Gaussian Convolutions Numerical Approximations Based on Interpolation

Abstract

Gaussian convolutions are perhaps the most often used image operators in low-level computer vision tasks. Surprisingly though, there are precious few articles that describe efficient and accurate implementations of these operators.

In this paper we describe numerical approximations of Gaussian convolutions based on interpolation. We start with the continuous convolution integral and use an interpolation technique to approximate the continuous image f from its sampled version F.

Based on the interpolation a numerical approximation of the continuous convolution integral that can be calculated as a discrete convolution sum is obtained. The discrete convolution kernel is not equal to the sampled version of the continuous convolution kernel. Instead the convolution of the continuous kernel and the interpolation kernel has to be sampled to serve as the discrete convolution kernel.

Some preliminary experiments are shown based on zero order (nearest neighbor) interpolation, first order (linear) interpolation, third order (cubic) interpolations and sinc-interpolation. These experiments show that the proposed algorithm is more accurate for small scales, especially for Gaussian derivative convolutions when compared to the classical way of discretizing the Gaussian convolution.