Generalized Gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space

Tony Lindeberg

Journal of Mathematical Imaging and Vision, Volume 40, Number 1, 36-81, May 2011.

Digitally published with DOI: 10.1007/s10851-010-0242-2 in December 2010.

Longer version available as Technical report ISRN KTH/CSC/CV--2010/3--SE, ISSN 1653-7092, School of Computer Science and Communication, KTH (Royal Institute of Technology), SE-100 44 Stockholm, Sweden, November 2010.


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 type of 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.

Keywords: 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.

PDF: (shorter journal version 47 pages 17.3 Mb) (longer technical report version 76 pages 17.4 Mb)

On-line version: (At the official site of the journal)

Earlier treatments on this topic can be found in:

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