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Uncertainty in Geometric Computations

Computer simulations and modelling are used frequently inscience and engineering, in applications ranging from theunderstanding of natural and artificial phenomena to the design, testand manufacturing stages of production. This widespread usenecessarily implies that a detailed knowledge of the limitations ofcomputer simulations is required. In particular, the usefulness of acomputer simulation is directly dependent on the user's knowledge ofthe uncertainty in the simulation. Typical limitations of computersimulations include uncertainty in the data, parameter uncertainty, errors in the initial data, modelling errors, unmodelled phenomena, reduced order models, and approximations and numerical errors.Although an improvement in the physical understanding of the phenomenabeing modelled is an important requirement of a good computersimulation, the simulation will be plagued by deficiencies if thelimitations listed above are not considered when analyzing itsresults. Since uncertainties can never be completely eliminated, theymust be quantified and their propagation through the computations mustbe considered. The uses of computer modelling are diverse, and oneparticular application, the effect of uncertainty in geometriccomputations, is considered in this book. In particular, geometriccomputations occur extensively in geometric modelling, computervision, computer graphics and pattern recognition."Uncertainty in Geometric Computations" contains the proceedingsof a workshop that was held in Sheffield, United Kingdom, in which themanagement and assessment of uncertainty in geometric computations wasconsidered. The theme that unites these four subject areas is therequirement to performcomputations on real geometric data, which (i)may have errors, for example, the tolerance of a coordinate measuringmachine that is used in reverse engineering, and/or (ii) is incompletebecause of occlusion, which may occur in computer vision, for example, a face recognition system. These characteristics of real geometricdata impose tight constraints on the methods and algorithms that areused for their processing and interrogation, and this workshopprovided a forum for their discussion.One of the novel features of the workshop was the wide background ofthe audience and invited speakers - applied mathematicians, computer scientists and engineers - and this provided a forumfor the establishment of new collaborative links betweenmathematicians and engineers, thereby emphasizing theinterdisciplinary nature of the many outstanding problems.
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Table of Contents

Contributors and Participants. Preface. 1. Affine Intervals in a CSG Geometric Modeller; A. Bowyer, et al. 2. Fast and Reliable Plotting of Implicit Curves; K. Buehler. 3. Data Assimilation with Sequential Gaussian Processes; L. Csato, et al. 4. Conformal Geometry, Euclidean Space and Geometric Algebra; C. Doran, et al. 5. Towards the Robust Intersection of Implicit Quadrics; L. Dupont, et al. 6. Computational Geometry and Uncertainty; A.R. Forrest. 7. Geometric Uncertainty in Solid Modeling; P.J. Malraison, W.A. Denker. 8. Reliable Geometric Computations with Algebraic Primitives and Predicates; M. Foskey, et al. 9. Feature Localization Error in 3D Computer Vision; D.D. Morris, T. Kanade. 10. Bayesian Analysis of Computer Model Outputs; J. Oakley, A. O'Hagan. 11. b.; T. Poggio, et al. 12. Affine Arithmetic and Bernstein Hull Methods for Algebraic Curve Drawing; H. Shou, et al. 13. Local Polynomial Metrics for kappa Nearest Neighbor Classifiers; R.R. Snapp. 14. Visualisation of Incomplete Data Using Class Information Constraints; Y. Sun, et al. 15. Towards Videorealistic Synthetic Visual Speech; B. Theobald, et al. 16. Properties of the Companion Matrix Resultant for Bernstein Polynomials; J.R. Winkler. 17. Computation with a Number of Neurons; S. Wu, D. Chen. Index.

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