Source: https://www.bennoack.net/publications/
Timestamp: 2019-04-26 00:37:35+00:00

Document:
S. Radtke, B. Noack, and U. D. Hanebeck, Distributed Estimation with Unequal State Vectors using Deterministic Sample-based Fusion (to appear), in Proceedings of the 2019 European Control Conference (ECC 2019), Naples, Italy, Jun. 2019.
K. Li, D. Frisch, B. Noack, and U. Hanebeck, Geometry-Driven Deterministic Sampling for Nonlinear Bingham Filtering (to appear), in Proceedings of the 2019 European Control Conference (ECC 2019), Naples, Italy, Jun. 2019.
S. Özgen, S. Kohn, B. Noack, and U. D. Hanebeck, State Estimation with Model-Mismatch-Based Secrecy against Eavesdroppers (to appear), in Proceedings of the 2019 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2019), Taipei, Taiwan, May 2019.
S. Radtke, K. Li, B. Noack, and U. D. Hanebeck, Comparative Study of Track-to-Track Fusion Methods for Cooperative Tracking with Bearing-only Measurements (to appear), in Proceedings of the 2019 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2019), Taipei, Taiwan, May 2019.
C. Pieper, F. Pfaff, G. Maier, H. Kruggel-Emden, S. Wirtz, B. Noack, R. Gruna, V. Scherer, U. D. Hanebeck, T. Längle, and J. Beyerer, Numerical Modelling of an Optical Belt Sorter Using a DEM–CFD Approach Coupled with Particle Tracking and Comparison with Experiments, Powder Technology, vol. 370, pp. 181–193, Dec. 2018.
Abstract — State-of-the-art optical sorting systems suffer from delays between the particle detection and separation stage, during which the material movement is not accounted for. Commonly line scan cameras, using simple assumptions to predict the future particle movement, are employed. In this study, a novel prediction approach is presented, where an area scan camera records the particle movement over multiple time steps and a tracking algorithm is used to reconstruct the corresponding paths to determine the time and position at which the material reaches the separation stage. In order to assess the benefit of such a model at different operating parameters, an automated optical belt sorter is numerically modelled and coupled with the tracking procedure. The Discrete Element Method (DEM) is used to describe the particle–particle as well as particle–wall interactions, while the air nozzles required for deflecting undesired material fractions are described with Computational Fluid Dynamics (CFD). The accuracy of the employed numerical approach is ensured by comparing the separation results of a predefined sorting task with experimental investigations. The quality of the aforementioned prediction models is compared when utilizing different belt lengths, nozzle activation durations, particle types, sampling frequencies and detection windows. Results show that the numerical model of the optical belt sorter is able to accurately describe the sorting system and is suitable for detailed investigation of various operational parameters. The proposed tracking prediction model was found to be superior to the common line scan camera method in all investigated scenarios. Its advantage is especially profound when difficult sorting conditions, e.g. short conveyor belt lengths or uncooperative moving bulk solids, apply.
K. Li, D. Frisch, S. Radtke, B. Noack, and U. D. Hanebeck, Wavefront Orientation Estimation Based on Progressive Bingham Filtering, in Proceedings of the IEEE ISIF Workshop on Sensor Data Fusion: Trends, Solutions, Applications (SDF 2018), Oct. 2018.
Abstract — In this paper, we propose the Progressive Bingham Filter (PBF), a novel stochastic filtering algorithm for nonlinear spatial orientation estimation. As an extension of the orientation filter previously proposed only for the identity measurement model based on the Bingham distribution, our method is able to handle arbitrary measurement models. Instead of the sampling- approximation scheme used in the prediction step, a closed- form solution is possible when the system equation is based on the Hamilton product. Besides stochastic approaches, we also introduce the Spherical Averaging Method (SAM), which is an application of the Riemannian averaging technique. The two approaches are then applied to a specific problem where the wavefront orientation is estimated based on Time Differences of Arrival (TDOA) and evaluated in simulations. The results show theoretical competitiveness of the PBF.
S. Özgen, U. D. Hanebeck, B. Noack, M. Huber, F. Rosenthal, and J. Mayer, Retrodiction of Data Association Probabilities via Convex Optimization, in Proceedings of the 21st International Conference on Information Fusion (Fusion 2018), Cambridge, United Kingdom, Jul. 2018.
Abstract — In a surveillance environment with high clutter, finding the correct measurement to track associations becomes extremely important for efficient target tracking. This study offers a novel algorithm to retrodict the data association probabilities at any past time instant, when the batch set of measurements is kept in memory. For the retrodiction procedure, the batch association cost is first written explicitly as a binary integer optimization problem with a quadratic cost function and it is shown that the relaxed form of the problem is convex. From the relaxed problem, a lower bound for the optimal association cost is derived, and this lower bound is used as the data association probabilities pertaining to that selected time instant in the past. Due to its consideration of the batch set of data in a retrospective manner, we will call this algorithm as Retrodictive Probabilistic Data Association, RPDA. For simplification of the mathematical analysis, a single point target with no missing measurements, i.e., PD = 1, is taken into account.
S. Radtke, B. Noack, U. D. Hanebeck, and O. Straka, Reconstruction of Cross-Correlations with Constant Number of Deterministic Samples, in Proceedings of the 21st International Conference on Information Fusion (Fusion 2018), Cambridge, United Kingdom, Jul. 2018.
Abstract — Optimal fusion of estimates that are computed in a distributed fashion is a challenging task. In general, the sensor nodes cannot keep track of the cross-correlations required to fuse estimates optimally. In this paper, a novel technique is presented that provides the means to reconstruct the required correlation structure. For this purpose, each node computes a set of deterministic samples that provides all the information required to reassemble the cross-covariance matrix for each pair of estimates. As the number of samples is increasing over time, a method to reduce the size of the sample set is presented and studied. In doing so, communication expenses can be reduced significantly, but approximation errors are possibly introduced by neglecting past correlation terms. In order to keep approximation errors at a minimum, an appropriate set size can be determined and a trade-off between communication expenses and estimation quality can be found.
M. Aristov, B. Noack, U. D. Hanebeck, and J. Müller-Quade, Encrypted Multisensor Information Filtering, in Proceedings of the 21st International Conference on Information Fusion (Fusion 2018), Cambridge, United Kingdom, Jul. 2018.
Abstract — With the advent of cheap sensor technology, multisensor data fusion algorithms have been becoming a key enabler for efficient in-network processing of sensor data. The information filter, in particular, has proven useful due to its simple additive structure of the measurement update equations. In order to exploit this structure for an efficient in-network processing, each node in the network is supposed to locally process and combine data from its neighboring nodes. The aspired in-network processing, at first glance, prohibits efficient privacy-preserving communication protocols, and encryption schemes that allow for algebraic manipulations are often computationally too expensive. Partially homomorphic encryption schemes constitute far more practical solutions but are restricted to a single algebraic operation on the corresponding ciphertexts. In this paper, an additive-homomorphic encryption scheme is used to derive a privacy-preserving implementation of the information filter where additive operations are sufficient to distribute the workload among the sensor nodes. However, the encryption scheme requires the floating-point data to be quantized, which impairs the estimation quality. The proposed filter and the implications of the necessary quantization are analyzed in a simulated multisensor tracking scenario.
F. Rosenthal, B. Noack, and U. D. Hanebeck, State Estimation in Networked Control Systems with Delayed and Lossy Acknowledgments, in Multisensor Fusion and Integration in the Wake of Big Data, Deep Learning and Cyber Physical System, S. Lee, H. Ko, and S. Oh, Eds. Cham: Springer International Publishing, Jul. 2018, pp. 22–38.
Abstract — In this article, we are concerned with state estimation in Networked Control Systems where both control inputs and measurements are transmitted over networks which are lossy and introduce random transmission delays. We focus on the case where acknowledgment packets transmitted by the actuator upon reception of applicable control inputs are also subject to delays and losses, as opposed to the common notion of TCP-like communication where successful transmissions are acknowledged instantaneously and without losses. As a consequence, the state estimator in the considered setup has only partial and belated knowledge concerning the actually applied control inputs which results in additional uncertainty. We derive an estimator by extending an existing approach for the special case of UDP-like communication which maintains estimates of the applied control inputs that are incorporated into the estimation of the plant state. The presented estimator is compared to the original approach in terms of Monte Carlo simulations where its increased robustness towards imperfect knowledge of the underlying networks is indicated.
J. Duník, O. Straka, B. Noack, J. Steinbring, and U. D. Hanebeck, On Directional Splitting of Gaussian Density in Nonlinear Random Variable Transformation, IET Signal Processing, Jul. 2018.
Abstract — Transformation of a random variable is a common need in a design of many algorithms in signal processing, automatic control, and fault detection. Typically, the design is tied to an assumption on a probability density function of the random variable, often in the form of the Gaussian distribution. The assumption may be, however, difficult to be met in algorithms involving nonlinear transformation of the random variable. This paper focuses on techniques capable to ensure validity of the Gaussian assumption of the nonlinearly transformed Gaussian variable by approximating the to-be-transformed random variable distribution by a Gaussian mixture distribution. The stress is laid on an analysis and selection of design parameters of the approximate Gaussian mixture distribution to minimise the error imposed by the nonlinear transformation such as the location and number of the Gaussian mixture terms. A special attention is devoted to the definition of the novel Gaussian mixture splitting directions based on the measures of non-Gaussianity. The proposed splitting directions are analysed and illustrated in numerical simulations.
F. Rosenthal, B. Noack, and U. D. Hanebeck, Scheduling of Measurement Transmission in Networked Control Systems Subject to Communication Constraints, in Proceedings of the 2018 American Control Conference (ACC 2018), Milwaukee, Wisconsin, USA, Jun. 2018.
