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，这些方法基于归一化采样高斯核或积分高斯核与中心差分的卷积。研究这些离散化方法的动机在于：当需要在同一尺度水平上计算多个不同阶次的空间导数时，相较于基于采样高斯核或积分高斯核的显式卷积的直接导数逼近方法，它们可以显著提高计算效率。尽管基于离散高斯核与中心差分的卷积计算高斯导数离散模拟的真正离散方法也具有这些计算优势，但离散高斯核的数学基元（以整数阶修正贝塞尔函数表示）在某些图像处理框架中可能不可用，例如在基于高斯导数的尺度参数化滤波器进行深度学习并学习尺度水平时。本文通过定量性能指标表征了这些混合离散化方法的性质，涉及它们隐含的空间平滑量，以及基于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，重点关注尺度参数极小值下的行为——这些行为可能显著区别于完全连续尺度空间理论得出的相应结果，且不同离散化方法之间也存在差异。