Algorithm for algebraic curve intersection
Computer-Aided Design
Base points, resultants, and the implicit representation of rational surfaces
Base points, resultants, and the implicit representation of rational surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves
IEEE Transactions on Pattern Analysis and Machine Intelligence
Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
Algebraic and numeric techniques in modeling and robotics
Algebraic and numeric techniques in modeling and robotics
Remark on algorithms to find roots of polynomials
SIAM Journal on Scientific Computing
Using Geometric Distance Fits for 3-D Object Modeling and Recognition
IEEE Transactions on Pattern Analysis and Machine Intelligence
Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms
ACM Transactions on Mathematical Software (TOMS)
Solving algebraic systems using matrix computations
ACM SIGSAM Bulletin
MARS: a MAPLE/MATLAB/C resultant-based solver
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
Statistical Optimization for Geometric Computation: Theory and Practice
Statistical Optimization for Geometric Computation: Theory and Practice
A Survey of Numerical Mathematics
A Survey of Numerical Mathematics
Solving Systems of Polynomial Equations
IEEE Computer Graphics and Applications
Parameterized Families of Polynomials for Bounded Algebraic Curve and Surface Fitting
IEEE Transactions on Pattern Analysis and Machine Intelligence
Statistical Bias of Conic Fitting and Renormalization
IEEE Transactions on Pattern Analysis and Machine Intelligence
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
Sketch based interfaces: early processing for sketch understanding
Proceedings of the 2001 workshop on Perceptive user interfaces
Sketch based interfaces: early processing for sketch understanding
ACM SIGGRAPH 2006 Courses
Sketch based interfaces: early processing for sketch understanding
ACM SIGGRAPH 2007 courses
Stable Algebraic Surfaces for 3D Object Representation
Journal of Mathematical Imaging and Vision
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Implicit polynomials (i.e., multinomials) have a number of properties that make them attractive for modeling curves and surfaces in computer vision. This paper considers the problem of finding the best fitting implicit polynomial (or algebraic curve) to a collection of points in the plane using an orthogonal distance metric. Approximate methods for orthogonal distance regression have been shown by others to be prone to the problem of cusps in the solution and this is confirmed here. Consequently, this work focuses on exact methods for orthogonal distance regression. The most difficult and costly part of exact methods is computing the closest point on the algebraic curve to an arbitrary point in the plane. This paper considers three methods for achieving this in detail. The first is the standard Newton's method, the second is based on resultants which are recently making a resurgence in computer graphics, and the third is a novel technique based on successive circular approximations to the curve. It is shown that Newton's method is the quickest, but that it can fail sometimes even with a good initial guess. The successive circular approximation algorithm is not as fast, but is robust. The resultant method is the slowest of the three, but does not require an initial guess. The driving application of this work was the fitting of implicit quartics in two variables to thinned oblique ionogram traces.