A group-theoretic approach to the triple correlation

The triple correlation is a useful tool for averaging multiple observations of a signal in noise, in particular when the signal is translating by unknown amounts in between observations. What makes the triple correlation attractive for such a task are three properties: it is invariant under translation of the underlying signal; it is unbiased in additive Gaussian noise; (3) it retains enough phase information to permit recovery of the underlying signal. The author investigates the extent to which all three properties generalize to signals on arbitrary groups. He aims is to develop a theory for averaging observations of signals that are undergoing not just translation, but also rotation, scaling, or any other type of geometric transformation. To that end, he describes the basic theoretical foundations of triple correlation on groups, and also describes several uniqueness results that establish the relationship between two signals on a group that have the same triple correlation.