Using Invariant Theory to Obtain Estimates of Unknown Shape and Motion, and Imaging Moving Objects in 3D from Single Aperture Synthetic Aperture Radar

October 18, 2005
When a moving object is imaged with conventional synthetic aperture radar (SAR) the result is a displaced smear. This is due to the extra information the objectmotion is imparting to the radar return. When a sensor collects data from a moving extended object, estimation of the direction vectors from the object to the sensor is often essential to the extraction of useful information from the sensor data. If the object or the sensor moves as result of uncontrolled or unknown forces, simple parametric models for the angular motions often rapidly loose fidelity. So, even if the object can be modeled parametrically, nonparametric motion estimates are desirable. In one example of such a problem, a direct approach to estimating all the unknowns leads to difficult nonlinear optimization problems. But a characterization of the shape of the object, using the right choice of geometric invariants, can decouple the problem, temporarily isolating the object shape estimation from the motion estimation. This facilitates the extraction of nonparametric motion estimates both by subdividing the parameter space, and by enabling parts of the problem to be solved using linear methods. If the motion is rich enough there should be a possibility of forming a 3D image of the object. This involves understanding the way the radar data is arranged in phase space. The data lies on a convoluted surface that occupies three dimensions rather than the two dimensional plane used in conventional SAR. To achieve three dimensional images the data must be extrapolated from the surface into a volume. In this complex space, there is a great deal of structure and therefore the possibility of extrapolating to a volume of data.