It is well known that both shape and motion can be factorized directly from the measurement matrix constructed from feature points trajectories under orthographic camera model. In practical applications, the measurement matrix might be contaminated by noises and contains outliers. A direct SVD (Singular Value Decomposition) to the measurement matrix with outliers would yield erroneous result. This paper presents a novel algorithm for computing SVD with outliers. We decompose the SVD computation as a set of alternate linear regression subproblems. The linear regression subproblems are solved robustly by applying the RANSAC strategy. The proposed robust factorization method with outliers can improve the reconstruction result remarkably. Quantitative and qualitative experiments illustrate the good performance of the proposed method.

The problem of multi-body motion segmentation is important in many computer vision applications. In this paper, we propose a novel algorithm called fuzzy k-subspace clustering for robust segmentation. The proposed method exploits the property that under orthographic camera model the tracked feature points of moving objects reside in multiple subspaces. We compute a partition of feature points into corresponding subspace clusters. First, we find a "soft partition" of feature points based on fuzzy k-subspace algorithm. The proposed fuzzy k-subspace algorithm iteratively minimizes the objective function using Weighted Singular Value Decomposition. Then the points with high partition confidence are gathered to form the subspace bases and the remaining points are classified using their distance to the bases. The proposed method can handle the case of missing data naturally, meaning that the feature points do not have to be visible throughout the sequence. The method is robust to noise and insensitive to initialization. Extensive experiments on synthetic and real data show the effectiveness of the proposed fuzzy k-subspace clustering algorithm.