Abstract — In this paper, we consider Networked Control Systems where the transmission of sensor data is restricted in terms of a fixed communication budget due to the limited capacity of the underlying network. Therefore, the remote estimator cannot be supplied with measurements every time step, which impacts the accuracy of the estimates and consequently the achievable control performance. In order to trade off estimation accuracy against the costs of measurement transmission, we formulate the considered task as an optimal control problem, so that it fits into the broader class of sensor and measurement scheduling problems. As the optimal solution of such problems is generally intractable, we derive a suboptimal algorithm based on randomized rounding. In two numerical examples, we show that the proposed approach can outperfom a state-of-the-art sensor selection algorithm.
K. Dormann, B. Noack, and U. D. Hanebeck, Optimally Distributed Kalman Filtering with Data-Driven Communication, Sensors, vol. 18, no. 4, Apr. 2018.
Abstract — For multisensor data fusion, distributed state estimation techniques that enable a local processing of sensor data are the means of choice in order to minimize storage and communication costs. In particular, a distributed implementation of the optimal Kalman filter has recently been developed. A significant disadvantage of this algorithm is that the fusion center needs access to each node so as to compute a consistent state estimate, which requires full communication each time an estimate is requested. In this article, different extensions of the optimally distributed Kalman filter are proposed that employ data-driven transmission schemes in order to reduce communication expenses. As a first relaxation of the full-rate communication scheme, it can be shown that each node only has to transmit every second time step without endangering consistency of the fusion result. Also, two data-driven algorithms are introduced that even allow for lower transmission rates, and bounds are derived to guarantee consistent fusion results. Simulations demonstrate that the data-driven distributed filtering schemes can outperform a centralized Kalman filter that requires each measurement to be sent to the center node.
G. Maier, F. Pfaff, C. Pieper, R. Gruna, B. Noack, H. Kruggel-Emden, T. Längle, U. D. Hanebeck, S. Wirtz, V. Scherer, and J. Beyerer, Application of Area-Scan Sensors in Sensor-Based Sorting, in Proceedings of the Eighth Conference on Sensor-Based Sorting & Control 2018 (SBSC 2018), Aachen, Germany, Mar. 2018.
Abstract — In the field of machine vision, sensor-based sorting is an important real-time application that enables the separation of a material feed into different classes. While state-of-the-art systems utilize scanning sensors such as line-scan cameras, advances in sensor technology have made application of area scanning sensors feasible. Provided a sufficiently high frame rate, objects can be observed at multiple points in time. By applying multiobject tracking, information about the objects contained in the material stream can be fused over time. Based on this information, our approach further allows predicting the position of each object for future points in time. While conventional systems typically apply a global, rather simple motion model, our approach includes an individual motion model for each object, which in turn allows estimating the point in time as well as the position when reaching the separation stage. In this contribution, we present results from our collaborative research project and summarize the present advances by discussing the potential of the application of area-scan sensors for sensor-based sorting. Among others, we introduce our simulation-driven approach and present results for physical separation efficiency for simulation-generated data, demonstrate the potential of using motion-based features for material classification and discuss real-time related challenges.
F. Rosenthal, B. Noack, and U. D. Hanebeck, State Estimation in Networked Control Systems With Delayed And Lossy Acknowledgments, in Proceedings of the 2017 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2017), Daegu, Korea, Nov. 2017.
Abstract — In this paper, we consider state estimation in Networked Control Systems where both control inputs and measurements are transmitted via networks which are lossy and introduce random transmission delays. In contrast to the common notion of TCP-like communication, where successful transmissions are acknowledged instantaneously and without losses, we focus on the case where the acknowledgment packets provided by the actuator upon reception of applicable control inputs are also subject to delays and losses. Consequently, the estimator has only partial and belated knowledge on the actually applied control inputs, which results in additional uncertainty. We derive an estimator for the considered setup by generalizing an existing approach for UDP-like communication which integrates estimates of the applied control inputs into the overall state estimation. The presented estimator is assessed in terms of Monte Carlo simulations.
F. Pfaff, G. Kurz, C. Pieper, G. Maier, B. Noack, H. Kruggel-Emden, R. Gruna, U. D. Hanebeck, S. Wirtz, V. Scherer, T. Längle, and J. Beyerer, Improving Multitarget Tracking Using Orientation Estimates for Sorting Bulk Materials, in Proceedings of the 2017 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2017), Daegu, Korea, Nov. 2017.
Abstract — Optical belt sorters can be used to sort a large variety of bulk materials. By the use of sophisticated algo- rithms, the performance of the complex machinery can be further improved. Recently, we have proposed an extension to industrial optical belt sorters that involves tracking the individual particles on the belt using an area scan camera. If the estimated behavior of the particles matches the true behavior, the reliability of the separation process can be improved. The approach relies on multitarget tracking using hard association decisions between the tracks and the measurements. In this paper, we propose to include the orientation in the assessment of the compatibility of a track and a measurement. This allows us to achieve more reliable associations, facilitating a higher accuracy of the tracking results.
K. Dormann, B. Noack, and U. D. Hanebeck, Distributed Kalman Filtering With Reduced Transmission Rate, in Proceedings of the 2017 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2017), Daegu, Korea, Nov. 2017.
Abstract — The centralized Kalman filter can be implemented in such a way that the required calculations can be distributed over multiple nodes in a network, each of which processes only the locally acquired sensor data. The main downside of this implementation is that it requires each distributed sensor node to communicate with the fusion center in every time step so as to compute the optimal state estimate. In this paper, two distributed Kalman filtering algorithms are proposed to overcome these limitations. The first algorithm merely requires communication of each local sensor node with the fusion center in every other time step. The second algorithm even allows for a lower communicate rate. Both algorithms apply event-based communication to compute consistent estimates and to reduce the estimation error for a fixed communication rate. Simulations demonstrate that both algorithms perform better in terms of the mean squared estimation error than the centralized Kalman filter.
G. Maier, F. Pfaff, M. Wagner, C. Pieper, R. Gruna, B. Noack, H. Kruggel-Emden, T. Längle, U. D. Hanebeck, S. Wirtz, V. Scherer, and J. Beyerer, Real-Time Multitarget Tracking for Sensor-Based Sorting, Journal of Real-Time Image Processing, Nov. 2017.
Abstract — Utilizing parallel algorithms is an established way of increasing performance in systems that are bound to real-time restrictions. Sensor-based sorting is a machine vision application for which firm real-time requirements need to be respected in order to reliably remove potentially harmful entities from a material feed. Recently, employing a predictive tracking approach using multitarget tracking in order to decrease the error in the physical separation in optical sorting has been proposed. For implementations that use hard associations between measurements and tracks, a linear assignment problem has to be solved for each frame recorded by a camera. The auction algorithm can be utilized for this purpose, which also has the advantage of being well suited for parallel architectures. In this paper, an improved implementation of this algorithm for a graphics processing unit (GPU) is presented. The resulting algorithm is implemented in both an OpenCL and a CUDA based environment. By using an optimized data structure, the presented algorithm outperforms recently proposed implementations in terms of speed while retaining the quality of output of the algorithm. Furthermore, memory requirements are significantly decreased, which is important for embedded systems. Experimental results are provided for two different GPUs and six datasets. It is shown that the proposed approach is of particular interest for applications dealing with comparatively large problem sizes.
G. Maier, F. Pfaff, F. Becker, C. Pieper, R. Gruna, B. Noack, H. Kruggel-Emden, T. Längle, U. D. Hanebeck, S. Wirtz, V. Scherer, and J. Beyerer, Motion-Based Material Characterization in Sensor-Based Sorting, tm - Technisches Messen, De Gruyter, Oct. 2017.
Abstract — Sensor-based sorting provides state-of-the-art solutions for sorting cohesive, granular materials. Typically, involved sensors, illumination, implementation of data analysis and other components are designed and chosen according to the sorting task at hand. A common property of conventional systems is the utilization of scanning sensors. However, the usage of area-scan cameras has recently been proposed. When observing objects at multiple time points, the corresponding paths can be reconstructed by using multiobject tracking. This in turn allows to accurately estimate the point in time and position at which any object will reach the separation stage of the optical sorter and hence contributes to decreasing the error in physical separation. In this paper, it is proposed to further exploit motion information for the purpose of material characterization. By deriving suitable features from the motion information, we show that high classification performance is obtained for an exemplary classification task. The approach therefore contributes towards decreasing the detection error of sorting systems.
C. Pieper, G. Maier, F. Pfaff, H. Kruggel-Emden, R. Gruna, B. Noack, S. Wirtz, V. Scherer, T. Längle, U. D. Hanebeck, and J. Beyerer, Numerical Modelling of the Separation of Complex Shaped Particles in an Optical Belt Sorter Using a DEM–CFD Approach and Comparison with Experiments, in V International Conference on Particle-based Methods. Fundamentals and Applications (PARTICLES 2017), Hannover, Germany, Sep. 2017.
Abstract — In the growing field of bulk solids handling, automated optical sorting systems are of increasing importance. However, the initial sorter calibration is still very time consuming and the precise optical sorting of many materials still remains challenging. In order to investigate the impact of different operating parameters on the sorting quality, a numerical model of an existing modular optical belt sorter is presented in this study. The sorter and particle interaction is described with the Discrete Element Method (DEM) while the air nozzles required for deflecting undesired material fractions are modelled with Computation Fluid Dynamics (CFD). The correct representation of the resulting particle–fluid interaction is realized through a one–way coupling of the DEM with CFD. Complex shaped particle clusters are employed to model peppercorns also used in experimental investigations. To test the correct implementation of the utilized models, the particle mass flow within the sorter is compared between experiment and simulation. The particle separation results of the developed numerical model of the optical sorting system are compared with matching experimental investigations. The findings show that the numerical model is able to predict the sorting quality of the optical sorting system with reasonable accuracy.
