This paper presents an analysis of properties of two hybrid discretization methods for Gaussian derivatives, based on convolutions with either the normalized sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretization methods is that in situations when multiple spatial derivatives of different order are needed at the same scale level, they can be computed significantly more efficiently compared to more direct derivative approximations based on explicit convolutions with either sampled Gaussian kernels or integrated Gaussian kernels. While these computational benefits do also hold for the genuinely discrete approach for computing discrete analogues of Gaussian derivatives, based on convolution with the discrete analogue of the Gaussian kernel followed by central differences, the underlying mathematical primitives for the discrete analogue of the Gaussian kernel, in terms of modified Bessel functions of integer order, may not be available in certain frameworks for image processing, such as when performing deep learning based on scale-parameterized filters in terms of Gaussian derivatives, with learning of the scale levels. In this paper, we present a characterization of the properties of these hybrid discretization methods, in terms of quantitative performance measures concerning the amount of spatial smoothing that they imply, as well as the relative consistency of scale estimates obtained from scale-invariant feature detectors with automatic scale selection, with an emphasis on the behaviour for very small values of the scale parameter, which may differ significantly from corresponding results obtained from the fully continuous scale-space theory, as well as between different types of discretization methods.

翻译：本文分析了两种高斯导数混合离散化方法的性质，这些方法基于对归一化采样高斯核或积分高斯核进行卷积后，再执行中心差分。研究这些离散化方法的动机在于，当需要在同一尺度层级上计算多个不同阶空间导数时，与基于采样高斯核或积分高斯核的直接显式卷积导数逼近相比，它们能显著提高计算效率。尽管这些计算优势也同样适用于基于离散高斯核卷积后执行中心差分的纯离散方法（以计算高斯导数的离散模拟），但离散高斯核的底层数学原语——即基于整数阶修正贝塞尔函数的表达——在某些图像处理框架中可能无法直接使用，例如在基于尺度参数化高斯导数滤波器进行深度学习并同时学习尺度层级时。本文通过量化性能指标，对这些混合离散化方法所隐含的空间平滑量及其性质进行了表征，并评估了采用自动尺度选择的尺度不变特征检测器获得的尺度估计的相对一致性。特别关注了尺度参数取值极小情况下的行为，该行为可能与完全连续尺度空间理论的结果存在显著差异，且在不同离散化方法之间也存在差异。