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Tight bounds in the quadtree complexity theorem and the maximal number of pixels crossed by a curve of given length

Abstract : The main purpose of this work is to determine the exact maximum number of pixels (a bi-dimensional sequence of unit squares tiling a plane) that a rectifiable curve of given length l can cross. In other words, given l ∈ R, we provide the value N (l) of the maximal cardinality of the digital cover of a rectifiable curve of length l. The optimal curves are polygonal curves with integer vertices, 0, 1 or 2 vertical or horizontal steps and an arbitrary number of diagonal steps. We also report the properties of the staircase function N (l), which is affinely periodic in the sense that N (l + √ 2) = N (l) + 3 and a bound N (l) ≤ 4 + 3 √ 2 l. Our second aim is to look at the restricted class of closed curves and offer some conjectures on the maximum number N closed (l) of pixels that a closed curve of length l can cross. This work finds its application in the quadtree complexity theorem. This well-known result bounds the number of quads with a shape of perimeter p by 16q − 11 + 16p. However, this linear bound is not tight. From our new upper bound N closed (l) ≤ N (l) ≤ 4 + 3 √ 2 l we derive a new improved multiresolution complexity theorem: Number(quads) ≤ 16q − 11 + 6 √ 2p. Lastly, we show that this new bound is tight up to a maximal error of 16(q − 1). quadtree complexity theorem, digital cover cardinality, rectifiable curve, length
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Yan Gerard, Antoine Vacavant, Jean-Marie Favreau. Tight bounds in the quadtree complexity theorem and the maximal number of pixels crossed by a curve of given length. Theoretical Computer Science, Elsevier, 2016, 624, pp.41-55. ⟨10.1016/j.tcs.2015.12.015⟩. ⟨hal-02023856⟩

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