Direct Segmentation of Multiple 2-D Motion Models of Different Types

We propose a closed form solution for segmenting mixtures of 2-D translational and 2-D affine motion models directly from the image intensities. Our approach exploits the fact that the spatial-temporal image derivatives generated by a mixture of these motion models must satisfy a bi-homogeneous polynomial called the multibody brightness constancy constraint (MBCC). We show that the degrees of the MBCC are related to the number of motions models of each kind. Such degrees can be automatically computed using a one-dimensional search. We then demonstrate that a sub-matrix of the Hessian of the MBCC encodes information about the type of motion models. For instance, the matrix is rank-1 for 2-D translational models and rank-3 for 2-D affine models. Once the type of motion model has been identified, one can obtain the parameters of each type of motion model at every image measurement from the cross products of the derivatives of the MBCC. We then demonstrate that accounting for a 2-D translational motion model as a 2-D affine one would result in erroneous estimation of the motion models, thus motivating our aim to account for different types of motion models. We apply our method to segmenting various dynamic scenes.