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Iterative circle fitting based on circular attracting factor

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Abstract

An intuitive method for circle fitting is proposed. Assuming an approximate circle (C A,n ) for the fitting of some scattered points, it can be imagined that every point would apply a force to C A,n , which all together form an overall effect that “draws” C A,n towards best fitting to the group of points. The basic element of the force is called circular attracting factor (CAF) which is defined as a real scalar in a radial direction of C A,n . An iterative algorithm based on this idea is proposed, and the convergence and accuracy are analyzed. The algorithm converges uniformly which is proved by the analysis of Lyapunov function, and the accuracy of the algorithm is in accord with that of geometric least squares of circle fitting. The algorithm is adopted to circle detection in grayscale images, in which the transferring to binary images is not required, and thus the algorithm is less sensitive to lightening and background noise. The main point for the adaption is the calculation of CAF which is extended in radial directions of C A,n for the whole image. All pixels would apply forces to C A,n , and the overall effect of forces would be equivalent to a force from the centroid of pixels to C A,n . The forces from would-be edge pixels would overweigh that from noisy pixels, so the following approximate circle would be of better fitting. To reduce the amount of calculation, pixels are only used in an annular area including the boundary of C A,n just in between for the calculation of CAF. Examples are given, showing the process of circle fitting of scattered points around a circle from an initial assuming circle, comparing the fitting results for scattered points from some related literature, applying the method proposed for circular edge detection in grayscale images with noise, and/or with only partial arc of a circle, and for circle detection in BGA inspection.

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Authors and Affiliations

  1. College of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, China

    Heng-sheng Wang  (王恒升), Qiang Zhang  (张强) & Fu-liang Wang  (王福亮)

  2. State Key Laboratory for High Performance Complex Manufacturing (Central South University), Changsha, 410083, China

    Heng-sheng Wang  (王恒升) & Fu-liang Wang  (王福亮)

Authors
  1. Heng-sheng Wang  (王恒升)
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  2. Qiang Zhang  (张强)
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  3. Fu-liang Wang  (王福亮)
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Corresponding author

Correspondence to Heng-sheng Wang  (王恒升).

Additional information

Foundation item: Project(2013CB035504) supported by the National Basic Research Program of China; Project(2012zzts078) supported by the Fundamental Research Funds for the Central Universities of Central South University, China; Project(2009ZX02038) supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China

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Wang, Hs., Zhang, Q. & Wang, Fl. Iterative circle fitting based on circular attracting factor. J. Cent. South Univ. 20, 2663–2675 (2013). https://doi.org/10.1007/s11771-013-1782-6

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  • DOI: https://doi.org/10.1007/s11771-013-1782-6

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