Shape and Motion Estimation from Near-Field Echo-Based Sensor Data

This paper presents a new approach for reconstructing both shape and motion from data collected by echo-based ranging sensors. The approach is based on geometric invariant theory and exploits object-image relations for near-field (spherical-wavefront) range data. These object-image equations relate the data to a unique matrix of Euclidean invariants that completely describe the object shape. The object-image relations can be used to determine the shape of a scene viewed from unknown vantage points. Specifically, the object-image equations form a linear system of equations whose solution determines the relevant shape parameters for a configuration of features within the scene. Once the shape parameters are estimated, a single shape exemplar from the point in shape space can be used to determine the relative motion (up to an arbitrary rotation) between the sensor and the object. One advantage of this motion-estimation approach is that the geometric-invariant-based strategy allows us to uniquely solve the optimization problem without the need to introduce coordinate-system-dependent “nuisance” parameters. The theorems stated in this paper hold for any range-measurement sensor scenario. As an example of the utility of the given theorems, the object-image relations are used to augment noisy GPS measurements in a circular synthetic aperture radar geometry.