476
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 5, MAY 1999
[22] M.K. Hu, “Visual Pattern Recognition by Moment Invariants,”
IRE Trans. Information Theory, vol. 8, 179-187, 1962.
[23] 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.
[24] H. Wely, Symmetry, Princeton Univ. Press, 1952.
[25] W. Miller, Symmetry Groups and their Application, Academic
Press, 1972.
[26] 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.
[27] 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.
[28] 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
implement.
Index Terms—Algebraic models, ellipse fitting, least squares fitting,
constrained minimization, generalized eigenvalue problem.
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1
INTRODUCTION
THE fitting of primitive models to image data is a basic task in
pattern