Computing the Camera Motion Direction from Many Images

We analyze the problem of estimating a camera's motion direction from a calibrated multi-image sequence. We assume that the camera moves roughly along a line and that its velocity and orientation are unknown and can vary over time. For infinitesimal camera motion (multiple flows rather than multiple images), we give a closed-form expression for the result of minimizing the true least-squares error over all variables but the camera's motion direction. Our result includes the rigidity constraint that the scene stays fixed over time. For finite motion, we present a noniterative algorithm that approximates the exact multi-image coplanarity error to better than a percent. Also, we define a new error contribution that incorporates the rigidity constraint and is analogous to the rigidity component of the error for infinitesimal motion. By adding this to the coplanarity error, we obtain a noniterative algorithm that approximates the complete finite motion least-squares error - including rigidity - as a function just of the translation direction.