A Fast Algorithm for Multidimensional Ellipsoid-Specific Fitting by Minimizing a New Defined Vector Norm of Residuals Using Semidefinite Programming
Issue No. 09 - Sept. (2012 vol. 34)
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
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.
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
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. , pp. 1856-1863, Sept. 2012, doi:10.1109/TPAMI.2012.109