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

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
$n$
data points into a
$p$
dimensional ellipsoid, our complexity is
$O(p^6 + np^4)+O(p^3)$
, where the former
$O$
results from preprocessing data
once
, while that of the state-of-the-art SDP method is
$O(p^6 + np^4 + n^{\frac{3}{2}}p^2)$

for each iteration
. The storage complexity of our algorithm is about
$\frac{1}{2}np^2$
, which is at most
$1/4$
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.