54 research outputs found
Visual analytics of delays and interaction in movement data
Maximilian Konzack, Tim Ophelders, Michel A. Westenberg and Kevin Buchin are supported by the Netherlands Organisation for Scientific Research (NWO) under grant no. 612.001.207 (Maximilian Konzack, Michel A. Westenberg and Kevin Buchin) and grant no. 639.023.208 (Tim Ophelders).The analysis of interaction between movement trajectories is of interest for various domains when movement of multiple objects is concerned. Interaction often includes a delayed response, making it difficult to detect interaction with current methods that compare movement at specific time intervals. We propose analyses and visualizations, on a local and global scale, of delayed movement responses, where an action is followed by a reaction over time, on trajectories recorded simultaneously. We developed a novel approach to compute the global delay in subquadratic time using a fast Fourier transform (FFT). Central to our local analysis of delays is the computation of a matching between the trajectories in a so-called delay space. It encodes the similarities between all pairs of points of the trajectories. In the visualization, the edges of the matching are bundled into patches, such that shape and color of a patch help to encode changes in an interaction pattern. To evaluate our approach experimentally, we have implemented it as a prototype visual analytics tool and have applied the tool on three bidimensional data sets. For this we used various measures to compute the delay space, including the directional distance, a new similarity measure, which captures more complex interactions by combining directional and spatial characteristics. We compare matchings of various methods computing similarity between trajectories. We also compare various procedures to compute the matching in the delay space, specifically the Fréchet distance, dynamic time warping (DTW), and edit distance (ED). Finally, we demonstrate how to validate the consistency of pairwise matchings by computing matchings between more than two trajectories.Publisher PDFPeer reviewe
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
Computing the Fréchet Distance with a Retractable Leash
All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet distance between polygonal curves in (Formula presented.) under polyhedral distance functions (e.g., (Formula presented.) and (Formula presented.)). We also get a (Formula presented.)-approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed (Formula presented.). For the exact Euclidean case, our framework currently yields an algorithm with running time (Formula presented.). However, we conjecture that it may eventually lead to a faster exact algorithm
Abstract Morphing Using the Hausdorff Distance and Voronoi Diagrams
This paper introduces two new abstract morphs for two 2-dimensional shapes. The intermediate shapes gradually reduce the Hausdorff distance to the goal shape and increase the Hausdorff distance to the initial shape. The morphs are conceptually simple and apply to shapes with multiple components and/or holes. We prove some basic properties relating to continuity, containment, and area. Then we give an experimental analysis that includes the two new morphs and a recently introduced abstract morph that is also based on the Hausdorff distance [Van Kreveld et al., 2022]. We show results on the area and perimeter development throughout the morph, and also the number of components and holes. A visual comparison shows that one of the new morphs appears most attractive
A discrete framework to find the optimal matching between manifold-valued curves
The aim of this paper is to find an optimal matching between manifold-valued
curves, and thereby adequately compare their shapes, seen as equivalent classes
with respect to the action of reparameterization. Using a canonical
decomposition of a path in a principal bundle, we introduce a simple algorithm
that finds an optimal matching between two curves by computing the geodesic of
the infinite-dimensional manifold of curves that is at all time horizontal to
the fibers of the shape bundle. We focus on the elastic metric studied in the
so-called square root velocity framework. The quotient structure of the shape
bundle is examined, and in particular horizontality with respect to the fibers.
These results are more generally given for any elastic metric. We then
introduce a comprehensive discrete framework which correctly approximates the
smooth setting when the base manifold has constant sectional curvature. It is
itself a Riemannian structure on the product manifold of "discrete curves"
given by a finite number of points, and we show its convergence to the
continuous model as the size of the discretization goes to infinity.
Illustrations of optimal matching between discrete curves are given in the
hyperbolic plane, the plane and the sphere, for synthetic and real data, and
comparison with dynamic programming is established
Benjamini-Schramm Convergence of Normalized Characteristic Numbers of Riemannian Manifolds
We study a weak form of Gromov-Hausdorff convergence
of Riemannian manifolds, also known as
Benjamini-Schramm convergence. This concept
is also applicable to other areas and has widely
been studied in the context of graphs.
The main result is the continuity of characteristic
numbers normalized by the volume with respect
to the Benjamini-Schramm topology on the class
of Riemannian manifolds with a uniform lower
bound on injectivity radius and Ricci curvature.
An immediate consequence is a comparison theorem
that gives for any characteristic number a
linear bound in terms of the volume on the entire
class of manifolds mentioned. We give another
interpretation of the result showing that characteristic
numbers can be reconstructed with some
accuracy from local random information
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