Digital Convex + Unimodular Mapping =8-Connected (All Points but One 4-Connected)
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In two dimensional digital geometry, two lattice points are 4-connected (resp. 8-connected) if their Euclidean distance is at most one (resp. √(2)). A set S ⊂ Z^2 is 4-connected (resp. 8-connected) if for all pair of points p_1, p_2 in S there is a path connecting p_1 to p_2 such that every edge consists of a 4-connected (resp. 8-connected) pair of points. The original definition of digital convexity which states that a set S ⊂ Z^d is digital convex if (S) ∩ Z^d= S, where (S) denotes the convex hull of S does not guarantee connectivity. However, multiple algorithms assume connectivity. In this paper, we show that in two dimensional space, any digital convex set S of n points is unimodularly equivalent to a 8-connected digital convex set C. In fact, the resulting digital convex set C is 4-connected except for at most one point which is 8-connected to the rest of the set. The matrix of SL_2(Z) defining the affine isomorphism of Z^2 between the two unimodularly equivalent lattice polytopes S and C can be computed in roughly O(n) time. We also show that no similar result is possible in higher dimension.
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