not sure I understand all their steps correctly, but the curves part can be approximated by -level. Here is a close approximation of it in IM
convert beardman.gif \
\( -clone 0 -sharpen 0x5 \) \
\( -clone 0 -clone 1 -compose hardlight -composite -sharpen 0x5 \) \
\( -clone 1 -clone 2 -compose blend -set option:compose:args 50 -composite \) \
-delete 0-2 -modulate 100,60,100 -level 0x80% beardman_proc.gif
You can play with the parameters for -sharpen, -compose blend, -modulate (to desaturate) and -level.