Deconvolution of 3-D Gaussian kernels

TL;DR

This paper extends existing formulas for deconvolving 1D Gaussian kernels to 3D by utilizing multivariate Hermite polynomials, enabling more accurate analysis of three-dimensional Gaussian data.

Contribution

It introduces a generalized deconvolution formula for 3D Gaussian kernels using scalar Grad's multivariate Hermite polynomials, expanding the mathematical toolkit for Gaussian analysis.

Findings

Derived a 3D deconvolution formula for Gaussian kernels

Connected multivariate Hermite polynomials to ordinary Hermite polynomials

Provides a mathematical foundation for 3D Gaussian deconvolution

Abstract

Ulmer and Kaissl formulas for the deconvolution of one-dimensional Gaussian kernels are generalized to the three-dimensional case. The generalization is based on the use of the scalar version of the Grad's multivariate Hermite polynomials which can be expressed through ordinary Hermite polynomials.