A unified algebraic approach to 2-D and 3-D motion segmentation and estimation

In this paper, we present an analytic solution to the problem of estimating an unknown number of 2-D and 3-D motion models from two-view point correspondences or optical flow. The key to our approach is to view the estimation of multiple motion models as the estimation of a single multibody motion model. This is possible thanks to two important algebraic facts. First, we show that all the image measurements, regardless of their associated motion model, can be fit with a single real or complex polynomial. Second, we show that the parameters of the individual motion model associated with an image measurement can be obtained from the derivatives of the polynomial at that measurement. This leads to an algebraic motion segmentation and estimation algorithm that applies to most of the two-view motion models that have been adopted in computer vision. Our experiments show that the proposed algorithm out-performs existing algebraic and factorization-based methods in terms of efficiency and robustness, and provides a good initialization for iterative techniques, such as Expectation Maximization, whose performance strongly depends on good initialization.