Obtaining Range from Visual Motion Using Local Image Derivatives

INTRODUCTION In many robotic and remote operation applications, depth (or range) information at various points in the scene is required. These include autonomous navigation, landing site selection, scene reconstruction, or postprocessing interpretation of video footage. Theoretically, depth, motion, and optical flow (generated by the relative movement between the camera and scene) are three parameters of a circular problem: Given the camera motion and three-dimensional structure of the scene, we can generate the optical flow, and hence construct a sequence of images of the moving scene (3-D simulation). Given knowledge of the three-dimensional scene and the optical flow, we can compute the motion which generated it (motion recovery). And finally, given the camera motion and the resulting optical flow, we can extract depth information and reconstruct the three-dimensional scene that gave rise to the flow (scene reconstruction). When only the optical flow is available, the problem becomes much harder, and the exact depth and motion cannot be determined [Horn, 1986]. Only the relative depth between various scene points, and the relative motion (or direction) can be recovered. Recovering depth and motion from a sequence of images only is an active research area. For example, Horn [1986] described a least-squares method wherein an iterative process may be used to solve a set of seven simultaneous equations involving the optical flow. Fermuller presented a tracking technique [1991] and a pattern-matching technique [in Aloimonos, 1993] to estimate motion parameters. This general problem is outside the scope of this report. We concentrate on finding the depth given a sequence of images and known motion or direction of motion. This problem is appropriate for many real-world applications, where robot (and camera) motion can be dictated by open-loop control, or motion information can be supplied by non-visual means, such as wheel encoders, accelerometers, gyroscopes, or other non-visual feedback control schemes. Albus [1990] computed range given known motion under various conditions using the optical flow. However, the optical flow itself is difficult to compute from a sequence of images (only a component of it, the normal flow, is easily computable). We will show how to obtain range data without having to find the optical flow itself, and analyze the method's sensitivity to inaccuracies in the known motion. Since the method uses only temporal and spatial first derivatives, which can be computed easily from any two consecutive frames, the depth map can be computed quickly in one pass, and …