F. Pfaff, B. Noack, U. D. Hanebeck, F. Govaers, and W. Koch, Information Form Distributed Kalman Filtering (IDKF) with Explicit Inputs, in Proceedings of the 20th International Conference on Information Fusion (Fusion 2017), Xi’an, China, Jul. 2017.
Abstract — With the ubiquity of information distributed in networks, performing recursive Bayesian estimation using distributed calculations is becoming more and more important. There are a wide variety of algorithms catering to different applications and requiring different degrees of knowledge about the other nodes involved. One recently developed algorithm is the distributed Kalman filter (DKF), which assumes that all knowledge about the measurements, except the measurements themselves, are known to all nodes. If this condition is met, the DKF allows deriving the optimal estimate if all information is combined in one node at an arbitrary time step. In this paper, we present an information form of the distributed Kalman filter (IDKF) that allows the use of explicit system inputs at the individual nodes while still yielding the same results as a centralized Kalman filter.
F. Pfaff, B. Noack, and U. D. Hanebeck, Optimal Distributed Combined Stochastic and Set-Membership State Estimation, in Proceedings of the 20th International Conference on Information Fusion (Fusion 2017), Xi’an, China, Jul. 2017.
Abstract — For distributed estimation, algorithms have to be specifically crafted to minimize communication between the sensor nodes. As an adjusted version of the regular Kalman filter, the distributed Kalman filter (DKF) allows for deriving optimal results while not requiring regular communication. To achieve this, the DKF requires that each node has full knowledge about the system model and measurement models of all nodes. However, the DKF is not sufficient if the characteristics of the errors in the system and measurement models are not purely stochastic. In this paper, we present a distributed version of a combined stochastic and set-membership Kalman filter. The proposed filter optimizes the approximations of the set-membership uncertainties and can even yield better results than the regular centralized filter.
B. Noack, J. Sijs, and U. D. Hanebeck, Inverse Covariance Intersection: New Insights and Properties, in Proceedings of the 20th International Conference on Information Fusion (Fusion 2017), Xi’an, China, Jul. 2017.
Abstract — Decentralized data fusion is a challenging task. Either it is too difficult to maintain and track the information required to perform fusion optimally, or too much information is discarded to obtain informative fusion results. A well-known solution is Covariance Intersection, which may provide too conservative fusion results. A less conservative alternative is discussed in this paper, and generalizations are proposed in order to apply it to a wide class of fusion problems. The Inverse Covariance Intersection algorithm is about finding the maximum possible common information shared by the estimates to be fused. A bound on the possibly shared common information is derived and removed from the fusion result in order to guarantee consistency. It is shown that the conditions required for consistency can be significantly relaxed, and also other causes of correlations, such as common process noise, can be treated.
J. Sijs and B. Noack, Event-Based Estimation in a Feedback Loop Anticipating on Imperfect Communication, in Proceedings of the 20th IFAC World Congress (IFAC 2017), Toulouse, France, Jul. 2017.
Abstract — Event-based sampling of sensor signals has become a mature alternative to time-periodic sampling completed with solutions for event-based estimation and control. Among those solutions there is a class of estimators exploiting the fact that an event was not triggered. Not receiving a new measurement is then interpreted as a sensor signal that has not violated the event criteria, which means that the signal is still within the triggering set defining the event. Such implied measurement information is exploited by the estimator, though it is only valid when no event occurred. The approach is thus sensitive to package loss and latency, as the estimator might be incorrect in assuming that no event took place. A solution to anticipate for package loss on the estimation error is studied in this article, and it is further turned into a first solution when the estimator is part of a feedback-control loop.
F. Pfaff, G. Maier, M. Aristov, B. Noack, R. Gruna, U. D. Hanebeck, T. Längle, J. Beyerer, C. Pieper, H. Kruggel-Emden, S. Wirtz, and V. Scherer, Real-Time Motion Prediction Using the Chromatic Offset of Line Scan Cameras, at - Automatisierungstechnik, De Gruyter, Jun. 2017.
Abstract — State-of-the-art optical belt sorters commonly employ line scan cameras and use simple assumptions to predict each particle’s movement, which is required for the separation process. Previously, we have equipped an experimental optical belt sorter with an area scan camera and were able to show that tracking the particles of the bulk material results in an improvement of the predictions and thus also the sorting process. In this paper, we use the slight gap between the sensor lines of an RGB line scan camera to derive information about the particles’ movements in real-time. This approach allows improving the predictions in optical belt sorters without necessitating any hardware modifications.
B. Noack, J. Sijs, M. Reinhardt, and U. D. Hanebeck, Decentralized Data Fusion with Inverse Covariance Intersection, Automatica, vol. 79, pp. 35–41, May 2017.
Abstract — In distributed and decentralized state estimation systems, fusion methods are employed to systematically combine multiple estimates of the state into a single, more accurate estimate. An often encountered problem in the fusion process relates to unknown common information that is shared by the estimates to be fused and is responsible for correlations. If the correlation structure is unknown to the fusion method, conservative strategies are typically pursued. As such, the parameterization introduced by the ellipsoidal intersection method has been a novel approach to describe unknown correlations, though suitable values for these parameters with proven consistency have not been identified yet. In this article, an extension of ellipsoidal intersection is proposed that guarantees consistent fusion results in the presence of unknown common information. The bound used by the novel approach corresponds to computing an outer ellipsoidal bound on the intersection of inverse covariance ellipsoids. As a major advantage of this inverse covariance intersection method, fusion results prove to be more accurate than those provided by the well-known covariance intersection method.
G. Maier, F. Pfaff, F. Becker, C. Pieper, R. Gruna, B. Noack, H. Kruggel-Emden, T. Längle, U. D. Hanebeck, S. Wirtz, V. Scherer, and J. Beyerer, Improving Material Characterization in Sensor-Based Sorting by Utilizing Motion Information, in Proceedings of the 3rd Conference on Optical Characterization of Materials (OCM 2017), Karlsruhe, Germany, Mar. 2017.
Abstract — Sensor-based sorting provides state-of-the-art solutions for sorting of cohesive, granular materials. Systems tailored to a task at hand, for instance by means of sensors and implementations of data analysis. Conventional systems utilize scanning sensors which do not allow for extraction of motion-related information of objects contained in a material feed. Recently, usage of area-scan cameras to overcome this disadvantage has been proposed. Multitarget tracking can then be used in order to accurately estimate the point in time and position at which any object will reach the separation stage. In this paper, utilizing motion information of objects which can be retrieved from multitarget tracking for the purpose of classification is proposed. Results show that corresponding features can significantly increase classification performance and eventually decrease the detection error of a sorting system.
F. Pfaff, C. Pieper, G. Maier, B. Noack, H. Kruggel-Emden, R. Gruna, U. D. Hanebeck, S. Wirtz, V. Scherer, T. Längle, and J. Beyerer, Simulation-based Evaluation of Predictive Tracking for Sorting Bulk Materials, in Proceedings of the 2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2016), Baden-Baden, Germany, Sep. 2016.
Abstract — Multitarget tracking problems arise in many real-world applications. The performance of the utilized algorithm strongly depends both on how the data association problem is handled and on the suitability of the motion models employed. Especially the motion models can be hard to validate. Previously, we have proposed to use multitarget tracking to improve optical belt sorters. In this paper, we evaluate both the suitability of our model and the tracking and then of our entire system incorporating the image processing component via the use of highly realistic numerical simulations. We first assess the model using noise-free measurements generated by the simulation and then evaluate the entire system by using synthetically generated image data.
B. Noack, J. Sijs, and U. D. Hanebeck, Algebraic Analysis of Data Fusion with Ellipsoidal Intersection, in Proceedings of the 2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2016), Baden-Baden, Germany, Sep. 2016.
Abstract — For decentralized fusion problems, ellipsoidal intersection has been proposed as an efficient fusion rule that provides less conservative results as compared to the well-know covariance intersection method. Ellipsoidal intersection relies on the computation of a common estimate that is shared by the estimates to be fused. In this paper, an algebraic reformulation of ellipsoidal intersection is discussed that circumvents the computation of the common estimate. It is shown that ellipsoidal intersection corresponds to an internal ellipsoidal approximation of the intersection of covariance ellipsoids. An interesting result is that ellipsoidal intersection can be computed with the aid of the Bar-Shalom/Campo fusion formulae. This is achieved by assuming a specific correlation structure between the estimates to be fused.
G. Maier, F. Pfaff, C. Pieper, R. Gruna, B. Noack, H. Kruggel-Emden, T. Längle, U. D. Hanebeck, S. Wirtz, V. Scherer, and J. Beyerer, Fast Multitarget Tracking via Strategy Switching for Sensor-Based Sorting, in Proceedings of the 2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2016), Baden-Baden, Germany, Sep. 2016.
Abstract — State-of-the-art sensor-based sorting systems provide solutions to sort various products according to quality aspects. Such systems face the challenge of an existing delay between perception and separation of the material. To reliably predict an object’s position when reaching the separation stage, information regarding its movement needs to be derived. Multitarget tracking offers approaches through which this can be achieved. However, processing time is typically limited since the sorting decision for each object needs to be derived sufficiently early before it reaches the separation stage. In this paper, an approach for multitarget tracking in sensor-based sorting is proposed which supports establishing an upper bound regarding processing time required for solving the measurement to track association problem. To demonstrate the success of the proposed method, experiments are conducted for data-sets obtained via simulation of a sorting system. This way, it is possible to not only demonstrate the impact on required runtime but also on the quality of the association.
F. Faion, A. Zea, B. Noack, J. Steinbring, and U. D. Hanebeck, Camera- and IMU-based Pose Tracking for Augmented Reality, in Proceedings of the 2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2016), Baden-Baden, Germany, Sep. 2016.
