Fast Multidimensional Ellipsoid-Specific Fitting by Alternating Direction Method of Multipliers

Zhouchen Lin, Yameng Huang

Abstract: Many problems in computer vision can be formulated as multidimensional ellipsoid-specific fitting, which is to minimize the residual error such that the underlying quadratic surface is a multidimensional ellipsoid. In this paper, we present a fast and robust algorithm for solving ellipsoid-specific fitting directly. Our method is based on the alternating direction method of multipliers, which does not introduce extra positive semi-definiteness constraints. The computation complexity is thus significantly lower than those of semi-definite programming (SDP) based methods. More specifically, to fit <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> data points into a <inline-formula> <tex-math notation="LaTeX">$p$</tex-math></inline-formula> dimensional ellipsoid, our complexity is <inline-formula><tex-math notation="LaTeX"> $O(p^6 + np^4)+O(p^3)$</tex-math> </inline-formula> , where the former <inline-formula><tex-math notation="LaTeX">$O$</tex-math></inline-formula> results from preprocessing data <b>once</b> , while that of the state-of-the-art SDP method is <inline-formula><tex-math notation="LaTeX">$O(p^6 + np^4 + n^{\frac{3}{2}}p^2)$</tex-math> </inline-formula> <i>for each iteration</i> . The storage complexity of our algorithm is about <inline-formula> <tex-math notation="LaTeX">$\frac{1}{2}np^2$</tex-math></inline-formula> , which is at most <inline-formula><tex-math notation="LaTeX">$1/4$</tex-math> </inline-formula> of those of SDP methods. Extensive experiments testify to the great speed and accuracy advantages of our method over the state-of-the-art approaches. The implementation of our method is also much simpler than SDP based methods.