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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.

E • • • • o. o- o. • o. • • • • • o- • • • • • • o. (::~iCL"" -9:°"": .... • • ,, • • • • • • • • • .. 21 Neighbourhoods corresponding to Farey sequences. (A) Order 3. (B) Order 4 Based on these moves, a general definition of chamfer distances can be given as follows. ), the chamfer distance between p and q relative to this neighbourhood is the length of the shortest digital arc from p to q. General conditions on move lengths can be defined for chamfer distances to satisfy the metric properties [162].

15. 30" Note that there exists a strong similarity between the n o r m s IlUill - I X f f l + lYffl, Ilgl12 -- v/IX~I 2 + ly~l 2 and Ilgll~ - sup(Ix~l, lY~I) defined in the continuous space IR 2 and d4, dE and ds on the digital space. Recalling that I1~111 ~ IlUl12 ~ IlUll~ Vff c IR 2, this property is mapped in the digital space as dn(p, q) <_ dE(p,q) <_ ds(p, q ) Vp, q E ~Z 2. The knight-neighbourhood is studied because of its analogy with the moves of a knight on the chess board. 31. 31" Knight-distance The knight-distance between p and q is the length of the shortest knight-arc joining p and q when the move lengths are all set to unity.