colorized.tex

% \documentclass{article}
\documentclass[preview]{standalone}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{color}

\renewcommand{\familydefault}{\sfdefault}

\definecolor{c1}{RGB}{114,0,172}   % primary
\definecolor{c2}{RGB}{45,177,93}   % true
\definecolor{c3}{RGB}{251,0,29}    % false
\definecolor{c4}{RGB}{18,110,213}  % secondary
\definecolor{c5}{RGB}{255,160,109} % tertiary
\definecolor{c6}{RGB}{219,78,158}  % alt-primary 


\begin{document}


The forward Abel transform is given by
\begin{equation} \label{eq:forward}
\color{c1} F(y,z)\color{black} = \color{c6}2\color{c2} \int_y^{\infty} \color{black} \frac{\color{c4}f(r,z)\color{black}\,\color{c6}r}{\color{c6}\sqrt{r^2-y^2}}\,\color{c6}dr\color{black}.
\end{equation}

To find the \color{c1} 2D projection\color{black}, break the \color{c4} cylindrically symmetric 3D object \color{black} into \color{c6}thin cylindrical shells\color{black}, and \color{c2} add them all together\color{black}. 
\\ \\



The inverse Abel transform is given by
\begin{equation} \label{eq:inverse}
\color{c4}f(r,z)\color{black} = \color{c6}-\frac{1}{\pi} \color{c2}\int_r^{\infty} \color{black} \frac{\color{c5}d\color{c1}F(y,z)\color{black}}{\color{c5}dy}\, \color{c6}\frac{1}{\sqrt{y^2-r^2}}\,dy\color{black}.
\end{equation}

To  reconstruct the \color{c4} 3D object\color{black}, \color{c2} add together \color{c6} thin cylindrical shells \color{black} proportional to the  \color{c5}slope \color{black} of the \color{c1} 2D image\color{black}.
		
\end{document}