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psf_detection.tex
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psf_detection.tex
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\documentclass[11pt]{article}
\usepackage[breakable]{tcolorbox}
\usepackage{parskip} % Stop auto-indenting (to mimic markdown behaviour)
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\fi
% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts  syntax from Markdown).
\usepackage{graphicx}
% Maintain compatibility with old templates. Remove in nbconvert 6.0
\let\Oldincludegraphics\includegraphics
% Ensure that by default, figures have no caption (until we provide a
% proper Figure object with a Caption API and a way to capture that
% in the conversion process - todo).
\usepackage{caption}
\DeclareCaptionFormat{nocaption}{}
\captionsetup{format=nocaption,aboveskip=0pt,belowskip=0pt}
\usepackage{float}
\floatplacement{figure}{H} % forces figures to be placed at the correct location
\usepackage{xcolor} % Allow colors to be defined
\usepackage{enumerate} % Needed for markdown enumerations to work
\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
\usepackage{textcomp} % defines textquotesingle
% Hack from http://tex.stackexchange.com/a/47451/13684:
\AtBeginDocument{%
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}
\usepackage{upquote} % Upright quotes for verbatim code
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% to support a larger range
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\@ifpackagelater{grffile}{2019/11/01}
{
% Do nothing on new versions
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{
\def\Gread@@xetex#1{%
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{\Gread@eps{\[email protected]}}%
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}
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% The hyperref package gives us a pdf with properly built
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% The default LaTeX title has an obnoxious amount of whitespace. By default,
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\usepackage{titling}
\usepackage{longtable} % longtable support required by pandoc >1.10
\usepackage{booktabs} % table support for pandoc > 1.12.2
\usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
\usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
% normalem makes italics be italics, not underlines
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% Colors for the hyperref package
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% ANSI colors
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% common color for the border for error outputs.
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% commands and environments needed by pandoc snippets
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% Define a nice break command that doesn't care if a line doesn't already
% exist.
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% Math Jax compatibility definitions
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\def\lt{<}
\let\Oldtex\TeX
\let\Oldlatex\LaTeX
\renewcommand{\TeX}{\textrm{\Oldtex}}
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% Document parameters
% Document title
\title{Detekce natočení DH PSF}
\author{Jakub Dokulil}
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\begin{document}
\maketitle
\hypertarget{automatizovanuxe1-detekce-rotace-dh-psf}{%
\section{Automatizovaná detekce rotace DH
PSF}\label{automatizovanuxe1-detekce-rotace-dh-psf}}
\hypertarget{axiuxe1lnuxed-lokalizace-pomocuxed-vuxedrovuxfdch-svazkux16f}{%
\subsection{Axiální lokalizace pomocí vírových
svazků}\label{axiuxe1lnuxed-lokalizace-pomocuxed-vuxedrovuxfdch-svazkux16f}}
Vírové svazky jsou svazky se šroubovitou vlnoplochou. Takový svazek může
být generován například při průchodu rovinné vlnoplochy fázovým prvkem,
například spirální maskou (SM). Při volbě vhodných vírových svazků může
dojít k samozobrazení a formování dvoučetné rozptylové funkce bodu
(Double Helix Point Spread Function -- DH PSF). Toho lze využít
například při detekci axiální polohy bodového zdroje světla.
Experiment pro detekci axiílní polohy lze vidět například na obrázku
níže. Bodový zdroj světla je umístěn do ohniska čočky. Svazek je
kolimován a do Fourierovské roviny je umístěna SM vytvářející optické
víry. Ty jsou následně zobrazeny pomocí další čočky na kamaru, kde je
detekována DH PSF. Pokud dojde k axiálnímu posunu bodového zdroje, tj.
dojde k \emph{defokusaci} \(\Delta z\) dochází ke změně tvaru vlnoplochy
z roviny na kulový povrch. Tím vzniká fázové zpoždění mezi jednotlivými
výry \(\Delta \beta_m\). Toto fázové zpoždění způsobí rotaci DH PSF.
Rotace je přímo úměrná defokusaci.
\begin{figure}
\centering
\includegraphics{umisteni_masky.png}
\caption{Schéma experimentu}
\end{figure}
Véce například v
Bouchal, P., \& Bouchal, Z. (2017). Flexible non-diffractive vortex
microscope for three-dimensional depth-enhanced super-localization of
dielectric, metal and fluorescent nanoparticles. Journal of Optics
(United Kingdom), 19(10). https://doi.org/10.1088/2040-8986/aa87fb
\hypertarget{zpracovuxe1nuxed-obrazu-dh-psf}{%
\subsection{Zpracování obrazu DH
PSF}\label{zpracovuxe1nuxed-obrazu-dh-psf}}
Pro zpracování byl původně použit následující postup.
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
Prahování obrazu.
