GEOMETRY AND MULTIPLE DIRECTION ESTIMATION

We consider the problem of the estimation of multiple target directions, given the output of a phased array of identical elements. The natural mathematical setting for this problem is the Grassmannian manifold. In order to understand the problems considered here, a knowledge of exterior forms, lie groups and differential geometry is essential. We review some of the essential ideas relevant to the problem, including the Maurer-Cartan equations, algebraic varieties in projective space and the invariant volume integral for symmetric spaces. Our purpose is to study probability measures on Grassmannians, in order to pose and solve the Bayes problem on Grassmannians.