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On the axiomatic foundations of linear scale-space: Combining semi-group structure with causality vs. scale invariance
KTH, School of Computer Science and Communication (CSC), Computational Biology, CB.ORCID iD: 0000-0002-9081-2170
1996 (English)In: Gaussian Scale-Space Theory: Proceedings of PhD School on Scale-Space (Copenhagen, Denmark) May 1996 / [ed] J. Sporring, M. Nielsen, L. Florack and P. Johansen, Kluwer Academic Publishers, 1996Chapter in book (Other academic)
Abstract [en]

The notion of multi-scale representation is essential to many aspects of early visual processing. This article deals with the axiomatic formulation of the special type of multi-scale representation known as scale-space representation. Specifically, this work is concerned with the problem of how different choices of basic assumptions (scale-space axioms) restrict the class of permissible smoothing operations.

A scale-space formulation previously expressed for discrete signals is adapted to the continuous domain. The basic assumptions are that the scale-space family should be generated by convolution with a one-parameter family of rotationally symmetric smoothing kernels that satisfy a semi-group structure and obey a causality condition expressed as a non-enhancement requirement of local extrema. Under these assumptions, it is shown that the smoothing kernel is uniquely determined to be a Gaussian.

Relations between this scale scale-space formulation and recent formulations based on scale invariance are explained in detail. Connections are also pointed out to approaches based on non-uniform smoothing.

Place, publisher, year, edition, pages
Kluwer Academic Publishers, 1996.
Keyword [en]
scale-space, Gaussian filtering, causality, diffusion, scale invariance, multi-scale representation, computer vision, signal processing
National Category
Computer and Information Science Computer Vision and Robotics (Autonomous Systems) Mathematics
URN: urn:nbn:se:kth:diva-40221OAI: diva2:456533

QC 20111115

Available from: 2013-04-19 Created: 2011-09-13 Last updated: 2013-04-19Bibliographically approved

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Lindeberg, Tony
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