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106
Computing Exact Aspect Graphs of Curved Objects: Algebraic Surfaces
"... This paper presents an algorithm for computing the exact aspect graph of an opaque solid bounded by a smooth algebraic surface and observed under orthographic projection. The algorithm uses curve tracing, cell decomposition, and ray tracing to construct the regions of the view sphere delineated by ..."
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Cited by 81 (10 self)
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This paper presents an algorithm for computing the exact aspect graph of an opaque solid bounded by a smooth algebraic surface and observed under orthographic projection. The algorithm uses curve tracing, cell decomposition, and ray tracing to construct the regions of the view sphere delineated by visual events. It has been fully implemented, and examples are presented.
Continuation and Path Following
, 1992
"... CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful ..."
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Cited by 70 (6 self)
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CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (18811886), Klein (1882 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
, 2001
"... In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposi ..."
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Cited by 56 (26 self)
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In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As byproducts, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.
Efficient Synthesis of Stringed Musical Instruments
, 1993
"... Techniques are described for reducing complexity in stringed instrument simulation for purposes of digital synthesis. These include commuting losses and dispersion to consolidate them into a single lter, replacing body resonators by lookup tables, simplied bowstring interaction, and singlelter ..."
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Cited by 52 (1 self)
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Techniques are described for reducing complexity in stringed instrument simulation for purposes of digital synthesis. These include commuting losses and dispersion to consolidate them into a single lter, replacing body resonators by lookup tables, simplied bowstring interaction, and singlelter, multiplyfree coupled strings implementation. Contents 1 Digital Waveguide Theory 2 2 The Terminated String 4 3 Simplied Body Filters 5 4 Simplied Bowed Strings 8 5 Coupled Strings 10 6 Summary 14 7 Appendix 14 1 Page 2 1 Digital Waveguide Theory This section summarizes the digital waveguide model for vibrating strings. Further details can be found in [Smith 1992]. Position y (t,x) 0 x . . . . . . 0 K String Tension e = Mass/Length Figure 1: The ideal vibrating string. The wave equation for the ideal (lossless, linear, exible) vibrating string, depicted in Fig. 1, is given by Ky 00 = y where K = string tension y = y(t; x) = linear mass density _ y...
Numerical Homotopies to compute generic Points on positive dimensional Algebraic Sets
 Journal of Complexity
, 1999
"... Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for fourbar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the com ..."
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Cited by 50 (24 self)
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Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for fourbar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to...
Some tests of generalized bisection
 ACM Trans. Math. Software
, 1987
"... This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically co ..."
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Cited by 48 (2 self)
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This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically compares such generalized bisection algorithms to themselves, to continuation methods, and to hybrid steepest descent/quasiNewton methods. A specific algorithm containing novel “expansion ” and “exclusion ” steps is fully described, and the effectiveness of these steps is evaluated. A test problem consisting of a small, highdegree polynomial system that is appropriate for generalized bisection, but very difticult for continuation methods, is presented. This problem forms part of a set of 17 test problems from published literature on the methods being compared; this test set is fully described here.
Relative orientation revisited
 Journal of the Optical Society of America A
, 1991
"... Relative Orientation is the recovery of the position and orientation of one imaging system relative to another from correspondences between five or more ray pairs. It is one of four core problems in photogrammetry and is of central importance in binocular stereo, as well as in long range motion visi ..."
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Cited by 34 (1 self)
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Relative Orientation is the recovery of the position and orientation of one imaging system relative to another from correspondences between five or more ray pairs. It is one of four core problems in photogrammetry and is of central importance in binocular stereo, as well as in long range motion vision. While five ray correspondences are sufficient to yield a finite number of solutions, more than five correspondences are used in practice to ensure an accurate solution using least squares methods. Most iterative schemes for minimizing the sum of squares of weighted errors require a good guess as a starting value. The author has previously published a method that finds the best solution without requiring an initial guess. In this paper an even simpler method is presented that utilizes the representation of rotations by unit quaternions. 1.
See also: ``Relative Orientation,''
{\it International Journal of Computer Vision},
Vol.~4, No.~1, pp.~5978, January 1990.
Game Networks
 In Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence (UAI
, 2000
"... We introduce Game networks (G nets), a novel representation for multiagent decision problems. Compared to other gametheoretic representations, such as strategic or extensive forms, G nets are more structured and more compact; more fundamentally, G nets constitute a computationally advantageo ..."
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Cited by 33 (0 self)
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We introduce Game networks (G nets), a novel representation for multiagent decision problems. Compared to other gametheoretic representations, such as strategic or extensive forms, G nets are more structured and more compact; more fundamentally, G nets constitute a computationally advantageous framework for strategic inference, as both probability and utility independencies are captured in the structure of the network and can be exploited in order to simplify the inference process. An important aspect of multiagent reasoning is the identification of some or all of the strategic equilibria in a game; we present original convergence methods for strategic equilibrium which can take advantage of strategic separabilities in the G net structure in order to simplify the computations. Specifically, we describe a method which identifies a unique equilibrium as a function of the game payo#s, and one which identifies all equilibria. 1 Introduction The formal analysis of m...
PHoM  a Polyhedral Homotopy Continuation Method for Polynomial Systems
 Computing
, 2003
"... PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedrallinear homotopy functions, based on the polyhedral ..."
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Cited by 31 (13 self)
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PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedrallinear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f (x) = 0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f (x) = 0. The third module Verify checks whether all isolated solutions of f (x) = 0 have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.