I am trying to find the edges of the centered box in this image:

I have tried using a H
Try using a preprocessing pass with the erosion filter. It will give you the same effect as the downscaling - the lines will become thinner and will not disappear at the same time.
The "Blur" filter is also a good idea, as chaiy says.
This way (with blur) it will become something like http://www.ic.uff.br/~laffernandes/projects/kht/index.html (Kernel Based Hough Transform)