Using ImageMagick, this is very similar to hackerb9 code and result, but is a little simpler command line. It does assume that the top left pixel is the background color. I just flood fill the background with transparency, then select the alpha channel and blur it and remove half of the blurred area using -level 50x100%. Then turn back on all the channels and flatten it against the brown color. The -blur 0x1 -level 50x100% acts to antialias the boundaries of the alpha channel transparency. You can adjust the fuzz value, blur amount and the -level 50% value to change the degree of antialiasing.
convert logo: -fuzz 25% -fill none -draw "matte 0,0 floodfill" -channel alpha -blur 0x1 -level 50x100% +channel -background saddlebrown -flatten result.jpg