Image Analysis: Identification of Objects via Polynomial Systems

Abstract

The problem is to identify a movable object that is in some sense known, if it is encountered later. Suppose we have a sensor, on a fixed radar station or a moving platform. We have an object, say object A, previously measured, with certain distinct identifiable points $$p_i$$ . We know the distances between these points. We later encounter a similar object B and want to know if it is A. We have a sensor that sends and receives electronic signals, and so we measure the distances $$t_i$$ from the sensor to the distinguished points on B. We first consider the two-dimensional case. Assume there are three distinct points on A. We have our measured distances $$t_1, t_2, t_3$$ and previously known distances between the points on A, $$d_1, d_2, d_3$$ . We derive a polynomial system relating these quantities and show that it is easy to solve yielding a resultant that is the “signature” for A. Its use will eliminate B if B is not A. The generalization to three dimensions is immediate. We need a fourth point. The polynomial system contains many parameters, but we solve it symbolically. We then discuss generalizations involving flexibility. In those cases we need five points and the systems are much more complex. In section five we consider the same question of recognition but using a different type of sensor, one that can only receive. We compare a Grobner basis approach with the Dixon resultant method. We compare solutions on Magma, Maple, and Fermat computer algebra systems.