Having read Paulhk's answer, it occurred to me that a possible alternative to the 'bleed', based on premultiplied alpha, would be to initialise your unfilled area to *transparent black* rather than opaque green. 

When you then perform the bilinear filtering, if the returned alpha is non-zero (*not sure if this test is necessary or not as I can't remember the specifics of the shader rcp() function*), multiply the returned filtered (premultiplied) RGB components by the reciprocal of the alpha component to get back a more appropriate colour.

I would hope the alpha is non-zero, but there may be inaccuracies in the pipeline so you might need to special case it.