How to reproduce this graphic using only tikz and pgfplots?

Question

I have this cool looking image from a blog post and I would like to reproduce it using TikZ and pgfplots.

I have found the property point meta but I do not think it's valid in this case. This plot has been generated applying a function $f(t)\in[0,1]$ if $t\in[-1,1]$ and assign a gradient from black ($f(t)=0$) to white ($f(t)=1$).

The squares are $[-1,1]x[-1,1]$.

Any idea aboud how to make any of those three plots?

Answer 1

You can draw squares with shading inside or you can draw functions and use pgfplots heatmaps to get the shading. This is probably total overkill and way slower than it should be:

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{
  compat=newest,
  heatmap/.style={
    view={0}{90},
    scale only axis=true,
    domain=-1:1,
    domain y=-1:1,
    width=3cm,
    height=3cm,
    xtick={0},
    xticklabels={},
    ytick={0},
    yticklabels={},
    yticklabel style={overlay},
    grid=major,
    grid style={thick,purple},
    axis line style={purple},
    colormap/blackwhite,
  },
}
\begin{document}
\begin{equation}
  \begin{tikzpicture}[baseline=(C.base)]
    \begin{axis}[heatmap,xlabel={$f(x)$}]
      \node (C) at (axis cs:0,0) {\strut};
      \addplot3 [surf,shader=interp] {exp(-x^2)};
    \end{axis}
  \end{tikzpicture}
  \times
  \begin{tikzpicture}[baseline=(C.base)]
    \begin{axis}[heatmap,xlabel={$f(y)$}]
      \node (C) at (axis cs:0,0) {\strut};
      \addplot3 [surf,shader=interp] {exp(-y^2)};
    \end{axis}
  \end{tikzpicture}
  =
  \begin{tikzpicture}[baseline=(C.base)]
    \begin{axis}[heatmap,xlabel={$f(x) f(y)$}]
      \node (C) at (axis cs:0,0) {\strut};
      \addplot3 [surf,shader=interp] {exp(-x^2-y^2)};
    \end{axis}
  \end{tikzpicture}
\end{equation}
\end{document}