\item
Převod obrazu na binární mmasku.
\item
Pomocí funkce pro hledání kruhů najít kruhy (funkce
\texttt{hough\_circle} knihovny \texttt{skimage}).
\item
Spořítat úhel spojnice středů kruhů pomocí funkce \texttt{arctan2}
\end{enumerate}
Postup však ne vždy dokázal najít správně kruhy. Proto byla použita změa
v segmentaci u bodů 3 a 4. Postup vypadá následovně:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
Prahování obrazu.
\item
Převod obrazu na binární mmasku.
\item
Označení jednotlivých oblastí (metoda \texttt{label} balíku
\texttt{scipy.ndimage}).
\item
Výpočet těžiště jednotlivých oblastí (metoda \texttt{centre\_of\_mass}
balíku \texttt{scipy.ndimage})
\item
Výpočet úhlu pomocí těžiště.
\end{enumerate}
\hypertarget{implemetace}{%
\subsection{Implemetace}\label{implemetace}}
Pro implementaci byl vytvořen externí balík \texttt{PSFrot}, obsahující
funkce pro zpřehlednění kódu. Balík obsahuje následující funkce.
\texttt{readImage} - funkce načte obraz, může jej normalizovat (výchozí
nastavení) a vrátí pole.
\texttt{threshold} - funkce pro prahování. Výhoczí hodnota prahu je 0,4
(pro hodnoty od 0 do 1).
\texttt{findCircles} - funkce která najde kruhy v obraze. Lze nastavit
maximální a miniimální poloměr hledaných kruhů.
\texttt{centresOfMass} - funkce která najde težiště dvou světlých
oblastí a spočítá jejich úhel.
Algoritmus byl testován na testovacích obrazcích, které lze vidět níže.
\hypertarget{import-baluxedkux16f-a-knihoven}{%
\subsubsection{Import balíků a
knihoven}\label{import-baluxedkux16f-a-knihoven}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{1}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{from} \PY{n+nn}{PSFrot} \PY{k+kn}{import} \PY{n}{PSFrot}
\PY{k+kn}{import} \PY{n+nn}{proplot} \PY{k}{as} \PY{n+nn}{pplt}
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
\PY{k+kn}{from} \PY{n+nn}{skimage}\PY{n+nn}{.}\PY{n+nn}{draw} \PY{k+kn}{import} \PY{n}{circle\PYZus{}perimeter}
\PY{k+kn}{from} \PY{n+nn}{skimage}\PY{n+nn}{.}\PY{n+nn}{color} \PY{k+kn}{import} \PY{n}{rgb2gray}\PY{p}{,} \PY{n}{gray2rgb}
\PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{pyplot} \PY{k}{as} \PY{n+nn}{plt}
\PY{k+kn}{from} \PY{n+nn}{scipy} \PY{k+kn}{import} \PY{n}{ndimage} \PY{k}{as} \PY{n}{ndi}
\end{Verbatim}
\end{tcolorbox}
\hypertarget{vykreslenuxed-testovacuxedch-obrazcux16f}{%
\subsubsection{Vykreslení testovacích
obrazců}\label{vykreslenuxed-testovacuxedch-obrazcux16f}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{2}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
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\PY{n}{suptitle}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Testovací obrazce}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{)}\PY{p}{:}
\PY{n}{filename} \PY{o}{=} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{test\PYZus{}imgs/}\PY{l+s+s2}{\PYZdq{}} \PY{o}{+} \PY{n+nb}{str}\PY{p}{(}\PY{n}{i}\PY{p}{)} \PY{o}{+} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{.tiff}\PY{l+s+s2}{\PYZdq{}}
\PY{n}{image} \PY{o}{=} \PY{n}{PSFrot}\PY{o}{.}\PY{n}{readImage}\PY{p}{(}\PY{n}{path}\PY{o}{=}\PY{n}{filename}\PY{p}{)}
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\PY{n}{fig}\PY{o}{.}\PY{n}{save}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{test.png}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{dpi}\PY{o}{=}\PY{l+m+mi}{333}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_3_0.png}
\end{center}
{ \hspace*{\fill} \\}
\hypertarget{zpracovuxe1nuxed-pomocuxed-prvnuxed-metody}{%
\subsubsection{Zpracování pomocí první
metody}\label{zpracovuxe1nuxed-pomocuxed-prvnuxed-metody}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{3}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{fig}\PY{p}{,} \PY{n}{axs} \PY{o}{=} \PY{n}{pplt}\PY{o}{.}\PY{n}{subplots}\PY{p}{(}\PY{n}{nrows}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{,} \PY{n}{ncols}\PY{o}{=}\PY{l+m+mi}{5}\PY{p}{,} \PY{n}{share}\PY{o}{=}\PY{k+kc}{False}\PY{p}{,}
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\PY{n}{cx}\PY{p}{,} \PY{n}{cy}\PY{p}{,} \PY{n}{rad} \PY{o}{=} \PY{n}{PSFrot}\PY{o}{.}\PY{n}{findCircles}\PY{p}{(}\PY{n}{image}\PY{p}{)}
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\end{Verbatim}
\end{tcolorbox}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_5_0.png}
\end{center}
{ \hspace*{\fill} \\}
Lze vidět, že pro defokusaci -1 \(\mathrm{\mu m}\) a méně metoda není
schopna najít jednotlivé bloby. Ani pro defokusaci 0 \(\mathrm{\mu m}\)
metoda nenalezla střed.
