A note on the least squares fitting of ellipses

Rosin, P.L., A note on the least squares fitting of ellipses, Pattern Recognition Letters 14 (1993) 799-808. The characteristics of two normalisations for the general conic equation are investigated for use in least squares fitting: either setting F= 1 or A + C= 1. The normalisations vary in three main areas: curvature bias, singularities, transformational invariance. It is shown that setting F= 1 is the more appropriate for ellipse fitting since it is less heavily curvature biased. Setting A + C= 1 produces more eccentric conics, resulting either in over-elongated ellipses or hyperbolae. Although the F= 1 normalisation is less well suited than the A + C= 1 normalisation with respect to singularities and transformational invariance both these problems are solved by normalising the data, shifting it so that it is centred on the origin before fitting, and then re-expressing the fit in the original frame of reference.

[1]  Paul L. Rosin.  Acquiring information from cues , 1993, Pattern Recognit. Lett..

[2]  Geoff A. W. West,et al.  Segmenting curves into elliptic arcs and straight lines , 1990, [1990] Proceedings Third International Conference on Computer Vision.

[3]  O. Faugeras,et al.  Computer Vision — ECCV 90 , 1990, Lecture Notes in Computer Science.

[4]  Azriel Rosenfeld,et al.  A note on polygonal and elliptical approximation of mechanical parts , 1979, Pattern Recognit..

[5]  Arthur C. Sanderson,et al.  Determining Object Orientation Using Ellipse Fitting , 1985, Other Conferences.

[6]  John Porrill,et al.  Fitting ellipses and predicting confidence envelopes using a bias corrected Kalman filter , 1990, Image Vis. Comput..

[7]  Theodosios Pavlidis,et al.  Curve Fitting with Conic Splines , 1983, TOGS.

[8]  Allen R. Hanson,et al.  Computer Vision Systems , 1978 .

[9]  Allen R. Hanson,et al.  Fast extraction of ellipses , 1988, [1988 Proceedings] 9th International Conference on Pattern Recognition.

[10]  PAUL D. SAMPSON,et al.  Fitting conic sections to "very scattered" data: An iterative refinement of the bookstein algorithm , 1982, Comput. Graph. Image Process..

[11]  F. Bookstein.  Fitting conic sections to scattered data , 1979 .

[12]  John Porrill.  Fitting ellipses and predicting confidence envelopes using a bias corrected Kalman filter , 1990, Image Vis. Comput..

[13]  Tadashi Nagata,et al.  Detection of an ellipse by use of a recursive least-squares estimator , 1985, J. Field Robotics.

[14]  Geoff A. W. West,et al.  Perceptual grouping of circular arcs under projection , 1990, BMVC.

[15]  Vishvjit S. Nalwa,et al.  Edgel aggregation and edge description , 1987, Comput. Vis. Graph. Image Process..

[16]  Gerald J. Agin.  Fitting Ellipses and General Second-Order Curves. , 1981 .

[17]  Dana H. Ballard,et al.  Computer Vision , 1982 .

[18]  D. Cooper,et al.  On the cost of approximating and recognizing a noise perturbed straight line or a quadratic curve segment in the plane. [central processing units] , 1975 .

[19]  Beno Benhabib,et al.  Accurate parameter estimation of quadratic curves from grey-level images , 1991, CVGIP Image Underst..

[20]  Antonio Albano,et al.  Representation of Digitized Contours in Terms of Conic Arcs and Straight-Line Segments , 1974, Comput. Graph. Image Process..

[21]  Terry Caelli,et al.  Model-Based 3D Object Localization and Recognization from a single Intensity Image , 1989, Computer Vision and Shape Recognition.

[22]  C. L. Huang,et al.  Elliptical feature extraction via an improved Hough transform , 1989, Pattern Recognit. Lett..

[23]  Robert C. Bolles,et al.  A RANSAC-Based Approach to Model Fitting and Its Application to Finding Cylinders in Range Data , 1981, IJCAI.

[24]  Saburo Tsuji,et al.  Detection of Ellipses by a Modified Hough Transformation , 1978, IEEE Transactions on Computers.

[25]  David A. Forsyth,et al.  Projectively invariant representations using implicit algebraic curves , 1991, Image Vis. Comput..

[26]  David B. Cooper,et al.  On the Computational Cost of Approximating and Recognizing Noise-Perturbed Straight Lines and Quadratic Arcs in the Plane , 1976, IEEE Transactions on Computers.