A Buyer's Guide to Euclidean Elliptical Cylindrical and Conical Surface Fitting

Faber, Petko; Fisher, Robert B.

doi:10.5244/c.15.54

Cited by 7 publications

(2 citation statements)

References 17 publications

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“…It uses the internal ellipse parameter t as described in Section 11. Another is the most popular (see, e.g., [31]) Levenberg-Marquardt algorithm using the geometric parameters of the ellipse, we call it LMG. One more is the Levenberg-Marquardt algorithm using our algebraic parameters, we call it LMA.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Fitting Quadratic Curves to Data Points

Chernov

Huang

2014

BJMCS

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Fitting quadratic curves (a.k.a. conic sections, or conics) to data points (digitized images) is a fundamental task in image processing and computer vision. This problem reduces to minimization of a certain function over the parameter space of conics. Here we undertake a thorough investigation of that space and the properties of the objective function on it. We determine under what conditions that function is continuous and differentiable. We identify its discontinuities and other singularities and determined what effect those have on the performance of minimization algorithms. Our analysis shows that algebraic parameters of conics are more suitable for minimization procedures than more popular geometric parameters, for a number of reasons. First, the space of parameters is naturally compact, thus their estimated values cannot grow indefinitely causing divergence. Second, with algebraic parameters minimization procedures can move freely and smoothly between conics of different types allowing shortcuts and faster convergence. Third, with algebraic parameters one avoids known issues occurring when the fitting conic becomes a circle. To support our conclusions we prove a dozen of mathematical theorems and provide a plenty of illustrations.

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Section: Numerical Experimentsmentioning

confidence: 99%

Fitting Quadratic Curves to Data Points

Chernov

Huang

2014

BJMCS

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“…(Closed forms exist for planes, elliptical cylinders and cones, which are a very practical subset of the quadric surfaces.) Recently we have reinvestigated this question because of the recent dramatic increase in computational power [17,16]. As well as exposing the great difference in fit quality, we have investigated the computational costs.…”

Section: Euclidean Distance Is Better and Fastmentioning

confidence: 99%

Applying knowledge to reverse engineering problems

Fisher

2004

Computer-Aided Design

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:This paper summarizes a series of recent research results made at Edinburgh University based on projects that apply domain knowledge of standard shapes and relationships to solve or improve reverse engineering problems. The problems considered are how to enforce known relationships when data fitting, how to extract features even in very noisy data, how to get better shape parameter estimates and how to infer data about unseen features.Keywords : range data, reverse engineering, constraints Copyright c 2004 by The University of Edinburgh. All Rights ReservedThe authors and the University of Edinburgh retain the right to reproduce and publish this paper for non-commercial purposes.Permission is granted for this report to be reproduced by others for non-commercial purposes as long as this copyright notice is reprinted in full in any reproduction. Applications to make other use of the material should be addressed in the first instance to Copyright Permissions, School of Informatics, The University of Edinburgh, 2 Buccleuch Place, Edinburgh EH8 9LW, Scotland. DRAFT: Applying knowledge to reverse engineering problemsRobert B. Fisher School of Informatics, University of Edinburgh rbf@inf.ed.ac.uk AbstractThis paper summarizes a series of recent research results made at Edinburgh University based on projects that apply domain knowledge of standard shapes and relationships to solve or improve reverse engineering problems. The problems considered are how to enforce known relationships when data fitting, how to extract features even in very noisy data, how to get better shape parameter estimates and how to infer data about unseen features.

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