Projected free fall trajectories. I. Theory and simulation

How we manage to reconstruct the three-dimensional character of the world from the two-dimensional representations on our retinae has been a lively subject of research in the last ten or fifteen years. One principle that has emerged unifying many of these ideas is the need for constraints to allow the visual system to interpret the images it receives as three-dimensional. These constraints come from assumptions about the nature of the situation that produced the image. We have looked at how gravity can be used as a constraint in the case of a free fall trajectory projected onto an image plane by central projection. We have examined several possible methods for deriving the initial conditions of the trajectory from the two-dimensional projection, and examined their behavior under noisy and noiseless conditions, using both image simulations and videotapes of a real ball. We show that there are several ways to robustly compute the initial conditions of the parabolic trajectory from the image data in the presence of noise.