Abstract — In this paper, we propose an algorithm for tracking mobile devices (such as smartphones, tablets, or smartglasses) in a known environment for augmented reality applications. For this purpose, we interpret the environment as an extended object with a known shape, and design likelihoods for different types of image features, using association models from extended object tracking. Based on these likelihoods, and together with sensor information of the inertial measurement unit of the mobile device, we design a recursive Bayesian tracking algorithm. We present results of our first prototype and discuss the lessons we learned from its implementation. In particular, we set up a “pick-by-vision” scenario, where the location of objects in a shelf is to be highlighted in a camera image. Our experiments confirm that the proposed tracking approach achieves accurate and robust tracking results even in scenarios with fast motion.
C. Pieper, H. Kruggel-Emden, S. Wirtz, V. Scherer, F. Pfaff, B. Noack, U. D. Hanebeck, G. Maier, R. Gruna, T. Längle, and J. Beyerer, Numerical Investigation of Optical Sorting using the Discrete Element Method, in Proceedings of the 7th International Conference on Discrete Element Methods (DEM7), Dalian, China, Aug. 2016.
Abstract — Automated optical sorting systems are important devices in the growing field of bulk solids handling. The initial sorter calibration and the precise optical sorting of many materials is still very time consuming and difficult. A numerical model of an automated optical belt sorter is presented in this study. The sorter and particle interaction is described with the Discrete Element Method (DEM) while the separation phase is considered in a post processing step. Different operating parameters and their influence on sorting quality are investigated. In addition, two models for detecting and predicting the particle movement between the detection point and the separation step are presented and compared, namely a conventional line scan camera model and a new approach combining an area scan camera model with particle tracking.
J. Steinbring, B. Noack, M. Reinhardt, and U. D. Hanebeck, Optimal Sample-Based Fusion for Distributed State Estimation, in Proceedings of the 19th International Conference on Information Fusion (Fusion 2016), Heidelberg, Germany, Jul. 2016.
Abstract — In this paper, we present a novel approach to optimally fuse estimates in distributed state estimation for linear and nonlinear systems. An optimal fusion requires the knowledge of the correct correlations between locally obtained estimates. The naive and intractable way of calculating the correct correlations would be to exchange information about every processed measurement between all nodes. Instead, we propose to obtain the correct correlations by keeping and processing a small set of deterministic samples on each node in parallel to the actual local state estimation. Sending these samples in addition to the local state estimate to the fusion center allows for correctly reconstructing the desired correlations between all estimates. In doing so, each node does not need any information about measurements processed on other nodes. We show the optimality of the proposed method by means of tracking an extended object in a multi-camera network.
B. Noack, F. Pfaff, M. Baum, and U. D. Hanebeck, State Estimation Considering Negative Information with Switching Kalman and Ellipsoidal Filtering, in Proceedings of the 19th International Conference on Information Fusion (Fusion 2016), Heidelberg, Germany, Jul. 2016.
Abstract — State estimation concepts like the Kalman filter heavily rely on potentially noisy sensor data. In general, the estimation quality depends on the amount of sensor data that can be exploited. However, missing observations do not necessarily impair the estimation quality but may also convey exploitable information on the system state. This type of information - noted as negative information - often requires specific measurement and noise models in order to take advantage of it. In this paper, a hybrid Kalman filter concept is employed that allows using both stochastic and set-membership representations of information. In particular, the latter representation is intended to account for negative information, which can often be easily described as a bounded set in the measurement space. Depending on the type of information, the filtering step of the proposed estimator adaptively switches between Gaussian and ellipsoidal noise representations. A target tracking scenario is studied to evaluate and discuss the proposed concept.
C. Pieper, G. Maier, F. Pfaff, H. Kruggel-Emden, S. Wirtz, R. Gruna, B. Noack, V. Scherer, T. Längle, J. Beyerer, and U. D. Hanebeck, Numerical Modeling of an Automated Optical Belt Sorter using the Discrete Element Method, Powder Technology, Jul. 2016.
Abstract — Optical sorters are important devices in the processing and handling of the globally growing material streams. The precise optical sorting of many bulk solids is still difficult due to the great technical effort necessary for transport and flow control. In this study, particle separation with an automated optical belt sorter is modeled numerically. The Discrete Element Method (DEM) is used to model the sorter and calculate the particle movement as well as particle – particle and particle – wall interactions. The particle ejection stage with air valves is described with the help of a MATLAB script utilizing particle movement information obtained with the DEM. Two models for predicting the particle movement between the detection and separation phase are implemented and compared. In the first model, it is assumed that the particles are moving with belt velocity and without any cross movements and a conventional line scan camera is used for particle detection. In the second model, a more sophisticated approach is employed where the particle motion is predicted with an area scan camera combined with a tracking algorithm. In addition, the influence of different operating parameters like particle shape or conveyor belt length on the separation quality of the system is investigated. Results show that numerical simulations can offer detailed insight into the operation performance of optical sorters and help to optimize operating parameters. The area scan camera approach was found to be superior to the standard line scan camera model in almost all investigated categories.
B. Noack and U. D. Hanebeck, State Estimation Using Virtual Measurement Information, in Proceedings of the 18. GMA/ITG Fachtagung Sensoren und Messsysteme 2016, May 2016.
Abstract — The computation of an estimate for the unknown state of a dynamical system is a central challenge in many disciplines and applications. In general, the estimation quality is directly tied to the amount of sensor data available to the state estimation system. However, insights from virtual or missing observations may also convey exploitable information on the system’s state. Such virtual measurement information may relate to constraints to which the state is subject. For instance, constraints to acceleration and turn rate of a mobile robot may apply and can be exploited. Analogously, missing observations that are attributable to obstacles can be translated into usable information, which is often referred to as negative sensor evidence. Such implicit information has to be reformulated into virtual measurement data in order to take advantage of it. As the Kalman filter and its derivatives are most widely used in state estimation applications, specific measurement and noise models for virtual observations are to be derived that can easily be integrated into the prediction-correction cycle of the Kalman filter. In this work, a set-membership representation of virtual measurement information is discussed.
F. Pfaff, C. Pieper, G. Maier, B. Noack, H. Kruggel-Emden, R. Gruna, U. D. Hanebeck, S. Wirtz, V. Scherer, T. Längle, and J. Beyerer, Improving Optical Sorting of Bulk Materials Using Sophisticated Motion Models, tm - Technisches Messen, De Gruyter, vol. 83, no. 2, pp. 77–84, Feb. 2016.
Abstract — Visual properties are powerful features to reliably classify bulk materials, thereby allowing to detect defect or low quality particles. Optical belt sorters are an established technology to sort based on these properties, but they suffer from delays between the simultaneous classification and localization step and the subsequent separation step. Therefore, accurate models to predict the particles’ motions are a necessity to bridge this gap. In this paper, we explicate our concept to use sophisticated simulations to derive accurate models and optimize the flow of bulk solids via adjustments of the sorter design. This allows us to improve overall sorting accuracy and cost efficiency. Lastly, initial results are presented.
J. Sijs, B. Noack, M. Lazar, and U. D. Hanebeck, Time-Periodic State Estimation with Event-Based Measurement Updates, in Event-Based Control and Signal Processing, M. Miskowicz, Ed. CRC Press, Nov. 2015, pp. 261–279.
Abstract — To reduce the amount of data transfers in networked systems, measurements can be taken at an event on the sensor value rather than periodically in time. Yet, this could lead to a divergence of estimation results when only the received measurement values are exploited in a state estimation procedure. A solution to this issue has been found by developing estimators that perform a state update at both the event instants as well as periodically in time: when an event occurs the estimated state is updated using the measurement received, while at periodic instants the update is based on knowledge that the sensor value lies within a bounded subset of the measurement space. Several solutions for event-based state estimation will be presented in this chapter, either based on stochastic representations of random vectors, on deterministic representations of random vectors or on a mixture of the two. All solutions aim to limit the required computational resources by deriving explicit solutions for computing estimation results. Yet, the main achievement for each estimation solution is that stability of the estimation results are (not directly) dependent on the employed event sampling strategy. As such, changing the event sampling strategy does not imply to change the event-based estimator as well. This aspect is also illustrated in a case study of tracking the distribution of a chemical compound effected by wind via a wireless sensor network.
B. Noack, M. Baum, and U. D. Hanebeck, State Estimation for Ellipsoidally Constrained Dynamic Systems with Set-membership Pseudo Measurements, in Proceedings of the 2015 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2015), San Diego, California, USA, Sep. 2015.
Abstract — In many dynamic systems, the evolution of the state is subject to specific constraints. In general, constraints cannot easily be integrated into the prediction-correction structure of the Kalman filter algorithm. Linear equality constraints are an exception to this rule and have been widely used and studied as they allow for simple closed-form expressions. A common approach is to reformulate equality constraints into pseudo measurements of the state to be estimated. However, equality constraints define deterministic relationships between state components which is an undesirable property in Kalman filtering as this leads to singular covariance matrices. A second problem relates to the knowledge required to identify and define precise constraints, which are met by the system state. In this article, ellipsoidal constraints are introduced that can be employed to model a bounded region, to which the system state is constrained. This concept constitutes an easy-to-use relaxation of equality constraints. In order to integrate ellipsoidal constraints into the Kalman filter structure, a generalized filter framework is utilized that relies on a combined stochastic and set-membership uncertainty representation.
M. Baum, B. Noack, and U. D. Hanebeck, Kalman Filter-based SLAM with Unknown Data Association using Symmetric Measurement Equations, in Proceedings of the 2015 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2015), San Diego, California, USA, Sep. 2015.
