A closed form solution to direct motion segmentation

We present a closed form solution to the problem of segmenting multiple 2D motion models of the same type directly from the partial derivatives of an image sequence. We introduce the multibody brightness constancy constraint (MBCC), a polynomial equation relating motion models, image derivatives and pixel coordinates that is independent of the segmentation of the image measurements. We first show that the optical flow at a pixel can be obtained analytically as the derivative of the MBCC at the corresponding image measurement, without knowing the motion model associated with that pixel. We then show that the parameters of the multiple motion models can be obtained from the cross products of the derivatives of the MBCC at a set of image measurements that minimize a suitable distance function. Our approach requires no feature tracking, point correspondences or optical flow, and provides a global non-iterative solution that can be used to initialize more expensive iterative approaches to motion segmentation. Experiments on real and synthetic sequences are also presented.