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：本文分析了两种基于高斯导数的混合离散化方法的性质，这两种方法分别基于与归一化采样高斯核或积分高斯核的卷积，随后进行中心差分。研究这些离散化方法的动机在于，当在同一尺度水平上需要多个不同阶数的空间导数时，与基于采样高斯核或积分高斯核的显式卷积的直接导数近似方法相比，这些方法可以显著提高计算效率。虽然这些计算优势同样适用于基于与高斯核离散模拟的卷积随后进行中心差分的真正离散方法（用于计算高斯导数的离散模拟），但高斯核离散模拟的底层数学原语（涉及整数阶修正贝塞尔函数）在某些图像处理框架中可能不可用，例如在执行基于尺度参数化滤波器（以高斯导数表示）的深度学习时，其中涉及尺度水平的学习。在本文中，我们通过定量性能度量来表征这些混合离散化方法的性质，包括它们所隐含的空间平滑量，以及从具有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，重点关注尺度参数值非常小时的行为，这可能与完全连续尺度空间理论得到的结果有显著差异，并且在不同类型的离散化方法之间也可能存在显著差异。