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Nicholas J. Redding, "Implicit Polynomials, Orthogonal Distance Regression, and the Closest Point on a Curve," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 2, pp. 191199, February, 2000.  
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@article{ 10.1109/34.825757, author = {Nicholas J. Redding}, title = {Implicit Polynomials, Orthogonal Distance Regression, and the Closest Point on a Curve}, journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence}, volume = {22}, number = {2}, issn = {01628828}, year = {2000}, pages = {191199}, doi = {http://doi.ieeecomputersociety.org/10.1109/34.825757}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Pattern Analysis and Machine Intelligence TI  Implicit Polynomials, Orthogonal Distance Regression, and the Closest Point on a Curve IS  2 SN  01628828 SP191 EP199 EPD  191199 A1  Nicholas J. Redding, PY  2000 KW  Fitting KW  orthogonal distance regression KW  implicit polynomials KW  algebraic curve KW  successive circular approximation KW  resultants KW  ionograms. VL  22 JA  IEEE Transactions on Pattern Analysis and Machine Intelligence ER   
Abstract—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.
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