How to remove weak edges in a binary photo?
When you are dealing with a binary image (black and white with 1 bit per pixel intensity) you loose all the continuous gradient information, and therefore cannot make use of edge detection (or removal) algortihms that rely on it. 

Instead you can return to geometric features such as line length, edge thickness (indeed line thikness), surface area etc to classify features found in the image. 

A weak edge should have a corresponding definition in terms of feature geometries so that it can be detected and removed (replaced) if necessary.