Analysis and synthesis of motion patterns using the projective plane

We first review theoretical results for the problem of estimating single and multiple transparent motions. For N motions we obtain a M × M generalized structure tensor J_N with M = 3 for one, M = 6 for two, and M = 10 for three motions. The analysis of motion patterns is based on the ranks of J_N and is thus not only conceptual but provides computable confidence measures for the different types of motions. To resolve the correspondence between the ranks of the tensors and the motion patterns, we introduce the projective plane as a new way of describing motion patterns. In the projective plane, intrinsically 2D spatial patterns (e.g. corners and line ends) that move correspond to points that represent the only admissible velocity, and ID spatial patterns (e.g. straight edges) that move correspond to lines that represent, as a set of points, the set of admissible velocities. We then show a few examples for how the projective plane can be used to generate novel motion patterns and explain the perception of these patterns. We believe that our results will be useful for designing new stimuli for visual psychophysics and neuroscience and thereby contribute to the understanding of the dynamical properties of human vision.