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29
On Entropy Approximation for Gaussian Mixture Random Vectors
"... Abstract — For many practical probability density representations such as for the widely used Gaussian mixture densities, an analytic evaluation of the differential entropy is not possible and thus, approximate calculations are inevitable. For this purpose, the first contribution of this paper deals ..."
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Cited by 15 (1 self)
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Abstract — For many practical probability density representations such as for the widely used Gaussian mixture densities, an analytic evaluation of the differential entropy is not possible and thus, approximate calculations are inevitable. For this purpose, the first contribution of this paper deals with a novel entropy approximation method for Gaussian mixture random vectors, which is based on a componentwise Taylorseries expansion of the logarithm of a Gaussian mixture and on a splitting method of Gaussian mixture components. The employed order of the Taylorseries expansion and the number of components used for splitting allows balancing between accuracy and computational demand. The second contribution is the determination of meaningful and efficiently to calculate lower and upper bounds of the entropy, which can be also used for approximation purposes. In addition, a refinement method for the more important upper bound is proposed in order to approach the true entropy value. I.
ClosedForm Prediction of Nonlinear Dynamic Systems by Means of Gaussian Mixture Approximation of the Transition Density
 in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems
"... Abstract — Recursive prediction of the state of a nonlinear stochastic dynamic system cannot be efficiently performed in general, since the complexity of the probability density function characterizing the system state increases with every prediction step. Thus, representing the density in an exact ..."
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Cited by 15 (13 self)
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Abstract — Recursive prediction of the state of a nonlinear stochastic dynamic system cannot be efficiently performed in general, since the complexity of the probability density function characterizing the system state increases with every prediction step. Thus, representing the density in an exact closedform manner is too complex or even impossible. So, an appropriate approximation of the density is required. Instead of directly approximating the predicted density, we propose the approximation of the transition density by means of Gaussian mixtures. We treat the approximation task as an optimization problem that is solved offline via progressive processing to bypass initialization problems and to achieve high quality approximations. Once having calculated the transition density approximation offline, prediction can be performed efficiently resulting in a closedform density representation with constant complexity. I.
Bayesian Estimation of Distributed Phenomena using Discretized Representations of Partial Differential Equations
 in 3rd International Conference on Informatics in Control, Automation and Robotics (ICINCO’06), Setubal
, 2006
"... This paper addresses a systematic method for the reconstruction and the prediction of a distributed phenomenon characterized by partial differential equations, which is monitored by a sensor network. In the first step, the infinitedimensional partial differential equation, i.e. distributedparamete ..."
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Cited by 15 (10 self)
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This paper addresses a systematic method for the reconstruction and the prediction of a distributed phenomenon characterized by partial differential equations, which is monitored by a sensor network. In the first step, the infinitedimensional partial differential equation, i.e. distributedparameter system, is spatially and temporally decomposed leading to a finitedimensional state space form. In the next step, the state of the resulting lumpedparameter system, which provides an approximation of the solution of the underlying partial differential equations, is dynamically estimated under consideration of uncertainties both occurring in the system and arising from noisy measurements. By using the estimation results, several additional tasks can be achieved by the sensor network, e.g. optimal sensor placement, optimal scheduling, and model improvement. The performance of the proposed modelbased reconstruction method is demonstrated by means of simulations. 1
Modelbased Motion Estimation of Elastic Surfaces for Minimally Invasive Cardiac Surgery
 in IEEE International Conference on Robotics and Automation (ICRA
, 2007
"... Abstract — In order to assist surgeons during surgery on moving organs, e.g. minimally invasive beating heart bypass surgery, a masterslave system which synchronizes surgical instruments with the organ’s motion is desired. This synchronization requires reliable estimation of the organ’s motion. In ..."
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Cited by 8 (5 self)
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Abstract — In order to assist surgeons during surgery on moving organs, e.g. minimally invasive beating heart bypass surgery, a masterslave system which synchronizes surgical instruments with the organ’s motion is desired. This synchronization requires reliable estimation of the organ’s motion. In this paper, we present a new approach to motion estimation based on a state motion model for a partition of the heart’s surface. Its motion behavior is described by a partial differential equation whose input function is assumed to be periodic. An estimator is used on one hand to predict future model states based on reconstruction of the input function and on the other hand to incorporate noisy spatially discrete measurements in order to improve state estimation. The modelbased motion estimation is evaluated using a simple heart simulator. Measurements are obtained by reconstructing 3D position of markers on a pulsating membrane by means of a stereo camera system. head mounted display View on the stagnant heart haptical interface Fig. 1. image processing estimation of heart motion manipulator control View on the beating heart manipulator Overview of the system architecture. camera system I.
The Hybrid Density Filter for Nonlinear Estimation based on Hybrid Conditional Density Approximation
 In Proceedings of the 10th International Conference on Information Fusion (Fusion
, 2007
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Efficient Nonlinear Bayesian Estimation based on Fourier Densities
 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems
, 2006
"... Abstract — Efficiently implementing nonlinear Bayesian estimators is still not a fully solved problem. For practical applications, a tradeoff between estimation quality and demand on computational resources has to be found. In this paper, the use of nonnegative Fourier series, socalled Fourier den ..."
