Distance bounds for high dimensional consistent digital rays and 2-D partially-consistent digital rays

Abstract

We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in 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 Θ(log N ) error, where resemblance between segments is measured with the Hausdorff distance, and N is the L1 distance between the two points. This construction was considered tight because of a Ω(log N ) lower bound that applies to any consistent construction in 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 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 ) 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. In order to show our lower bound, we also consider a colored variation of the concept of discrepancy of a set of points that we find of independent interest.

Optimal Bounds for Weak Consistent Digital Rays in 2D

“…This implies that the slope of ℓ(r) would need to be less than 1 in order for r d−1 to be above or on q. However, Lemma 7 states that M (ℓ(r)) > 3 3−2δ > 1. It follows that r d−1 is below q and therefore r d−1 x − p ← x > 1.5 − δ.…”

“…Indeed, 24.57−236.16(0.1)+140.58(0.1) 2 +22.86(0.1) 3 = 2.38266. Thus M (p) > 3 3−2δ for all p ∈ P which implies that for every p ∈ Subtree(s) it must be that M (p) > 3 3−2δ . Next we will show that M (p) < 1 + 4δ 3−2δ = 3+2δ 3−2δ for every point p that is 3 above ℓ(t) such that N ′′ ≤ D(p) < N ′ .…”

“…They also consider what is the minimum number of inner leaves needed to obtain a WCDR with error e. In the full version of their paper [3], they give a system with error 2.5 under the L ∞ metric establishing that O(1) error is in fact possible. Note that their Ω( N log N N +k ) lower bound implies that k ∈ Ω(N log N ) in order to achieve O(1) error, but their construction has k ∈ Θ(N 2 ).…”

“…Given a fixed M (s), we can see that M (h) grows as D(s) becomes larger. Recall that M (s) > 3 3−2δ and D(s) ≥ N ′′ . It follows that M (h) is at least the slope of the point, q, that is 1.5 diagonal distance below Inter(ℓ(3 − 2δ, 3), N ′′ ).…”

“…Figure13 An illustration of P under ℓ(r). The Euclidean line with slope3 3−2δ is represented by the purple line on the bottom.…”