My solution:

  #!/bin/sh
  wget https://www.mi5.gov.uk/sites/default/files/u2/puzzle.png
  convert puzzle.png puzzle.xpm # Convert the image to XPM with ImageMagick.
  # Here is what the contents of puzzle.xpm looks like:
  # http://pastebin.com/0PySSAXx. The spaces represent one color and the dots
  # the other. We'll replace the spaces with underscores for readability.
  <puzzle.xpm tr ' ' _ \
  | tr -d "\n" \
  | sed -e 's|","||g;   s|^.*\*/"||;   s|".*$||   # Remove metadata, C framing.' \
  | sed -e 's|_\.|_\n.|g;   s|\._|.\n_|g   # New line on color change. ' \
  | awk '{printf "%c", length($0)}' \
  | xxd -r -p

The XPM image file format (https://en.wikipedia.org/wiki/X_PixMap) is great for doing things like this. I find it easier to read and to edit by hand than the Netpbm format family. It seems ImageMagick can't write XPM2, though, which makes extra preprocessing necessary.