This question is linked to previous question but significantly different: Get cell boundaries in a noisy image? I am using Mathematica 10.2. I have a binarized image (see below) and wish to get a deformable mesh from it where I can add lines, delete lines or move nodes that are connecting edges (see the bottom most image). The point is to discretize the white lines into segments (mesh white lines in a way) that can be moved, removed or edited so as to ensure edge correction while not affecting the overall topology. Is that even possible? Any strategy or solution will be appreciated.

Same image as above but with the alterations that I wish to possibly make.