A tensor decomposition for geometric grouping and segmentation

While spectral clustering has been applied successfully to problems in computer vision, their applicability is limited to pairwise similarity measures that form a probability matrix. However many geometric problems with parametric forms require more than two observations to estimate a similarity measure, e.g. epipolar geometry. In such cases we can only define the probability of belonging to the same cluster for an n-tuple of points and not just a pair, leading to an n-dimensional probability tensor. However spectral clustering methods are not available for tensors. In this paper we present an algorithm to infer a similarity matrix by decomposing the n-dimensional probability tensor. Our method exploits the super-symmetry of the probability tensor to provide a randomised scheme that does not require the explicit computation of the probability tensor. Our approach is fast and accurate and its applicability is illustrated on two significant problems, namely perceptually salient geometric grouping and parametric motion segmentation (like affine, epipolar etc).