Mathematical aspects of shape analysis for object recognition In this paper we survey some of the mathematical techniques that have led to useful new results in shape analysis and their application to a variety of object recognition tasks. In particular, we will show how these techniques allow one to solve a number of fundamental problems related to object recognition for configurations of point features under a generalized weak perspective model of image formation. Our approach makes use of progress in shape theory and includes the development of objectimage equations for shape matching and the exploitation of shape space metrices (especially objectimage metrics) to measure matching up to certain transformations. This theory is built on advanced mathematical techniques from algebraic and differential geometry which are used to construct generalized shape spaces for various projection and sensor models. That construction in turn is used to find natural metrics that express the distance (geometric difference) between two configurations of object features, two configurations of image features, or an object and an image pair. Such metrics are believed to produce the most robust tests for object identification; at least as far as the objectu0027s geometry is concerned. Moreover, these metrics provide a basis for efficient hashing schemes to do identification quickly, and they provide a rigorous foundation for error and statistical analysis in any recognition system. The most important feature of a shape theoretic approach is that all of the matching tests and metrics are independent of the choice of coordinates used to express the feature locations on the object or in the image. In addition, the approach is independent of the camerasensor position and any camerasensor parameters. Finally, the method is also independent of object pose or image orientation. This is what makes the results so powerful.

Image Analysis: Identification of Objects via Polynomial Systems 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 twodimensional 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 xe2x80x9csignaturexe2x80x9d 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.