9 research outputs found
LOCALIS: Locally-adaptive Line Simplification for GPU-based Geographic Vector Data Visualization
Visualization of large vector line data is a core task in geographic and
cartographic systems. Vector maps are often displayed at different cartographic
generalization levels, traditionally by using several discrete levels-of-detail
(LODs). This limits the generalization levels to a fixed and predefined set of
LODs, and generally does not support smooth LOD transitions. However, fast GPUs
and novel line rendering techniques can be exploited to integrate dynamic
vector map LOD management into GPU-based algorithms for locally-adaptive line
simplification and real-time rendering. We propose a new technique that
interactively visualizes large line vector datasets at variable LODs. It is
based on the Douglas-Peucker line simplification principle, generating an
exhaustive set of line segments whose specific subsets represent the lines at
any variable LOD. At run time, an appropriate and view-dependent error metric
supports screen-space adaptive LOD levels and the display of the correct subset
of line segments accordingly. Our implementation shows that we can simplify and
display large line datasets interactively. We can successfully apply line style
patterns, dynamic LOD selection lenses, and anti-aliasing techniques to our
line rendering
On Optimal Polyline Simplification using the Hausdorff and Fréchet Distance
We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given epsilon>0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most epsilon. We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than epsilon. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of n vertices under the Fréchet distance, we give an O(kn^5) time algorithm that requires O(kn^2) space, where k is the output complexity of the simplification
On Optimal Polyline Simplification using the Hausdorff and Fréchet Distance
We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given epsilon>0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most epsilon. We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than epsilon. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of n vertices under the Fréchet distance, we give an O(kn^5) time algorithm that requires O(kn^2) space, where k is the output complexity of the simplification
On Optimal Polyline Simplification using the Hausdorff and Fréchet Distance
We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given epsilon>0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most epsilon. We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than epsilon. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of n vertices under the Fréchet distance, we give an O(kn^5) time algorithm that requires O(kn^2) space, where k is the output complexity of the simplification
On optimal polyline simplification using the Hausdorff and Fréchet distance
We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given ε > 0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most ε.We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than we require for the optimum solution. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of n vertices under the (continuous) Fréchet distance, we give an time algorithm that requires space, where is the output complexity of the simplification
Fr\'echet Distance for Uncertain Curves
In this paper we study a wide range of variants for computing the (discrete
and continuous) Fr\'echet 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 Fr\'echet distance, which are the minimum and maximum Fr\'echet distance
for any realisations of the curves.
We prove that both the upper and lower bound problems are NP-hard for the
continuous Fr\'echet distance in several uncertainty models, and that the upper
bound problem remains hard for the discrete Fr\'echet distance. In contrast,
the lower bound (discrete and continuous) Fr\'echet distance can be computed in
polynomial time. Furthermore, we show that computing the expected discrete
Fr\'echet 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 Fr\'echet 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 is polynomially
bounded, where is the Fr\'echet distance and 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 -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 Fr\'echet distance for uncertainty regions modelled as
point sets.Comment: 48 pages, 11 figures. This is the full version of the paper to be
published in ICALP 202