A Tensor Decomposition for Geometric Grouping and Segmentation

Abstract

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.

Spectral Curvature Clustering (SCC)

“…First of all, as the size of the data and the intrinsic dimension d increase, it is computationally prohibitive to calculate or store, not to mention process, the affinity tensor. Approximating this tensor by uniformly sampling a small subset of its "fibers" (Govindu 2005) is insufficient for large d and data of moderate size. Better numerical techniques have to be developed while maintaining both reasonable performance and fast speed.…”

“…We emphasize the clustering component, and thus refer to this special case as d-flats clustering. We follow Govindu's framework of multi-way spectral clustering (Govindu 2005) and Ng et al's framework of spectral clustering (Ng et al 2002). In our setting, the former framework starts by assigning to any d + 2 points in the data an affinity measure quantifying d-dimensional flatness, thus forming a (d + 2)-way affinity tensor.…”

“…Some of them emphasize modeling of the underlying flats and then use the models to infer the clusters (see e.g., Independent Component Analysis (Hyvärinen and Oja 2000), Subspace Separation (Kanatani 2001(Kanatani , 2002, Generalized Principal Component Analysis (GPCA) (Vidal et al 2005;Ma et al 2008)). A few others address the clustering part and then use its output to estimate the parameters of the underlying flats (see e.g., Multi-way Clustering algorithms (Agarwal et al 2005(Agarwal et al , 2006Govindu 2005;Shashua et al 2006), Tensor Voting (Medioni et al 2000), k-Manifolds (Souvenir and Pless 2005), Grassmann Clustering (Gruber and Theis 2006), Poisson Mixture Model (Haro et al 2006)). There are also algorithms that iterate between the two components of data clustering and subspace modeling (see e.g., Mixtures of PPCA (MoPPCA) (Tipping and Bishop 1999), K-Subspaces (Ho et al 2003)/kPlanes (Bradley and Mangasarian 2000;Tseng 1999)).…”

“…• It introduces an iterative sampling procedure to significantly improve accuracy over the standard random sampling scheme used in Govindu (2005) (see Sect. 3.1.1).…”

Multiple structure recovery with maximum coverage

“…Finally, we can ascribe to preference analysis also all the approaches based on higher order clustering [19,20,21,22], where higher order similarity tensors are defined between n-tuple of points as the probability of points to be clustered together measured in terms of residual errors with respect to provisional models. In this way preferences give rise to a hypergraph whose hyperedges encode the existence of a structure able to explain the incident vertices.…”