end{block}}
\only<3>{\img[0.8]{scat01}}
\only<4>{\img[0.8]{scatFFT}}
\only<3,4>{\centerline{$T=111.5\quad\Arr\quad \nu\approx0.00897$}}
\end{frame}

% \begin{frame}{}
% \only<1>{
% \begin{block}{}
% \end{block}
% }\only<2>{
% \img[0.8]{}
% }
% \end{frame}



\section{òÏÂÁÓÔÎÙÅ ÍÅÔÏÄÙ}
\begin{frame}{òÏÂÁÓÔÎÙÅ ÍÅÔÏÄÙ}
\begin{block}{}
òÏÂÁÓÔÎÁÑ ÎÁÄÅÖÎÏÓÔØ íîë~--- 0!

ðÒÏÓÔÅÊÛÁÑ ÒÏÂÁÓÔÎÁÑ ÏÃÅÎËÁ~--- ÍÅÄÉÁÎÁ (ÍÏÖÎÏ <<ÚÁÓÏÒÉÔØ>> ÄÏ 50\% ÄÁÎÎÙÈ!).
ïÃÅÎËÁ <<ÐÌÏÈÏÓÔÉ>>: $MAD=\med(x_i-\med(x))$.

çÒÕÐÐÉÒÏ×ËÁ ÄÁÎÎÙÈ É ÍÅÔÏÄ ÕÓÅÞÅÎÎÙÈ Ë×ÁÄÒÁÔÏ×.

íÅÔÏÄ ÎÁÉÍÅÎØÛÉÈ ÍÅÄÉÁÎ Ë×ÁÄÒÁÔÏ×: $\min\bigl(\med(x^2)\bigr)$.

ïÃÅÎËÁ ÄÉÓÐÅÒÓÉÉ: $\med(|x_i-\med(x)|)\approx0.67\sigma$.

M-, R-, S-, Q- ÏÃÅÎËÉ.


\end{block}
\end{frame}




\begin{frame}{ðÒÏÇÒÁÍÍÎÏÅ ÏÂÅÓÐÅÞÅÎÉÅ}
\begin{block}{\url{http://heasarc.gsfc.nasa.gov/docs/heasarc/astro-update/}}
\begin{itemize}
\item ASTROPY: A single core package for Astronomy in Python
\item Aladin: An interactive software sky atlas
\item CFITSIO: FITS file access subroutine library
\item GSL: GNU Scientific Library
\item IDLAUL: IDL Astronomical Users Library
\item IRAF: Image Reduction and Analysis Facility
\item MIDAS: Munich Image Data Analysis System
\item PyRAF: Run IRAF tasks in Python
\item SAOImage ds9: FITS image viewer and analyzer
\item SEXTRACTOR: Builds catalogue of objects from an astronomical image
\item WCSLIB: World Coordinate System software library
\item \dots~\url{http://tdc-www.harvard.edu/astro.software.html}
\end{itemize}
\end{block}
\end{frame}

\begin{frame}{ìÉÔÅÒÁÔÕÒÁ}
\begin{itemize}
\item W. Romanishin. An Introduction to Astronomical Photometry Using CCDs.
\item Jean-Luc Starck and Fionn Murtagh. Handbook of Astronomical Data Analysis.
\item E.D.~Feigelson, G.J.~Babu. Modern Statistical Methods for Astronomy With R Applications.
\item R.A.~Edelson, J.H.~Krolik. The discrete correlation function --- A new method for analyzing
unevenly sampled variability data. ApJ, {\bf 333},1988, 646--659.
\end{itemize}
\end{frame}

\begin{frame}{óÐÁÓÉÂÏ ÚÁ ×ÎÉÍÁÎÉÅ!}
\centering
\begin{minipage}{5cm}
\begin{block}{mailto}
eddy@sao.ru\\
edward.emelianoff@gmail.com
\end{block}\end{minipage}
\end{frame}
\end{document}