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.

翻译：本文深入研究了尺度空间理论中，针对离散数据应用高斯平滑与高斯导数计算时的近似问题。结合连续与离散尺度空间理论的前期公理化研究，我们主要探讨了通过显式离散卷积实现这些尺度空间操作的三种主要离散化方法，分别基于：(i) 对高斯核及高斯导数核进行采样；(ii) 在每个像素支撑区域上对高斯核及高斯导数核进行局部积分；(iii) 将尺度空间分析建立在高斯核的离散模拟基础上，随后通过对空间平滑后的图像数据施加小支撑中心差分算子来计算导数近似。我们从理论和实验两方面研究了这三种主要离散化方法的性质，并通过定量指标表征其性能，包括它们对尺度选择任务产生的效果。该任务针对四种不同用例进行了研究，并重点关注精细尺度下的行为。结果表明，采样高斯核与导数以及积分高斯核与导数在极细尺度上表现非常差。而在极细尺度上，高斯核的离散模拟及其相应的离散导数近似则表现显著更优。另一方面，本文实验表明，当尺度参数足够大时——在本文实验中，当尺度参数大于约1（以网格间距为单位）时——采样高斯核与采样高斯导数能够对相应的连续结果产生数值上非常良好的近似。