Linear Spatio-Temporal Scale-Space

This article shows how a linear scale-space formulation previously expressed for spatial domains extends to spatio-temporal data. Starting from the main assumptions that: (i) the scale-space should be generated by convolution with a semi-group of filter kernels and that (ii) local extrema must not be enhanced when the scale parameter increases, a complete taxonomy is given of the linear scale-space concepts that satisfy these conditions on spatial, temporal and spatio-temporal domains, including the cases with continuous as well as discrete data. Key aspects captured by this theory include that: (i) time-causal scale-space kernels must not extend into the future, (ii) filter shapes can be tuned from specific context information, permitting mechanisms such local shifting, shape adaptation and velocity adaptation, all expressed in terms of local diffusion operations. Receptive field profiles generated by the proposed theory show high qualitative similarities to receptive field profiles recorded from biological vision. Earlier versions of this manuscript have been presented at the PhD School on Scale-Space Theory in Copenhagen, Denmark, May 1996 and in B. ter Haar Romeny et al (eds) Proc. 1st International Conference on ScaleSpace Theory in Computer Vision, (Utrecht, Netherlands), July 2-4, 1997, Springer-Verlag Lecture Notes in Computer Science, volume 1252. The support from the Swedish Research Council for Engineering Sciences, TFR, and and from the Royal Swedish Academy of Sciences as well as the Knut and Alice Wallenberg Foundation is gratefully acknowledged.