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 object-image equations for shape matching and the exploitation of shape space metrices (especially object-image 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 object's 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 camera/sensor position and any camera/sensor parameters. Finally, the method is also independent of object pose or image orientation. This is what makes the results so powerful.