IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 5, MAY 1999
 M.K. Hu, “Visual Pattern Recognition by Moment Invariants,”
IRE Trans. Information Theory, vol. 8, 179-187, 1962.
 J. Bigun and J.M.H. du Buf, “N-Folded Symmetries by Complex
Moments in Gabor Space and their Application to Unsupervised
Texture Segmentation,” IEEE Trans. Pattern Analysis and Machine
Intelligence, vol. 16, no. 1, pp. 80-87, 1994.
 H. Wely, Symmetry, Princeton Univ. Press, 1952.
 W. Miller, Symmetry Groups and their Application, Academic
 D. Shen, H.H.S. Ip, and E.K. Teoh, “Robust Detection of Skewed
Symmetries by Combining Local and Semi-Local Affine Invari-
ants,” IEEE Trans. Pattern Analysis and Machine Intelligence, sub-
mitted for publication.
 H.H.S. Ip, D. Shen, and K.K.T. Cheung, “Indexing and Retrieval
of Binary Patterns Using Generalized Complex Moments,” Proc.
Second Int’l Conf. Visual Information System, California, Mar. 1997.
 C. Sun, “Symmetry Detection Using Gradient Information,”
Pattern Recognition Letters, vol. 16, no. 9, pp. 987-996, 1995.
Direct Least Square Fitting of Ellipses
Andrew Fitzgibbon, Maurizio Pilu, and Robert B. Fisher
Abstract—This work presents a new efficient method for fitting ellipses
to scattered data. Previous algorithms either fitted general conics or
were computationally expensive. By minimizing the algebraic distance
subject to the constraint 4ac - b2 = 1, the new method incorporates the
ellipticity constraint into the normalization factor. The proposed method
combines several advantages: It is ellipse-specific, so that even bad
data will always return an ellipse. It can be solved naturally by a
generalized eigensystem. It is extremely robust, efficient, and easy to
Index Terms—Algebraic models, ellipse fitting, least squares fitting,
constrained minimization, generalized eigenvalue problem.
———————— F ————————
THE fitting of primitive models to image data is a basic task in