Applications of algebraic geometry to object/image recognition

Abstract

In recent years, new approaches to the problem of Automated Target Recognition using techniques of shape theory and algebraic geometry have been explored. The power of this shape theoretic approach is that it allows one to develop tests for object/image matching that do not require knowledge of the object’s position in relation to the sensor nor the internal parameters of the sensor. Furthermore, these methods do not depend on the choice of coordinate systems in which the objects and images are represented. In this dissertation, we will expand on existing shape theoretic techniques and adapt these techniques to new sensor models. In each model, we develop an appropriate notion of shape for our objects and images and define the spaces of such shapes. The goal in each case is to develop tests for matching object and image shapes under an appropriate class of projections. The first tests we develop take the form of systems of polynomial equations (the so-called object/image relations) that check for exact matches of object/image pairs. Later, a more robust approach to matching is obtained by defining metrics on the shape spaces. This allows us in each model to develop a measure of “how close” an object is to being able to produce a given image. We conclude this dissertation by computing a number of examples using these tests for object/image matching.