High Dimensional Consistent Digital Segments

Chiu, Man Kwun; Korman, Matias

doi:10.1137/17m1136572

Cited by 2 publications

(4 citation statements)

References 12 publications

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“…A key property for Håstad's lower bound argument is that the intersection of a CDR in two dimensions with line x + y = c (for any c ∈ Z) contains a single split vertex. 4 Then, he maps the relative position of the split vertex into the [0, 1) interval and links the error of the CDR to the discrepancy of the transformed pointset. Unfortunately, the key property does not hold when we look at two dimensional subspaces of a CDR in three or more dimensions.…”

Section: Need Of Bichromatic Discrepancymentioning

confidence: 99%

“…Although the error is small, the resulting segments are far from what we would consider similar to the Euclidean segments (because they loop around many times). Recently, Chiu and Korman [ 4 ] showed that the problem in higher dimensions behaves very differently from the two dimensional case. Specifically, they studied how to extend the CDS construction of Christ et al [ 7 ] and showed that it is very limiting in three (and higher) dimensions.…”

Section: Introductionmentioning

confidence: 99%

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Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays

Chiu

Korman²,

Suderland

et al. 2022

Discrete Comput Geom

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We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in $$\mathbb {Z}^d$$ Z d . The construction must be consistent (that is, satisfy the natural extension of the Euclidean axioms) while resembling them as much as possible. Previous work has shown asymptotically tight results in two dimensions with $$\varTheta (\log N)$$ Θ ( log N ) error, where resemblance between segments is measured with the Hausdorff distance, and N is the $$L_1$$ L 1 distance between the two points. This construction was considered tight because of a $$\varOmega (\log N)$$ Ω ( log N ) lower bound that applies to any consistent construction in $$\mathbb {Z}^2$$ Z 2 . In this paper we observe that the lower bound does not directly extend to higher dimensions. We give an alternative argument showing that any consistent construction in d dimensions must have $$\varOmega (\log ^{1/(d-1)}\!N)$$ Ω ( log 1 / ( d - 1 ) N ) error. We tie the error of a consistent construction in high dimensions to the error of similar weak constructions in two dimensions (constructions for which some points need not satisfy all the axioms). This not only opens the possibility for having constructions with $$o(\log N)$$ o ( log N ) error in high dimensions, but also opens up an interesting line of research in the tradeoff between the number of axiom violations and the error of the construction. A side result, that we find of independent interest, is the introduction of the bichromatic discrepancy: a natural extension of the concept of discrepancy of a set of points. In this paper, we define this concept and extend known results to the chromatic setting.

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Section: Need Of Bichromatic Discrepancymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays

Chiu

Korman²,

Suderland

et al. 2022

Discrete Comput Geom

Self Cite

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“…To deal with these issues, past researchers have considered systems of digital rays and digital line segments that collectively satisfy some properties that are also satisfied by their Euclidean counterparts. In particular [7,6,4,5,1] has considered systems that satisfy the following five properties.…”

Section: Consistent Digital Line Segmentsmentioning

confidence: 99%

“…The construction of Chun et al [7] for two-dimensional CDRs can be extended to obtain an O(log n) construction for a CDR in a three-dimensional grid. More recently Chiu and Korman [1] have considered extending the two-dimensional results of [6] to three dimensions, and they show that at times they are able to obtain three-dimensional CDRs with error Ω(log n), and even at times they can obtain a three-dimensional CDS, but unfortunately these systems have error Ω(n).…”

Section: Previous Work On Cdses and Cdrsmentioning

confidence: 99%

Optimal Bounds for Weak Consistent Digital Rays in 2D

Gibson-Lopez¹,

Zamarripa²

2022

Preprint

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Representation of Euclidean objects in a digital space has been a focus of research for over 30 years. Digital line segments are particularly important as other digital objects depend on their definition (e.g., digital convex objects or digital star-shaped objects). It may be desirable for the digital line segment systems to satisfy some nice properties that their Euclidean counterparts also satisfy. The system is a consistent digital line segment system (CDS) if it satisfies five properties, most notably the subsegment property (the intersection of any two digital line segments should be connected) and the prolongation property (any digital line segment should be able to be extended into a digital line). It is known that any CDS must have Ω(log n) Hausdorff distance to their Euclidean counterparts, where n is the number of grid points on a segment. In fact this lower bound even applies to consistent digital rays (CDR) where for a fixed p ∈ Z 2 , we consider the digital segments from p to q for each q ∈ Z 2 . In this paper, we consider families of weak consistent digital rays (WCDR) where we maintain four of the CDR properties but exclude the prolongation property. In this paper, we give a WCDR construction that has optimal Hausdorff distance to the exact constant. That is, we give a construction whose Hausdorff distance is 1.5 under the L∞ metric, and we show that for every ϵ > 0, it is not possible to have a WCDR with Hausdorff distance at most 1.5 − ϵ.

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High Dimensional Consistent Digital Segments

Cited by 2 publications

References 12 publications

Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays

Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays

Optimal Bounds for Weak Consistent Digital Rays in 2D

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