Energy filters,motion uncertainty,and motion sensitive cells in the visual cortex: a mathematical analysis

Energy filters are tuned to space-time frequency orientations. In order to compute velocity it is necessary to use a collection of filters, each tuned to a different space-time frequency. Here we analyze, in a probabilistic framework, the properties of the motion uncertainty. Its lower bound, which can be explicitly computed through the Cramér-Rao inequality, will have different values depending on the filter parameters. We show for the Gabor filter that, in order to minimize the motion uncertainty, the spatial and temporal filter sizes cannot be arbitrarily chosen; they are only allowed to vary over a limited range of values such that the temporal filter bandwidth is larger than the spatial bandwidth. This property is shared by motion sensitive cells in the primary visual cortex of the cat, which are known to be direction selective and are tuned to spacetime frequency orientations. We conjecture that these cells have larger temporal bandwidth relative to their spatial bandwidth because they compute velocity with maximum efficiency, that is, with a minimum motion uncertainty.