Abstract — This work investigates a novel method for dealing with unknown data associations in Kalman filter-based Simultaneous Localization and Mapping (SLAM) problems. The key idea is to employ the concept of Symmetric Measurement Equations (SMEs) in order to remove the data association uncertainty from the original measurement equation. Based on the resulting modified measurement equation, standard nonlinear Kalman filters can estimate the full joint state vector of the robot and landmarks without explicitly calculating data association hypotheses.
F. Pfaff, M. Baum, B. Noack, U. D. Hanebeck, R. Gruna, T. Längle, and J. Beyerer, TrackSort: Predictive Tracking for Sorting Uncooperative Bulk Materials, in Proceedings of the 2015 IEEE International Conference on Multisensor Fusion and Information Integration (MFI 2015), San Diego, California, USA, Sep. 2015.
M. Reinhardt, B. Noack, P. O. Arambel, and U. D. Hanebeck, Minimum Covariance Bounds for the Fusion under Unknown Correlations, IEEE Signal Processing Letters, vol. 22, no. 9, pp. 1210–1214, Sep. 2015.
Abstract — One of the key challenges in distributed linear estimation is the systematic fusion of estimates. While the fusion gains that minimize the mean squared error of the fused estimate for known correlations have been established, no analogous statement could be obtained so far for unknown correlations. In this contribution, we derive the gains that minimize the bound on the true covariance of the fused estimate and prove that Covariance Intersection (CI) is the optimal bounding algorithm for two estimates under completely unknown correlations. When combining three or more variables, the CI equations are not necessarily optimal, as shown by a counterexample.
B. Noack, J. Sijs, M. Reinhardt, and U. D. Hanebeck, Treatment of Dependent Information in Multisensor Kalman Filtering and Data Fusion, in Multisensor Data Fusion: From Algorithms and Architectural Design to Applications, H. Fourati, Ed. CRC Press, Aug. 2015, pp. 169–192.
Abstract — Distributed and decentralized processing and fusion of sensor data are becoming increasingly important. In view of the Internet of Things and the vision of ubiquitous sensing, designing and implementing multisensor state estimation algorithm have already become a key issue. A network of interconnected sensor devices is usually characterized by the idea to process and collect data locally and independently on the sensor nodes. However, this does not imply that the data are independent of each other, and the state estimation algorithms have to address possible interdependencies so as to avoid erroneous data fusion results. Dependencies among local estimates generally can be traced back to common sensor information and common process noise. A wide variety of Kalman filtering schemes allow for the treatment of dependent data in centralized, distributed, and decentralized networks of sensor nodes, but making the right choice is itself dependent upon analyzing and weighing up the different advantages and disadvantages. This chapter discusses different strategies to identify and treat dependencies among Kalman filter estimates while pointing out advantages and challenges.
B. Noack, S. J. Julier, and U. D. Hanebeck, Treatment of Biased and Dependent Sensor Data in Graph-based SLAM, in Proceedings of the 18th International Conference on Information Fusion (Fusion 2015), Washington D. C., USA, Jul. 2015.
Abstract — A common approach to attack the simultaneous localization and mapping problem (SLAM) is to consider factor-graph formulations of the underlying filtering and estimation setup. While Kalman filter-based methods provide an estimate for the current pose of a robot and all landmark positions, graph-based approaches take not only the current pose into account but also the entire trajectory of the robot and have to solve a nonlinear least-squares optimization problem. Using graph-based representations has proven to be highly scalable and very accurate as compared with traditional filter-based approaches. However, biased measurements as well as unmodeled correlations can lead to a sharp deterioration in the estimation quality and hence require careful consideration. In this paper, a method to incorporate biased or dependent measurement information is proposed that can easily be integrated into existing optimization algorithms for graph-based SLAM. For biased sensor data, techniques from ellipsoidal calculus are employed to compute the corresponding information matrices. Dependencies among noise terms are treated by a generalization of the covariance intersection concept. The treatment of both biased and correlated sensor data rest upon the inflation of the involved error matrices. Simulations are used to discuss and evaluate the proposed method.
B. Noack, J. Sijs, and U. D. Hanebeck, Fusion Strategies for Unequal State Vectors in Distributed Kalman Filtering, in Proceedings of the 19th IFAC World Congress (IFAC 2014), Cape Town, South Africa, Aug. 2014.
Abstract — Distributed implementations of state estimation algorithms generally have in common that each node in a networked system computes an estimate on the entire global state. Accordingly, each node has to store and compute an estimate of the same state vector irrespective of whether its sensors can only observe a small part of it. In particular, the task of monitoring large-scale phenomena renders such distributed estimation approaches impractical due to the sheer size of the corresponding state vector. In order to reduce the workload of the nodes, the state vector to be estimated is subdivided into smaller, possibly overlapping parts. In this situation, fusion does not only refer to the computation of an improved estimate but also to the task of reassembling an estimate for the entire state from the locally computed estimates of unequal state vectors. However, existing fusion methods require equal state representations and, hence, cannot be employed. For that reason, a fusion strategy for estimates of unequal and possibly overlapping state vectors is derived that minimizes the mean squared estimation error. For the situation of unknown cross-correlations between local estimation errors, also a conservative fusion strategy is proposed.
J. Sijs, L. Kester, and B. Noack, A Study on Event Triggering Criteria for Estimation, in Proceedings of the 17th International Conference on Information Fusion (Fusion 2014), Salamanca, Spain, Jul. 2014.
Abstract — To reduce the amount of data transfer in networked systems measurements are usually taken only when an event occurs rather than periodically in time. However, a fundamental assessment on the response of estimation algorithms receiving event sampled measurements is not available. This research presents such an analysis when new measurements are sampled at well-designed events and sent to a Luenberger observer. Conditions are then derived under which the estimation error is bounded, followed by an assessment of two event sampling strategies when the estimator encounters two different types of disturbances: an impulse and a step disturbance. The sampling strategies are compared via four performance measures, such as estimation-error and communication resources. The result is a clear insight of the estimation response in an event-based setup.
M. Reinhardt, B. Noack, S. Kulkarni, and U. D. Hanebeck, Distributed Kalman Filtering in the Presence of Packet Delays and Losses, in Proceedings of the 17th International Conference on Information Fusion (Fusion 2014), Salamanca, Spain, Jul. 2014.
Abstract — Distributed Kalman filtering aims at optimizing an estimate at a fusion center based on information that is gathered in a sensor network. Recently, an exact solution based on local estimation tracks has been proposed and an extension to cope with packet losses has been derived. In this contribution, we generalize both algorithms to packet delays. The key idea is to introduce augmented measurement vectors in the sensors that permit the optimization of local filter gains according to time-dependent measurement capabilities at the fusion center. In the most general form, the algorithm provides optimized estimates in sensor networks with packets delays and losses. The precision depends on the actual arrival patterns, and the results correspond to those of the centralized Kalman filter when specific assumptions about the measurement capability are satisfied.
B. Noack, M. Reinhardt, and U. D. Hanebeck, On Nonlinear Track-to-track Fusion with Gaussian Mixtures, in Proceedings of the 17th International Conference on Information Fusion (Fusion 2014), Salamanca, Spain, Jul. 2014.
Abstract — The problem of fusing state estimates is encountered in many network-based multi-sensor applications. The majority of distributed state estimation algorithms are designed to provide multiple estimates on the same state, and track-to-track fusion then refers to the task of combining these estimates. While linear fusion only requires the joint cross-covariance matrix to be known, dependencies between estimates in nonlinear estimation problems have to be represented by high-dimensional probability density functions. In general, storing and keeping track of nonlinear dependencies is too cumbersome. However, this paper demonstrates that estimates represented by Gaussian mixtures prove to be an important exception to this rule. The dependency structure can as well be characterized in terms of a higher-dimensional Gaussian mixture. The different processing steps of distributed nonlinear state estimation, i.e., prediction, filtering, and fusion, are studied in light of the joint density representation. The presented concept is complemented with different simpler suboptimal representations of the dependency structure between Gaussian mixture densities.
J. Ajgl, M. Šimandl, M. Reinhardt, B. Noack, and U. D. Hanebeck, Covariance Intersection in State Estimation of Dynamical Systems, in Proceedings of the 17th International Conference on Information Fusion (Fusion 2014), Salamanca, Spain, Jul. 2014.
Abstract — The Covariance Intersection algorithm linearly combines estimates when the cross-correlations between their errors are unknown. It provides a fused estimate and an upper bound of the corresponding mean square error matrix. The weights of the linear combination are designed in order to minimise the upper bound. This paper analyses the optimal weights in relation to state estimation of dynamical systems. It is shown that the use of the optimal upper bound in a standard recursive filtering does not lead to optimal upper bounds in subsequent processing steps. Unlike the fusion under full knowledge, the fusion under unknown cross-correlations can fuse the same information differently, depending on the independent information that will be available in the future.
M. Reinhardt, B. Noack, and U. D. Hanebeck, Reconstruction of Joint Covariance Matrices in Networked Linear Systems, in Proceedings of the 48th Annual Conference on Information Sciences and Systems (CISS 2014), Princeton, New Jersey, USA, Mar. 2014.
Abstract — In this paper, a sample representation of the estimation error is utilized to reconstruct the joint covariance matrix in a distributed estimation system. The key idea is to sample uncorrelated and fully correlated noise according to different techniques at local estimators without knowledge about the processing of other nodes in the network. This way, the correlation between estimates is inherently linked to the representation of the corresponding sample sets. We discuss the noise processing, derive key attributes, and evaluate the precision of the covariance estimates.
J. Sijs, U. D. Hanebeck, and B. Noack, An Empirical Method to Fuse Partially Overlapping State Vectors for Distributed State Estimation, in Proceedings of the 2013 European Control Conference (ECC 2013), Zürich, Switzerland, Jul. 2013.
