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：本文深入探讨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先验公理化处理方法，我们主要考虑了三种基于显式离散卷积的离散化方法：（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内对高斯核及高斯导数核进行局部积分；（iii）基于高斯核的离散模拟建立尺度空间分析，然后通过使用小支撑中心差分算子对空间平滑图像数据计算导数近似。我们从理论和实验两方面研究了这三种离散化方法的特性，并通过定量指标（包括针对尺度选择任务所产生的结果）评估其性能。我们针对四种不同使用场景进行了分析，特别关注精细尺度下的行为表现。结果表明：在非常精细的尺度下，采样高斯核与导数以及积分高斯核与导数的表现极差；而基于高斯核离散模拟及其对应离散导数近似的方法在精细尺度下表现显著更优。不过，当尺度参数足够大时（本文实验中当尺度参数大于约1个网格间距单位时），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精确的近似。