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arXiv · 1211.3038

Gradient density estimation in arbitrary finite dimensions using the method of stationary phase

Karthik S. Gurumoorthy·Anand Rangarajan·John Corring
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Abstract

We prove that the density function of the gradient of a sufficiently smooth function $S : \Omega \subset \mathbb{R}^d \rightarrow \mathbb{R}$, obtained via a random variable transformation of a uniformly distributed random variable, is increasingly closely approximated by the normalized power spectrum of $\phi=\exp\left(\frac{iS}{\tau}\right)$ as the free parameter $\tau \rightarrow 0$. 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 approach.

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Karthik S. Gurumoorthy, Anand Rangarajan, John Corring. 2012-11-13. Gradient density estimation in arbitrary finite dimensions using the method of stationary phase. https://arxiv.org/abs/1211.3038

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