Draw dotted background

Here's a pattern I defined by taking the coordinates from your image. One could probably use some external program such as octave or the scripting capabilities of luatex to generate new points.

Just taking purely random points doesn't really work, because those won't be distributed evenly enough.

\documentclass[tikz,margin=1cm]{standalone}
\usetikzlibrary{patterns}

\pgfmathsetmacro\sprayRadius{.2pt}
\pgfmathsetmacro\sprayPeriod{.5cm}

\pgfdeclarepatternformonly{spray}{\pgfpoint{-\sprayRadius}{-\sprayRadius}}{\pgfpoint{1cm + \sprayRadius}{1cm + \sprayRadius}}{\pgfpoint{\sprayPeriod}{\sprayPeriod}}{
    \foreach \x/\y in {2/53,6/52,11/48,23/49,20/47,32/46,41/47,47/51,56/52,46/44,4/43,16/42,33/41,41/37,49/35,55/31,37/35,44/30,28/37,24/36,17/37,7/38,0/31,8/29,18/31,28/30,37/28,30/27,46/24,51/21,24/23,12/24,4/21,18/19,12/16,31/21,38/18,26/16,46/16,56/12,52/10,45/8,51/4,37/12,35/7,24/9,14/9,2/12,8/6,15/4,27/0,34/1,40/1} {
        \pgfpathcircle{\pgfpoint{(\x + random()) / 57 * \sprayPeriod}{\sprayPeriod - (\y + random()) / 55 * \sprayPeriod}}{\sprayRadius}
    }
    %\foreach \i in {1,...,100} {
    %   \pgfpathcircle{\pgfpoint{1cm * random()}{1cm * random()}}{\sprayRadius}
    %}
    \pgfusepath{fill}
}

\begin{document}
    \begin{tikzpicture}
        \filldraw[pattern=spray] (0,0) -- (3,3) -- (2,3) -- cycle;
    \end{tikzpicture}
\end{document}