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：本文深入研究了在尺度空间理论中，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处理，我们考虑了三种基于显式离散卷积对该尺度空间运算进行离散化的主要方法，具体包括：(i) 对高斯核及高斯导数核进行采样；(ii) 在每个像素支持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，然后通过对空间平滑图像数据应用小支撑中心差分算子来计算导数近似。我们从理论和实验角度研究了这三种主要离散化方法的特性，并通过定量指标对其性能进行刻画，其中包括在四个不同应用场景下针对尺度选择任务所产生的结果，重点关注细尺度下的行为。结果表明：采样高斯核及其导数、以及积分高斯核及其导数在极细尺度下表现非常差。而在极细尺度下，高斯核的离散模拟及其对应的离散导数近似则表现出显著更优的性能。另一方面，当尺度参数足够大时——在本文实验中，当尺度参数（以网格间距为单位）大于约1时——采样高斯核及采样高斯导数在数值上能够对相应连续结果产生非常好的近似。