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arXiv · 1108.1783

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

Karthik S. Gurumoorthy·Anand Rangarajan·Arunava Banerjee
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Abstract

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

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Karthik S. Gurumoorthy, Anand Rangarajan, Arunava Banerjee. 2011-08-08. An application of the stationary phase method for estimating probability densities of function derivatives. https://arxiv.org/abs/1108.1783

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