There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

The simultaneous localization and map building (SLAM) problem asks if it is possible for an autonomous vehicle to start in an unknown location in an unknown environment and then to incrementally build a map of this environment while simultaneously using this map to compute absolute vehicle location. Starting from estimation-theoretic foundations of this problem, the paper proves that a solution to the SLAM problem is indeed possible. The underlying structure of the SLAM problem is first elucidated. A proof that the estimated map converges monotonically to a relative map with zero uncertainty is then developed. It is then shown that the absolute accuracy of the map and the vehicle location reach a lower bound defined only by the initial vehicle uncertainty. Together, these results show that it is possible for an autonomous vehicle to start in an unknown location in an unknown environment and, using relative observations only, incrementally build a perfect map of the world and to compute simultaneously a bounded estimate of vehicle location. The paper also describes a substantial implementation of the SLAM algorithm on a vehicle operating in an outdoor environment using millimeter-wave radar to provide relative map observations. This implementation is used to demonstrate how some key issues such as map management and data association can be handled in a practical environment. The results obtained are cross-compared with absolute locations of the map landmarks obtained by surveying. In conclusion, the paper discusses a number of key issues raised by the solution to the SLAM problem including suboptimal map-building algorithms and map management.

/pdf/a-solution-to-the-simultaneous-localization-and-map-building-2pp3i7uhdj.pdf

A solution to the simultaneous localization and map building (SLAM) problem

Fusing two random vectors is simple, when they are characterized by continuous probability density functions. According to Bayes' law, fusion then consists of multiplying the two densities. When only empirical distributions are given and a resulting empirical distribution is desired, Bayes' law is no longer applicable. Obviously, fusion could now be performed by reconstructing the underlying continuous densities, subsequent multiplication, and sampling of the result. As this is overly complicated, our goal is to perform a direct Bayesian fusion of the two given empirical distributions.We devise a generalized multiplication procedure that mutually reweights appropriate points of one density by local density values of the other density. The density values are efficiently estimated locally by nearest neighbor operations. The method is symmetric in the sense that it uses points from both densities.

Bayesian fusion of empirical distributions based on local density reconstruction

Abstract Optical belt sorters are a versatile means to sort bulk materials. In previous work, we presented a novel design of an optical belt sorter, which includes an area scan camera instead of a line scan camera. Line scan cameras, which are well-established in optical belt sorting, only allow for a single observation of each particle. Using multitarget tracking, the data of the area scan camera can be used to derive a part of the trajectory of each particle. The knowledge of the trajectories can be used to generate accurate predictions as to when and where each particle passes the separation mechanism. Accurate predictions are key to achieve high quality sorting results. The accuracy of the trajectories and the predictions heavily depends on the motion model used. In an evaluation based on a simulation that provides us with ground truth trajectories, we previously identified a bias in the temporal component of the prediction. In this paper, we analyze the simulation-based ground truth data of the motion of different bulk materials and derive models specifically tailored to the generation of accurate predictions for particles traveling on a conveyor belt. The derived models are evaluated using simulation data involving three different bulk materials. The evaluation shows that the constant velocity model and constant acceleration model can be outperformed by utilizing the similarities in the motion behavior of particles of the same type.

/pdf/predictive-tracking-with-improved-motion-models-for-optical-38zlhuj5p9.pdf

Predictive tracking with improved motion models for optical belt sorting

We assume that a finite set of moments of a random vector is given. Its underlying density is unknown. An algorithm is proposed for efficiently calculating Dirac mixture densities maintaining these moments while providing a homogeneous coverage of the state space.

Truncated Moment Problem for Dirac Mixture Densities with Entropy Regularization.

In this paper, a novel measurement model based on spherical double Fourier series (DFS) for estimating the 3D shape of a target concurrently with its kinematic state is introduced. Here, the shape is represented as a star-convex radial function, decomposed as spherical DFS. In comparison to ordinary DFS, spherical DFS do not suffer from ambiguities at the poles. Details will be given in the paper. The shape representation is integrated into a Bayesian state estimator framework via a measurement equation. As range sensors only generate measurements from the target side facing the sensor, the shape representation is modified to enable application of shape symmetries during the estimation process. The model is analyzed in simulations and compared to a shape estimation procedure using spherical harmonics. Finally, shape estimation using spherical and ordinary DFS is compared to analyze the effect of the pole problem in extended object tracking (EOT) scenarios.

Shape Estimation and Tracking using Spherical Double Fourier Series for Three-Dimensional Range Sensors

This work presents a filter for estimating the state of nonlinear dynamic systems. It is based on optimal recursive approximation the state densities by means of Dirac mixture functions in order to allow for a closed form solution of the prediction and filter step. The approximation approach is based on a systematic minimization of a distance measure and is hence optimal and deterministic. In contrast to non-deterministic methods we are able to determine the optimal number of components in the Dirac mixture. A further benefit of the proposed approach is the consideration of measurements during the approximation process in order to avoid parameter degradation.

Uwe D. Hanebeck

Papers

Bayesian fusion of empirical distributions based on local density reconstruction

Predictive tracking with improved motion models for optical belt sorting

Truncated Moment Problem for Dirac Mixture Densities with Entropy Regularization.

Shape Estimation and Tracking using Spherical Double Fourier Series for Three-Dimensional Range Sensors

Dirac Mixture Approximation for Nonlinear Stochastic Filtering