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Abstract: This paper presents a method to reconstruct symmetric geometric models from point cloud with inherent symmetric structure. symmetry types commonly found in engineering parts, i.e., translational, reflectional and rotational symmetries are considered. The reconstruction problem is formulated as a constrained optimization, where the objective function is the sum of squared distances of points to the model, and constraints are enforced to keep geometric relationships in the model. First, the explicit representations of symmetric models are presented. Then, by using the concept of parameterized points (where the coordinate components are represented as functions rather than constants), the distances of points to symmetric models are deduced. With these distance functions, symmetry information, for both 2D and 3D models, is uniformly represented in the process of reconstruction. The constrained optimization problem is solved by a standard nonlinear optimization method. Owing to the explicit representation of symmetry information, the computational complexity of our method is reduced greatly. Finally, examples are given to demonstrate the application of the proposed method.

[1] Benkő, P., Kós, G., Várady, T., Andor, L., Martin, R.R., 2002. Constrained fitting in reverse engineering. Computer Aided Geometric Design, 19(3):173-205.

[2] Faber, P., Fisher, R.B., 2001. Pros and cons of Euclidean fitting. LNCS, 2191:414-420.

[3] Fiacco, A., McCormick, G., 1968. Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York, p.593-594.

[4] Ke, Y., Zhu, W., Liu, F., Shi, X., 2006. Constrained fitting for 2D profile-based reverse engineering. Computer-Aided Design, 38(2):101-114.

[5] Kós, G., Martin, R.R.,Várady, T., 1999. Methods to recover constant radius rolling ball blends in reverse engineering. Computer Aided Geometric Design, 17(2):127-160.

[6] Langbein, F.C., 2003. Beautification of Reverse Engineered Geometric Models. PhD Thesis, Department of Computer Science, Cardiff University, Cardiff, Wales, UK.

[7] Langbein, F.C., Marshall, A.D., Martin, R.R., 2004. Choosing consistent constraints for beautification of reverse engineered geometric models. Computer-Aided Design, 36(3):261-278.

[8] Li, M., Langbein, F.C., Martin, R.R., 2007. Detecting Approximate Incomplete Symmetries in Discrete Point Sets. Proc. ACM Symp. on Solid and Physical Modeling. ACM Press, p.335-340.

[9] Li, M., Langbein, F.C., Martin, R.R., 2008. Detecting approximate symmetries of discrete point subsets. Computer-Aided Design, 40(1):76-93.

[10] Marshall, D., Lukacs, G., Matin, R.R., 2001. Robust segmentation of primitives from range data in the presence of geometric degeneracy. IEEE Trans. on Pattern Anal. Machine Intell., 23(3):304-314.

[11] Mills, B.I., Langbein, F.C., Marshall, A.D., Martin, R.R., 2001. Approximate Symmetry Detection for Reverse Engineering. Proc. 6th ACM Symp. on Solid Modelling and Applications. ACM Press, p.241-248.

[12] Mitra, N.J., Guibas, L.J., Pauly, M., 2006. Partial and approximate symmetry detection for 3D geometry. ACM Trans. on Graph., 25(3):560-568.

[13] Pratt, V., 1987. Direct least-squares fitting of algebraic surfaces. Comput. Graph., 21(4):145-152.

[14] Shen, D., Ip, H.H.S., Cheung, K.K.T., Teoh, E.K., 1999. Symmetry detection by generalized complex (GC) moments: a close-form solution. IEEE Trans. on Pattern Anal. Machine Intell., 21(5):466-476.

[15] Sun, C., Sherrah, J., 1997. 3D symmetry detection using the extended Gaussian image. IEEE Trans. on Pattern Anal. Machine Intell., 19(2):164-168.

[16] Thompson, W.B., Owen, J.C., James, H., Stark, S.R., Henderson, T.C., 1999. Feature-based reverse engineering of mechanical parts. IEEE Trans. on Rob. Autom., 15(1):57-66.

[17] Várady, T., Martin, R.R., Coxt, J., 1997. Reverse engineering of geometric models—an introduction. Computer-Aided Design, 29(4):255-268.

[18] Werghi, N., Fisher, R.B., Robertson, C., Ashbrook, A., 1999. Object reconstruction by incorporating geometric constraints in reverse engineering. Computer-Aided Design, 31(6):363-399.

[19] Weyl, H., 1952. Symmetry. Princeton University Press, Princeton, New Jersey.