Abstract — State fusion is a method for merging multiple estimates of the same state into a single fused estimate. Dealing with multiple estimates is one of the main concerns in distributed state estimation, where an estimated value of the desired state vector is computed in each node of a networked system. Most solutions for distributed state estimation currently available assume that every node computes an estimate of the (same) global state vector. This assumption is impractical for systems observing large-area processes, due to the sheer size of the process state. A more feasible solutions is one where each node estimates a part of the global state vector, allowing different nodes in the network to have overlapping state elements. Although such an approach should be accompanied by a corresponding state fusion method, existing solutions cannot be employed as they merely consider fusion of two different estimates with equal state representations. Therefore, an empirical solution is presented for fusing two state estimates that have partially overlapping state elements. A justification of the proposed fusion method is presented, along with an illustrative case study for observing the temperature profile of a large rod, though a formal derivation is future research.
J. Sijs, B. Noack, and U. D. Hanebeck, Event-based State Estimation with Negative Information, in Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey, Jul. 2013.
Abstract — To reduce the amount of data transfer in networked systems, measurements are usually taken only when an event occurs rather than periodically in time. However, this complicates estimation problems considerably as it is not guaranteed that new sensor data will be sampled. Therefore, an existing state estimator is extended so to cope with event-based measurements successfully, i.e., curtail any diverging behavior in the estimation results. To that extent, a general formulation of event sampling is proposed. This formulation is used to set up a state estimator combining stochastic as well as set-membership measurement information according to a hybrid update: when an event occurs the estimated state is updated using the stochastic measurement received (positive information), while at periodic time instants no measurement is received (negative information) and the update is based on knowledge that the sensor value lies within a bounded subset of the measurement space. An illustrative example further shows that the developed estimator has an improved representation of estimation errors compared to a purely stochastic estimator for various event sampling strategies.
M. Reinhardt, B. Noack, and U. D. Hanebeck, Advances in Hypothesizing Distributed Kalman Filtering, in Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey, Jul. 2013.
Abstract — In this paper, linear distributed estimation is revisited on the basis of the hypothesizing distributed Kalman filter and equations for a flexible application of the algorithm are derived. We propose a new approximation for the mean-squared-error matrix and present techniques for automatically improving the hypothesis about the global measurement model. Utilizing these extensions, the precision of the filter is improved so that it asymptotically yields optimal results for time-invariant models. Pseudo-code for the implementation of the algorithm is provided and the lossless inclusion of out-of-sequence measurements is discussed. An evaluation demonstrates the effect of the new extensions and compares the results to state-of-the-art methods.
F. Pfaff, B. Noack, and U. D. Hanebeck, Data Validation in the Presence of Stochastic and Set-membership Uncertainties, in Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey, Jul. 2013.
Abstract — For systems suffering from different types of uncertainties, finding criteria for validating measurements can be challenging. In this paper, we regard both stochastic Gaussian noise with full or imprecise knowledge about correlations and unknown but bounded errors. The validation problems arising in the individual and combined cases are illustrated to convey different perspectives on the proposed conditions. Furthermore, hints are provided for the algorithmic implementation of the validation tests. Particular focus is put on ensuring a predefined lower bound for the probability of correctly classifying valid data.
B. Noack, S. J. Julier, M. Reinhardt, and U. D. Hanebeck, Nonlinear Federated Filtering, in Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey, Jul. 2013.
Abstract — The federated Kalman filter embodies an efficient and easy-to-implement solution for linear distributed estimation problems. Data from independent sensors can be processed locally and in parallel on different nodes without running the risk of erroneously ignoring possible dependencies. The underlying idea is to counteract the common process noise issue by inflating the joint process noise matrix. In this paper, the same trick is generalized to nonlinear models and non-Gaussian process noise. The probability density of the joint process noise is split into an exponential mixture of transition densities. By this means, the process noise is modeled to independently affect the local system models. The estimation results provided by the sensor devices can then be fused, just as if they were indeed independent.
B. Noack, State Estimation for Distributed Systems with Stochastic and Set-membership Uncertainties, Dissertation, Karlsruhe Institute of Technology (KIT), Karlsruhe Series on Intelligent Sensor-Actuator-Systems 14, Jan. 2013.
Abstract — State estimation methods provide the means to extract usable information from sensor data – even in the presence of severe uncertainties. Their results are supposed to be informative and uncertainty-aware estimates at once. In particular, the clear tendency towards networked sensor systems gives rise to many additional challenges in developing state estimation methods. This book focuses on a comprehensive study of state estimation techniques in centralized, distributed, and decentralized systems with uncertainties being modeled and incorporated in a systematic fashion. We study an easy-to-implement estimation concept that generalizes and combines basic principles of Kalman filter theory and ellipsoidal calculus. By means of this method, stochastic and set-membership uncertainties can be taken into consideration simultaneously. We discuss different solutions to meet the increasing demand for implementing these estimation algorithms in distributed networked systems and investigate the problem of fusing estimates under unknown cross-correlations and nonlinear dependencies.
M. Reinhardt, B. Noack, and U. D. Hanebeck, Decentralized Control Based on Globally Optimal Estimation, in Proceedings of the 51st IEEE Conference on Decision and Control (CDC 2012), Maui, Hawaii, USA, Dec. 2012.
Abstract — A new method for globally optimal estimation in decentralized sensor-networks is applied to the decentralized control problem. The resulting approach is proven to be optimal when the nodes have access to all information in the network. More precisely, we utilize an algorithm for optimal distributed estimation in order to obtain local estimates whose combination yields the globally optimal estimate. When the interconnectivity is high, the local estimates are almost optimal, which motivates the application of the principle of separation. Thus, we optimize the controller and finally obtain a flexible algorithm, whose quality is evaluated in different scenarios. In applications where the strong requirements on a perfect communication cannot be guaranteed, we derive quality bounds by help of a detailed evaluation of the algorithm. When information is regularly exchanged, it is demonstrated that the algorithm performs almost optimally and therefore, offers system designers a flexible and easy to implement approach. The field of applications lies within the area of strongly networked systems, in particular, when communication disturbances cannot be foreseen or when the network structure is too complicated to apply optimized regulators.
B. Noack, F. Pfaff, and U. D. Hanebeck, Optimal Kalman Gains for Combined Stochastic and Set-Membership State Estimation, in Proceedings of the 51st IEEE Conference on Decision and Control (CDC 2012), Maui, Hawaii, USA, Dec. 2012.
Abstract — In state estimation theory, two directions are mainly followed in order to model disturbances and errors. Either uncertainties are modeled as stochastic quantities or they are characterized by their membership to a set. Both approaches have distinct advantages and disadvantages making each one inherently better suited to model different sources of estimation uncertainty. This paper is dedicated to the task of combining stochastic and set-membership estimation methods. A Kalman gain is derived that minimizes the mean squared error in the presence of both stochastic and additional unknown but bounded uncertainties, which are represented by Gaussian random variables and ellipsoidal sets, respectively. As a result, a generalization of the well-known Kalman filtering scheme is attained that reduces to the standard Kalman filter in the absence of set-membership uncertainty and that otherwise becomes the intersection of sets in case of vanishing stochastic uncertainty. The proposed concept also allows to prioritize either the minimization of the stochastic uncertainty or the minimization of the set-membership uncertainty.
M. Reinhardt, B. Noack, and U. D. Hanebeck, The Hypothesizing Distributed Kalman Filter, in Proceedings of the IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2012), Hamburg, Germany, Sep. 2012.
Abstract — This paper deals with distributed information processing in sensor networks. We propose the Hypothesizing Distributed Kalman Filter that incorporates an assumption of the global measurement model into the distributed estimation process. The procedure is based on the Distributed Kalman Filter and inherits its optimality when the assumption about the global measurement uncertainty is met. Recursive formulas for local processing as well as for fusion are derived. We show that the proposed algorithm yields the same results, no matter whether the measurements are processed locally or globally, even when the process noise is not negligible. For further processing of the estimates, a consistent bound for the error covariance matrix is derived. All derivations and explanations are illustrated by means of a new classification scheme for estimation processes.
M. Reinhardt, B. Noack, and U. D. Hanebeck, Closed-form Optimization of Covariance Intersection for Low-dimensional Matrices, in Proceedings of the 15th International Conference on Information Fusion (Fusion 2012), Singapore, Jul. 2012.
Abstract — The fusion under unknown correlations is an important technique in sensor-network information processing as the cross-correlations between different estimates remain often unknown to the nodes. Covariance intersection is a wide-spread and efficient algorithm to fuse estimates under such uncertain conditions. Although different optimization criteria have been developed, the trace or determinant minimization of the fused covariance matrix seems to be most meaningful. However, this minimization requires numeric solutions of a convex optimization problem. We derive an algorithm to reduce this nonlinear optimization to the well-known polynomial root-finding problem. This allows us to present closed-form solutions for the determinant criterion when the dimension of the occurring covariance matrices is at most 4 and for the trace criterion when the dimension of the covariance matrices is at most 3. We demonstrate the effectiveness of the approach by means of a speed evaluation.
M. Reinhardt, B. Noack, and U. D. Hanebeck, On Optimal Distributed Kalman Filtering in Non-ideal Situations, in Proceedings of the 15th International Conference on Information Fusion (Fusion 2012), Singapore, Jul. 2012.
Abstract — The distributed processing of measurements and the subsequent data fusion is called Track-to-Track fusion. Although a solution for the Track-to-Track fusion that is equivalent to a central processing scheme has been proposed, this algorithm suffers from strict requirements regarding the local availability of knowledge about utilized models of the remote nodes. By means of simple examples, we investigate the effects of incorrectly assumed models and trace the errors back to a bias, which we derive in closed form. We propose an extension to the exact Track-to-Track fusion algorithm that corrects the bias after arbitrarily many time steps. This new approach yields optimal results when the assumptions about the measurement models are correct and otherwise still provides the exact value for the mean-squared-error matrix. The performance of this algorithm is demonstrated and applications are presented that, e.g., allow the employment of nonlinear filter methods.
