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Suppose I have a picture and let two people mark regions within it. How can I measure the (dis)similarity of the two marked regions?

More mathematically, suppose I have a (filled) rectangle $A\subset \mathbb{R}$ and two "marked regions" $B\subset A$ and $C\subset A$. How can I measure the (dis)similarity between $B$ and $C$? To make it easier, $A, B, C$ can be made discrete because I'm working with images (that is, pixels).

I've had two ideas and written down their shortcomings to me:

  • Minimal Euclidean distance: this is zero as soon as the regions intersect somewhere, but this does not mean that the two sets are the same.
  • Jaccard index: this does not take into account the Euclidean geometry. If $B$ and $C$ do not intersect, the Jaccard index is 0, but this does not take into account whether they are close to each other in the plane or not.

Do you perhaps have a better idea for a distance metric?

Karolis Koncevičius
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Dasherman
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  • If you want to calculate some distance between two sets, group of points, you could use distances that Hierarchical cluster analysis [compute](https://stats.stackexchange.com/a/217742/3277). There also are other good set distances, including among them Hausdorff and point-centroid cross-distance. – ttnphns Sep 26 '21 at 14:48

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