Harmonious Simplification of Isolines

Current techniques for simplification focus on reducing complexity while maintaining the geometric similarity to the input. For isolines that jointly describe a scalar field, however, we postulate that geometric similarity of each isoline separately is not sufficient. Rather, we need to maintain the harmony between these isolines to make them visually relate and describe the structures of the underlying terrain. Based on principles of manual cartography, we propose an algorithm for simplifying isolines while considering harmony explicitly. Our preliminary visual and quantitative results suggest that our algorithm is effective

Uncertain Curve Simplification

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 Fr\'echet distance. For both these distance measures, we present polynomial-time algorithms for this problem.Comment: 25 pages, 5 figure

On kk-means for segments and polylines

We study the problem of k-means clustering in the space of straight-line segments in R2 under the Hausdorff distance. For this problem, we give a (1+ϵ)-approximation algorithm that, for an input of n segments, for any fixed k, and with constant success probability, runs in time O(n+ϵ−O(k)+ϵ−O(k)⋅logO(k)(ϵ−1)). The algorithm has two main ingredients. Firstly, we express the k-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron~\cite{Vigneron14} to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg~\cite{Feldman11, Feldman2020}. Our results can be extended to polylines of constant complexity with a running time of O(n+ϵ−O(k))

On k-means for segments and polylines

We study the problem of k-means clustering in the space of straight-line segments in R² under the Hausdorff distance. For this problem, we give a (1 + ε)-approximation algorithm that, for an input of n segments, for any fixed k, and with constant success probability, runs in time O(n + ε−O(k) + ε−O(k)· logO(k)(ε−1)). The algorithm has two main ingredients. Firstly, we express the k-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron [40] to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg [21, 22]. Our results can be extended to polylines of constant complexity with a runningtime of O(n + ε−O(k))

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