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By Stéphane Marchand-Maillet, Yazid M. Sharaiha

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In this case, the only possible definition of a discrete distance that is geometrically consistent is that of the d4 distance, where a - 1. A simple extension of the 4-neighbourhood leads to the 8-neighbourhood. Diagonal moves are added to the horizontal and vertical moves. The length of such diagonal moves is denoted b. In this respect, diagonal moves are called b-moves and the chamfer distance obtained in the 8-neighbourhood is denoted da,b. e. the length for all 4-moves), in order to preserve a geometrical consistency within the 8-neighbourhood, the diagonal moves should be associated with a length b larger than a.

38. Therefore, as mentioned earlier, an extension of ds in the 16-neighbourhood by setting all move lengths to unity is not possible. 20. Typically, the convex hull of a chamfer disc in the 8-neighbourhood is an octagon that approximates the Euclidean circle depending on the values of a and b. More generally, a chamfer disc is a polygon with as many sides as there are different moves in the neighbourhood on which the chamfer distance is defined. 20 0 0 0 0 0 0 0 0 (B) (C) Chamfer discs. (A) A3,4(27).

The topic of computing discrete and Euclidean distances will be revisited in Chapter 5 when dealing with distance transformations. Algorithmic solutions to these problems will then be presented. 1 GEOMETRY Introduction Discrete geometry aims for a characterisation of the geometrical properties of a set of discrete points. Geometrical properties of a set are understood to be global properties. In discrete geometry, points are grouped, thus forming discrete objects, and it is the properties of these discrete objects that are under study.

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