Temporal Scale Spaces

A computational theory of visual receptive fields

“…Figures 14 and 15 show examples of such time-causal spatio-temporal kernels with their partial spatio-temporal derivatives in the space–time separable case with and for the velocity-adapted case with The time-causal smoothing kernel has been previously used for modelling heat conduction in solids by (Carslaw and Jaeger (1959), section 14.2) and also been derived by Fagerström (2005) as one member in a family of self-similar kernels obtained from the assumption of scale invariance.

Fig. 14 Space–time separable kernels up to order two obtained from the time-causal spatio-temporal scale space in the case of a 1+1D space–time (, ) ( horizontal axis : space , vertical axis : time )

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“…Fagerström (2005) investigated self-similar temporal scale-space concepts derived from the assumptions of a semigroup structure combined with scale invariance, with an extension to the spatio-temporal domain in Fagerström (2007) that also comprises the notion of velocity-adapted filters. Lindeberg (2011) gives a unified treatment of the scale-space axiomatics of linear, affine, and spatio-temporal scale space for continuous images based on the assumption of non-enhancement of local extrema over spatial and spatio-temporal domains, including more explicit statements of the uniqueness results regarding the Gaussian spatio-temporal scale space earlier outlined in Lindeberg (2001) and the application of non-enhancement of local extrema to a continuous time-causal and time-recursive spatio-temporal scale space.…”