Estimating and quantifying uncertainties on level sets using the Vorob'ev expectation and deviation with Gaussian process models

Several methods based on Kriging have recently been proposed for calculating a probability of failure involving costly-to-evaluate functions. A closely related problem is to estimate the set of inputs leading to a response exceeding a given threshold. Now, estimating such a level set—and not solely its volume—and quantifying uncertainties on it are not straightforward. Here we use notions from random set theory to obtain an estimate of the level set, together with a quantification of estimation uncertainty. We give explicit formulae in the Gaussian process set-up and provide a consistency result. We then illustrate how space-filling versus adaptive design strategies may sequentially reduce level set estimation uncertainty.

Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

Association of optical tracklets from a geosynchronous belt survey via the direct Bayesian admissible region approach

The direct Bayesian admissible region approach is an a priori state free measurement association and initial orbit determination technique for optical tracks. In this paper, we test a hybrid approach that appends a least squares estimator to the direct Bayesian method on measurements taken at the Zimmerwald Observatory of the Astronomical Institute at the University of Bern. Over half of the association pairs agreed with conventional geometric track correlation and least squares techniques. The remaining pairs cast light on the fundamental limits of conducting tracklet association based solely on dynamical and geometrical information.

Automated 3D Lumbar Intervertebral Disc Segmentation from MRI Data Sets

This paper proposed an automated 3D lumbar intervertebral disc (IVD) segmentation strategy from MRI data. Starting from two user supplied landmarks, the geometrical parameters of all lumbar vertebral bodies and intervertebral discs are automatically extracted from a mid-sagittal slice using a graphical model based approach. After that, a three-dimensional (3D) variable-radius soft tube model of the lumbar spine column is built to guide the 3D disc segmentation. The disc segmentation is achieved as a multi-kernel diffeomorphic registration between a 3D template of the disc and the observed MRI data. Experiments on 15 patient data sets showed the robustness and the accuracy of the proposed algorithm.

Von Fuzzy-Sets zu Computing-with-Words

Dieser Artikel bietet einen Überblick über die Entwicklung und Zusammenhänge der einzelnen Elemente der Fuzzy-Logik, wovon Fuzzy-Set-Theorie die Grundlage bildet. Die Grundproblematik besteht in der Handhabung von linguistischen Informationen, die häufig durch Ungenauigkeit gekennzeichnet sind. Die verschiedenen technischen Anwendungen von Fuzzy-Logik bieten eine Möglichkeit, intelligentere Computersysteme zu konstruieren, die mit unpräzisen Informationen umgehen können. Solche Systeme sind Indizien für die Entstehung einer neuen Ära des Cognitive-Computing, di in diesemArtikel ebenfalls zur Sprache kommt. Für das bessere Verständnis wird der Artikel mit einem Beispiel aus der Meteorologie (d. h. Schnee in Adelboden) begleitet.

Regularity conditions in the realisability problem in applications to point processes and random closed sets

We study existence of random elements with partially specified distributions. The technique relies on the existence of a positive ex-tension for linear functionals accompanied by additional conditions that ensure the regularity of the extension needed for interpreting it as a probability measure. It is shown in which case the extens ion can be chosen to possess some invariance properties. The results are applied to the existence of point processes with given correlation measure and random closed sets with given two-point covering function or contact distribution function. It is shown that the regularity condition can be efficiently checked in many cases in order to ensure that the obtained point processes are indeed locally finite and random sets have closed realisations.

Continued fractions built from convex sets and convex functions

In a partially ordered semigroup with the duality (or polarity) transform, it is pos- sible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

3D documentation and visualization of external injury findings by integration of simple photography in CT/MRI data sets (IprojeCT).

This study evaluated the feasibility of documenting patterned injury using three dimensions and true colour photography without complex 3D surface documentation methods. This method is based on a generated 3D surface model using radiologic slice images (CT) while the colour information is derived from photographs taken with commercially available cameras. The external patterned injuries were documented in 16 cases using digital photography as well as highly precise photogrammetry-supported 3D structured light scanning. The internal findings of these deceased were recorded using CT and MRI. For registration of the internal with the external data, two different types of radiographic markers were used and compared. The 3D surface model generated from CT slice images was linked with the photographs, and thereby digital true-colour 3D models of the patterned injuries could be created (Image projection onto CT/IprojeCT). In addition, these external models were merged with the models of the somatic interior. We demonstrated that 3D documentation and visualization of external injury findings by integration of digital photography in CT/MRI data sets is suitable for the 3D documentation of individual patterned injuries to a body. Nevertheless, this documentation method is not a substitution for photogrammetry and surface scanning, especially when the entire bodily surface is to be recorded in three dimensions including all external findings, and when precise data is required for comparing highly detailed injury features with the injury-inflicting tool.