\hypertarget{pracovuxe1nuxed-pomocuxed-druhuxe9-metody}{%
\subsubsection{Pracování pomocí druhé
metody}\label{pracovuxe1nuxed-pomocuxed-druhuxe9-metody}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{4}{\boxspacing}
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\PY{n}{axisTitle} \PY{o}{=} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{Delta z = }\PY{l+s+s2}{\PYZdq{}} \PY{o}{+} \PYZbs{}
\PY{n+nb}{str}\PY{p}{(}\PY{n}{i}\PY{p}{)} \PY{o}{+} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ }\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{,}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mu m\PYZdl{}, }\PY{l+s+se}{\PYZbs{}t}\PY{l+s+s2}{ \PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{phi\PYZdl{}}\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{ = }\PY{l+s+s2}{\PYZdq{}} \PY{o}{+} \PY{n+nb}{str}\PY{p}{(}\PY{n+nb}{round}\PY{p}{(}\PY{n}{angles}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{)}\PY{p}{)}
\PY{n}{axs}\PY{p}{[}\PY{n}{i}\PY{o}{/}\PY{o}{/}\PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{4}\PY{o}{\PYZhy{}}\PY{n}{i} \PY{o}{\PYZpc{}} \PY{l+m+mi}{5}\PY{p}{]}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{title}\PY{o}{=}\PY{n}{axisTitle}\PY{p}{)}
\PY{n}{fig}\PY{o}{.}\PY{n}{save}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{vysl\PYZus{}CoM.png}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{dpi}\PY{o}{=}\PY{l+m+mi}{333}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_7_0.png}
\end{center}
{ \hspace*{\fill} \\}
Následně je vykreslena závislost defokusace na natočení PSF, která je ve
vybraných bodech proložena regresní přímkou.
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{5}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{cond} \PY{o}{=} \PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{defocus}\PY{p}{)} \PY{o}{\PYZlt{}} \PY{l+m+mi}{4}\PY{p}{)}
\PY{n}{params} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{polyfit}\PY{p}{(}\PY{n}{defocus}\PY{p}{[}\PY{n}{cond}\PY{p}{]}\PY{p}{,} \PY{n}{angles}\PY{p}{[}\PY{n}{cond}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}
\PY{n}{yFit} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{polyval}\PY{p}{(}\PY{n}{params}\PY{p}{,} \PY{n}{defocus}\PY{p}{)}
\PY{n}{fig} \PY{o}{=} \PY{n}{pplt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{n}{suptitle}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Závislost \PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{phi }\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{\PYZdl{} na \PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{Delta z\PYZdl{} }\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{ax} \PY{o}{=} \PY{n}{fig}\PY{o}{.}\PY{n}{subplot}\PY{p}{(}\PY{n}{xlabel}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{Delta z\PYZdl{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{ylabel}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{phi }\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{\PYZdl{} }\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{defocus}\PY{p}{,} \PY{n}{angles}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{.}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{lw}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{defocus}\PY{p}{[}\PY{n}{cond}\PY{p}{]}\PY{p}{,} \PY{n}{angles}\PY{p}{[}\PY{n}{cond}\PY{p}{]}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{.}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{lw}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{defocus}\PY{p}{,} \PY{n}{yFit}\PY{p}{,} \PY{n}{lw}\PY{o}{=}\PY{l+m+mf}{0.5}\PY{p}{,} \PY{n}{color}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{grey}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{fig}\PY{o}{.}\PY{n}{save}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{defocus.png}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{dpi}\PY{o}{=}\PY{l+m+mi}{333}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_9_0.png}
\end{center}
{ \hspace*{\fill} \\}
\hypertarget{zuxe1vux11br}{%
\subsection{Závěr}\label{zuxe1vux11br}}
Při použití metody založené na hledání těžiště jednotlivých blobů lze
vidět, žebyly poměrně dobře nalezeny středy a rotace DH PSF má lineární
průběh vzhledem k defokusaci. Metoda přitozeně nefunguje pro velké
defokusace u níchž nelze v obraze vidět dva jasné bloby.
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\end{document}