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Generalized Gaussian Scale-Space Axiomatics Comprising Linear Scale-Space, Affine Scale-Space and Spatio-Temporal Scale-Space
KTH, School of Computer Science and Communication (CSC), Computational Biology, CB.ORCID iD: 0000-0002-9081-2170
2011 (English)In: Journal of Mathematical Imaging and Vision, ISSN 0924-9907, E-ISSN 1573-7683, Vol. 40, no 1, 36-81 p.Article in journal (Refereed) Published
Abstract [en]

This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms). The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives. The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and/or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area.

Place, publisher, year, edition, pages
Springer Science+Business Media B.V., 2011. Vol. 40, no 1, 36-81 p.
Keyword [en]
Scale-space, Multi-scale representation, Scale-space axioms, Non-enhancement of local extrema, Causality, Scale invariance, Gaussian kernel, Gaussian derivative, Spatio-temporal, Affine, Spatial, Temporal, Time-recursive, Receptive field, Diffusion, Computer vision, Image processing
National Category
Computer Science Computer Vision and Robotics (Autonomous Systems) Mathematics
URN: urn:nbn:se:kth:diva-40136DOI: 10.1007/s10851-010-0242-2ISI: 000288260500003ScopusID: 2-s2.0-79953187911OAI: diva2:440633
Swedish Research Council, 2004-4680Knut and Alice Wallenberg Foundation

QC 20110913

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

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