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.
Highlights
Density estimation methods provide a faithful estimate of a non-observable probability density function based on a given collection of observed data [1] [2] [3] [4]
We prove that the density function of the gradient of a sufficiently smooth function S : Ω ⊂ d →, obtained via a random variable transformation of a uniformly distributed random variable, is increasingly closely approximated by the normalized power spectrum of φ as the free parameter τ → 0
We show that the computation of the joint density function of Y = ∇S may be approximated by the power spectrum of φ, with the approximation becoming increasingly tight point-wise as τ → 0
Summary
Density estimation methods provide a faithful estimate of a non-observable probability density function based on a given collection of observed data [1] [2] [3] [4]. In a recent article [16], an adaption of the HOG descriptor called the Gradient Field HOG (GF-HOG) is used for sketch-based image retrieval In these systems, the image intensity is treated as a function S ( X ) over a 2D domain, and the distribution of intensity gradients or edge directions is used as the feature descriptor to characterize the object appearance or shape within an image. As we will show, the core of our density estimation approach is based on evaluating interval measures of the squared magnitude of a wave function in the frequency domain. For this reason, our approach is deemed a wave function approach to density estimation and we refer to it as such. In contrast to our previous work, we regard our current effort as a generalization of the gradient density estimation result, established for arbitrary smooth functions in arbitrary finite dimensions
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