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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?

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    $\begingroup$ Can you give us some context, e.g., a citation to the source where you saw the phrase used and the surrounding context? $\endgroup$
    – D.W.
    Feb 22, 2016 at 21:01

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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.

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