"The triple correlation of an ordinary function on the real line is the integral of the
product of that function with two independently shifted copies of itself. Triple
correlation methods are frequently used in signal processing for treating signals
that are corrupted by additive Gaussian noise; in particular, triple correlation
techniques perform well when multiple observations of the signal are available
and the signal may be translating in between the observations,e.g.,a sequence of
images of an object translating on a noisy background. What makes the triple
correlation particularly suitable for such tasks are three properties: (1) it is
invariant under translation of the underlying signal; (2) it is insensitive to
additive Gaussian noise; and (3) it retains most of the phase information in the
underlying signal. This thesis investigates whether properties (1)-(3) of the
triple correlation extend to functions on arbitrary locally compact groups, in
particular the groups of rotations and rigid motions of euclidean space that arise
in computer vision and signal processing.After defining the triple correlation for any locally compact group by using
the group's left-invariant Haar measure, it is easily shown that the resulting
object is invariant under left translation of the underlying function and
insensitive to additive Gaussian noise. What is more interesting is the question
of uniqueness : when two functions have the same triple correlation, how are the
functions related? Our results show that for most cases of practical interest, the
triple correlation of a function on an abstract group uniquely identifies that
function up to a group translation. We show how our results utilize the duality
theorems of Pontryagin, Tannaka-Krein, Iwahori-Sugiura, and Tatsuuma. We
also develop explict algorithms for recovering bandlimited functions from the
triple correlation on the rotation groups in two and three dimensions. Finally
we describe the formal relationship between our triple correlation analysis and
the Tauberian theorem of N. Wiener concerning the span of translates of a
function. "
R. Kakarala, Triple correlation on groups , Ph.D. Thesis, University of California, Irvine. 1992.
Written 3 May 2007
