By Topic

A Fast Algorithm for Multidimensional Ellipsoid-Specific Fitting by Minimizing a New Defined Vector Norm of Residuals Using Semidefinite Programming

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$33 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

3 Author(s)
Xianghua Ying ; Peking University, Beijing ; Li Yang ; Hongbin Zha

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.

Published in:

IEEE Transactions on Pattern Analysis and Machine Intelligence  (Volume:34 ,  Issue: 9 )