  The generalized density is a product of a density function and a weight
function. For example, the average local brightness of an astronomical image is
the probability of finding a galaxy times the mean brightness of the galaxy. We
propose a method for studying the geometric structure of generalized densities.
In particular, we show how to find the modes and ridges of a generalized
density function using a modification of the mean shift algorithm and its
variant, subspace constrained mean shift. Our method can be used to perform
clustering and to calculate a measure of connectivity between clusters. We
establish consistency and rates of convergence for our estimator and apply the
methods to data from two astronomical problems.
