I came across the notion of topology and topological distance in the context of image processing several times (especially when it came to mathematical morphology). I looked for a not too abstract explanation of topological distance but haven't found one yet. Can you please explain (in layaman's terms) the notion of topological distance between 2 pixels? And what's the difference between a euclidean distance and a topological distance in image-processing?
Meaning of topological distance between 2 pixels
image processing;topology
Nonlinear Signal and Image Processing: Theory, Methods, and Applications defines topological distance as follows. First, you have to define when two pixels are neighbors. The book offers two possibilities:The neighbors of a pixel are the 4 cardinal pixels around it.The neighbors of a pixel are the 8 pixels surrounding it.This defines a graph (two pixels are connected by an edge if they are neighbors). The topological distance between two pixels $p,q$ is then the graph distance between $p$ and $q$.The first choice above corresponds to $L^1$ (Manhattan) distance, and the second choice to $L^\infty$ distance. This contrasts with the usual $L^2$ (Euclidean) distance.