The integer medial axis and the distance between closest points

Lecturer: Wilberd van der Kallen (Utrecht University)

We consider a geometric problem about lattice points. It originates in image processing. Say we have an image consisting of black and white pixels, and we consider the set of pairs of neighboring pixels P, Q so that the/a black pixel X closest to P is rather far from the/a black pixel Y closest to Q. Such a set of pairs P, Q is known as an integer medial axis. It is the analogue of a conflict set. One wishes to have a definition that does not yield any spurious pairs for the class of test pictures in which the black pixels are the integer lattice points in a half plane. A conjecture of Hesselink provides a formula. We can prove it in some relevant cases using classical geometry and continued fractions. We also present examples showing that in three dimensions the situation is different.

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