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Cited by 5 (4 self)
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Abstract — Efficiently implementing nonlinear Bayesian estimators is still not a fully solved problem. For practical applications, a tradeoff between estimation quality and demand on computational resources has to be found. In this paper, the use of nonnegative Fourier series, socalled Fourier densities, for Bayesian estimation is proposed. By using the absolute square of Fourier series for the density representation, it is ensured that the density stays nonnegative. Nonetheless, approximation of arbitrary probability density functions can be made by using the Fourier integral formula. An efficient Bayesian estimator algorithm with constant complexity for nonnegative Fourier series is derived and demonstrated by means of an example. I.
Progressive Gaussian Mixture Reduction
 in Proceedings of the 11th International Conference on Information Fusion (Fusion 2008
, 2008
"... Abstract—For estimation and fusion tasks it is inevitable to approximate a Gaussian mixture by one with fewer components to keep the complexity bounded. Appropriate approximations can be typically generated by exploiting the redundancy in the shape description of the original mixture. In contrast to ..."
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Cited by 4 (3 self)
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Abstract—For estimation and fusion tasks it is inevitable to approximate a Gaussian mixture by one with fewer components to keep the complexity bounded. Appropriate approximations can be typically generated by exploiting the redundancy in the shape description of the original mixture. In contrast to the common approach of successively merging pairs of components to maintain a desired complexity, the novel Gaussian mixture reduction algorithm introduced in this paper avoids to directly reduce the original Gaussian mixture. Instead, an approximate mixture is generated from scratch by employing homotopy continuation. This allows starting the approximation with a single Gaussian, which is constantly adapted to the progressively incorporated true Gaussian mixture. Whenever a userdefined bound on the deviation of the approximation cannot be maintained during the continuation, further components are added to the approximation. This facilitates significantly reducing the number of components even for complex Gaussian mixtures.
Nonlinear Multidimensional Bayesian Estimation with Fourier Densities
"... Abstract — Efficiently implementing nonlinear Bayesian estimators is still an unsolved problem, especially for the multidimensional case. A tradeoff between estimation quality and demand on computational resources has to be found. Using multidimensional Fourier series as representation for probabil ..."
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Cited by 4 (4 self)
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Abstract — Efficiently implementing nonlinear Bayesian estimators is still an unsolved problem, especially for the multidimensional case. A tradeoff between estimation quality and demand on computational resources has to be found. Using multidimensional Fourier series as representation for probability density functions, so called Fourier densities, is proposed. To ensure nonnegativity, the approximation is performed indirectly via Ψdensities, of which the absolute square represent the Fourier density. It is shown that Ψdensities can be determined using the efficient fast Fourier transform algorithm and their coefficients have an ordering with respect to the Hellinger metric. Furthermore, the multidimensional Bayesian estimator based on Fourier densities is derived in closed form. That allows an efficient realization of the Bayesian estimator where the demands on computational resources are adjustable. I.
Parameterized Joint Densities with Gaussian and Gaussian Mixture Marginals
 Proc. of the 9th International Conference on Information Fusion 2006
, 2006
"... Abstract In this paper we attempt to lay the foundation for a novel filtering technique for the fusion of two random vectors with imprecisely known stochastic dependency. This problem mainly occurs in decentralized estimation, e.g., of a distributed phenomenon, where the stochastic dependencies bet ..."
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Cited by 4 (2 self)
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Abstract In this paper we attempt to lay the foundation for a novel filtering technique for the fusion of two random vectors with imprecisely known stochastic dependency. This problem mainly occurs in decentralized estimation, e.g., of a distributed phenomenon, where the stochastic dependencies between the individual states are not stored. Thus, we derive parameterized joint densities with both Gaussian marginals and Gaussian mixture marginals. These parameterized joint densities contain all information about the stochastic dependencies between their marginal densities in terms of a parameter vector ξ, which can be regarded as a generalized correlation parameter. Unlike the classical correlation coefficient, this parameter is a sufficient measure for the stochastic dependency even characterized by more complex density functions such as Gaussian mixtures. Once this structure and the bounds of these parameters are known, bounding densities containing all possible density functions could be found.
A State Estimator for Nonlinear Stochastic Systems Based on Dirac Mixture Approximations
 in Proceedings of the 4th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2007
, 2007
"... This paper presents a filter approach for estimating the state of nonlinear dynamic systems based on recursive approximation of posterior densities by means of Dirac mixture functions. The filter consists of a prediction step and a filter step. The approximation approach is based on a systematic min ..."
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Cited by 4 (3 self)
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This paper presents a filter approach for estimating the state of nonlinear dynamic systems based on recursive approximation of posterior densities by means of Dirac mixture functions. The filter consists of a prediction step and a filter step. The approximation approach is based on a systematic minimization of a distance measure and is hence optimal and deterministic. In contrast to nondeterministic methods we are able to determine the optimal number of components in the Dirac mixture. A further benefit of the proposed approach is the consideration of measurements during the approximation process in order to avoid parameter degradation. NOTATION k xk yk