B. Noack, F. Pfaff, and U. D. Hanebeck, Combined Stochastic and Set-membership Information Filtering in Multisensor Systems, in Proceedings of the 15th International Conference on Information Fusion (Fusion 2012), Singapore, Jul. 2012.
Abstract — In state estimation theory, stochastic and set-membership approaches are generally considered separately from each other. Both concepts have distinct advantages and disadvantages making each one inherently better suited to model different sources of estimation uncertainty. In order to better utilize the potentials of both concepts, the core element of this paper is a Kalman filtering scheme that allows for a simultaneous treatment of stochastic and set-membership uncertainties. An uncertain quantity is herein modeled by a set of Gaussian densities. Since many modern applications operate in networked systems that may consist of a multitude of local processing units and sensor nodes, estimates have to be computed in a distributed manner and measurements may arrive at high frequency. An algebraic reformulation of the Kalman filter, the information filter, significantly eases the implementation of such distributed fusion architectures. This paper explicates how stochastic and set-membership uncertainties can simultaneously be treated within this information form and compared to the Kalman filter, it becomes apparent that the quality of some required approximations is enhanced.
A. Benavoli and B. Noack, Pushing Kalman’s Idea to the Extremes, in Proceedings of the 15th International Conference on Information Fusion (Fusion 2012), Singapore, Jul. 2012.
Abstract — The paper focuses on the fundamental idea of Kalman’s seminal paper: how to solve the filtering problem from the only knowledge of the first two moments of the noise terms. In this paper, by exploiting set of distributions based filtering, we solve this problem without introducing additional assumptions on the distributions of the noise terms (e.g., Gaussianity) or on the final form of the estimator (e.g., linear estimator). Given the moments (e.g., mean and variance) of random variable X, it is possible to define the set of all distributions that are compatible with the moments information. This set of distributions can be equivalently characterized by its extreme distributions which is a family of mixtures of Dirac’s deltas. The lower and upper expectation of any function g of X are obtained in correspondence of these extremes and can be computed by solving a linear programming problem. The filtering problem can then be solved by running iteratively this linear programming problem.
M. Baum, B. Noack, and U. D. Hanebeck, Random Hypersurface Mixture Models for Tracking Multiple Extended Objects, in Proceedings of the 50th IEEE Conference on Decision and Control (CDC 2011), Orlando, Florida, USA, Dec. 2011.
Abstract — This paper presents a novel method for tracking multiple extended objects. The shape of a single extended object is modeled with a recently developed approach called Random Hypersurface Model (RHM) that assumes a varying number of measurement sources to lie on scaled versions of the shape boundaries. This approach is extended by introducing a so-called Mixture Random Hypersurface Model (Mixture RHM), which allows for modeling multiple extended targets. Based on this model, a Gaussian-assumed Bayesian tracking method that provides the means to track and estimate shapes of multiple extended targets is derived. Simulations demonstrate the performance of the new approach.
B. Noack, M. Baum, and U. D. Hanebeck, Automatic Exploitation of Independencies for Covariance Bounding in Fully Decentralized Estimation, in Proceedings of the 18th IFAC World Congress (IFAC 2011), Milan, Italy, Aug. 2011.
Abstract — Especially in the field of sensor networks and multi-robot systems, fully decentralized estimation techniques are of particular interest. As the required elimination of the complex dependencies between estimates generally yields inconsistent results, several approaches, e.g., covariance intersection, maintain consistency by providing conservative estimates. Unfortunately, these estimates are often too conservative and therefore, much less informative than a corresponding centralized approach. In this paper, we provide a concept that conservatively decorrelates the estimates while bounding the unknown correlations as closely as possible. For this purpose, known independent quantities, such as measurement noise, are explicitly identified and exploited. Based on tight covariance bounds, the new approach allows for an intuitive and systematic derivation of appropriate tailor-made filter equations and does not require heuristics. Its performance is demonstrated in a comparative study within a typical SLAM scenario.
M. Reinhardt, B. Noack, M. Baum, and U. D. Hanebeck, Analysis of Set-theoretic and Stochastic Models for Fusion under Unknown Correlations, in Proceedings of the 14th International Conference on Information Fusion (Fusion 2011), Chicago, Illinois, USA, Jul. 2011.
Abstract — In data fusion theory, multiple estimates are combined to yield an optimal result. In this paper, the set of all possible results is investigated, when two random variables with unknown correlations are fused. As a first step, recursive processing of the set of estimates is examined. Besides set-theoretic considerations, the lack of knowledge about the unknown correlation coefficient is modeled as a stochastic quantity. Especially, a uniform model is analyzed, which provides a new optimization criterion for the covariance intersection algorithm in scalar state spaces. This approach is also generalized to multi-dimensional state spaces in an approximative, but fast and scalable way, so that consistent estimates are obtained.
B. Noack, M. Baum, and U. D. Hanebeck, Covariance Intersection in Nonlinear Estimation Based on Pseudo Gaussian Densities, in Proceedings of the 14th International Conference on Information Fusion (Fusion 2011), Chicago, Illinois, USA, Jul. 2011.
Abstract — Many modern fusion architectures are designed to process and fuse data in networked systems. Alongside the advantages, such as scalability and robustness, distributed fusion techniques particularly have to tackle the problem of dependencies between locally processed data. In linear estimation problems, uncertain quantities with unknown cross-correlations can be fused by means of the covariance intersection algorithm, which avoids overconfident fusion results. However, for nonlinear system dynamics and sensor models perturbed by arbitrary noise, it is not only a problem to characterize and parameterize dependencies between estimates, but also to find a proper notion of consistency. This paper addresses these issues by transforming the state estimates to a different state space, where the corresponding densities are Gaussian and only linear dependencies between estimates, i.e., correlations, can arise. These pseudo Gaussian densities then allow the notion of covariance consistency to be used in distributed nonlinear state estimation.
M. Baum, B. Noack, F. Beutler, D. Itte, and U. D. Hanebeck, Optimal Gaussian Filtering for Polynomial Systems Applied to Association-free Multi-Target Tracking, in Proceedings of the 14th International Conference on Information Fusion (Fusion 2011), Chicago, Illinois, USA, Jul. 2011.
Abstract — This paper is about tracking multiple targets with the so-called Symmetric Measurement Equation (SME) filter. The SME filter uses symmetric functions, e.g., symmetric polynomials, in order to remove the data association uncertainty from the measurement equation. By this means, the data association problem is converted to a nonlinear state estimation problem. In this work, an efficient optimal Gaussian filter based on analytic moment calculation for discrete-time multi-dimensional polynomial systems corrupted with Gaussian noise is derived, and then applied to the polynomial system resulting from the SME filter. The performance of the new method is compared to an UKF implementation by means of typical multiple target tracking scenarios.
B. Noack, D. Lyons, M. Nagel, and U. D. Hanebeck, Nonlinear Information Filtering for Distributed Multisensor Data Fusion, in Proceedings of the 2011 American Control Conference (ACC 2011), San Francisco, California, USA, Jun. 2011.
Abstract — 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.
J. Schmid, F. Beutler, B. Noack, U. D. Hanebeck, and K. D. Müller-Glaser, An Experimental Evaluation of Position Estimation Methods for Person Localization in Wireless Sensor Networks, in Proceedings of the 8th European Conference on Wireless Sensor Networks (EWSN 2011), Bonn, Germany, Feb. 2011, vol. 6567, pp. 147–162.
Abstract — In this paper, the localization of persons by means of a Wireless Sensor Network (WSN) is considered. Persons carry on-body sensor nodes and move within a WSN. The location of each person is calculated on this node and communicated through the network to a central data sink for visualization. Applications of such a system could be found in mass casualty events, firefighter scenarios, hospitals or retirement homes for example. For the location estimation on the sensor node, three derivatives of the Kalman Filter and a closed-form solution (CFS) are applied, compared, and evaluated in a real-world scenario. A prototype 65-node ZigBee WSN is implemented and data are collected in in- and outdoor environments with differently positioned on-body nodes. The described estimators are then evaluated off-line on the experimentally collected data. The goal of this paper is to present a comprehensive real-world evaluation of methods for person localization in a WSN based on received signal strength (RSS) range measurements. It is concluded that person localization in in- and outdoor environments is possible under the considered conditions with the considered filters. The compared methods allow for suffciently accurate localization results and are robust against inaccurate range measurements.
E. Bogatyrenko, B. Noack, and U. D. Hanebeck, Reliable Estimation of Heart Surface Motion under Stochastic and Unknown but Bounded Systematic Uncertainties, in Proceedings of the 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2010), Taipei, Taiwan, Oct. 2010.
Abstract — A reliable estimation of heart surface motion is an important prerequisite for the synchronization of surgical instruments in robotic beating heart surgery. In general, only an imprecise description of the heart dynamics and measurement systems is available. This means that the estimation of heart motion is corrupted by stochastic and systematic uncertainties. Without consideration of these uncertainties, the obtained results will be inaccurate and a safe robotic operation cannot be guaranteed. Until now, existing approaches for estimating the motion of the heart surface are either deterministic or treat only stochastic uncertainties. The proposed method extends the heart motion estimation to the simultaneous consideration of stochastic and unknown but bounded systematic uncertainties. It computes dynamic bounds in order to provide the surgeon with a guidance by constraining the motion of the surgical instruments and thereby protecting sensitive tissue.
