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Recognition of 3D Objects Based on Implicit Polynomials
May 2010 (vol. 32 no. 5)
pp. 954-960
Hilla Ben-Yaacov, Technion-Israel Institute of Technology, Haifa
David Malah, Technion-Israel Institute of Technology, Haifa
Meir Barzohar, Technion-Israel Institute of Technology, Haifa
Closed-form expressions for a new set of 3D rotation invariants that are linear, quadratic, and angular combinations of implicit polynomial (IP) coefficients are developed. Based on these invariants, we propose a 3D object recognition method that outperforms recognition based on IP fitting after pose estimation, and the MPEG-7 SSD technique.

[1] J. Subrahmonia, D.B. Cooper, and D. Keren, "Practical Reliable Bayesian Recognition of 2D and 3D Objects Using Implicit Polynomials and Algebraic Invariants," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 5, pp. 505-519, May 1996.
[2] M. Barzohar, D. Keren, and D. Cooper, "Recognizing Groups of Curves Based on New Affine Mutual Geometric Invariants, with Applications to Recognizing Intersecting Roads in Aerial Images," Proc. IAPR Int'l Conf. Pattern Recognition, vol. 1, pp. 205-209, Oct. 1994.
[3] J. Tarel, H. Civi, and D. Cooper, "Estimation of Free-Form 3D Objects Without Point Matching Using Algebraic Surface Models," Proc. IEEE Workshop Model-Based 3D Image Analysis, pp. 13-21, 1998.
[4] G. Taubin, "Estimation of Planar Curves, Surfaces and Nonplanar Space Curves Defined by Implicit Equations, with Applications to Edge and Range Image Segmentation," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 1115-1138, Nov. 1991.
[5] M.M. Blane, Z. Lei, H. Civi, and D.B. Cooper, "The 3L Algorithm for Fitting Implicit Polynomial Curves and Surfaces to Data," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 3, pp. 298-313, Mar. 2000.
[6] T. Tasdizen, J. Tarel, and D. Cooper, "Improving the Stability of Algebraic Curves for Applications," IEEE Trans. Image Processing, vol. 9, no. 3, pp. 405-416, Mar. 2000.
[7] D. Keren and C. Gotsman, "Fitting Curves and Surfaces with Constrained Implicit Polynomials," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 1, pp. 31-41, Jan. 1999.
[8] D. Keren, "Topologically Faithful Fitting of Simple Closed Curves," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 1, pp. 118-123, Jan. 2004.
[9] A. Helzer, M. Barzohar, and D. Malah, "Stable Fitting of 2D Curves and 3d Surfaces by Implicit Polynomials," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 10, pp. 1283-1294, Oct. 2004.
[10] T. Sahin and M. Unel, "Stable Algebraic Surfaces for 3D Object Representation," J. Math. Imaging and Vision, vol. 32, no. 2, pp. 127-137, Oct. 2008.
[11] Z. Landa, "2D Object Description and Classification Based on Contour Matching by Implicit Polynomials," MSc thesis, The Technion-Israel Inst. of Technology, http://sipl.technion.ac.il/siglib/FPZoya_Landa.pdf , Aug. 2006.
[12] Z. Landa, D. Malah, and M. Barzohar, "2D Object Description and Recognition Based on Contour Matching by Implicit Polynomials," CCIT Report #755, Electrical Eng. Dept., Technion, Feb. 2010.
[13] P. Besl and N. McKay, "A Method for Registration of 3D Shapes," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 2, pp. 239-256, Feb. 1992.
[14] P. Mordohai and G. Medioni, "Dimensionality Estimation and Manifold Learning Using Tensor Voting," http://iris.usc.edu/medioni/downloadndmanual.htm , 2009.
[15] A.E. Johnson and M. Hebert, "Using Spin Images for Efficient Object Recognition in Cluttered 3D Scenes," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 5, pp. 433-449, May 1999.
[16] T. Zaharia and F. Preteux, "3D Shape-Based Retrieval within the Mpeg-7 Framework," Proc. SPIE Conf. Nonlinear Image Processing and Pattern Analysis, pp. 133-145, Jan. 2001.
[17] C. Dorai and A.K. Jain, "Shape Spectrum Based View Grouping and Matching of 3D Free-Form Objects," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 10, pp. 1139-1145, Oct. 1997.
[18] J. Tarel and D. Cooper, "The Complex Representation of Algebraic Curves and Its Simple Exploitation for Pose Estimation and Invariant Recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 7, pp. 663-674, July 2000.
[19] D. Keren, "Using Symbolic Computation to Find Algebraic Invariants," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 11, pp. 1143-1149, Nov. 1994.
[20] M. Unel and W.A. Wolovich, "A New Representation for Quartic Curves and Complete Sets of Geometric Invariants," Int'l J. Pattern Recognition and Artificial Intelligence, vol. 13, no. 8, pp. 1137-1149, Dec. 1999.
[21] M. Unel and W. Wolovich, "On the Construction of Complete Sets of Geometric Invariants for Algebraic Curves," Advances in Applied Math., vol. 24, no. 1, pp. 65-87, Jan. 2000.
[22] G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists. Elsevier, 2005.
[23] H. Ben-Yaacov, "3D Object Description and Classification by Implicit Polynomials," MSc thesis, The Technion-Israel Inst. of Technology, http://sipl.technion.ac.il/siglib/FPHilla-Ben-Yaacov.pdf , Sept. 2008.
[24] "Stuttgart Range Image Database," http://range.informatik.uni- stuttgart.de/ htdocshtml/, 2009.
[25] S. Rusinkiewicz, D. DeCarlo, A. Finkelstein, and A. Santella, "Suggestive Contour Gallery," http://www.cs.princeton.edu/gfx/proj/sugcon models /, 2009.
[26] "The Digital Michelangelo Project," www.graphics.stanford.edu/datadmich-public /, 2009.
[27] D. Scharstein and R. Szeliski, "High-Accuracy Stereo Depth Maps Using Structured Light," Proc. 2003 IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 1, I-195-I-202, June 2003.
[28] MPEG-7 Implementation Studies Group, "Technology—Multimedia Content Description Interface—Part 6: Reference Software, ISO/IEC FCD 15938-6/N4006, MPEG-7," Mar. 2001.
[29] B. Horn, "Closed-Form Solution of Absolute Orientation Using Unit Quaternions," J. Optical Soc. Am., vol. 4, pp. 629-642, Apr. 1987.

Index Terms:
Implicit polynomials, 3D object recognition, tensor contraction, rotation-invariant, 3D object fitting.
Citation:
Hilla Ben-Yaacov, David Malah, Meir Barzohar, "Recognition of 3D Objects Based on Implicit Polynomials," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 32, no. 5, pp. 954-960, May 2010, doi:10.1109/TPAMI.2009.197
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