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

Modeling 2D extended targets with star-convex Random Hypersurface Models (RHMs) allows for accurate object pose and shape estimation. A star-convex RHM models the shape of an object with the aid of a radial function that describes the distance from the object center to any point on its boundary. However, up to now only linear estimators, i.e., Kalman Filters, are used due to the lack of a explicit likelihood function. In this paper, we propose a closed-form and easy to implement likelihood function for tracking extended targets with star-convex RHMs. This makes it possible to apply nonlinear estimators such as Particle Filters to estimate a detailed shape of a target.We compared the proposed likelihood against the usual Kalman filter approaches with tracking pose and shape of an airplane in 2D. The evaluations showed that the combination of the Progressive Gaussian Filter (PGF) and the new likelihood function delivers the best estimation performance and can outperform the usually employed Kalman Filters.

A closed-form likelihood for Particle Filters to track extended objects with star-convex RHMs

The information filter has evolved into a key tool for distributed and decentralized multisensor estimation and control. Essentially, it is an algebraical reformulation of the Kalman filter and provides estimates on the information about an uncertain state rather than on a state itself. Whereas many practicable Kalman filtering techniques for nonlinear system and sensor models have been developed, approaches towards nonlinear information filtering are still scarce and limited. In order to deal with nonlinear systems and sensors, this paper derives an approximation technique for arbitrary probability densities that provides the same distributable fusion structure as the linear information filter. The presented approach not only constitutes a nonlinear version of the information filter, but it also points the direction to a Hilbert space structure on probability densities, whose vector space operations correspond to the fusion and weighting of information.

/pdf/nonlinear-information-filtering-for-distributed-multisensor-4t2tr2g7c0.pdf

Nonlinear information filtering for distributed multisensor data fusion

The problem of estimating a vector x_ of unknown quantities based on a set of measurements depending nonlinearly on x_ is considered. The measurements are assumed to be taken sequentially and are corrupted by unknown but bounded uncertainties. For this uncertainty model, a systematic design approach is introduced, which yields closed-form expressions for the desired nonlinear estimates. The estimates are recursively calculated and provide solution sets /spl chi/ containing the feasible sets, i.e., the sets of all x_ consistent with all the measurements available and their associated bounds. The sets /spl chi/ are tight upper bounds for the exact feasible sets and are in general not convex and not connected. The proposed design approach is versatile and the resulting nonlinear filter algorithms are both easy to implement and efficient.

Recursive nonlinear set-theoretic estimation based on pseudo ellipsoids

We propose a geometry-driven deterministic sampling method for Bingham distributions in arbitrary dimensions. With flexibly adjustable sampling sizes, the novel scheme can generate equally weighted samples that satisfy requirements of the unscented transform and approximate higher-order shape information of the Bingham distribution. By leveraging retraction techniques from Riemannian geometry, the sigma points are constrained to preserve the second-order moment. Meanwhile, samples in each principal direction are located in a way that minimizes a distance measure between the on-tangent-plane Dirac mixtures and the underlying on-manifold density. For that, the modified Cramer-von Mises distance based on the localized cumulative distribution (LCD) is employed. We further integrate the proposed approach into a quaternion-based orientation estimation framework. Compared to the existing unscented sampling approach drawing only fixed and limited numbers of sigma points, simulation results show that the proposed scheme enables better accuracy and robustness for nonlinear Bingham filtering.

Geometry-driven Deterministic Sampling for Nonlinear Bingham Filtering

Telepresence aims at giving a human user the impression of being present in a remote environment. However, the user is actually situated in a user environment and his motion is tracked. A mobile teleoperator in the remote environment replicates this motion. The user can thus control the mobile teleoperator's locomotion by walking. A stereo-camera system mounted on the mobile teleoperator constantly records live camera images and transfers them to the user environment, where they are presented to the user on a head-mounted display. This paper presents a long distance experiment, in which a mobile teleoperator was controlled over a standard Internet connection by natural locomotion. Without further processing of the user's motion data, however, only exploration of a remote environment of the same size or smaller than the user environment is possible. As this is not desirable, we use motion compression, an optimal nonlinear transformation of the user's path. This algorithm allows controlling free motion in an arbitrarily large target environment from a limited user environment.

Uwe D. Hanebeck

Papers

A closed-form likelihood for Particle Filters to track extended objects with star-convex RHMs

Nonlinear information filtering for distributed multisensor data fusion

Recursive nonlinear set-theoretic estimation based on pseudo ellipsoids

Geometry-driven Deterministic Sampling for Nonlinear Bingham Filtering

Motion compression applied to guidance of a mobile teleoperator