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Register a new userUncertain Curve Simplification
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Added by
Aleksandr Popov
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We study the problem of polygonal curve simplification under uncertainty, where instead of a sequence of exact points, each uncertain point is represented by a region, which contains the (unknown) true location of the vertex. The regions we consider are disks, line segments, convex polygons, and discrete sets of points. We are interested in finding the shortest subsequence of uncertain points such that no matter what the true location of each uncertain point is, the resulting polygonal curve is a valid simplification of the original polygonal curve under the Hausdorff or the Frechet distance. For both these distance measures, we present polynomial-time algorithms for this problem.
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Simplification is one of the fundamental operations used in geoinformation science (GIS) to reduce size or representation complexity of geometric objects. Although different simplification methods can be applied depending on ones purpose, a simplification that many applications employ is designed to preserve their spatial properties after simplification. This article addresses one of the 2D simplification methods, especially working well on human-made structures such as 2D footprints of buildings and indoor spaces. The method simplifies polygons in an iterative manner. The simplification is segment-wise and takes account of intrusion, extrusion, offset, and corner portions of 2D structures preserving its dominant frame.
In the classic polyline simplification problem we want to replace a given polygonal curve $P$, consisting of $n$ vertices, by a subsequence $P$ of $k$ vertices from $P$ such that the polygonal curves $P$ and $P$ are as close as possible. Closeness is usually measured using the Hausdorff or Frechet distance. These distance measures can be applied globally, i.e., to the whole curves $P$ and $P$, or locally, i.e., to each simplified subcurve and the line segment that it was replaced with separately (and then taking the maximum). This gives rise to four problem variants: Global-Hausdorff (known to be NP-hard), Local-Hausdorff (in time $O(n^3)$), Global-Frechet (in time $O(k n^5)$), and Local-Frechet (in time $O(n^3)$). Our contribution is as follows. - Cubic time for all variants: For Global-Frechet we design an algorithm running in time $O(n^3)$. This shows that all three problems (Local-Hausdorff, Local-Frechet, and Global-Frechet) can be solved in cubic time. All these algorithms work over a general metric space such as $(mathbb{R}^d,L_p)$, but the hidden constant depends on $p$ and (linearly) on $d$. - Cubic conditional lower bound: We provide evidence that in high dimensions cubic time is essentially optimal for all three problems (Local-Hausdorff, Local-Frechet, and Global-Frechet). Specifically, improving the cubic time to $O(n^{3-epsilon} textrm{poly}(d))$ for polyline simplification over $(mathbb{R}^d,L_p)$ for $p = 1$ would violate plausible conjectures. We obtain similar results for all $p in [1,infty), p e 2$. In total, in high dimensions and over general $L_p$-norms we resolve the complexity of polyline simplification with respect to Local-Hausdorff, Local-Frechet, and Global-Frechet, by providing new algorithms and conditional lower bounds.
In this paper we study a wide range of variants for computing the (discrete and continuous) Frechet distance between uncertain curves. We define an uncertain curve as a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Frechet distance, which are the minimum and maximum Frechet distance for any realisations of the curves. We prove that both the upper and lower bound problems are NP-hard for the continuous Frechet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Frechet distance. In contrast, the lower bound (discrete and continuous) Frechet distance can be computed in polynomial time. Furthermore, we show that computing the expected discrete Frechet distance is #P-hard when the uncertainty regions are modelled as point sets or line segments. The construction also extends to show #P-hardness for computing the continuous Frechet distance when regions are modelled as point sets. On the positive side, we argue that in any constant dimension there is a FPTAS for the lower bound problem when $Delta / delta$ is polynomially bounded, where $delta$ is the Frechet distance and $Delta$ bounds the diameter of the regions. We then argue there is a near-linear-time 3-approximation for the decision problem when the regions are convex and roughly $delta$-separated. Finally, we also study the setting with Sakoe--Chiba time bands, where we restrict the alignment between the two curves, and give polynomial-time algorithms for upper bound and expected discrete and continuous Frechet distance for uncertainty regions modelled as point sets.
We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call homotopic curve shortening (HCS), starts with a closed curve (which might self-intersect) in the presence of a set $Psubset mathbb R^2$ of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent. We find experimentally that, if the initial curve is held fixed and $P$ is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between ACSF and HCS generalizes the link between ACSF and convex-layer decomposition (Eppstein et al., 2017; Calder and Smart, 2020), which is restricted to convex curves. We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.
We study the complexity of clustering curves under $k$-median and $k$-center objectives in the metric space of the Frechet distance and related distance measures. Building upon recent hardness results for the minimum-enclosing-ball problem under the Frechet distance, we show that also the $1$-median problem is NP-hard. Furthermore, we show that the $1$-median problem is W[1]-hard with the number of curves as parameter. We show this under the discrete and continuous Frechet and Dynamic Time Warping (DTW) distance. This yields an independent proof of an earlier result by Bulteau et al. from 2018 for a variant of DTW that uses squared distances, where the new proof is both simpler and more general. On the positive side, we give approximation algorithms for problem variants where the center curve may have complexity at most $ell$ under the discrete Frechet distance. In particular, for fixed $k,ell$ and $varepsilon$, we give $(1+varepsilon)$-approximation algorithms for the $(k,ell)$-median and $(k,ell)$-center objectives and a polynomial-time exact algorithm for the $(k,ell)$-center objective.
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