Subscribe
Issue No.09 - Sept. (2012 vol.34)
pp: 1856-1863
Xianghua Ying , Key Lab. of Machine Perception (Minist. of Educ.), Peking Univ., Beijing, China
Li Yang , Key Lab. of Machine Perception (Minist. of Educ.), Peking Univ., Beijing, China
Hongbin Zha , Key Lab. of Machine Perception (Minist. of Educ.), Peking Univ., Beijing, China
ABSTRACT
A quadratic surface in n-dimensional space is defined as the locus of zeros of a quadratic polynomial. The quadratic polynomial may be compactly written in notation by an (n+1)-vector and a real symmetric matrix of order n+1, where the vector represents homogenous coordinates of an n-D point, and the symmetric matrix is constructed from the quadratic coefficients. If an n-D quadratic surface is an n-D ellipsoid, the leading n \times n principal submatrix of the symmetric matrix would be positive or opposite definite. As we know, to impose a matrix being positive or opposite definite, perhaps the best choice may be to employ semidefinite programming (SDP). From such straightforward and intuitive knowledge, in the literature until 2002, Calafiore first proposed a feasible method for multidimensional ellipsoid-specific fitting using SDP, which minimizes the 2--norm of the algebraic residual vector. However, the runtime of the method is significantly long and memory is often out when the number of fitted points is greater than several thousand. In this paper, we propose a fast and easily implemented algorithm for multidimensional ellipsoid-specific fitting by minimizing a new defined vector norm of the algebraic residual vector using SDP, which drastically decreases the size of the SDP problem while preserving accuracy. The proposed fast method can handle several million fitted points without any difficulty.
INDEX TERMS
vectors, mathematical programming, matrix algebra, polynomials, quadratic programming, symmetric matrix, multidimensional ellipsoid-specific fitting, vector norm, semidefinite programming, quadratic surface, n-dimensional space, quadratic polynomial, Vectors, Fitting, Minimization, Symmetric matrices, Programming, Surface fitting, Eigenvalues and eigenfunctions, new defined vector norm., Multidimensional ellipsoid, ellipsoid-specific fitting, semidefinite programming
CITATION
Xianghua Ying, Li Yang, Hongbin Zha, "A Fast Algorithm for Multidimensional Ellipsoid-Specific Fitting by Minimizing a New Defined Vector Norm of Residuals Using Semidefinite Programming", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.34, no. 9, pp. 1856-1863, Sept. 2012, doi:10.1109/TPAMI.2012.109
REFERENCES
 [1] S. Ahn, W. Rauh, H.S. Cho, and H. Warnecke, "Orthogonal Distance Fitting of Implicit Curves and Surfaces," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 620-638, May 2002. [2] M. Blane, Z. Lei, H. Civil, and D. Cooper, "The 3L Algorithm for Fitting Implicit Polynomials Curves and Surface to Data," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 3, pp. 298-313, Mar. 2000. [3] G. Calafiore, "Approximation of N-Dimensional Data Using Spherical and Ellipsoidal Primitives," IEEE Trans. Systems, Man, and Cybernetics, Part A, vol. 32, no. 2, pp. 269-278, Mar. 2002. [4] X. Cao and N. Shrikhande, "Quadratic Surface Fitting for Sparse Range Data," Proc. IEEE Int'l Conf. Systems, Man, and Cybernetics, pp. 123-128, 1991. [5] Y. Chen and C. Liu, "Quadratic Surface Extraction Using Genetic Algorithms," Computer-Aided Design, vol. 31, no. 2, pp. 101-110, Feb. 1999. [6] G. Cross and A. Zisserman, "Quadric Reconstruction from Dual-Space Geometry," Proc. Sixth IEEE Int'l Conf. Computer Vision, pp. 25-31, 1998. [7] A. Fitzgibbon, M. Pilu, and R. Fisher, "Direct Least Square Fitting of Ellipses," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 5, pp. 476-480, May 1999. [8] W. Gander, G.H. Golub, and R. Strebel, "Least-Square Fitting of Circles and Ellipses," BIT Numerical Math., vol. 34, no. 4, pp. 558-578, 1994. [9] G.H. Golub and C.F. Van Loan, Matrix Computations, pp. 52-54, third ed. Johns Hopkins Univ. Press, 1996. [10] P. Gotardo, O. Bellon, K. Boyer, and L. Silva, "Range Image Segmentation into Planar and Quadratic Surfaces Using an Improved Robust Estimator and Genetic Algorithm," IEEE Trans. Systems, Man, and Cybernetics, Part B, vol. 34, no. 6, pp. 2303-2316, Dec. 2004. [11] 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. [12] O.M. van Kaick, M.V.G. da Silva, W.R. Schwartz, and H. Pedrini, "Fitting Smooth Surfaces to Scattered 3D Data Using Piecewise Quadratic Approximation," Proc. IEEE Ninth Int'l Conf. Image Processing, pp. 493-496, 2002. [13] P.Y. Lee and J.B. Moore, "Geometric Optimization for 3D Pose Estimation of Quadratic Surfaces," Proc. 38th Asilomar Conf. Signals, Systems, and Computers, pp. 131-135, 2004. [14] A.D. Sappa and M. Rouhani, "Efficient Distance Estimation for Fitting Implicit Quadric Surfaces," Proc. IEEE 16th Int'l Conf. Image Processing, pp. 3521-3524, 2009. [15] P. Sturm and P. Gargallo, "Conic Fitting Using the Geometric Distance," Proc. Asian Conf. Computer Vision, pp. 784-795, 2007. [16] K.C. Toh, M.J. Todd, and R.H. Tutuncu, "SDPT3-A Matlab Software Package for Semidefinite Programming," Optimization Methods and Software, vol. 11, nos. 1-4, pp. 545-581, 1999. [17] L. Vandenberghe and S. Boyd, "Semidefinite Programming," SIAM Rev., vol. 38, no. 1, pp. 49-95, Mar. 1996.