Data in many image and video analysis tasks can be viewed as points drawn from multiple low-dimensional subspaces with each subspace corresponding to one category or class. One basic task for processing such kind of data is to separate the points according to the underlying subspace, referred to as subspace clustering. Extensive studies have been made on this subject, and nearly all of them use unconstrained subspace models, meaning the points can be drawn from everywhere of a subspace, to represent the data. In this paper, we attempt to do subspace clustering based on a constrained subspace assumption that the data is further restricted in the corresponding subspaces, e.g., belonging to a submanifold or satisfying the spatial regularity constraint. This assumption usually describes the real data better, such as differently moving objects in a video scene and face images of different subjects under varying illumination. A unified integer linear programming optimization framework is used to approach subspace clustering, which can be efficiently solved by a branch-and-bound (BB) method. We also show that various kinds of supervised information, such as subspace number, outlier ratio, pairwise constraints, size prior and etc., can be conveniently incorporated into the proposed framework. Experiments on real data show that the proposed method outperforms the state-of-the-art algorithms significantly in clustering accuracy. The effectiveness of the proposed method in exploiting supervised information is also demonstrated.
