Robust Algebraic Segmentation of Mixed Rigid-Body and Planar Motions from Two Views

Abstract

This paper studies segmentation of multiple rigidbody motions in a 3-D dynamic scene under perspective camera projection. We consider dynamic scenes that contain both 3-D rigid-body structures and 2-D planar structures. Based on the well-known epipolar and homography constraints between two views, we propose a hybrid perspective constraint (HPC) to unify the representation of rigidbody and planar motions. Given a mixture of K hybrid perspective constraints, we propose an algebraic process to partition image correspondences to the individual 3-D motions, called Robust Algebraic Segmentation (RAS). Particularly, we prove that the joint distribution of image correspondences is uniquely determined by a set of (2K)-th degree polynomials, a global signature for the union of K motions of possibly mixed type. The first and second deriva- tives of these polynomials provide a means to recover the association of the individual image samples to their respective motions. Finally, using robust statistics, we show that the polynomials can be robustly estimated in the presence of moderate image noise and outliers. We conduct extensive simulations and real experiments to validate the performance of the new algorithm. The results demonstrate that RAS achieves notably higher accuracy than most existing robust motion-segmentation methods, including random sample consensus (RANSAC) and its variations. The implementation of the algorithm is also two to three times faster than the existing methods. The implementation of the algorithm and the benchmark scripts are available at http:// perception.csl.illinois.edu/ras/.

Tensor LRR based subspace clustering

“…Nevertheless, the performance of algebraic based methods in the presence of noise deteriorates as the number of subspaces increases. Robust Algebraic Segmentation (RAS) [8] is proposed to improve robustness performance, but the complexity issue still exists.Iterative methods improve the performance of algebraic based algorithms to handle noisy data in a repeated refinement. The k-subspace method [7], [13] extends the k-means clustering algorithm from data distributed around cluster centers to data drawn from subspaces of any dimensions.…”

“…Both [3], [8] and [14] are representatives of spectral clustering-based methods. They aim to find a linear representation Z for all the samples in terms of all other samples, which is solved by finding the optimal solution of the following objective function:…”

“…We use the Augmented Lagrange Multiplier (ALM) method [21] to solve the constrained optimization problem (8). The reason we choose ALM to solve this optimization problem is threefold: (1) Superior convergence property of ALM makes it very attractive; (2) Parameter tuning is much easier than the iterative thresholding algorithm; and (3) It converges to an exact optimal solution.…”

“…In the last decade, subspace clustering (SC) has been widely applied to many real-world applications, including motion segmentation [2], [3], social community identification [4], and image clustering [5]. A famous survey on subspace clustering [6] classifies most existing SC algorithms into three categories: statistical methods [7] , algebraic methods [8] and spectral clustering-based methods [3], [8]. 1 …”

A Review of Subspace Segmentation: Problem, Nonlinear Approximations, and Applications to Motion Segmentation

“…The subspace segmentation problem has connections to several active areas of research, including learning theory, compressed sampling, and signal processing in general [2,3,17,21,[29][30][31][32]. Moreover, it is relevant to several computer vision applications including motion tracking of rigid objects in videos and facial recognition [1,4,14,[33][34][35][36][37][38].…”