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

TL;DR

This paper analyzes two hybrid discretization methods for Gaussian derivatives, focusing on their properties, efficiency, and behavior at small scales, especially in contexts like deep learning where discrete kernels may not be available.

Contribution

It provides a detailed characterization of hybrid discretization methods, highlighting their performance and consistency in scale-space analysis, especially for small scale parameters.

Findings

Hybrid methods offer computational efficiency for multiple derivatives at the same scale.

Performance varies significantly at very small scales compared to continuous theory.

Discretization choice impacts the accuracy of scale estimates in feature detection.

Abstract

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.