This paper deals with the problem of estimating multiple transparent motions that can occur in computer vision applications, e.g. in the case of semi-transparencies and occlusions, and also in medical imaging when different layers of tissue move independently. Methods based on the known optical-flow equation for two motions are extended in three ways. Firstly, we include a regularization term to cope with sparse flow fields. We obtain an Euler-Lagrange system of differential equations that becomes linear due to the use of the mixed motion parameters. The system of equations is solved for the mixed-motion parameters in analogy to the case of only one motion. To extract the motion parameters, the velocity vectors are treated as complex numbers and are obtained as the roots of a complex polynomial of a degree that is equal to the number of overlaid motions. Secondly, we extend a Fourier-Transform based method proposed by Vernon such as to obtain analytic solutions for more than two motions. Thirdly, we not only solve for the overlaid motions but also separate the moving layers. Performance is demonstrated by using synthetic and real sequences.
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