# Differential Evolution for Reverse Engineering Using Cubic Ball Curve

## Keywords:

differential evolution, cubic Ball function, reverse engineering, curve fitting## Abstract

In this study, some of the pictures were processed or extracted to obtain the boundary points. Then, the image was matched by using cubic Ball function which consisted of two control points that could be used to modify the curve constructed. This study used the approximation method automatically. The “Differential Evolution” was used to optimize the two points found on the Ball cubic functions. This process was done iteratively until the curves were successfully built to approximate with the original picture. Some results and numerical examples are illustrated in this study.

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## References

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Sarfraz, M., Irshad, M. & Hussain, M.Z. (2012). Vectorization of image outlines using rational spline and genetic algorithm. The Proceedings of The International Conference on Image,Vision and Computing (ICIVC 2012). Shanghai, China, August 25-26, 2012, IASCIT Press Singapore.

Storn, R. & Price, K. (1995). Differential evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces, 1–12.

Tien, H. L. (1999). Rational Ball curves. International Journal of Mathematical Education in Science and Technology. doi:10.1080/002073999288021.

Zainor , R.Y., Piah A.R.M.P & Majid, A.A. (2012). Conic curve fitting particle swarm optimization: Parameter tuning. Communication in Computer and Informatione Science (295) 379-382.

Felipe, J., Saldarriaga, I., Vélez, S. C., Andrés, M. E. C. & Valencia, T. (2011). Design and manufacturing of a custom skull implant, 4(1), 169–174.

Hasan, Z. A., Piah, A. R. M. & Yahya, Z. R. (2014). Monotonicity preserving C1 rational cubic Ball interpolation. Proceedings of The 21st National Symposium on Mathematical Sciences, 1605, 34–39. doi:10.1063/1.4887561.

Hou, Z. J. & Wei, G. W. (2002). A new approach to edge detection. Pattern Recognition, 35(7), 1559–1570. doi:10.1016/S0031-3203(01)00147-9.

Ho¨ lzle, G.E. (1983). Knot placement for piecewise polynomial approximation of curves. Computer Aided Design, 15(5): 295-296.

Lü, W. (2009). Curves with chord length parameterization. Computer Aided Geometric Design, 26 (3), 342–350. doi:10.1016/j.cagd.2008.08.001.

Pandunata, P. & Shamsuddin, S. M. H. (2010). Differential evolution optimization for Bezier curve fitting. In Proceedings - 2010 7th International Conference on Computer Graphics, Imaging and Visualization, CGIV 2010 (pp. 68–72). doi:10.1109/CGIV.2010.18.

Piah, A. R. M. & Unsworth, K. (2011). Improved sufficient conditions for monotonic piecewise rational quartic interpolation. Sains Malaysiana, 40(10), 1173–1178.

Ramers U. (1972), An interative procedure for the polygonal approximation of plane curves. Computers Graphics and Image Processing. 1(3). 244-256.

Sarfraz, M., Irshad, M. & Hussain, M. (2013). Reverse engineering of planar objects using gas. Sains Malaysiana, 42(8), 1167–1179. Retrieved from http://core.kmi.open.ac.uk/download/pdf/16388677.pdf.

Sarfraz, M. & Khan, M.A. (2004). An automatic algorithm for approximating boundary of bitmap characters. Future Generation Computing Systems 20(8): 1327-1336.

Sarfraz, M. & Khan, M. (2000). Towards automation of capturing outlines of Arabic fonts. Proceedings of the third KFUPM workshop on information and computer science: Software development for the new millennium, Saudi Arabia, October22–23.

Sarfraz, M. & Rasheed, A. (2007). A randomized knot insertion algorithm for outline capture of planar images using cubic spline. The Proceedings of the 22nd ACM Symposium on Applied Computing, Seoul, Korea, March 11-15.

Sarfraz, M. & Razzak, M.F.A. (2002). An algorithm for automatic capturing of font outlines. Computers and Graphics 26(5): 795-804.

Sarfraz, M., Irshad, M. & Hussain, M.Z. (2012). Vectorization of image outlines using rational spline and genetic algorithm. The Proceedings of The International Conference on Image,Vision and Computing (ICIVC 2012). Shanghai, China, August 25-26, 2012, IASCIT Press Singapore.

Storn, R. & Price, K. (1995). Differential evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces, 1–12.

Tien, H. L. (1999). Rational Ball curves. International Journal of Mathematical Education in Science and Technology. doi:10.1080/002073999288021.

Zainor , R.Y., Piah A.R.M.P & Majid, A.A. (2012). Conic curve fitting particle swarm optimization: Parameter tuning. Communication in Computer and Informatione Science (295) 379-382.

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## Published

2015-12-17

## How to Cite

*EDUCATUM Journal of Science, Mathematics and Technology*,

*2*(2), 57–67. Retrieved from https://ojs.upsi.edu.my/index.php/EJSMT/article/view/35

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Articles