[Paper Review] Closed Form for Some Gaussian Convolutions

Hossein Mobahi|arXiv (Cornell University)|Feb 10, 2016

Image and Object Detection Techniques6 references3 citations

TL;DR

This paper derives closed-form expressions for the Weierstrass transform (Gaussian convolution) of two important function classes: multivariate polynomials and Gaussian radial basis functions. It shows that the convolution of a polynomial with an isotropic Gaussian yields another polynomial whose coefficients are determined by Hermite polynomials, while the convolution of a Gaussian RBF with a Gaussian kernel results in a scaled, broader Gaussian RBF. These results enable exact analytical computation in applications involving scale-space filtering, feature extraction, and optimization.

ABSTRACT

The convolution of a function with an isotropic Gaussian appears in many contexts such as differential equations, computer vision, signal processing, and numerical optimization. Although this convolution does not always have a closed form expression, there are important family of functions for which closed form exists. This article investigates some of such cases.

Motivation & Objective

• To derive exact closed-form expressions for the Weierstrass transform of multivariate polynomials and Gaussian radial basis functions.
• To provide analytical tools for efficient computation in applications requiring Gaussian smoothing, such as image processing, signal filtering, and nonconvex optimization.
• To unify the treatment of Gaussian convolutions across different function classes using Hermite polynomials and convolution identities.
• To enable exact computation instead of numerical approximation in contexts where closed forms exist, improving accuracy and efficiency.

Proposed method

• Uses Hermite polynomials to express the convolution of monomials with an isotropic Gaussian kernel, leveraging the identity $ u(x,p, au) = (i au)^p h_p(x/(i au)) $, where $ h_p $ is the $ p $-th Hermite polynomial.
• Applies the convolution theorem and properties of delta functions and Gaussians to derive the closed form for functions of the form $ f(oldsymbol{x}^T oldsymbol{w}) $, showing that the transform depends on the norm of the weight vector.
• Derives the convolution of a Gaussian RBF $ e^{-rac{\|\boldsymbol{x}-\boldsymbol{x}_j\|^2}{2\delta_j^2}} $ with a kernel $ k_\sigma $, resulting in a scaled and broader RBF with variance $ \delta_j^2 + \sigma^2 $.
• Uses the identity $ [f(h(\cdot)) \star k_\sigma](\boldsymbol{w}) = \tilde{f}(\boldsymbol{w}^T \boldsymbol{x}; \sigma \|\boldsymbol{x}\|) $ to generalize results to linear functionals of vectors.
• Employs the convolution of two Gaussians: $ k_{\delta}(at+b) \star k_{\sigma}(x-t) = k_{\sqrt{\delta^2 + a^2\sigma^2}}(at+b) $, which underpins the derivation of the linear argument case.
• Provides explicit tables of $ u(x,p,\sigma) $ for $ p = 0 $ to $ 10 $, enabling direct computation of polynomial convolutions.

Experimental results

Research questions

• RQ1 Can the Weierstrass transform of a multivariate polynomial be expressed in closed form, and if so, what is its analytical structure?
• RQ2 What is the closed-form expression for the convolution of a Gaussian radial basis function with an isotropic Gaussian kernel?
• RQ3 How does the convolution of a function $ f(\boldsymbol{x}^T \boldsymbol{w}) $ with a Gaussian kernel depend on the norm of the vector $ \boldsymbol{x} $?
• RQ4 What is the analytical form of the convolution of trigonometric functions like $ \sin(x) $ and $ \cos(x) $ with a Gaussian?
• RQ5 How can the convolution of a function with a linear argument, such as $ f(ax+b) $, be simplified using Gaussian convolution identities?

Key findings

• The convolution of a monomial $ x_d^{p_{d,j}} $ with a 1D Gaussian kernel is given by $ u(x_d, p_{d,j}, \sigma) = (i\sigma)^{p_{d,j}} h_{p_{d,j}}(x_d/(i\sigma)) $, where $ h_p $ is the $ p $-th Hermite polynomial.
• For a multivariate polynomial $ f(\boldsymbol{x}) = \sum_j a_j \prod_d x_d^{p_{d,j}} $, the full convolution is $ [f \star k_\sigma](\boldsymbol{x}) = \sum_j a_j \prod_d u(x_d, p_{d,j}, \sigma) $, enabling exact analytical evaluation.
• The convolution of a Gaussian RBF $ e^{-\|\boldsymbol{x}-\boldsymbol{x}_j\|^2 / (2\delta_j^2)} $ with $ k_\sigma $ yields $ \left(\frac{\delta_j}{\sqrt{\delta_j^2 + \sigma^2}}\right)^n e^{-\|\boldsymbol{x}-\boldsymbol{x}_j\|^2 / (2(\delta_j^2 + \sigma^2))} $, a scaled and broader RBF.
• The convolution of $ f(\boldsymbol{x}^T \boldsymbol{w}) $ with $ k_\sigma $ is $ \tilde{f}(\boldsymbol{w}^T \boldsymbol{x}; \sigma \|\boldsymbol{x}\|) $, showing that the effective smoothing scale depends on the norm of the weight vector.
• For trigonometric functions like $ \sin(x) $ and $ \cos(x) $, the convolution results in $ e^{-\sigma^2/2} \sin(x) $ and $ e^{-\sigma^2/2} \cos(x) $, respectively, with higher-order trigonometric functions yielding more complex but closed-form expressions.
• The convolution of $ f(ax+b) $ with a Gaussian kernel results in $ k_{\sqrt{\delta^2 + a^2\sigma^2}}(ax+b) $, demonstrating that linear transformations of the argument lead to rescaling of the kernel width.