Gradient Density Estimation in Arbitrary Finite Dimensions Using the Method of Stationary Phase

Abstract

We prove that the density function of the gradient of a sufficiently smooth function , obtained via a random variable transformation of a uniformly distributed random variable, is increasingly closely approximated by the normalized power spectrum of as the free parameter . The frequencies act as gradient histogram bins. The result is shown using the stationary phase approximation and standard integration techniques and requires proper ordering of limits. We highlight a relationship with the well-known characteristic function approach to density estimation, and detail why our result is distinct from this method. Our framework for computing the joint density of gradients is extremely fast and straightforward to implement requiring a single Fourier transform operation without explicitly computing the gradients.