Approximation properties relative to continuous scale space for hybrid discretisations of Gaussian derivative operators

This paper presents an analysis of properties of two hybrid discretisation methods for Gaussian derivatives, based on convolutions with either the normalised sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretisation methods is that in situations when multiple spatial derivatives of different orders 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 derivative kernels or integrated Gaussian derivative kernels. We characterise the properties of these hybrid discretisation methods in terms of quantitative performance measures, concerning the amount of spatial smoothing that they imply, as well as the relative consistency of the 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 discretisation methods. The presented results are intended as a guide, when designing as well as interpreting the experimental results of scale-space algorithms that operate at very fine scale levels.