Generalized Gaussian scalespace axiomatics comprising linear scalespace, affine scalespace and spatiotemporal scalespaceTony LindebergJournal of Mathematical Imaging and Vision, Volume 40, Number 1, 3681, May 2011. Digitally published with DOI: 10.1007/s1085101002422 in December 2010. Longer version available as Technical report ISRN KTH/CSC/CV2010/3SE, ISSN 16537092, School of Computer Science and Communication, KTH (Royal Institute of Technology), SE100 44 Stockholm, Sweden, November 2010. AbstractThis paper describes a generalized axiomatic scalespace theory that makes it possible to derive the notions of linear scalespace, affine Gaussian scalespace and linear spatiotemporal scalespace using a similar set of assumptions (scalespace axioms).The notion of nonenhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general nonisotropic kernels over both spatial and spatiotemporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scalespace concepts that satisfy these conditions on isotropic spatial, nonisotropic spatial and spatiotemporal domains, which results in a general taxonomy of Gaussian scalespaces 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 timecausal spatiotemporal scalespaces can be derived from similar assumptions. The mathematical structure of these scalespaces 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 multiscale spatiotemporal 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 timecausal scalespaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatiotemporal scalespace as well as the timecausal spatiotemporal scalespace relate to spatiotemporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to spacetime separable cells in the LGN as well as to both spacetime separable and nonseparable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasiquadrature measures expressed in terms of scale normalized spatiotemporal 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 timecausal scalespaces using a unified framework, which generalizes and complements previously presented scalespace formulations in this area. Keywords: scalespace, multiscale representation, scalespace axioms, nonenhancement of local extrema, causality, scale invariance, Gaussian kernel, Gaussian derivative, spatiotemporal, affine, spatial, temporal, timerecursive, 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) Online version: (At the official site of the journal) Earlier treatments on this topic can be found in:
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