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Approximation properties relative to continuous scale space for hybrid discretizations of Gaussian derivative operators
KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Computational Science and Technology (CST). (Computational Brain Science Lab)ORCID iD: 0000-0002-9081-2170
2024 (English)Report (Other academic)
Abstract [en]

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. The hybrid discretizations studied in this paper do, from this perspective, offer a computationally more efficient way of implementing deep networks based on Gaussian derivatives for such use cases.  

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.

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.

Place, publisher, year, edition, pages
2024.
Keywords [en]
scale, discrete, continuous, smoothing, Gaussian kernel, Gaussian derivative, scale space
National Category
Computer Vision and Robotics (Autonomous Systems) Computational Mathematics
Research subject
Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-346267OAI: oai:DiVA.org:kth-346267DiVA, id: diva2:1857071
Projects
Covariant and invariant deep networks
Funder
Swedish Research Council, 2022-02969
Note

QC 20240514

Available from: 2024-05-10 Created: 2024-05-10 Last updated: 2024-06-13Bibliographically approved

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arXiv:2405.05095

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CiteExportLink to record
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Citation style
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