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The definition of 8-adjacency and m-adjacency are clear. But is there a case where two pixels p and q can be 8-adjacent, but not m-adjacent. I have given the definition of 8-adjacent and m-adjacent below:
8-adjacency: two pixels p and q with values from V are 8-adjacent if q is in the set N8(p).
m-adjacency: two pixels p and q with values from V are m-adjacent if
i) q is in N4(p), OR
ii) q is in ND(p) AND the set N4(p) N4(q) has no pixels whose values are from V.
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b) 8-connectivity: Two or more pixels are said to be 8-connected if they are 8-adjacent with each others. c) m-connectivity: Two or more pixels are said to be m-connected if they are m-adjacent with each others.
It is introduced to eliminate ambiguities of 8-adjacency. The role of m-adjacency is to define a single path between pixels.
Common adjacency relations on sets of points arranged in a grid are 4- adjacency, 8-adjacency and 6-adjacency. Two points are 4-adjacent if they are vertical or horizontal neighbours in the grid, 8-adjacent if they are either 4- adjacent or are diagonal neighbours.
Definition of an 8-neighbor:A pixel, Q, is an 8-neighbor (or simply a neighbor) of a given pixel, P, if Q and P either share an edge or a vertex. The 8-neighbors of a given pixel P make up the Moore neighborhood of that pixel.
There is a clear mathematical defition for these adjacencies. So no, there can be no difference other than that definition.
m adjacendy is used to resolve abiguities in 8-adjacency. m-adjacency is a special case of 8-adjacency. So yes there are cases where 2 pixels are 8-adjacent but not m-adjacent. Otherwise if they were the same, you wouldn't need a separate m-adjacency right?
May pixels p and q have the value 1 (from V).
Then in both examples p and q are 8-adjacent.
0 0 0
0 p 0
q 0 0
0 0 0
0 p 0
q 1 0
But only in the first one they are m-adjacent.
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