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Discrete approximations of Gaussian smoothing and Gaussian derivatives
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)In: Journal of Mathematical Imaging and Vision, ISSN 0924-9907, E-ISSN 1573-7683Article in journal (Refereed) Accepted
Abstract [en]

This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region and (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data.

We study the properties of these three main discretization methods both theoretically and experimentally, and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and derivatives as well as the integrated Gaussian kernels and derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing. Below a standard deviation of about 0.75, the results from the sampled Gaussian kernel or the sampled Gaussian derivatives are, however, not really usable, while the results obtained from the discrete analogue of the Gaussian kernel with its associated central difference operators applied to the spatially smoothed image data is then a much better choice.

Place, publisher, year, edition, pages
Springer Science+Business Media B.V., 2024.
Keywords [en]
discrete, continuous, Gaussian kernel, Gaussian derivative, directional derivative, scale-normalized derivative, steerable filter, filter bank, scale-space properties, scale space
National Category
Computer Vision and Robotics (Autonomous Systems)
Research subject
Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-346033OAI: oai:DiVA.org:kth-346033DiVA, id: diva2:1855169
Projects
Covariant and invariant deep networks
Funder
Swedish Research Council, 2022-02969
Note

QC 20240430

Available from: 2024-04-30 Created: 2024-04-30 Last updated: 2024-05-02Bibliographically approved

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