B. Noack, V. Klumpp, D. Lyons, and U. D. Hanebeck, Modellierung von Unsicherheiten und Zustandsschätzung mit Mengen von Wahrscheinlichkeitsdichten, tm - Technisches Messen, Oldenbourg Verlag, vol. 77, no. 10, pp. 544–550, Oct. 2010.
Abstract — Die systematische Behandlung von Unsicherheiten stellt eine wesentliche Herausforderung in der Informationsfusion dar. Einerseits müssen geeignete Darstellungsformen für die Unsicherheiten bestimmt werden und andererseits darauf aufbauend effiziente Schätzverfahren hergeleitet werden. Im Allgemeinen wird zwischen stochastischen und mengenbasierten Unsicherheitsbeschreibungen unterschieden. Dieser Beitrag stellt ein Verfahren zur Zustandsschätzung vor, welches simultan stochastische und mengenbasierte Fehlergrößen berücksichtigen kann, indem unsichere Größen nicht mehr durch eine einzelne Wahrscheinlichkeitsdichte, sondern durch eine Menge von Dichten repräsentiert werden. Besonderes Augenmerk liegt hier auf den Vorteilen und Anwendungsmöglichkeiten dieser Unsicherheitsbeschreibung.
A. Hekler, D. Lyons, B. Noack, and U. D. Hanebeck, Nonlinear Model Predictive Control Considering Stochastic and Systematic Uncertainties with Sets of Densities, in Proceedings of the IEEE Multi-Conference on Systems and Control (MSC 2010), Yokohama, Japan, Sep. 2010.
Abstract — In Model Predictive Control, the quality of control is highly dependent upon the model of the system under control. Therefore, a precise deterministic model is desirable. However, in real-world applications, modeling accuracy is typically limited and systems are generally affected by disturbances. Hence, it is important to systematically consider these uncertainties and to model them correctly. In this paper, we present a novel Nonlinear Model Predictive Control method for systems affected by two different types of perturbations that are modeled as being either stochastic or unknown but bounded quantities. We derive a formal generalization of the Nonlinear Model Predictive Control principle for considering both types of uncertainties simultaneously, which is achieved by using sets of probability densities. In doing so, a more robust and reliable control is obtained. The capabilities and benefits of our approach are demonstrated in real-world experiments with miniature walking robots.
B. Noack, V. Klumpp, N. Petkov, and U. D. Hanebeck, Bounding Linearization Errors with Sets of Densities in Approximate Kalman Filtering, in Proceedings of the 13th International Conference on Information Fusion (Fusion 2010), Edinburgh, United Kingdom, Jul. 2010.
Abstract — Applying the Kalman filtering scheme to linearized system dynamics and observation models does in general not yield optimal state estimates. More precisely, inconsistent state estimates and covariance matrices are caused by neglected linearization errors. This paper introduces a concept for systematically predicting and updating bounds for the linearization errors within the Kalman filtering framework. To achieve this, an uncertain quantity is not characterized by a single probability density anymore, but rather by a set of densities and accordingly, the linear estimation framework is generalized in order to process sets of probability densities. By means of this generalization, the Kalman filter may then not only be applied to stochastic quantities, but also to unknown but bounded quantities. In order to improve the reliability of Kalman filtering results, the last-mentioned quantities are utilized to bound the typically neglected nonlinear parts of a linearized mapping.
V. Klumpp, B. Noack, M. Baum, and U. D. Hanebeck, Combined Set-Theoretic and Stochastic Estimation: A Comparison of the SSI and the CS Filter, in Proceedings of the 13th International Conference on Information Fusion (Fusion 2010), Edinburgh, United Kingdom, Jul. 2010.
Abstract — In estimation theory, mainly set-theoretic or stochastic uncertainty is considered. In some cases, especially when some statistics of a distribution are not known or additional stochastic information is used in a set-theoretic estimator, both types of uncertainty have to be considered. In this paper, two estimators that cope with combined stoachastic and set-theoretic uncertainty are compared, namely the Set-theoretic and Statistical Information filter, which represents the uncertainty by means of random sets, and the Credal State filter, in which the state information is given by sets of probability density functions. The different uncertainty assessment in both estimators leads to different estimation results, even when the prior information and the measurement and system models are equal. This paper explains these differences and states directions, when which estimator should be applied to a given estimation problem.
M. Baum, B. Noack, and U. D. Hanebeck, Extended Object and Group Tracking with Elliptic Random Hypersurface Models, in Proceedings of the 13th International Conference on Information Fusion (Fusion 2010), Edinburgh, United Kingdom, Jul. 2010.
Abstract — This paper provides new results and insights for tracking an extended target object modeled with an Elliptic Random Hypersurface Model (RHM). An Elliptic RHM specifies the relative squared Mahalanobis distance of a measurement source to the center of the target object by means of a one-dimensional random scaling factor. It is shown that uniformly distributed measurement sources on an ellipse lead to a uniformly distributed squared scaling factor. Furthermore, a Bayesian inference mechanisms tailored to elliptic shapes is introduced, which is also suitable for scenarios with high measurement noise. Closed-form expressions for the measurement update in case of Gaussian and uniformly distributed squared scaling factors are derived.
D. Lyons, B. Noack, and U. D. Hanebeck, A Log-Ratio Information Measure for Stochastic Sensor Management, in Proceedings of the IEEE International Conference on Sensor Networks, Ubiquitous, and Trustworthy Computing (SUTC 2010), Newport Beach, California, USA, Jun. 2010.
Abstract — In distributed sensor networks, computational and energy resources are in general limited. Therefore, an intelligent selection of sensors for measurements is of great importance to ensure both high estimation quality and an extended lifetime of the network. Methods from the theory of model predictive control together with information theoretic measures have been employed to pick sensors yielding measurements with high information value. We present a novel information measure that originates from a scalar product on a class of continuous probability densities and apply it to the field of sensor management. Aside from its mathematical justifications for quantifying the information content of probability densities, the most remarkable property of the measure, an analogon of the triangle inequality under Bayesian information fusion, is deduced. This allows for deriving computationally cheap upper bounds for the model predictive sensor selection algorithm and for comparing the performance of planning over different lengths of time horizons.
B. Noack, V. Klumpp, D. Lyons, and U. D. Hanebeck, Systematische Beschreibung von Unsicherheiten in der Informationsfusion mit Mengen von Wahrscheinlichkeitsdichten, in Verteilte Messsysteme, F. Puente León, K.-D. Sommer, and M. Heizmann, Eds. KIT Scientific Publishing, Mar. 2010, pp. 167–178.
D. Lyons, A. Hekler, B. Noack, and U. D. Hanebeck, Maße für Wahrscheinlichkeitsdichten in der informationstheoretischen Sensoreinsatzplanung, in Verteilte Messsysteme, F. Puente León, K.-D. Sommer, and M. Heizmann, Eds. KIT Scientific Publishing, Mar. 2010, pp. 121–132.
Abstract — Bei der Beobachtung eines räumlich verteilten Phänomens mit einer Vielzahl von Sensoren ist die intelligente Auswahl von Messkonfigurationen aufgrund von beschränkten Rechen- und Kommunikationskapazitäten entscheidend für die Lebensdauer des Sensornetzes. Mit der Sensoreinsatzplanung kann die im nächsten Zeitschritt anzusteuernde Messkonfiguration dynamisch mittels einer stochastischen modell-prädiktiven Planung über einen endlichen Zeithorizont bestimmt werden. Dabei wird als Gütekriterium die Maximierung des zu erwartenden Informationsgewinns durch zukünftige Messungen unter sparsamer Verwendung der Energieressourcen gewählt. In diesem Artikel wird ein neues Maß für kontinuierliche Wahrscheinlichkeitsdichten vorgestellt, das sich kanonisch aus der Konstruktion eines Vektorraums für Wahrscheinlichkeitsdichten ergibt. Dieses Maß wird als Gütefunktion in der vorausschauenden Sensoreinsatzplanung zur Bewertung des informationstheoretischen Einfluß von Messungen auf die aktuelle Zustandsschätzung verwendet.
B. Noack, V. Klumpp, and U. D. Hanebeck, State Estimation with Sets of Densities considering Stochastic and Systematic Errors, in Proceedings of the 12th International Conference on Information Fusion (Fusion 2009), Seattle, Washington, USA, Jul. 2009.
Abstract — In practical applications, state estimation requires the consideration of stochastic and systematic errors. If both error types are present, an exact probabilistic description of the state estimate is not possible, so that common Bayesian estimators have to be questioned. This paper introduces a theoretical concept, which allows for incorporating unknown but bounded errors into a Bayesian inference scheme by utilizing sets of densities. In order to derive a tractable estimator, the Kalman filter is applied to ellipsoidal sets of means, which are used to bound additive systematic errors. Also, an extension to nonlinear system and observation models with ellipsoidal error bounds is presented. The derived estimator is motivated by means of two example applications.
B. Noack, V. Klumpp, D. Brunn, and U. D. Hanebeck, Nonlinear Bayesian Estimation with Convex Sets of Probability Densities, in Proceedings of the 11th International Conference on Information Fusion (Fusion 2008), Cologne, Germany, Jul. 2008, pp. 1–8.
Abstract — This paper presents a theoretical framework for Bayesian estimation in the case of imprecisely known probability density functions. The lack of knowledge about the true density functions is represented by sets of densities. A formal Bayesian estimator for these sets is introduced, which is intractable for infinite sets. To obtain a tractable filter, properties of convex sets in form of convex polytopes of densities are investigated. It is shown that pathwise connected sets and their convex hulls describe the same ignorance. Thus, an exact algorithm is derived, which only needs to process the hull, delivering tractable results in the case of a proper parametrization. Since the estimator delivers a convex hull of densities as output, the theoretical grounds are laid for deriving efficient Bayesian estimators for sets of densities. The derived filter is illustrated by means of an example.

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