Motion Competition: A Variational Approach to Piecewise Parametric Motion Segmentation

We present a novel variational approach for segmenting the image plane into a set of regions of parametric motion on the basis of two consecutive frames from an image sequence. Our model is based on a conditional probability for the spatio-temporal image gradient, given a particular velocity model, and on a geometric prior on the estimated motion field favoring motion boundaries of minimal length. Exploiting the Bayesian framework, we derive a cost functional which depends on parametric motion models for each of a set of regions and on the boundary separating these regions. The resulting functional can be interpreted as an extension of the Mumford-Shah functional from intensity segmentation to motion segmentation. In contrast to most alternative approaches, the problems of segmentation and motion estimation are jointly solved by continuous minimization of a single functional. Minimizing this functional with respect to its dynamic variables results in an eigenvalue problem for the motion parameters and in a gradient descent evolution for the motion discontinuity set. We propose two different representations of this motion boundary: an explicit spline-based implementation which can be applied to the motion-based tracking of a single moving object, and an implicit multiphase level set implementation which allows for the segmentation of an arbitrary number of multiply connected moving objects. Numerical results both for simulated ground truth experiments and for real-world sequences demonstrate the capacity of our approach to segment objects based exclusively on their relative motion.