Discrete approximations of Gaussian smoothing and Gaussian derivatives

TL;DR

The paper analyzes how to discretize Gaussian smoothing and Gaussian derivatives while preserving scale-space properties on discrete data. It compares three discretization families—sampled, integrated, and the genuinely discrete analogue—across theoretical criteria and quantitative measures, identifying the discrete analogue as the most robust at fine scales and the sampled variant as highly accurate at coarser scales. For derivatives, convolving with the discrete Gaussian kernel followed by small-step central differences yields the most faithful derivatives at fine scales, while at larger scales the sampled derivatives perform best. The work extends to scale selection applications, directional derivatives, and affine extensions, offering a rigorous framework and practical masks for efficient, scale-consistent feature detection with potential use in deep learning. A comprehensive appendix provides exact formulas, diffusion connections, and directional-derivative constructions to support implementation and reproducibility.

Abstract

This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region and (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data. We study the properties of these three main discretization methods both theoretically and experimentally, and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and derivatives as well as the integrated Gaussian kernels and derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing.

Figures (18)

• Figure 1: Graphs of the main types of Gaussian smoothing kernels and Gaussian derivative kernels considered in this paper, here at the scale $\sigma = 1$, with the raw smoothing kernels in the top row and the order of spatial differentiation increasing downwards up to order 4: (left column) continuous Gaussian kernels and continuous Gaussian derivatives, (middle left column) sampled Gaussian kernels and sampled Gaussian derivatives, (middle right column) integrated Gaussian kernels and integrated Gaussian derivatives, and (right column) discrete Gaussian kernels and discrete analogues of Gaussian derivatives. Note that the scaling of the vertical axis may vary between the different subfigures. (Horizontal axis: the 1-D spatial coordinate $x \in [-5, 5]$.)
• Figure 2: Graphs of the $l_1$-norm-based normalization error $E_{\scriptsize\hbox{norm}}(T(\cdot;\; s))$, according to (\ref{['eq-gauss-L1norm-error']}), for the discrete analogue of the Gaussian kernel, the sampled Gaussian kernel and the integrated Gaussian kernel. Note that this error measure is equal to zero for both the discrete analogue of the Gaussian kernel, the normalized sampled Gaussian kernel and the integrated Gaussian kernel. (Horizontal axis: Scale parameter in units of $\sigma = \sqrt{s} \in [0.1, 2]$.)
• Figure 3: Graphs of the spatial standard deviations $\sqrt{V(T(\cdot;\; s))}$ for the discrete analogue of the Gaussian kernel, the sampled Gaussian kernel and the integrated Gaussian kernel. The standard deviation is exactly equal to the scale parameter $\sigma = \sqrt{s}$ for the discrete analogue of the Gaussian kernel. The standard deviation of the normalized sampled Gaussian kernel is equal to the standard deviation of the regular sampled Gaussian kernel. (Horizontal axis: Scale parameter in units of $\sigma = \sqrt{s} \in [0.1, 2]$.)
• Figure 4: Graphs of the absolute scale difference $E_{\Delta s}(T(\cdot;\; s))$, according to (\ref{['eq-scale-diff-gauss']}) and in units of the spatial variance $V(T(\cdot;\; s))$, for the discrete analogue of the Gaussian kernel, the sampled Gaussian kernel and the integrated Gaussian kernel. This scale difference is exactly equal to zero for the discrete analogue of the Gaussian kernel. For scale values $\sigma < 0.75$, the absolute scale difference is substantial for the sampled Gaussian kernel, and then rapidly tends to zero for larger scales. For the integrated Gaussian kernel, the absolute scale difference does, however, not approach zero with increasing scale. Instead, it approaches the numerical value $\Delta s \approx 0.0833$, close to the spatial variance $1/12$ of a box filter over each pixel support region. The spatial variance-based absolute scale difference for the normalized sampled Gaussian kernel is equal to the spatial variance-based absolute scale difference for the regular sampled Gaussian kernel. (Horizontal axis: Scale parameter in units of $\sigma = \sqrt{s} \in [0.1, 2]$.)
• Figure 5: Graphs of the relative scale difference $E_{\scriptsize\hbox{relscale}}(T(\cdot;\; s))$, according to (\ref{['eq-rel-scale-error-gauss']}) and in units of the spatial standard deviation of the discrete kernels, for the discrete analogue of the Gaussian kernel, the sampled Gaussian kernel and the integrated Gaussian kernel. This relative scale error is exactly equal to zero for the discrete analogue of the Gaussian kernel. For scale values $\sigma < 0.75$, the relative scale difference is substantial for sampled Gaussian kernel, and then rapidly tends to zero for larger scales. For the integrated Gaussian kernel, the relative scale difference is significantly larger, while approaching zero with increasing scale. The relative scale difference for the normalized sampled Gaussian kernel is equal to the relative scale difference for the regular sampled Gaussian kernel. (Horizontal axis: Scale parameter in units of $\sigma = \sqrt{s} \in [0.1, 2]$.)