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We prove a novel result wherein the density function of thegradients---corresponding to density function of the derivatives in onedimension---of a thrice differentiable function S (obtained via a randomvariable transformation of a uniformly distributed random variable) defined ona closed, bounded interval \Omega \subset R is accurately approximated by thenormalized power spectrum of \phi=exp(iS/\tau) as the free parameter \tau-->0.The result is shown using the well known stationary phase approximation andstandard integration techniques and requires proper ordering of limits.Experimental results provide anecdotal visual evidence corroborating theresult.

An application of the stationary phase method for estimating probability densities of function derivatives