Introduction - The aims of this book the features of this book organization and background the analytical mind: strengh and weakness. Fundamentals of linear algebra - Vector and matrix calculus Eigenvalue problem linear systems and optimization matrix and tensor algebra. Probabilities and statistical estimation - probability distributions manifolds and local distributions gaussian distributions and X2 distributions statistical estimation for gaussian models general statistical estimation maximum likelihood estimation Akaike information criterion. Representation of geometric objects - image points and image lines space points and space lines space planes conics space conics and quadrics coordinate transformation and projection. Geometric correction - general theory correction of image points and image lines correction of space points and space lines correction of space planes orthogonality correction conic incidence correction. 3-D computation by stereo vision - epipolar constraint optimal correction of correspondence 3-D reconstruction of points 3-D reconstruction of lines optimal back projection onto a space plane scenes infinitely far away camera calibration errors. Parametric fitting - general theory optimal fitting for image points optimal fitting for image lines optimal fitting for space points optimal fitting for space lines optimal fitting for space planes. Optimal filter - general theory iterative estimation scheme effective gradient approximation reduction from the klaman filter estimation from linear hypotheses. Renormalization - eigenvector fit unbiased eigenvector generalized eigenvalue fit renormalization lincarization second order renormalization. Applications of geometric estimation - image line fitting conic fitting space plane fitting by range sensing space plane fitting by stereo vision. 3-D motion analysis - general theory lincarization and renormalization optimal correction and decomposition reliability of 3-D reconstruction critical surfaces 3-D reconstruction from planar surface motion camera rotation and information. 3-D interpretation of optical flow - optical flow detection theoretical basis of 3-D interpretation optimal estimation of motion parameters. (Part contents).

Statistical optimization for geometric computation : theory and practice

In Bayesian multi-target filtering, we have to contend with two notable sources of uncertainty, clutter and detection. Knowledge of parameters such as clutter rate and detection profile are of critical importance in multi-target filters such as the probability hypothesis density (PHD) and cardinalized PHD (CPHD) filters. Significant mismatches in clutter and detection model parameters result in biased estimates. In practice, these model parameters are often manually tuned or estimated offline from training data. In this paper we propose PHD/CPHD filters that can accommodate model mismatch in clutter rate and detection profile. In particular we devise versions of the PHD/CPHD filters that can adaptively learn the clutter rate and detection profile while filtering. Moreover, closed-form solutions to these filtering recursions are derived using Beta and Gaussian mixtures. Simulations are presented to verify the proposed solutions.

CPHD Filtering With Unknown Clutter Rate and Detection Profile

Target tracking algorithms usually assume that the received measurements stem from a point source. However, in many scenarios this assumption is not feasible so that measurements may stem from different locations, named measurement sources, on the target surface. Then, it is necessary to incorporate the target extent into the estimation procedure in order to obtain robust and precise estimation results. This paper introduces the novel concept of Random Hypersurface Models for extended targets. A Random Hypersurface Model assumes that each measurement source is an element of a randomly generated hypersurface. The applicability of this approach is demonstrated by means of an elliptic target shape. In this case, a Random Hypersurface Model specifies the random (relative) Mahalanobis distance of a measurement source to the center of the target object. As a consequence, good estimation results can be obtained even if the true target shape significantly differs from the modeled shape. Additionally, Random Hypersurface Models are computationally tractable with standard nonlinear stochastic state estimators.

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Random Hypersurface Models for extended object tracking

The random hypersurface model (RHM) is introduced for estimating a shape approximation of an extended object in addition to its kinematic state. An RHM represents the spatial extent by means of randomly scaled versions of the shape boundary. In doing so, the shape parameters and the measurements are related via a measurement equation that serves as the basis for a Gaussian state estimator. Specific estimators are derived for elliptic and star-convex shapes.

Extended Object Tracking with Random Hypersurface Models

The Random Hypersurface Model (RHM) is introduced that allows for estimating a shape approximation of an extended object in addition to its kinematic state. An RHM represents the spatial extent by means of randomly scaled versions of the shape boundary. In doing so, the shape parameters and the measurements are related via a measurement equation that serves as the basis for a Gaussian state estimator. Specific estimators are derived for elliptic and star-convex shapes.

This paper is concerned with distances for comparing multivariate random vectors with a special focus on the case that at least one of the random vectors is of discrete type, i.e., assumes values from a discrete set only. The first contribution is a new type of characterization of multivariate random quantities, the so called localized cumulative distribution (LCD) that, in contrast to the conventional definition of a cumulative distribution, is unique and symmetric. Based on the LCDs of the random vectors under consideration, the second contribution is the definition of generalized distance measures that are suitable for the multivariate case. These distances are used for both analysis and synthesis purposes. Analysis is concerned with assessing whether a given sample stems from a given continuous distribution. Synthesis is concerned with both density estimation, i.e., calculating a suitable continuous approximation of a given sample, and density discretization, i.e., approximation of a given continuous random vector by a discrete one.

Localized Cumulative Distributions and a multivariate generalization of the Cramér-von Mises distance

For the optimal approximation of multivariate Gaussian densities by means of Dirac mixtures, i.e., by means of a sum of weighted Dirac distributions on a continuous domain, a novel systematic method is introduced. The parameters of this approximate density are calculated by minimizing a global distance measure, a generalization of the well-known Cramervon Mises distance to the multivariate case. This generalization is obtained by defining an alternative to the classical cumulative distribution, the Localized Cumulative Distribution (LCD). In contrast to the cumulative distribution, the LCD is unique and symmetric even in the multivariate case. The resulting deterministic approximation of Gaussian densities by means of discrete samples provides the basis for new types of Gaussian filters for estimating the state of nonlinear dynamic systems from noisy measurements.

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Dirac mixture approximation of multivariate Gaussian densities

This paper introduces a novel recursive Bayesian estimator for the center and radius of a circle based on noisy points. Each given point is assumed to be a noisy measurement of an unknown true point on the circle that is corrupted with known isotropic Gaussian noise. In contrast to existing approaches, the novel method does not make assumptions about the true points on the circle, where the measurements stem from. Closed-form expressions for the measurement update step are derived. Simulations show that the novel method outperforms standard Bayesian approaches for circle fitting.

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A novel Bayesian method for fitting a circle to noisy points

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.

State estimation with sets of densities considering stochastic and systematic errors

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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Vesa Klumpp

Papers

Localized Cumulative Distributions and a multivariate generalization of the Cramér-von Mises distance

Dirac mixture approximation of multivariate Gaussian densities

A novel Bayesian method for fitting a circle to noisy points

State estimation with sets of densities considering stochastic and systematic errors

Nonlinear Bayesian estimation with convex sets of probability densities