Approximations of the Wiener sausage and its curvature measures

A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.Comment: Published in at http://dx.doi.org/10.1214/09-AAP596 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Functional limit theorems for certain intrinsic volumes of excursion sets of random fields

In the first part of this thesis we develop a method to compute all d+1 intrinsic volumes multigrid convergently for the class of unions of compact sets with positive reach given a digitization. In the second part we prove two functional limit theorems for the volume of excursion sets and for the (d-1)-dimensional Hausdorff measure of level sets of random fields, respectively. As a corollary, we prove large deviation results and propose an asymptotical significance test

ERROR BOUNDS FOR SURFACE AREA ESTIMATORS BASED ON CROFTON’S FORMULA

According to Crofton's formula, the surface area S(A) of a sufficiently regular compact set A in Rd is proportional to the mean of all total projections pA (u) on a linear hyperplane with normal u, uniformly averaged over all unit vectors u. In applications, pA (u) is only measured in k directions and the mean is approximated by a finite weighted sum bS(A) of the total projections in these directions. The choice of the weights depends on the selected quadrature rule. We define an associated zonotope Z (depending only on the projection directions and the quadrature rule), and show that the relative error bS (A)/S (A) is bounded from below by the inradius of Z and from above by the circumradius of Z. Applying a strengthened isoperimetric inequality due to Bonnesen, we show that the rectangular quadrature rule does not give the best possible error bounds for d =2. In addition, we derive asymptotic behavior of the error (with increasing k) in the planar case. The paper concludes with applications to surface area estimation in design-based digital stereology where we show that the weights due to Bonnesen's inequality are better than the usual weights based on the rectangular rule and almost optimal in the sense that the relative error of the surface area estimator is very close to the minimal error

Regulation-Oriented Filtering in Web-Based Air Traffic Exploration

Airspace route planning relies on many regulations and individual factors that can be hard to understand for audiences without advanced domain knowledge. This aspect is problematic if regulations are discussed in complex debates about changing air traffic distributions, affecting the broad public in negative and positive ways. To increase accessibility and transparency, we propose a regulation-oriented scheme of trajectory filters that includes a fully automated detection component for regulation deviations. The scheme further includes filters by daytime, custom areas, MTOM, and is part of a client-independent web prototype. In this publication, we specify details on individual filters and their inter- play (1st contribution), while putting a particular emphasis on the deviation detector (2nd contribution)

Regulation-Oriented Filtering in Web-Based Air Traffic Exploration

Airspace route planning relies on many regulations and individual factors that can be hard to understand for audiences without advanced domain knowledge. This aspect is problematic if regulations are discussed in complex debates about changing air traffic distributions, affecting the broad public in negative and positive ways. To increase accessibility and transparency, we propose a regulation-oriented scheme of trajectory filters that includes a fully automated detection component for regulation deviations. The scheme further includes filters by daytime, custom areas, MTOM, and is part of a client-independent web prototype. In this publication, we specify details on individual filters and their inter- play (1st contribution), while putting a particular emphasis on the deviation detector (2nd contribution).publishe

CHARACTERIZATION OF MAMMARY GLAND TISSUE USING JOINT ESTIMATORS OF MINKOWSKI FUNCTIONALS

A theoretical approach to estimate the Minkowski functionals, i.e., area fraction, specifc boundary length and specifc Euler number in 2D, and their asymptotic covariance matrix proposed by Spodarev and Schmidt (2005) and Pantle et al. (2006a;b) is applied to real image data. These two-dimensional images show mammary gland tissue and should be classifed automatically as tumor-free or mammary cancer, respectively. The estimation procedure is illustrated step-by-step and the calculations are described in detail. To reduce dependencies from chosen parameters, a least-squares approach is considered as recommended by Klenk et al. (2006). Emphasis is placed on the detailed description of the estimation procedure and the application of the theory to real image data

MultiSegVA : Using Visual Analytics to Segment Biologging Time Series on Multiple Scales

Segmenting biologging time series of animals on multiple temporal scales is an essential step that requires complex techniques with careful parameterization and possibly cross-domain expertise. Yet, there is a lack of visual-interactive tools that strongly support such multi-scale segmentation. To close this gap, we present our MultiSegVA platform for interactively defining segmentation techniques and parameters on multiple temporal scales. MultiSegVA primarily contributes tailored, visual-interactive means and visual analytics paradigms for segmenting unlabeled time series on multiple scales. Further, to flexibly compose the multi-scale segmentation, the platform contributes a new visual query language that links a variety of segmentation techniques. To illustrate our approach, we present a domain-oriented set of segmentation techniques derived in collaboration with movement ecologists. We demonstrate the applicability and usefulness of MultiSegVA in two real-world use cases from movement ecology, related to behavior analysis after environment-aware segmentation, and after progressive clustering. Expert feedback from movement ecologists shows the effectiveness of tailored visual-interactive means and visual analytics paradigms at segmenting multi-scale data, enabling them to perform semantically meaningful analyses. A third use case demonstrates that MultiSegVA is generalizable to other domains.publishe

A Comparative Study of Orientation Support Tools in Virtual Reality Environments with Virtual Teleportation

Movement-compensating interactions like teleportation are commonly deployed techniques in virtual reality environments. Although practical, they tend to cause disorientation while navigating. Previous studies show the effectiveness of orientation-supporting tools, such as trails, in reducing such disorientation and reveal different strengths and weaknesses of individual tools. However, to date, there is a lack of a systematic comparison of those tools when teleportation is used as a movement-compensating technique, in particular under consideration of different tasks. In this paper, we compare the effects of three orientation-supporting tools, namely minimap, trail, and heatmap. We conducted a quantitative user study with 48 participants to investigate the accuracy and efficiency when executing four exploration and search tasks. As dependent variables, task performance, completion time, space coverage, amount of revisiting, retracing time, and memorability were measured. Overall, our results indicate that orientation-supporting tools improve task completion times and revisiting behavior. The trail and heatmap tools were particularly useful for speed-focused tasks, minimal revisiting, and space coverage. The minimap increased memorability and especially supported retracing tasks. These results suggest that virtual reality systems should provide orientation aid tailored to the specific tasks of the users.publishe

Tiled Stereoscopic 3D Display Wall - Concept, Applications and Evaluation

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