Time-Causal and Time-Recursive Spatio-Temporal Receptive Fields

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, obtained by a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain. Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about (i) parameterizing the intermediate temporal scale levels, (ii) analysing the resulting temporal dynamics, (iii) transferring the theory to a discrete implementation in terms of recursive filters over time, (iv) computing scale-normalized spatio-temporal derivative expressions for spatio-temporal feature detection and (v) computational modelling of receptive fields in the lateral geniculate nucleus (LGN) and the primary visual cortex (V1) in biological vision. We show that by distributing the intermediate temporal scale levels according to a logarithmic distribution, we obtain a new family of temporal scale-space kernels with better temporal characteristics compared to a more traditional approach of using a uniform distribution of the intermediate temporal scale levels. Specifically, the new family of time-causal kernels has much faster temporal response properties (shorter temporal delays) compared to the kernels obtained from a uniform distribution. When increasing the number of temporal scale levels, the temporal scale-space kernels in the new family do also converge very rapidly to a limit kernel possessing true self-similar scale-invariant properties over temporal scales. Thereby, the new representation allows for true scale invariance over variations in the temporal scale, although the underlying temporal scale-space representation is based on a discretized temporal scale parameter. We show how scale-normalized temporal derivatives can be defined for these time-causal scale-space kernels and how the composed theory can be used for computing basic types of scale-normalized spatio-temporal derivative expressions in a computationally efficient manner.